Single Higgs Boson Production at Electron-Positron Colliders
in Gauge-Higgs Unification

Shuichiro Funatsu1, Hisaki Hatanaka2, Yuta Orikasa3, and Naoki
<EMAIL_ADDRESS>
1Ushiku, Ibaraki 300-1234, Japan
2Osaka, Osaka 536-0014, Japan
3Institute of Experimental and Applied Physics, Czech Technical University in
Prague,
Husova 240/5, 110 00 Prague 1, Czech Republic
4Department of Physics, National Taiwan University, Taipei, Taiwan 10617,
R.O.C.

##  1 Introduction

The Standard Model (SM) in particle physics is well established at low
energies. However, it is not yet clear whether the properties of the observed
Higgs boson are exactly the same as those of the Higgs boson in the SM.
Current and future experiments such as the Large Hadron Collider (LHC) [1, 2],
the High-Luminosity Large Hadron Collider (HL-LHC) [3], the International
Linear Collider (ILC) [4, 5, 6, 7, 8, 9, 10], the Compact Linear Collider
(CLIC)[11], the Future Circular Collider (FCC-hh, FCC-ee)[12, 13], the Cool
Copper Collider (C3) [14], and Circular Electron Positron Collider (CEPC)[15]
are needed to more precisely determine the couplings of the Higgs boson to
quarks, leptons, and SM gauge bosons, as well as the self-coupling of the
Higgs boson.

The SM Higgs boson sector remains unsatisfactory: the dynamics of the SM gauge
bosons, the photon, the $W$ and $Z$ bosons, and the gluons, are governed by
the gauge principle, but the dynamics of the SM Higgs boson is not governed by
the gauge principle. The Higgs boson coupling of quarks and leptons and the
self-coupling of the Higgs boson are also not governed by the principle. At
the quantum level, there are large corrections to the mass of the Higgs boson.
To reproduce the observed mass of the Higgs boson $m_{h}=125.25\pm 0.17$ GeV
[16], an extremely strong cancellation between the bare mass and quantum
effects must be required. One known way to stabilize the mass of the Higgs
boson against quantum corrections is to identify the Higgs boson as the zero
mode of the five-dimensional component of the gauge potential. This scenario
is called gauge-Higgs unification (GHU) [17, 18, 19, 20, 21, 22].

In GHU models, the Higgs boson appears as a fluctuating mode in the
5-dimensional (5D) Aharonov-Bohm (AB) phase $\theta_{H}$; $SU(3)_{C}\times
SO(5)_{W}\times U(1)_{X}$ GHU models in the Randall-Sundrum (RS) warped space
are proposed in Refs. [23, 24, 25, 26]. The phenomena of the GHU models below
the electroweak (EW) scale are very close to those of the SM [26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36]. The gauge coupling constants of quarks and
leptons to the $Z$ and $W$ bosons in the GHU model differ by less than 0.1%
from the coupling constants in the SM for the parameter region that satisfies
the current experiments, i.e., the Kaluza-Klein (KK) mass $m_{\rm KK}\gtrsim
13$ TeV and the AB phase $\theta_{H}\lesssim 0.1$ [35]. The gauge coupling
constants of the Higgs boson to the $W$ and $Z$ bosons are approximately equal
to the corresponding coupling constants in the SM multiplied by
$\cos\theta_{H}$. The coupling constants of the Higgs boson to the quarks and
the leptons are approximately equal to those in the SM multiplied by
$\cos\theta_{H}$ or $\cos^{2}(\theta_{H}/2)$ depending on the $SO(5)_{W}$
representations of the quarks and the leptons. The deviation of the coupling
constants of the Higgs boson to the $W$ and $Z$ bosons, the quarks, and the
leptons from the SM in the GHU model is less than 1% for $\theta_{H}\leq 0.1$
[32]. The decay of the Higgs boson to two photons in the GHU model is finite
and very small, even when the contribution of KK excitation modes is taken
into account[27]. The electroweak phase transition (EWPT) in the GHU model is
a weakly first order phase transition [34], which is very similar to the EWPT
in the SM [37].

The GHU models predict massive vector bosons $Z^{\prime}$ and $W^{\prime}$
bosons. The $Z^{\prime}$ bosons are mixed vector bosons of $U(1)_{X}$,
$U(1)_{L}(\subset SU(2)_{L})$, and $U(1)_{R}(\subset SU(2)_{R})$, where
$SU(2)_{L}\times SU(2)_{R}\subset SO(5)$. The $W^{\prime}$ bosons are mixed
vector bosons of $SO(5)_{W}/(U(1)_{L}\times U(1)_{R})$. The $SU(3)_{C}\times
SO(5)_{W}\times U(1)_{X}$ GHU models can be roughly classified into two types
of models depending on whether the $SO(5)_{W}$ representations of the quarks
and the leptons are vector or spinor representations. The GHU model whose
fermions belong to the spinor representation of $SO(5)_{W}$ [26, 31, 32, 33,
34, 35, 36] can be regarded as a low-energy effective description of the GHU
model based on the $SO(11)$ grand unified gauge symmetry [38, 39, 40, 41, 42,
43, 44], where the SM gauge symmetry and field content are incorporated into
grand unified theory (GUT) [45, 46, 47, 48, 49, 50] in higher dimensional
framework [51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65]. We
focus on the spinor type model in this paper. In the GHU model,
$\theta_{H}\lesssim 0.10$ and $m_{\rm KK}\gtrsim 13$TeV are obtained as
parameter constraints of the GHU model by using the $Z^{\prime}$ and
$W^{\prime}$ boson search results for the $pp\to\ell\nu$ and
$pp\to\ell^{-}\ell^{+}$ processes at the LHC at $\sqrt{s}=13$ TeV with up to
140 fb-1 data [66, 67, 68, 69, 70, 71].

The $e^{-}e^{+}$ collider experiment is capable of searching for signals in
the GHU model up to the KK mass scale $m_{\rm KK}$ of tens of times the
center-of-mass energy $\sqrt{s}$ of $e^{-}e^{+}$ [72, 73, 74, 75, 33, 36, 76,
77, 78, 79, 80, 81, 82, 83]. Large parity violation appears in the couplings
of quarks and leptons to KK gauge bosons, especially to the first KK modes.
The sign of the bulk mass parameter of the fermions in the GHU model
determines whether the coupling constants of the $Z^{\prime}$ and $W^{\prime}$
bosons to right- or left-handed fermions are larger. In Ref. [33], the SM and
the GHU model are compared in detail for observables such as cross sections,
forward-backward asymmetries [84, 85], left-right asymmetries [84, 85, 86,
87], and left-right forward-backward asymmetries [88, 85, 89, 90, 91] in the
processes $e^{-}e^{+}\to f\bar{f}$
($f\bar{f}=\mu^{-}\mu^{+},c\bar{c},b\bar{b},t\bar{t})$. For example, the total
cross section of the $e^{-}e^{+}\to\mu^{-}\mu^{+}$ process at $\sqrt{s}=250$
GeV in the GHU model whose $m_{\rm KK}\simeq 13$ TeV and $\theta_{H}=0.10$ is
different at most about 3% compared to that in the SM, where it strongly
depends on the initial polarizations of the electron and the positron. Due to
the very large cross section of the fermion pair production processes, we can
clearly observe deviations from the SM in the early stage of the ILC
experiment $(\sqrt{s}=250$ GeV, integral luminosity $L_{\rm int}=250$ fb-1)
even when $m_{\rm KK}$ is slightly larger than the current experimental
constraint $m_{\rm KK}\simeq 13$ TeV. The cross section of the Bhabha
scattering $e^{-}e^{+}\to e^{-}e^{+}$ process [92, 93, 94, 95, 96] is also
very large, and the deviation from the SM in the GHU model can be also
observed in the early-stage of the ILC experiment when $m_{\rm KK}$ is
slightly larger than the current experimental constraint $m_{\rm KK}=13$ TeV
[77, 36]. Furthermore, since the cross sections of the fermion pair production
processes are very sensitive to the initial polarizations of the electron and
the positron, the sign of the corresponding bulk mass of each fermion in each
final state in the GHU model can also be determined by analyzing the
polarization dependence.

To clarify the nature of the Higgs boson in the SM and the GHU model, it is
essential to understand Higgs boson production processes. Since the mass of
the SM Higgs boson is about $125$ GeV, a Higgs strahlung process
$e^{-}e^{+}\to Zh$ is the main single Higgs boson production process for
$\sqrt{s}\lesssim 500$ GeV, and the $W^{-}W^{+}$ gauge boson fusion process
$e^{-}e^{+}\to\nu\bar{\nu}W^{-}{}^{*}W^{+}{}^{*}\to\nu\bar{\nu}h$ is the main
single Higgs boson production process for $\sqrt{s}\gtrsim 500$ GeV [97, 98,
10], where the contributions depend on the initial polarizations of the
electron and the positron. There are the $Z^{\prime}$ and $W^{\prime}$ bosons
in the GHU model, so these additional gauge bosons affect the Higgs boson
production processes. The Higgs boson production processes in the GHU model
have not been analyzed in detail. In particular, the dependence of the initial
polarizations of the electron and the positron has not been discussed at all.
In this paper, we analyze the cross sections and the left-right asymmetries of
the $e^{-}e^{+}\to Zh$ and $e^{-}e^{+}\to f\bar{f}h$
$(f\bar{f}=\nu\bar{\nu},e^{-}e^{+})$ in the SM and the GHU model within the
Born approximation. The Higgs boson production processes in the SM have been
analyzed not only in the Born approximation but also including quantum
corrections [99, 100, 101, 102]. The Higgs boson production processes in other
models beyond the SM have been also discussed in e.g., Refs. [103, 104, 105,
106].

In this paper, we consider three single Higgs boson production processes
$e^{-}e^{+}\to Zh$, $e^{-}e^{+}\to\nu\bar{\nu}h$, and $e^{-}e^{+}\to
e^{-}e^{+}h$ to clarify the difference between the predictions of the cross
sections in the SM and the GHU model. For the Higgs strahlung process
$e^{-}e^{+}\to Zh$, we show that for a parameter region satisfying the current
experimental constraints, a maximum deviation from the SM is about up to 6%
for $\sqrt{s}=250$ GeV and that from the SM is about up to 20% for
$\sqrt{s}=500$ GeV, depending on the initial polarization of the electron and
the positron. The deviation from the SM is monotonically increasing when
$\sqrt{s}$ is increasing for $\sqrt{s}\lesssim 1$ TeV. By using the Higgs
strahlung process, it is possible to explore up to the region of tens of TeV
in terms of the KK mass. We also show that the sign of the bulk mass of a
lepton multiplet in the GHU model can be determined by examining the deviation
of the left-right symmetry of the $e^{-}e^{+}\to Zh$ process from the SM. The
deviation of the left-right asymmetry of the $e^{-}e^{+}\to Zh$ process from
the SM in the GHU model can be large even when $\sqrt{s}$ is not so large. We
show that the deviations of the $e^{-}e^{+}\to\nu\bar{\nu}h$ and
$e^{-}e^{+}\to e^{-}e^{+}h$ processes from the SM in the GHU model are
relatively small compared to the deviations of the $e^{-}e^{+}\to Zh$ process
at least for $\sqrt{s}\leq 1$ TeV.

The paper is organized as follows. In Sec. 2, the $SU(3)_{C}\times
SO(5)_{W}\times U(1)_{X}$ GHU model is introduced. In Sec. 3, we give some
parameter sets of the GHU model. In Sec. 4, we present numerical results for
the cross sections of the three Higgs boson production processes
$e^{-}e^{+}\to Zh$, $e^{-}e^{+}\to\nu\bar{\nu}h$, and $e^{-}e^{+}\to
e^{-}e^{+}h$. Section 5 is devoted to summary and discussions. In Appendix A,
we give the formulas for the cross sections of the $e^{-}e^{+}\to Zh$ and
$e^{-}e^{+}\to f\bar{f}h$ $(f\bar{f}=\nu_{e}\bar{\nu}_{e},e^{-}e^{+})$
processes, involving the $Z^{\prime}$ and $W^{\prime}$ bosons as well as the
$Z$ and $W$ bosons.

##  2 Model

In this paper, we focus on observables related with the EW gauge bosons and
leptons at tree level. The $SU(3)_{C}$ gauge bosons and fermions except
leptons are not directly involved, so we omit them. For the full field content
in the GHU model, see Ref. [26], in which the $SU(3)_{C}\times SO(5)_{W}\times
U(1)_{X}$ GHU model was originally proposed.

The GHU model is defined in the RS warped space with the following [107]:

$\displaystyle
ds^{2}=g_{MN}dx^{M}dx^{N}=e^{-2\sigma(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2},$
(2.1)

where $M,N=0,1,2,3,5$, $\mu,\nu=0,1,2,3$, $y=x^{5}$,
$\eta_{\mu\nu}=\mbox{diag}(-1,+1,+1,+1)$, $\sigma(y)=\sigma(y+2L)=\sigma(-y)$,
and $\sigma(y)=ky$ for $0\leq y\leq L$. By using the conformal coordinate
$z=e^{ky}$ ($1\leq z\leq z_{L}=e^{kL}$) in the region $0\leq y\leq L$, the
metric is rewritten by

$\displaystyle
ds^{2}=\frac{1}{z^{2}}\bigg{(}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\frac{dz^{2}}{k^{2}}\bigg{)}.$
(2.2)

The bulk region $0<y<L$ ($1<z<z_{L}$) is anti-de Sitter (AdS) spacetime with a
cosmological constant $\Lambda=-6k^{2}$, which is sandwiched by the UV brane
at $y=0$ ($z=1$) and the IR brane at $y=L$ ($z=z_{L}$). The KK mass scale is
$m_{\rm KK}=\pi k/(z_{L}-1)$.

###  2.1 Field content

The EW symmetry $SU(2)_{L}\times U(1)_{Y}$ is embedded into $SO(5)_{W}\times
U(1)_{X}$ symmetry. The associated gauge fields of $SO(5)_{W}$ and $U(1)_{X}$
are denoted by $A_{M}^{SO(5)_{W}}$ and $A_{M}^{U(1)_{X}}$, respectively. The
orbifold boundary conditions (BCs) $P_{j}(j=0,1)$ of the gauge fields on the
UV brane $(y=0)$ and the IR brane $(y=L)$ are given by

$\displaystyle\begin{pmatrix}A_{\mu}\cr
A_{y}\end{pmatrix}(x,y_{j}-y)=P_{j}\begin{pmatrix}A_{\mu}\cr-
A_{y}\end{pmatrix}(x,y_{j}+y)P_{j}^{-1}$ (2.3)

for each gauge field, where $(y_{0},y_{1})=(0,L)$. For the $U(1)_{X}$ gauge
boson $A_{M}^{U(1)_{X}}$, $P_{0}=P_{1}=1$. For the $SO(5)_{W}$ gauge boson
$A_{M}^{SO(5)_{W}}$, $P_{0}=P_{1}=P_{\bf 5}^{SO(5)_{W}}$, where $P_{\bf
5}^{SO(5)_{W}}=\mbox{diag}\left(I_{4},-I_{1}\right)$. The orbifold BCs of the
$SO(5)_{W}$ symmetry break $SO(5)_{W}$ to $SO(4)_{W}\simeq SU(2)_{L}\times
SU(2)_{R}$. $W$, $Z$ bosons and $\gamma$ (photon) are zero modes in the
$SO(5)_{W}\times U(1)_{X}$ of 4 dimensional (4D) gauge bosons, whereas the 4D
Higgs boson is a zero mode in the $SO(5)_{W}/SO(4)_{W}$ part of the 5th
dimensional gauge boson. In the GHU model, extra neutral gauge bosons
$Z^{\prime}$ correspond to the KK photons $\gamma^{(n)}$, the KK $Z$ bosons
$Z^{(n)}$, and the KK $Z_{R}$ bosons $Z_{R}^{(n)}$ ($n\geq 1$), where the
$\gamma$, and $Z$, $Z_{R}$ bosons are the mass eigen states of the electro-
magnetic (EM) $U(1)_{\rm EM}$ neutral gauge bosons of $SU(2)_{L}$,
$SU(2)_{R}$, and $U(1)_{X}$. Extra charged gauge bosons $W^{\prime}$
correspond to the KK $W$ boson $W^{\pm(n)}$ ($n\geq 1$) and the KK $W_{R}$
bosons $W_{R}^{\pm(n)}$ ($n\geq 1$).

Leptons are introduced both in the 5D bulk and on the UV brane. They are
listed in Table 1. The SM lepton multiples are identified with the zero modes
of the lepton multiplets $\Psi_{({\bf 1,4})}^{\alpha}$ $(\alpha=1,2,3)$. The
bulk fields $\Psi_{({\bf 1,4})}^{\alpha}$ obey the following BCs:

$\displaystyle\Psi_{({\bf 1,4})}^{\alpha}(x,y_{j}-y)=-P_{\bf
4}^{SO(5)_{W}}\gamma^{5}\Psi_{({\bf 1,4})}^{\alpha}(x,y_{j}+y),$ (2.4)

where $P_{\bf 4}^{SO(5)_{W}}=\mbox{diag}\left(I_{2},-I_{2}\right)$. With the
BCs in Eq. (2.4), the parity assignment of the leptons is summarized in Table
2. Neutral fermions are introduced as the brane fermions $\chi^{\alpha}$
$(\alpha=1,2,3)$ on the UV brane, which are responsible for reproducing tiny
neutrino masses via the seesaw mechanism in the GHU model [42].

| $(SU(3)_{C}\times SO(5)_{W})_{X}$
---|---
Lepton | $({\bf 1},{\bf 4})_{-\frac{1}{2}}$
Brane fermion | $({\bf 1},{\bf 1})_{0}$
Brane scalar | $({\bf 1},{\bf 4})_{\frac{1}{2}}$
Table 1: The field content for the lepton sector in the GHU model is shown.  Field | $(SU(3)_{C}\times SO(5)_{W})_{X}$ | $G_{22}$ | Left-handed | Right-handed | Name
---|---|---|---|---|---
$\Psi_{({\bf 1,4})}^{\alpha}$ | $({\bf 1,4})_{-\frac{1}{2}}$ | $[{\bf 2,1}]$ | $(+,+)$ | $(-,-)$ | $\begin{matrix}\nu_{e}&\nu_{\mu}&\nu_{\tau}\cr e&\mu&\tau\end{matrix}$
|  | $[{\bf 1,2}]$ | $(-,-)$ | $(+,+)$ | $\begin{matrix}\nu_{e}^{\prime}&\nu_{\mu}^{\prime}&\nu_{\tau}^{\prime}\cr e^{\prime}&\mu^{\prime}&\tau^{\prime}\end{matrix}$
Table 2: Parity assignment $(P_{0}^{SO(5)_{W}},P_{1}^{SO(5)_{W}})$ of the
lepton multiplets in the bulk is shown. $G_{22}$ stands for $SU(2)_{L}\times
SU(2)_{R}(\subset SO(5)_{W})$.

The brane scalar field $\Phi_{({\bf 1},{\bf 4})}(x)$ in Table 1 is responsible
for breaking $SO(5)_{W}\times U(1)_{X}$ to $SU(2)_{L}\times U(1)_{Y}$. A
spinor 4 of $SO(5)_{W}$ is decomposed into $[{\bf 2},{\bf 1}]\oplus[{\bf
1},{\bf 2}]$ of $SO(4)_{W}\simeq SU(2)_{L}\times SU(2)_{R}$. We assume that
the brane scalar $\Phi_{({\bf 1},{\bf 4})}$ develops a nonvanishing vacuum
expectation value (VEV):

$\displaystyle\Phi_{({\bf 1,4})}=\begin{pmatrix}\Phi_{[{\bf
2,1}]}\cr\Phi_{[{\bf 1,2}]}\end{pmatrix},\ \ \langle\Phi_{[{\bf
1,2}]}\rangle=\begin{pmatrix}0\cr w\end{pmatrix},$ (2.5)

which reduces the symmetry $SO(4)_{W}\times U(1)_{X}$ to the EW gauge symmetry
$SU(2)_{L}\times U(1)_{Y}$. It is assumed that $w\gg m_{\rm KK}$, which
ensures that the orbifold BCs for the 4D components of the $SU(2)_{R}\times
U(1)_{X}/U(1)_{Y}$ gauge fields become effectively Dirichlet conditions at the
UV brane [40]. Accordingly the mass of the neutral physical mode of
$\Phi_{({\bf 1,4})}$ is much larger than $m_{\rm KK}$.

The $U(1)_{Y}$ gauge boson is a mixed state of $U(1)_{R}(\subset SU(2)_{R})$
and $U(1)_{X}$ gauge bosons. The $U(1)_{Y}$ gauge field $B_{M}^{Y}$ is given
in terms of the $SU(2)_{R}$ gauge fields $A_{M}^{a_{R}}$
$(a_{R}=1_{R},2_{R},3_{R})$ and the $U(1)_{X}$ gauge field $B_{M}$ by

$\displaystyle B_{M}^{Y}=\sin\phi A_{M}^{3_{R}}+\cos\phi B_{M}~{}.$ (2.6)

Here the mixing angle $\phi$ between $U(1)_{R}$ and $U(1)_{X}$ is given by
$\cos\phi={g_{A}}/{\sqrt{g_{A}^{2}+g_{B}^{2}}}$ and
$\sin\phi={g_{B}}/{\sqrt{g_{A}^{2}+g_{B}^{2}}}$, where $g_{A}$ and $g_{B}$ are
gauge couplings in $SO(5)_{W}$ and $U(1)_{X}$, respectively. The 4D
$SU(2)_{L}$ gauge coupling is given by $g_{w}=g_{A}/\sqrt{L}$. The 5D gauge
coupling constant $g_{Y}^{\rm 5D}$ of $U(1)_{Y}$ and the 4D bare Weinberg
angle at the tree level, $\theta_{W}^{0}$, are given by

$\displaystyle g_{Y}^{\rm 5D}=\frac{g_{A}g_{B}}{\sqrt{g_{A}^{2}+g_{B}^{2}}},\
\ \ \sin\theta_{W}^{0}=\frac{\sin\phi}{\sqrt{{1+\sin^{2}\phi}}}.$ (2.7)

The 4D Higgs boson $\phi_{H}(x)$ is the zero mode contained in the
$A_{z}=(kz)^{-1}A_{y}$ component:

$\displaystyle
A_{z}^{(j5)}(x,z)=\frac{1}{\sqrt{k}}\,\phi_{j}(x)u_{H}(z)+\cdots,\
u_{H}(z)=\sqrt{\frac{2}{z_{L}^{2}-1}}\,z,\ \
\phi_{H}(x)=\frac{1}{\sqrt{2}}\begin{pmatrix}\phi_{2}+i\phi_{1}\cr\phi_{4}-i\phi_{3}\end{pmatrix}.$
(2.8)

Without loss of generality, we assume that
$\langle\phi_{1}\rangle,\langle\phi_{2}\rangle,\langle\phi_{3}\rangle=0$ and
$\langle\phi_{4}\rangle\not=0$, which is related to the AB phase $\theta_{H}$
in the fifth dimension by $\langle\phi_{4}\rangle=\theta_{H}f_{H}$, where
$f_{H}=2g_{w}^{-1}k^{1/2}L^{-1/2}(z_{L}^{2}-1)^{-1/2}$.

###  2.2 Action

The bulk part of the action for the EW gauge and lepton sectors is given by

$\displaystyle S_{\rm bulk}^{\rm EW+lepton}$ $\displaystyle=S_{\rm bulk}^{\rm
EW\,gauge}+S_{\rm bulk}^{\rm lepton},$ (2.9)

where $S_{\rm bulk}^{\rm EW\,gauge}$ and $S_{\rm bulk}^{\rm lepton}$ are bulk
actions of the EW gauge bosons and the leptons, respectively. The action of
each gauge field, $A_{M}^{SO(5)_{W}}$ or $A_{M}^{U(1)_{X}}$, is given in the
form

$\displaystyle S_{\rm bulk}^{\rm EW\,gauge}$ $\displaystyle=\int
d^{5}x\sqrt{-\det
G}\,\bigg{[}-\mbox{tr}\left(\frac{1}{4}F^{MN}F_{MN}+\frac{1}{2\xi}(f_{\rm
gf})^{2}+{\cal L}_{\rm gh}\right)\bigg{]},$ (2.10)

where $\sqrt{-\det G}=1/kz^{5}$, $z=e^{ky}$, tr is a trace over all group
generators for each group, and $F_{MN}$ is a field strength defined by

$\displaystyle F_{MN}$
$\displaystyle:=\partial_{M}A_{N}-\partial_{N}A_{M}-ig[A_{M},A_{N}]$ (2.11)

with each 5D gauge coupling constant $g$. The second and third terms in Eq.
(2.10) are the gauge fixing term and the ghost term given in Ref. [26],
respectively.

Each lepton multiplet $\Psi_{({\bf 1,4})}^{\alpha}(x,y)$ in the bulk has its
own bulk-mass parameter $c_{L}^{\alpha}$ $(\alpha=1,2,3)$. The covariant
derivative is given by

$\displaystyle{\cal
D}(c_{L}^{\alpha})=\gamma^{A}{e_{A}}^{M}\bigg{(}D_{M}+\frac{1}{8}\omega_{MBC}[\gamma^{B},\gamma^{C}]\bigg{)}-c_{L}^{\alpha}\sigma^{\prime}(y),$
$\displaystyle
D_{M}=\partial_{M}-ig_{A}A_{M}^{SO(5)_{W}}+i\frac{1}{2}g_{B}A_{M}^{U(1)_{X}}.$
(2.12)

Here $\sigma^{\prime}(y):=d\sigma(y)/dy$ and $\sigma^{\prime}(y)=k$ for
$0<y<L$. Then the action for the lepton sector in the bulk is given by

$\displaystyle S_{\rm bulk}^{\rm lepton}=\int d^{5}x\sqrt{-\det
G}\,\sum_{\alpha=1}^{3}\overline{\Psi_{({\bf 1,4})}^{\alpha}}{\cal
D}(c_{L}^{\alpha})\Psi_{({\bf 1,4})}^{\alpha},$ (2.13)

where $\overline{\Psi_{({\bf 1,4})}^{\alpha}}=i\Psi_{({\bf
1,4})}^{\alpha}{}^{\dagger}\gamma^{0}$. By using $\check{\Psi}_{({\bf
1,4})}^{\alpha}:=\Psi_{({\bf 1,4})}^{\alpha}/z^{2}$, the bulk part of the
fermion action is rewritten by

$\displaystyle S_{\rm bulk}^{\rm lepton}=\int
d^{4}x\int_{1}^{z_{L}}\frac{dz}{k}\sum_{\alpha=1}^{3}\overline{\check{\Psi}_{({\bf
1,4})}^{\alpha}}\Big{[}\gamma^{\mu}D_{\mu}+k\Big{(}\gamma^{5}D_{z}-\frac{c_{L}^{\alpha}}{z}\Big{)}\Big{]}\check{\Psi}_{({\bf
1,4})}^{\alpha}.$ (2.14)

The action for the brane scalar field $\Phi_{({\bf 1,4})}(x)$ is given by

$\displaystyle S_{\rm brane}^{\Phi}$ $\displaystyle=\int d^{5}x\sqrt{-\det
G}\,\delta(y)\left\\{-(D_{\mu}\Phi_{({\bf
1,4})})^{{\dagger}}D^{\mu}\Phi_{({\bf 1,4})}-\lambda_{\Phi_{({\bf
1,4})}}\big{(}\Phi_{({\bf 1,4})}^{\dagger}\Phi_{({\bf
1,4})}-|w|^{2}\big{)}^{2}\right\\},$ (2.15)

where

$\displaystyle D_{\mu}\Phi_{({\bf 1,4})}$
$\displaystyle=\bigg{\\{}\partial_{\mu}-ig_{A}\sum_{\alpha=1}^{10}A_{\mu}^{\alpha}T^{\alpha}-i\frac{1}{2}g_{B}B_{\mu}\bigg{\\}}\Phi_{({\bf
1,4})}~{}.$ (2.16)

Here $SO(5)_{W}$ generators $\\{T^{\alpha}\\}$ consist of $SU(2)_{L}$ and
$SU(2)_{R}$ generators $\\{T^{a_{L}},T^{a_{R}}\\}$ ($a=1,2,3$) and
$SO(5)_{W}/SO(4)_{W}$ generators $\\{T^{\hat{p}}=T^{p5}/\sqrt{2}\\}$
($p=1,2,3,4$). The corresponding canonically normalized gauge fields are given
by

$\displaystyle
A_{M}^{a_{L}}=\frac{1}{\sqrt{2}}\Big{(}\frac{1}{2}\epsilon^{abc}A_{M}^{bc}+A_{M}^{a4}\Big{)},\
\
A_{M}^{a_{R}}=\frac{1}{\sqrt{2}}\Big{(}\frac{1}{2}\epsilon^{abc}A_{M}^{bc}-A_{M}^{a4}\Big{)},\
\ A_{M}^{\hat{p}}=A_{M}^{p5}.$ (2.17)

$B_{M}$ represents the $U(1)_{X}$ gauge field.

The action for the gauge-singlet brane fermion $\chi^{\alpha}(x)$ is given by

$\displaystyle S_{\rm brane}^{\chi}=$ $\displaystyle\int d^{5}x\sqrt{-\det
G}\,\delta(y)\bigg{\\{}\frac{1}{2}\overline{\chi}^{\alpha}\gamma^{\mu}\partial_{\mu}\chi^{\alpha}-\frac{1}{2}M^{\alpha\beta}\overline{\chi}^{\alpha}\chi^{\beta}\bigg{\\}},$
(2.18)

where $\chi^{\alpha}(x)$ satisfies the Majorana condition $\chi^{c}=\chi$;

$\displaystyle\chi=\begin{pmatrix}\xi\cr\eta\end{pmatrix},~{}~{}\chi^{c}=\begin{pmatrix}+\eta^{c}\cr-\xi^{c}\end{pmatrix}=e^{i\delta_{C}}\begin{pmatrix}+\sigma^{2}\eta^{*}\cr-\sigma^{2}\xi^{*}\end{pmatrix}.$
(2.19)

On the UV brane, the brane interaction terms among the bulk leptons, the brane
fermions, and the brane scalar are given by

$\displaystyle S_{\rm brane}^{\rm int}=-\int d^{5}x\sqrt{-\det
G}\,\delta(y)\Big{\\{}\widetilde{\kappa}_{\bf
1}^{\alpha\beta}\,\overline{\chi}^{\beta}\widetilde{\Phi}_{({\bf
1,4})}^{\dagger}\Psi_{({\bf 1,4})}^{\alpha}+{\rm h.c.}\Big{\\}}~{},$ (2.20)

where $\widetilde{\kappa}_{\bf 1}^{\alpha\beta}$ is a coupling constant.

The nonvanishing VEV $\langle\Phi_{({\bf 1,4})}\rangle\not=0$ generates brane
mass terms on the UV brane from the interaction term in Eq. (2.20). Together
with the Majorana mass term in Eq. (2.18) the brane fermion mass terms are
given by

$\displaystyle S_{\rm brane\ mass}^{\rm fermion}=\int d^{5}x\sqrt{-\det
G}\,\delta(y)\left[-\frac{m_{B}^{\alpha\beta}}{\sqrt{k}}\,(\overline{\chi}^{\beta}\check{\nu}_{R}^{\prime\alpha}+\overline{\check{\nu}}{}_{R}^{\prime\alpha}\chi^{\beta})-\frac{1}{2}M^{\alpha\beta}\overline{\chi}^{\alpha}\chi^{\beta}\right],$
(2.21)

where $m_{B}^{\alpha\beta}=\widetilde{\kappa}_{\bf 1}^{\alpha\beta}w\sqrt{k}$.

The VEV of the brane scalar field $\langle\Phi_{({\bf 1,4})}\rangle\not=0$
also generates additional brane mass terms for the 4D components of the
$SO(5)_{W}\times U(1)_{X}$ gauge fields. It follows from Eq. (2.15) that

$\displaystyle S_{\rm brane}^{\rm gauge}=\int d^{5}x\sqrt{-\det
G}\,\delta(y)\bigg{\\{}-\frac{g_{A}^{2}|w|^{2}}{4}\big{(}A_{\mu}^{1_{R}}A^{1_{R}\mu}+A_{\mu}^{2_{R}}A^{2_{R}\mu}\big{)}-\frac{(g_{A}^{2}+g_{B}^{2})|w|^{2}}{4}A_{\mu}^{3_{R}^{\prime}}A^{3_{R}^{\prime}\mu}\bigg{\\}}.$
(2.22)

##  3 Parameter sets

To evaluate cross sections and other observable quantities in single Higgs
boson production processes $e^{-}e^{+}\to Zh$, $e^{-}e^{+}\to\nu\bar{\nu}h$,
$e^{-}e^{+}\to e^{-}e^{+}h$ at tree level in the GHU model, we need to know
the masses, decay widths, and coupling constants of the gauge bosons, the
Higgs boson, and the leptons. Parameters of the model are determined in the
steps described in Refs. [33].

We present several parameter sets of the coupling constants of the leptons,
which are necessary for the present analysis. In Sec. 2, we gave only the 5D
Lagrangian of the GHU model, but in the analysis, we use a 4D effective theory
with KK mode expansion of the 5th dimension. By solving the equations of
motions derived from the BCs of each 5D multiplet, we can obtain the mass
spectra of 4D modes for each 5D field. Once mass spectra of a field are known,
wave functions of the zero mode and the KK modes of the field can be
determined by substituting mass spectra into the mode function of the field.
Furthermore, coupling constants can be obtained by performing overlap
integrals of the wave functions of the corresponding fields. For more details,
see e.g., Ref. [26] for gauge boson and fermion mass formulas and wave
functions and Ref. [32] for the Higgs boson mass and coupling formulas.

We will describe the steps to fix parameter sets in the GHU model, where we
will show parameter sets for leptons, the gauge bosons, and the Higgs boson.

  1. 1.

We pick the values of $\theta_{H}$ and $m_{\rm KK}=\pi k(z_{L}-1)^{-1}$. From
the constraints on $\theta_{H}$ and $m_{\rm KK}$ from the LHC-Run 2 results in
the GHU model [35], we only consider parameters satisfying $\theta_{H}\leq
0.10$ and $m_{\rm KK}\geq 13$ TeV.

(Note that possible values of $m_{\rm KK}$ are restricted with given
$\theta_{H}$ [32]. For example, for $\theta_{H}=0.10$ $z_{L}\geq 10^{8.1}$ to
reproduce the top quark mass, and $z_{L}\leq 10^{15.5}$ to realize the EW
symmetry breaking. They lead to $m_{\rm KK}\simeq[11,15]\,$TeV.)

  2. 2.

$k$ is determined in order for the $Z$ boson mass $m_{Z}$ to be reproduced,
which fixes the warped factor $z_{L}$ as well. (For the mass formula of $Z$
boson, see Ref. [26].)

  3. 3.

The bare Weinberg angle $\theta_{W}^{0}$ in the GHU model is given in Eq.
(2.7). For each value of $\theta_{H}$, the value of $\theta_{W}^{0}$ is
determined self-consistently to fit the observed forward-backward asymmetry
$A_{\rm FB}(e^{-}e^{+}\to\mu^{-}\mu^{+})=0.0169\pm 0.0013$ at
$\sqrt{s}=m_{Z}$[108, 16], after evaluating the coupling constants of the muon
to the $Z$ boson with the procedure described below. We have checked that each
self-consistent value of $\theta_{W}^{0}$ is found after a couple of
iterations of this process.

Name | $\theta_{H}$ | $m_{\rm KK}$ | $z_{L}$ | $k$ | $\sin^{2}\theta_{W}^{0}$
---|---|---|---|---|---
| [rad.] | [TeV] |  | [GeV] |
A- | 0.10 | 13.00 | 3.865$\times 10^{11}$ | 1.599$\times 10^{15}$ | 0.2306
A+ | 0.10 | 13.00 | 4.029$\times 10^{11}$ | 1.667$\times 10^{15}$ | 0.2318
B- | 0.07 | 19.00 | 1.420$\times 10^{12}$ | 8.589$\times 10^{15}$ | 0.2309
B+ | 0.07 | 19.00 | 1.452$\times 10^{12}$ | 8.779$\times 10^{15}$ | 0.2315
C- | 0.05 | 25.00 | 5.546$\times 10^{10}$ | 4.413$\times 10^{14}$ | 0.2310
C+ | 0.05 | 25.00 | 5.600$\times 10^{10}$ | 4.456$\times 10^{14}$ | 0.2313
Table 3: The name of the parameter set and the corresponding $z_{L}$, $k$, and
$\sin^{2}\theta_{W}^{0}$ for each $\theta_{H}$ and $m_{\rm KK}$ are listed. In
the SM, $\sin^{2}\theta_{W}(\overline{\mbox{MS}})=0.23122\pm 0.00004$ [16].
The column “Name” denotes each parameter set.

The parameter sets of $(\theta_{H},m_{\rm KK})$, named $A_{\pm}$,$B_{\pm}$ and
$C_{\pm}$, used in this analysis are summarized in Table 3, where the
subscripts denote the sign of the bulk masses of the leptons. For example,
$A_{+}$ denotes the case where the bulk mass of the lepton is positive and
$A_{-}$ denotes the case where the bulk mass of the lepton is negative.

  4. 4.

With given $\sin\theta_{W}^{0}$, wave functions of the gauge bosons are fixed.
Masses and widths of $Z^{\prime}$, $W$, $W^{\prime}$ bosons are listed for
each parameter set in Table 4. Although information on parameters of the quark
sector is also needed when we determine the decay widths of the gauge bosons,
the specific values of the parameters of the quark sector are omitted since
they are not directly used in this analysis. The parameters of the quark
sector are obtained by the same procedure as for the lepton sector. (For more
detail, see Ref. [26].)

Name | $m_{Z^{(1)}}$ | $\Gamma_{Z^{(1)}}$ | $m_{Z_{R}^{(1)}}$ | $\Gamma_{Z_{R}^{(1)}}$ | $m_{W}$ | $\Gamma_{W}$ | $m_{W^{(1)}}$ | $\Gamma_{W^{(1)}}$ | $m_{W_{R}^{(1)}}$ | $\Gamma_{W_{R}^{(1)}}$
---|---|---|---|---|---|---|---|---|---|---
| [TeV] | [TeV] | [TeV] | [TeV] | [GeV] | [GeV] | [TeV] | [TeV] | [TeV] | [TeV]
A- | 10.196 | 7.840 | 9.951 | 0.816 | 79.98 | 2.02 | 10.196 | 9.802 | 9.951 | 0.359
A+ | 10.196 | 5.982 | 9.951 | 0.775 | 79.92 | 2.01 | 10.196 | 7.163 | 9.951 | 0.358
B- | 14.886 | 11.962 | 14.544 | 1.231 | 79.97 | 2.02 | 14.886 | 14.970 | 14.544 | 0.549
B+ | 14.886 | 9.162 | 14.544 | 1.168 | 79.94 | 2.02 | 14.886 | 10.979 | 14.544 | 0.548
C- | 19.648 | 14.006 | 19.137 | 1.505 | 79.96 | 2.03 | 19.648 | 17.502 | 19.137 | 0.639
C+ | 19.648 | 10.700 | 19.137 | 1.432 | 79.95 | 2.02 | 19.648 | 12.784 | 19.137 | 0.638
Table 4: Masses and widths of $Z^{\prime}$, $W$, $W^{\prime}$ bosons are
listed for each parameter set listed in Table 3. In the SM, $\alpha_{\rm
EM}=1/128$ at the mass scale of $Z$ boson, $m_{Z}=91.1876$ GeV,
$\Gamma_{Z}=2.4952$ GeV, $m_{W}=80.377\pm 0.012$ GeV, $\Gamma_{W}=2.085\pm
0.042$ GeV, [16]. The column “Name” denotes each parameter set in Table 3.
$m_{V}$ and $\Gamma_{V}$ $(V=Z^{(1)},Z_{R}^{(1)},W,W^{(1)},W_{R}^{(1)})$ are
the mass and decay width of each $V$ boson, respectively. Decay widths are
calculated by using the formulas in Ref. [33], where all possible two-body
final states are taken into account.

  5. 5.

The bulk masses of the leptons $c_{L}^{\alpha}$ in Eq. (2.14) are determined
so as to reproduce the masses of charged leptons, where we denote
$c_{L}^{1},c_{L}^{2},c_{L}^{3}$ as $c_{e},c_{\mu},c_{\tau}$. We use the masses
of the electron, the muon, and the tau lepton given by $m_{e}=0.48657\,$MeV,
$m_{\mu}=102.72\,$MeV, and $m_{\tau}=1.7462\,$GeV at $\mu=m_{Z}$ [109]. The
parameters of the Majorana mass terms and brane interactions in the neutrino
sector in Eq. (2.21), $M^{\alpha\beta}$ and $m_{B}^{\alpha\beta}$, are
determined so as to reproduce the neutrino masses. The neutrino masses cannot
be completely fixed because only two mass-squared differences are known from
observations [16]. In our analysis, we take
$m_{\nu_{e}}=m_{\nu_{\mu}}=m_{\nu_{\tau}}=10^{-12}$ GeV for reference. For
simplicity we assume that $M^{\alpha\beta}$ and $m_{B}^{\alpha\beta}$ are
diagonal to the flavor. We take $M^{\alpha\beta}=M_{\ell}\delta_{\alpha\beta}$
$(\alpha,\beta=1,2,3)$, $m_{B}^{11}=m_{B_{e}}$, $m_{B}^{22}=m_{B_{\mu}}$,
$m_{B}^{33}=m_{B_{\tau}}$, and $m_{B}^{\alpha\beta}=0$ for $\alpha\not=\beta$.
The neutrino masses are so small that a difference of one to two orders of
magnitude in their masses will not affect the results of this analysis.
Neutrino mixings are ignored, but they do not affect the present analysis. The
bulk masses and the brane interaction parameters of the leptons are listed in
Table 5.

Name | $c_{e}$ | $c_{\mu}$ | $c_{\tau}$ | $M_{\ell}$ | $m_{Be}$ | $m_{B\mu}$ | $m_{B\tau}$
---|---|---|---|---|---|---|---
|  |  |  | [GeV] | [GeV] | [GeV] | [GeV]
A- |  $-$1.0067 |  $-$0.7929 |  $-$0.6753 | $10^{6}$ | 4.834$\times 10^{5}$ | 1.342$\times 10^{8}$ | 2.949$\times 10^{9}$
A+ |  $+$1.0058 |  $+$0.7924 |  $+$0.6750 | $10^{6}$ | 8.382$\times 10^{22}$ | 6.373$\times 10^{22}$ | 4.931$\times 10^{22}$
B- |  $-$0.9828 |  $-$0.7791 |  $-$0.6670 | $10^{6}$ | 4.952$\times 10^{5}$ | 1.375$\times 10^{8}$ | 3.021$\times 10^{9}$
B+ |  $+$0.9824 |  $+$0.7788 |  $+$0.6669 | $10^{6}$ | 3.017$\times 10^{23}$ | 2.294$\times 10^{23}$ | 1.775$\times 10^{23}$
C- |  $-$1.0469 |  $-$0.8162 |  $-$0.6893 | $10^{6}$ | 4.652$\times 10^{5}$ | 1.292$\times 10^{8}$ | 2.838$\times 10^{9}$
C+ |  $+$1.0467 |  $+$0.8161 |  $+$0.6982 | $10^{6}$ | 1.165$\times 10^{22}$ | 8.857$\times 10^{21}$ | 6.853$\times 10^{21}$
Table 5: The bulk masses of electron, muon, and tau lepton,
$c_{e},c_{\mu},c_{\tau}$ and the brane interaction parameters of leptons
$M_{\ell},m_{B_{e}},m_{B_{m}u},m_{B_{\tau}}$ are listed. In the SM,
$m_{e}=0.48657\,$MeV, $m_{\mu}=102.72\,$MeV, and $m_{\tau}=1.7462\,$GeV at
$\mu=m_{Z}$ [109]. We take the neutrino masses:
$m_{\nu_{e}}=m_{\nu_{\mu}}=m_{\nu_{\tau}}=10^{-12}$ GeV. The column “Name”
denotes each parameter set in Table 3.

In Table 5, it may be problematic that the mass parameters
$m_{B\ell}(\ell=e,\mu,\tau)$ of the brane fermions are larger than the Planck
mass for the cases of positive bulk masses of the leptons. Furthermore, it has
been pointed out in Ref. [26] that the appearance of extra vector-like
neutrinos with MeV-scale masses is for the case of the positive bulk masses of
the leptons. The sign of the bulk masses causes a qualitative difference in
the behavior of the coupling constants of fermions to the KK gauge bosons, so
in this paper we consider the parameter sets of the positive bulk masses of
the leptons as well as the negative bulk masses of the leptons for reference.

  6. 6.

With given the bulk masses and the brane interaction parameters, wave
functions of fermions are fixed.

  7. 7.

The mass and self-coupling constant of the Higgs boson can be obtained from
the effective potential of the Higgs boson [32]. The mass of the The Higgs
boson is determined by adjusting the bulk mass of the dark fermions so that
the mass of the Higgs boson is $m_{h}=125.25\pm 0.17$ GeV [16].

To evaluate cross sections for single Higgs boson production processes such as
$e^{-}e^{+}\to Zh$, $e^{-}e^{+}\to\nu\bar{\nu}h$, $e^{-}e^{+}\to e^{-}e^{+}h$,
we need to know not only the masses and the decay widths but also coupling
constants of the gauge bosons, the Higgs boson, and the leptons. They are
obtained from the five-dimensional gauge interaction terms by substituting the
wave functions of gauge bosons and fermions or the Higgs boson and integrating
over the fifth-dimensional coordinate[28, 29, 30].

  * •

The coupling constants of the gauge bosons to the leptons are obtained by
performing overlap integrals of the wave functions in the fifth dimension of
gauge bosons and leptons. Coupling constants of gauge bosons to leptons are
listed in Tables 6, 7, and 8.

Name | $g_{We\nu_{e}}^{L}$ | $g_{We\nu_{e}}^{R}$ | $g_{W^{(1)}e\nu_{e}}^{L}$ | $g_{W^{(1)}e\nu_{e}}^{R}$ | $g_{W_{R}^{(1)}e\nu_{e}}^{L}$ | $g_{W_{R}^{(1)}e\nu_{e}}^{R}$
---|---|---|---|---|---|---
A- |  $+$0.9976 | 0 |  $+$5.7451 | 0 |  $+$0.0145 | 0
A+ |  $+$0.9988 | 0 |  $-$0.2220 | 0 | 0 | 0
B- |  $+$0.9988 | 0 |  $+$5.8583 | 0 |  $+$0.0073 | 0
B+ |  $+$0.9994 | 0 |  $-$0.2171 | 0 | 0 | 0
C- |  $+$0.9994 | 0 |  $+$5.5706 | 0 |  $+$0.0035 | 0
C+ |  $+$0.9997 | 0 |  $-$0.2316 | 0 | 0 | 0

Name | $g_{W\mu\nu_{\mu}}^{L}$ | $g_{W\mu\nu_{\mu}}^{R}$ | $g_{W^{(1)}\mu\nu_{\mu}}^{L}$ | $g_{W^{(1)}\mu\nu_{\mu}}^{R}$ | $g_{W_{R}^{(1)}\mu\nu_{\mu}}^{L}$ | $g_{W_{R}^{(1)}\mu\nu_{\mu}}^{R}$
---|---|---|---|---|---|---
A- |  $+$0.9976 | 0 |  $+$5.4705 | 0 |  $+$0.0139 | 0
A+ |  $+$0.9988 | 0 |  $-$0.2222 | 0 | 0 | 0
B- |  $+$0.9988 | 0 |  $+$5.5850 | 0 |  $+$0.0069 | 0
B+ |  $+$0.9994 | 0 |  $-$0.2171 | 0 | 0 | 0
C- |  $+$0.9994 | 0 |  $+$5.2941 | 0 |  $+$0.0034 | 0
C+ |  $+$0.9997 | 0 |  $-$0.2316 | 0 | 0 | 0

Name | $g_{W\tau\nu_{\tau}}^{L}$ | $g_{W\tau\nu_{\tau}}^{R}$ | $g_{W^{(1)}\tau\nu_{\tau}}^{L}$ | $g_{W^{(1)}\tau\nu_{\tau}}^{R}$ | $g_{W_{R}^{(1)}\tau\nu_{\tau}}^{L}$ | $g_{W_{R}^{(1)}\tau\nu_{\tau}}^{R}$
---|---|---|---|---|---|---
A- |  $+$0.9976 | 0 |  $+$5.2878 | 0 |  $+$0.0134 | 0
A+ |  $+$0.9988 | 0 |  $-$0.2217 | 0 | 0 | 0
B- |  $+$0.9988 | 0 |  $+$5.4046 | 0 |  $+$0.0067 | 0
B+ |  $+$0.9994 | 0 |  $-$0.2169 | 0 | 0 | 0
C- |  $+$0.9994 | 0 |  $+$5.1077 | 0 |  $+$0.0032 | 0
C+ |  $+$0.9997 | 0 |  $-$0.2314 | 0 | 0 | 0

Table 6: Coupling constants of charged vector bosons,
$W,W^{\prime}(=W^{(1)},W_{R}^{(1)})$ bosons, to leptons in units of
$g_{w}/\sqrt{2}$ are listed. The fine structure constant is the same as in the
SM: $\alpha(M_{Z})^{-1}=127.951\pm 0.009$ [16]. In the SM,
$g_{w}=e/\sin\theta_{W}$, while in the GHU $g_{w}=e/\sin\theta_{W}^{0}$. The
coupling constants of $W$ boson to leptons in the SM are
$g_{We\nu_{e}}^{L}=1$. When the value is less than $10^{-4}$, we write $0$.

Name | $g_{Z\nu_{e}}^{L}$ | $g_{Z\nu_{e}}^{R}$ | $g_{Z^{(1)}\nu_{e}}^{L}$ | $g_{Z^{(1)}\nu_{e}}^{R}$ | $g_{Z_{R}^{(1)}\nu_{e}}^{L}$ | $g_{Z_{R}^{(1)}\nu_{e}}^{R}$
---|---|---|---|---|---|---
A- |  $+$0.5687 | 0 |  $+$3.2774 | 0 |  $-$1.0322 | 0
A+ |  $+$0.5691 | 0 |  $-$0.1264 | 0 | 0 | 0
B- |  $+$0.5695 | 0 |  $+$3.3413 | 0 |  $-$1.0586 | 0
B+ |  $+$0.5697 | 0 |  $-$0.1237 | 0 | 0 | 0
C- |  $+$0.5698 | 0 |  $+$3.1768 | 0 |  $-$1.0099 | 0
C+ |  $+$0.5699 | 0 |  $-$0.1320 | 0 | 0 | 0

Name | $g_{Z\nu_{\mu}}^{L}$ | $g_{Z\nu_{\mu}}^{R}$ | $g_{Z^{(1)}\nu_{\mu}}^{L}$ | $g_{Z^{(1)}\nu_{\mu}}^{R}$ | $g_{Z_{R}^{(1)}\nu_{\mu}}^{L}$ | $g_{Z_{R}^{(1)}\nu_{\mu}}^{R}$
---|---|---|---|---|---|---
A- |  $+$0.5687 | 0 |  $+$3.1207 | 0 |  $-$0.9852 | 0
A+ |  $+$0.5691 | 0 |  $-$0.1264 | 0 | 0 | 0
B- |  $+$0.5695 | 0 |  $+$3.1854 | 0 |  $-$1.0114 | 0
B+ |  $+$0.5697 | 0 |  $-$0.1237 | 0 | 0 | 0
C- |  $+$0.5698 | 0 |  $+$3.0191 | 0 |  $-$0.9623 | 0
C+ |  $+$0.5699 | 0 |  $-$0.1320 | 0 | 0 | 0

Name | $g_{Z\nu_{\tau}}^{L}$ | $g_{Z\nu_{\tau}}^{R}$ | $g_{Z^{(1)}\nu_{\tau}}^{L}$ | $g_{Z^{(1)}\nu_{\tau}}^{R}$ | $g_{Z_{R}^{(1)}\nu_{\tau}}^{L}$ | $g_{Z_{R}^{(1)}\nu_{\tau}}^{R}$
---|---|---|---|---|---|---
A- |  $+$0.5691 | 0 |  $+$3.0165 | 0 |  $-$0.9539 | 0
A+ |  $+$0.5687 | 0 |  $-$0.1262 | 0 | 0 | 0
B- |  $+$0.5695 | 0 |  $+$3.0825 | 0 |  $-$0.9803 | 0
B+ |  $+$0.5697 | 0 |  $-$0.1236 | 0 | 0 | 0
C- |  $+$0.5698 | 0 |  $+$2.9129 | 0 |  $-$0.9302 | 0
C+ |  $+$0.5699 | 0 |  $-$0.1319 | 0 | 0 | 0

Table 7: Coupling constants of neutral vector bosons,
$Z,Z^{\prime}(=Z^{(1)},Z_{R}^{(1)})$ bosons, to neutrinos in units of
$g_{w}=e/\sin\theta_{W}^{0}$ are listed. The coupling constants of $Z$ boson
to neutrinos in the SM are $(g_{Z_{\nu}}^{L},g_{Z_{\nu}}^{R})=(0.5703,0)$ in
units of $g_{w}$. Other information is the same as in Table 6.

Name | $g_{Ze}^{L}$ | $g_{Ze}^{R}$ | $g_{Z^{(1)}e}^{L}$ | $g_{Z^{(1)}e}^{R}$ | $g_{Z_{R}^{(1)}e}^{L}$ | $g_{Z_{R}^{(1)}e}^{R}$
---|---|---|---|---|---|---
A- |  $-$0.3058 |  $+$0.2629 |  $-$1.7621 |  $-$0.0584 |  $-$1.0444 | 0
A+ |  $-$0.3060 |  $+$0.2631 |  $+$0.0680 |  $+$1.5171 | 0 |  $+$1.3826
B- |  $-$0.3062 |  $+$0.2633 |  $-$1.7964 |  $-$0.0572 |  $-$1.0646 | 0
B+ |  $-$0.3063 |  $+$0.2634 |  $+$0.0665 |  $+$1.5459 | 0 |  $+$1.4102
C- |  $-$0.3064 |  $+$0.2634 |  $-$1.7082 |  $-$0.0610 |  $-$1.0128 | 0
C+ |  $-$0.3065 |  $+$0.2635 |  $+$0.0710 |  $+$1.4690 | 0 |  $+$1.3428

Name | $g_{Z\mu}^{L}$ | $g_{Z\mu}^{R}$ | $g_{Z^{(1)}\mu}^{L}$ | $g_{Z^{(1)}\mu}^{R}$ | $g_{Z_{R}^{(1)}\mu}^{L}$ | $g_{Z_{R}^{(1)}\mu}^{R}$
---|---|---|---|---|---|---
A- |  $-$0.3058 |  $+$0.2629 |  $-$1.6778 |  $-$0.0584 |  $-$0.9969 | 0
A+ |  $-$0.3060 |  $+$0.2631 |  $+$0.0680 |  $+$1.4447 | 0 |  $+$1.3197
B- |  $-$0.3062 |  $+$0.2633 |  $-$1.7126 |  $-$0.0572 |  $-$1.0173 | 0
B+ |  $-$0.3063 |  $+$0.2634 |  $+$0.0665 |  $+$1.4738 | 0 |  $+$1.3474
C- |  $-$0.3064 |  $+$0.2634 |  $-$1.6234 |  $-$0.0610 |  $-$0.9651 | 0
C+ |  $-$0.3065 |  $+$0.2635 |  $+$0.0710 |  $+$1.3961 | 0 |  $+$1.2796

Name | $g_{Z\tau}^{L}$ | $g_{Z\tau}^{R}$ | $g_{Z^{(1)}\tau}^{L}$ | $g_{Z^{(1)}\tau}^{R}$ | $g_{Z_{R}^{(1)}\tau}^{L}$ | $g_{Z_{R}^{(1)}\tau}^{R}$
---|---|---|---|---|---|---
A- |  $-$0.3058 |  $+$0.2629 |  $-$1.6218 |  $-$0.0584 |  $-$0.9652 | 0
A+ |  $-$0.3060 |  $+$0.2631 |  $+$0.0679 |  $+$1.3965 | 0 |  $+$1.2778
B- |  $-$0.3062 |  $+$0.2633 |  $-$1.6473 |  $-$0.0571 |  $-$0.9859 | 0
B+ |  $-$0.3063 |  $+$0.2634 |  $+$0.0664 |  $+$1.4262 | 0 |  $+$1.3060
C- |  $-$0.3064 |  $+$0.2634 |  $-$1.5663 |  $-$0.0610 |  $-$0.9330 | 0
C+ |  $-$0.3065 |  $+$0.2635 |  $+$0.0710 |  $+$1.3470 | 0 |  $+$1.2370

Table 8: Coupling constants of neutral vector bosons, $Z,Z^{\prime}$ bosons,
to charged leptons in units of $g_{w}=e/\sin\theta_{W}^{0}$ are listed. The
coupling constants of $Z$ boson to charged leptons in the SM are
$(g_{Z_{e}}^{L},g_{Z_{e}}^{R})=(-0.3065,0.2638)$ in units of $g_{w}$. Other
information is the same as in Table 6.

As can be seen from Tables 6, 7, and 8 the coupling constants of the zero
modes and the 1st KK modes of the $Z$ and $W$ bosons to the zero modes of the
leptons in the GHU are very close to those in the SM. More specifically, the
deviations from the SM are less than 1%. There is almost no difference in the
coupling constants of the $Z$ and $W$ bosons to the leptons for the positive
and negative bulk masses of the leptons, while there is a large difference in
the coupling constants of the KK gauge bosons,
$Z^{\prime}(=Z^{(1)},Z_{R}^{(1)})$ and
$W^{\prime}(=W^{\pm(1)},W_{R}^{\pm(1)})$ bosons, to the zero modes of the
leptons. When the bulk masses of leptons are negative, the coupling constants
of the 1st KK gauge bosons to the zero modes of the left-handed leptons are
large, and those of the 1st KK gauge bosons to the zero modes of the right-
handed leptons are small. On the other hand, when the bulk masses of the
leptons are positive, the coupling constants of the 1st KK gauge bosons to the
zero modes of the right-handed leptons are large and those of the 1st KK gauge
bosons to the zero modes of the left-handed leptons are small.

  * •

The coupling constants of gauge bosons to the Higgs boson are obtained by
performing overlap integrals of the wave functions in the fifth dimension of
gauge bosons and the Higgs boson. Coupling constants of gauge bosons to Higgs
boson are listed Table 9.

Name | $g_{WWh}$ | $g_{WW^{(1)}h}$ | $g_{W^{(1)}W^{(1)}h}$ | $g_{WW_{R}^{(1)}h}$ | $g_{W_{R}^{(1)}W_{R}^{(1)}h}$ | $g_{W^{(1)}W_{R}^{(1)}h}$
---|---|---|---|---|---|---
| [GeV] | [GeV] | [GeV] | [GeV] | [GeV] | [GeV]
A- |  $+$79.60 |  $+$395.80 |  $-$315.90 |  $+$405.14 | 0 |  $-$156.05
A+ |  $+$79.54 |  $+$395.81 |  $-$315.62 |  $+$405.14 | 0 |  $-$155.92
B- |  $+$79.78 |  $+$406.12 |  $-$315.67 |  $+$414.29 | 0 |  $-$155.67
B+ |  $+$79.75 |  $+$406.12 |  $-$315.53 |  $+$414.29 | 0 |  $-$155.60
C- |  $+$79.87 |  $+$381.16 |  $-$318.16 |  $+$389.29 | 0 |  $-$156.69
C+ |  $+$79.85 |  $+$381.17 |  $-$318.80 |  $+$389.29 | 0 |  $-$156.65

Name | $g_{ZZh}$ | $g_{ZZ^{(1)}h}$ | $g_{Z^{(1)}Z^{(1)}h}$ | $g_{ZZ_{R}^{(1)}h}$ | $g_{Z_{R}^{(1)}Z_{R}^{(1)}h}$ | $g_{Z^{(1)}Z_{R}^{(1)}h}$
---|---|---|---|---|---|---
| [GeV] | [GeV] | [GeV] | [GeV] | [GeV] | [GeV]
A- |  $+$103.47 |  $+$514.84 |  $-$410.55 |  $+$386.81 | 0 |  $-$148.87
A+ |  $+$103.55 |  $+$515.66 |  $-$410.82 |  $+$386.56 | 0 |  $-$148.64
B- |  $+$103.74 |  $+$528.25 |  $-$410.43 |  $+$395.33 | 0 |  $-$148.48
B+ |  $+$103.78 |  $+$528.67 |  $-$410.57 |  $+$395.20 | 0 |  $-$148.37
C- |  $+$103.86 |  $+$495.76 |  $-$414.64 |  $+$371.38 | 0 |  $-$149.45
C+ |  $+$103.88 |  $+$495.96 |  $-$414.71 |  $+$371.32 | 0 |  $-$149.39

Table 9: Coupling constants of gauge bosons to Higgs boson in units of $g_{w}$
are listed. In the SM, $g_{WWh}/g_{w}=m_{W}=80.38$ GeV and
$g_{ZZh}/g_{w}=m_{Z}/\cos\theta_{W}=104.00$ GeV. When the value is less than
$0.1$, we write $0$.

From Table 9, the coupling constants of the $Z$ and $W$ bosons to the Higgs
boson in the GHU model are very close to those in the SM. More specifically,
the deviations from the SM are less than 1%.

We comment on the expected improvement in the accuracy of the Weinberg angle
in the ILC experiment. At the moment we use $A_{\rm
FB}(e^{-}e^{+}\to\mu^{-}\mu^{+})$ to determine the value of the Weinberg angle
$\theta_{W}^{0}$. If the uncertainty of $A_{\rm
FB}(e^{-}e^{+}\to\mu^{-}\mu^{+})$ is reduced by future experiments, we can use
exactly the same method to determine the parameters. If the uncertainty of
$A_{\rm FB}(e^{-}e^{+}\to\mu^{-}\mu^{+})$ is reduced by a factor of 1/10, the
uncertainty of $\sin^{2}\theta_{W}^{0}$ is also reduced by a factor of about
1/10. However, the (unpolarized) forward-backward asymmetry $A_{\rm FB}$ is
not a direct observable in the ILC experiment with polarized electrons and
positrons. Although it is possible to obtain $A_{\rm FB}$ by combining
observables, it is known that the accuracy of Weinberg angle determination can
be improved by using more direct observables, polarization asymmetry
parameters $A_{f}(f=e,\mu,...)$ [10, 16]. The relations between the Weinberg
angle and the asymmetry parameters are approximately given as

$\displaystyle A_{f}\simeq 8\left(\frac{1}{4}-|Q_{f}|\sin^{2}\theta_{\rm
eff}^{f}\right).$ (3.1)

From this relationship, the accuracy of $\sin^{2}\theta_{\rm eff}^{f}$ is
estimated to be 1/8 times the decision accuracy of $A_{f}$. This is also true
for $\sin^{2}\theta_{W}^{0}$ in the GHU model. At present, the accuracy of
$\sin^{2}\theta_{W}^{0}$ is $O(0.1)\%$ by using the currently forward-backward
asymmetry $A_{\rm FB}(e^{-}e^{+}\to\mu^{-}\mu^{+})$. We expect that the
decision accuracy of $\sin^{2}\theta_{W}^{0}$ will be $O(0.01)\%$ by using the
expected accuracy of the asymmetry parameter $A_{e}$ at the ILC experiment
with $\sqrt{s}=250$ GeV and $L_{\rm int}=2$ ab-1 [10]; $O(0.001)\%$ by using
the expected accuracy of the asymmetry parameter $A_{e}$ at the ILC experiment
with $\sqrt{s}=m_{Z}\simeq 91.2$ GeV and $L_{\rm int}=100$ fb-1 [10].

##  4 Numerical analysis

We calculate cross sections of single Higgs boson production processes. More
specifically, we analyze three processes $e^{-}e^{+}\to Zh$,
$e^{-}e^{+}\to\nu\bar{\nu}h$, $e^{-}e^{+}\to e^{-}e^{+}h$ for the initial
states of unpolarized and polarized electrons and positrons, where we use the
formula of the cross sections given in Appendix A. For the values of the
initial polarizations, we mainly use $(P_{e^{-}},P_{e^{+}})=(\mp 0.8,\pm 0.3)$
with the ILC experiment in mind.

###  4.1 $e^{-}e^{+}\to Zh$

Here we evaluate observables of the $e^{-}e^{+}\to Zh$ process in the SM and
the GHU model at tree level. We use the parameter sets A±, B±, C± listed in
Tables 3, 4, 5, 6, 7, 8, and 9.

In Figure 1, we show the $\sqrt{s}$ dependence of the total cross sections of
the $e^{-}e^{+}\to Zh$ process in wider range of $\sqrt{s}$ in the SM and the
GHU model with unpolarized $e^{\pm}$ beams. The calculation incorporates
contributions up to the 1st KK mode. Above the first KK mass scale, the
contributions from higher KK modes cannot be neglected, and the region where
the calculation is reliable and valid for this analysis is well below the
first KK mass scale. The left figure in Figure 1 shows
$\sigma^{Zh}(P_{e^{-}},P_{e^{+}})$ given in Eq. (A.14) in the SM and the GHU
model with $(P_{e^{-}},P_{e^{+}})=(0,0)$, where $A_{\pm}$, $B_{\pm}$,
$C_{\pm}$ are the names of the parameter sets listed in Table 3. The right
figure shows $\sigma^{Zh}(P_{e^{-}},P_{e^{+}})$ inf the SM and GHU models with
the initial states of polarized and unpolarized electron and positron. U, L, R
stand $(P_{e^{-}},P_{e^{+}})=(0,0),(-0.8,+0.3),(+0.8,-0.3)$, respectively.

Figure 1: The total cross sections of the $e^{-}e^{+}\to Zh$ process in the SM
and the GHU model are shown in wider range of $\sqrt{s}$. The left figure
shows the $\sqrt{s}$ dependence of $\sigma^{Zh}(P_{e^{-}}=0,P_{e^{+}}=0)$ in
the SM and the GHU model with unpolarized electron and positron beams, where
$A_{\pm}$, $B_{\pm}$, $C_{\pm}$ are the names of the parameter sets listed in
Table 3. The right figure shows the $\sqrt{s}$ dependence of
$\sigma^{Zh}(P_{e^{-}},P_{e^{+}})$ in the SM and the GHU model whose parameter
sets are $A_{\pm}$ with the three different polarizations U, L, R, where U, L,
R stand for $(P_{e^{-}},P_{e^{+}})=(0,0),(-0.8,+0.3),(+0.8,-0.3)$,
respectively.

From Figure 1, in the $\sqrt{s}\lesssim O(1)$ TeV region, the cross sections
in the GHU model are smaller than the cross sections in the SM, independent of
the parameter sets and initial polarizations. From the right figure, we find
that the deviations from the SM for each initial polarization of $e^{\pm}$
beams depend on the sign of the bulk masses. The deviation from the SM is
monotonically increasing with respect to $\sqrt{s}$ for $\sqrt{s}\lesssim 1$
TeV. The cross sections in the GHU model are smaller than those in the SM
because of the interference effects between the $Z$ boson and the KK gauge
bosons, and the cross sections in the GHU begin to increase in the energy
scale region where the contributions to the cross sections from the KK gauge
bosons are larger than the contribution to the cross section from the
interference effects, indicating a turning point located at
$\sqrt{s}\simeq\sqrt{m_{Z}m_{\rm KK}}$, where $m_{\rm KK}=13,19,25$ TeV for
the parameter sets $A_{\pm}$, $B_{\pm}$, $C_{\pm}$, respectively.

Figure 2: The electron polarization dependence of the deviation from the SM in
the GHU models for the total cross sections,
$\Delta_{\sigma}^{Zh}(P_{e^{-}},P_{e^{+}})$, is shown, where we fix
$P_{e^{+}}=0$. The gray region represents the $1\sigma$ statistical error
estimated by using the decay mode of $Z$ to $\mu^{-}\mu^{+}$, where in the
first and second figures, we used the sets of the center-of-mass energies and
the integrated luminosity $(\sqrt{s},L_{\rm
int})=(250\,\mbox{GeV},1\,\mbox{ab}^{-1}),(500\,\mbox{GeV},2\,\mbox{ab}^{-1})$,
respectively. Note that $\mbox{Br}(Z\to\mu^{-}\mu^{+})=(3.3662\pm 0.0066)$%
[16].

In Figure 2, the electron polarization dependence of the deviation from the SM
in the GHU models for the total cross sections,
$\Delta_{\sigma}^{Zh}(P_{e^{-}},P_{e^{+}})$, is shown, where
$\Delta_{\sigma}^{Zh}(P_{e^{-}},P_{e^{+}})$ is given in Eq. (A.17). The
$1\sigma$ statistical errors are estimated by using the decay mode of the $Z$
boson to $\mu^{-}$ and $\mu^{+}$, where
$\mbox{Br}(Z\to\mu^{-}\mu^{+})=(3.3662\pm 0.0066)$% [16]. We use the sets of
the center-of-mass energies and the integrated luminosity $(\sqrt{s},L_{\rm
int})=(250\,\mbox{GeV},1\,\mbox{ab}^{-1}),(500\,\mbox{GeV},2\,\mbox{ab}^{-1})$,
respectively. For reference, here we consider only the decay channel of the
$Z$ boson to a pair of the muons [98]. In the ILC experiment, smaller
statistical errors may be available by using the decay channel of the $Z$
boson to hadron jets [98, 10].

Figure 3: The $\sqrt{s}$ dependence of the left-right asymmetry of the
$e^{-}e^{+}\to Zh$ process and the deviation from the SM are shown. The left
figure shows the $\sqrt{s}$ dependence of $A_{LR}^{Zh}$ in the SM and the GHU
model. The right figure shows the $\sqrt{s}$ dependence of
$\Delta_{A_{LR}}^{Zh}$. The energy ranges $\sqrt{s}$ in the first and second
figures are $\sqrt{s}=[200,3000]\,$GeV, $\sqrt{s}=[200,1000]\,$GeV,
respectively. The gray region represents the $1\sigma$ statistical error in
the SM at each $\sqrt{s}$ with $1$ ab-1 for each polarized initial states
$(P_{e^{-}},P_{e^{+}})=(\mp 0.8,\pm 0.3)$ by using the decay mode of $Z$ to
$\mu^{-}\mu^{+}$. Other information is the same as in Figure 2.

In Figure 3, the $\sqrt{s}$ dependence of the left-right asymmetry
$A_{LR}^{Zh}$ of the $e^{-}e^{+}\to Zh$ processes and the deviation from the
SM $\Delta_{A_{LR}}^{Zh}$ are shown, where $A_{LR}^{Zh}$ and
$\Delta_{A_{LR}}^{Zh}$ are given in Eqs. (A.18) and (A.23), respectively. The
$1\sigma$ statistical error in the SM at each $\sqrt{s}$ with $1$ ab-1 for
each polarized initial electron and positron $(P_{e^{-}},P_{e^{+}})=(\mp
0.8,\pm 0.3)$ is estimated by using the decay mode of $Z$ to $\mu^{-}\mu^{+}$.

From Figure 3, we find that for $\sqrt{s}\lesssim 2$ TeV, the left-right
asymmetry $A_{LR}^{Zh}$ in the GHU model is larger that that in the SM when
the bulk mass of the leptons is positive, while $A_{LR}^{Zh}$ in the GHU model
is smaller that in the SM when the bulk mass of the leptons is negative. As
can be seen from Table 9, there is almost no bulk mass dependence for the
coupling constants of the Higgs boson to the $W$ and $Z$ bosons in the SM and
the GHU model, and the cubic coupling constants of the $Z$ boson, the
$Z^{\prime}(=Z^{(1)},Z_{R}^{(1)})$ boson, and the Higgs boson, $g_{ZZ^{(1)}h}$
and $g_{ZZ_{R}^{(1)}h}$, are larger than the cubic coupling constant of the
$Z$ boson, the $Z$ boson, and the Higgs boson, $g_{ZZh}$. From Tables 6, 7,
and 8, there is almost no dependence on the sign of the bulk masses for the
coupling constants of zero modes of the charged leptons to the the zero mode
of the $Z$ boson, while there is large difference between the zero modes of
the right-handed and left-handed leptons to the 1st KK modes of the $Z$ and
$Z_{R}$ bosons. Some gauge coupling constants of the zero modes of the right-
handed or left-handed leptons to the 1st KK modes of the $Z^{\prime}$ bosons
$Z^{(1)}$ and $Z_{R}^{(1)}$ are larger than those of zero modes of the leptons
to the zero mode of the $Z$ boson.

In summary, for the $e^{-}e^{+}\to Zh$ process, some of the coupling constants
of the $Z^{\prime}$ bosons to the leptons and the Higgs boson are much larger
than the coupling constants of the $Z$ boson to the leptons and the Higgs
boson, so that even a center-of-mass energy $\sqrt{s}$ that is an order of
magnitude smaller than the masses of the $Z^{\prime}$ bosons can yield
experimentally verifiable predictions.

###  4.2 $e^{-}e^{+}\to\nu\bar{\nu}h$

Here we calculate the total cross sections of the $e^{-}e^{+}\to\nu\bar{\nu}h$
process given in Eq. (A.53), where if $\nu$ has no subscript, the contribution
of all flavors shall be considered. Since neutrinos are not observable
experimentally, we add up the contributions not only from the
$e^{-}e^{+}\to\nu_{e}\bar{\nu}_{e}h$ process but also from the
$e^{-}e^{+}\to\nu_{\mu}\bar{\nu}_{\mu}h$ and
$e^{-}e^{+}\to\nu_{\tau}\bar{\nu}_{\tau}h$ processes. In the SM, the
contributions to $\sigma(e^{-}e^{+}\to\nu_{\mu}\bar{\nu}_{\mu}h)$ and
$\sigma(e^{-}e^{+}\to\nu_{\tau}\bar{\nu}_{\tau}h)$ come only from the
processes $\sigma(e^{-}e^{+}\to Z\to Zh\to\nu_{\mu}\bar{\nu}_{\mu}h)$ and
$\sigma(e^{-}e^{+}\to Z\to Zh\to\nu_{\tau}\bar{\nu}_{\tau}h)$, respectively.
When we ignore the neutrino masses in the SM,
$\sum_{\ell=e,\mu,\tau}\sigma(e^{-}e^{+}\to Z\to
Zh\to\nu_{\ell}\bar{\nu}_{\ell}h)=3\sigma(e^{-}e^{+}\to Z\to
Zh\to\nu_{e}\bar{\nu}_{e}h)$. In the GHU model, the coupling constants of the
$Z$ boson to the neutrinos given in Table 7 are almost the same regardless of
neutrino flavor. Therefore, to a very good approximation, the following
relationship holds: $\sum_{\ell=e,\mu,\tau}\sigma(e^{-}e^{+}\to Z\to
Zh\to\nu_{\ell}\bar{\nu}_{\ell}h)\simeq 3\sigma(e^{-}e^{+}\to Z\to
Zh\to\nu_{e}\bar{\nu}_{e}h)$.

Figure 4: The $\sqrt{s}$ dependence of the total cross sections of the
$e^{-}e^{+}\to\nu\bar{\nu}h$ process with unpolarized and polarized electron
and positron beams in the SM and the GHU model is shown. The left figure shows
the $\sqrt{s}$ dependence of the total cross sections with unpolarized
electron and positron beams in the SM and the GHU model whose parameter sets
are $A_{\pm}$, $B_{\pm}$, $C_{\pm}$. The right figure shows the $\sqrt{s}$
dependence of the total cross sections with unpolarized and polarized electron
and positron beams in the SM and the GHU model whose parameter sets are
$A_{\pm}$. The energy ranges $\sqrt{s}$ in the left and right figures are
$\sqrt{s}=[200,21000]$ GeV and $\sqrt{s}=[200,15000]$ GeV, respectively. U, L,
R stand $(P_{e^{-}},P_{e^{+}})=(0,0),(-0.8,+0.3),(+0.8,-0.3)$, respectively.

In Figure 4, the total cross sections of the $e^{-}e^{+}\to\nu\bar{\nu}h$
process with unpolarized and polarized electron and positron beams in the SM
and the GHU model whose parameter sets are $A_{\pm}$, $B_{\pm}$, $C_{\pm}$ are
showed up to the energy scale $\sqrt{s}$ where resonances due to the KK gauge
boson can be confirmed. The left figure shows the $\sqrt{s}$ dependence of the
total cross sections with unpolarized electron and positron beams in the SM
and the GHU model whose parameter sets are $A_{\pm}$, $B_{\pm}$, $C_{\pm}$.
The right figure shows the $\sqrt{s}$ dependence of the total cross sections
with polarized electron and positron beams for the SM and the GHU model whose
parameter sets are $A_{\pm}$. In both the figures, the main contribution to
the peak around $\sqrt{s}\simeq 250$ GeV comes from the $e^{-}e^{+}\to Z\to
Zh\to\nu\bar{\nu}h$ process, and the main contributions to the peaks around
$O(10)$ TeV come from the $e^{-}e^{+}\to Z^{\prime}(=Z^{(1)},Z_{R}^{(1)})\to
Zh\to\nu\bar{\nu}h$ process for each parameter set. In our calculation, we
take into account the contribution to the cross section from the $W$ and $Z$
bosons and the 1st KK gauge bosons $W^{\pm(1)}$, $W_{R}^{\pm(1)}$, $Z^{(1)}$,
and $Z_{R}^{(1)}$ but the contribution from higher KK modes cannot be ignored
above $O(m_{\rm KK})$. We verify that for the $e^{-}e^{+}\to\nu\bar{\nu}h$
process the contribution from the KK gauge bosons is small, independent of the
KK masses and the initial polarization of the electron and the positron,
except around the $Z^{\prime}$ boson mass scale. Since the coupling constants
of the $W$ and $Z$ bosons to the left-handed leptons are different from that
of the $W$ and $Z$ bosons to the right-handed leptons, the cross section
depend on the initial polarizations of the electron and the positron.
Furthermore, since the parity violation of the $W$ boson to the leptons are
larger than that of the $Z$ boson to the leptons, the vector boson fusion
(VBF) process, in which the $W$ boson gives the main contribution, depends
more strongly on the initial polarizations than the $Zh$ process, in which the
$Z$ boson gives the main contribution. The coupling constants of the
$Z^{\prime}$ bosons to the leptons depend on the sign o the bulk masses. When
the bulk mass is negative, the coupling constants to the left-handed fermions
is larger, and when the bulk mass is positive, the coupling constant to the
right-handed fermions is larger. As a result, the magnitude of the cross
section by polarization at resonance energies strongly depends on the sign of
the bulk mass.

Figure 5: The $\sqrt{s}$ dependence of the total cross sections of the
$e^{-}e^{+}\to\nu\bar{\nu}h$ process with unpolarized electron and positron
beams is shown. The left figure shows the $\sqrt{s}$ dependence of the cross
sections for total, vector boson fusion (VBF) only, $Zh$ only, and
interference between VBF and $Zh$ processes only in the SM. The right figure
shows the $\sqrt{s}$ dependence of the cross sections for total, VBF from $Z$
boson, $Zh$ from $Z$ boson, and the others without VBF and $Zh$ only from $Z$
boson in the GHU model with the parameter set $A_{-}$. The others includes
such as VBF and $Zh$ including at least one KK mode. The energy ranges
$\sqrt{s}$ in the figures are $\sqrt{s}=[200,1000]$ GeV.

In Figure 5, the $\sqrt{s}$ dependence of the total cross sections of the
$e^{-}e^{+}\to\nu\bar{\nu}h$ process with unpolarized electron and positron
beams in the SM and the GHU model whose parameter set is $A_{-}$ is shown up
to $\sqrt{s}=1$ TeV. The contributions to the total cross sections from the
VBF, $Zh$, and interference (other) in the SM and the GHU model are decomposed
and displayed in the left and right figures, respectively. There is almost no
difference between the cross sections in the SM and the GHU model for
$\sqrt{s}\leq 1$ TeV. In both the SM and the GHU model, the main contribution
comes from the $Zh$ process below $\sqrt{s}\simeq 350$ GeV, while the main
contribution comes from the VBF process above $\sqrt{s}\simeq 350$ GeV, where
the contributions from the $Zh$ and VBF processes depend on the initial
polarizations of the electron and the positron.

Figure 6: The $\sqrt{s}$ dependence of the deviation from the SM in the GHU
model with six parameter sets $A_{\pm}$, $B_{\pm}$, $C_{\pm}$ for total cross
sections $\Delta_{\sigma}^{\nu\nu h}(P_{e^{-}},P_{e^{+}})$ is shown. The left
and right figures show the center-of-mass energies
$\sqrt{s}=[200,1000]\,\mbox{GeV}$ and $(P_{e^{-}},P_{e^{+}})=(-0.8,+0.3)$ and
$(+0.8,-0.3)$, respectively. The gray regions represent the $1\sigma$
statistical errors in the SM at each $\sqrt{s}$ with $1$ ab-1 by using the
decay mode of the Higgs boson $h$ to two photons $\gamma\gamma$, where the
branching ratio for the SM Higgs boson with $m_{H}=125$ GeV is given by
$\mbox{Br}(h\to\gamma\gamma)=2.27(1\pm 0.021)\times 10^{-3}$ [16].

In Figure 6, the $\sqrt{s}$ dependence of the deviation from the SM in the GHU
model with six parameter sets $A_{\pm}$, $B_{\pm}$, $C_{\pm}$ for the total
cross sections $\Delta_{\sigma}^{\nu\nu h}$ are shown, where
$\Delta_{\sigma}^{ffh}(P_{e^{-}},P_{e^{+}})$ is given in Eq. (A.55). The left
and right figures show the center-of-mass energies
$\sqrt{s}=[200,1000]\,\mbox{GeV}$ and $(P_{e^{-}},P_{e^{+}})=(-0.8,+0.3)$ and
$(+0.8,-0.3)$, respectively. The $1\sigma$ statistical errors are estimated in
the SM at each $\sqrt{s}$ with $1$ ab-1 by using the decay mode of the Higgs
boson $h$ to two photons $\gamma\gamma$, where the branching ratio for the SM
Higgs boson with $m_{H}=125$ GeV is given by
$\mbox{Br}(h\to\gamma\gamma)=2.27(1\pm 0.021)\times 10^{-3}$ [16]. The
branching ratio of the Higgs boson to the final state of the muon pair, whose
energy and momentum can be well observed experimentally with high precision,
is very small, where $\mbox{Br}(h\to\mu^{-}\mu^{+})=2.18(1\pm 0.017)\times
10^{-4}$ [16]. We refer to the decay of the Higgs boson to two photons, but
the analysis of the signal by hadron jets in the ILC experiment is currently
in progress. The mass reconstruction of the Higgs boson by using bottom quark
pairs and $\tau$ lepton pairs may be available, which have much larger
branching ratios of the Higgs boson, where $\mbox{Br}(h\to b\bar{b})=5.82(1\pm
0.013)\times 10^{-1}$ and $\mbox{Br}(h\to\tau^{-}\tau^{+})=6.27(1\pm
0.016)\times 10^{-2}$ [16]. If so, the deviation from the SM in the GHU model
can be explored by using the $e^{-}e^{+}\to\nu\bar{\nu}h$ process.

Figure 7: The left-right asymmetry of the $e^{-}e^{+}\to\nu\bar{\nu}h$ process
$A_{LR}^{\nu\nu h}(P_{e^{-}},P_{e^{+}})$ in the SM and the GHU model and the
deviation of the left-right asymmetry from the SM in the GHU model
$\Delta_{A_{LR}}^{\nu\nu h}(P_{e^{-}},P_{e^{+}})$ are shown. The left and
right figures show the $\sqrt{s}$ dependence of $A_{LR}^{\nu\nu
h}(P_{e^{-}},P_{e^{+}})$ and $\Delta_{A_{LR}}^{\nu\nu h}(P_{e^{-}},P_{e^{+}})$
with the polarization $(P_{e^{-}},P_{e^{+}})=(-0.8,+0.3)$ and the six
parameter sets $A_{\pm}$, $B_{\pm}$, $C_{\pm}$. The energy ranges $\sqrt{s}$
in the left and right figures are $\sqrt{s}=[200,3000]\,$GeV,
$\sqrt{s}=[200,1000]\,$GeV, respectively. The gray region represents the
statistical error explained in Table 6.

In Figure 7, the $\sqrt{s}$ dependence of the left-right asymmetry of the
$e^{-}e^{-}\to\nu\bar{\nu}h$ process $A_{LR}^{\nu\nu h}(P_{e^{-}},P_{e^{+}})$
and the deviation of the left-right asymmetry from the SM
$\Delta_{A_{LR}}^{\nu\nu h}(P_{e^{-}},P_{e^{+}})$ with
$(P_{e^{-}},P_{e^{+}})=(-0.8,+0.3)$ are shown, where $A_{LR}^{\nu\nu
h}(P_{e^{-}},P_{e^{+}})$ and $\Delta_{A_{LR}}^{\nu\nu h}(P_{e^{-}},P_{e^{+}})$
are given in Eqs. (A.56) and (A.58), respectively. From Eq. (A.57), when
$(P_{e^{-}},P_{e^{+}})=(\mp 0.8,\pm 0.3)$ and $\sigma_{LL}^{\nu\nu
h}\gg\sigma_{RR}^{\nu\nu h},\sigma_{LR}^{\nu\nu h},\sigma_{RL}^{\nu\nu h}$, we
find $A_{LR}^{\nu\nu h}(P_{e^{-}}=-0.8,P_{e^{+}}=+0.3)\simeq 0.887$. Thus, we
confirmed that when $\sqrt{s}$ is sufficiently large, the main contribution to
the $e^{-}e^{+}\to\nu\bar{\nu}h$ process comes almost exclusively from the VBF
processes. Furthermore, the deviation from the SM becomes smaller when
$\sqrt{s}$ is larger. Therefore, it it very difficult to confirm the deviation
from the SM by using the left-right asymmetry of the
$e^{-}e^{+}\to\nu\bar{\nu}h$ process.

###  4.3 $e^{-}e^{+}\to e^{-}e^{+}h$

Here we calculate the total cross section of the $e^{-}e^{+}\to e^{-}e^{+}h$
process given in Eq. (A.53). The cross section of the $e^{-}e^{+}\to
e^{-}e^{+}h$ process is smaller than that of the $e^{-}e^{+}\to\nu\bar{\nu}h$
process in all energy regions unless the initial polarization of the electron
(the positron) is extremely right-handed (left-handed). Therefore, this
process is usually a subdominant Higgs boson production process.

The following figures in this subsection are made by using exactly the same
formula and calculations as those in the previous subsection 4.2, except for
the difference between the final and intermediate state particles, and their
coupling constants. The explanation of the figures is basically the same as in
the previous section. We will omit duplicated explanations.

Figure 8: The $\sqrt{s}$ dependence of the total cross sections of the
$e^{-}e^{+}\to e^{-}e^{+}h$ process with unpolarized and polarized electron
and positron beams in the SM and the GHU model are shown. Other information is
the same as in Figure 4.

In Figure 8, the $\sqrt{s}$ dependence of the total cross sections of the
$e^{-}e^{+}\to e^{-}e^{+}h$ process with unpolarized and polarized electron
and positron beams in the SM and the GHU model are showed. The left figure
shows the $\sqrt{s}$ dependence of the total cross sections with unpolarized
electron and positron beams in the SM and the GHU model whose parameter sets
are $A_{\pm}$, $B_{\pm}$, $C_{\pm}$. The right figure shows the $\sqrt{s}$
dependence of the total cross sections with polarized electron and positron
beams in the SM and the GHU model whose parameter sets are $A_{\pm}$. In both
the figures, the main contribution to the peak around $\sqrt{s}\simeq 250$ GeV
comes from the $e^{-}e^{+}\to Z\to Zh\to e^{-}e^{+}h$ process, and the main
contributions to the peaks around $O(10)$ TeV come from the $e^{-}e^{+}\to
Z^{\prime}(=Z^{(1)},Z_{R}^{(1)})\to Zh\to e^{-}e^{+}h$ process for each
parameter set. We verify that the contribution to the $e^{-}e^{+}\to
e^{-}e^{+}h$ process from the KK gauge bosons is small, independent of the KK
masses and initial polarization of the electron and the positrons, except
around the $Z^{\prime}$ boson mass scale.

From Figures 4 and 8, we confirm that the cross section of the
$e^{-}e^{+}\to\nu\bar{\nu}h$ process is larger than that of the $e^{-}e^{+}\to
e^{-}e^{+}h$ process, unless the initial polarization is extremely right-
handed polarization since the coupling constants of the $W$ boson to the left-
handed leptons are larger than the coupling constants of the $Z$ boson to
leptons.

Figure 9: The $\sqrt{s}$ dependence of the total cross sections of the
$e^{-}e^{+}\to e^{-}e^{+}h$ process with unpolarized electron and positron
beams in the SM and the GHU model are shown for the left and right figures,
respectively. Other information is the same as in Figure 5.

In Figure 9, the $\sqrt{s}$ dependence of the total cross sections of the
$e^{-}e^{+}\to e^{-}e^{+}h$ process in the SM and the GHU model whose
parameter set is $A_{-}$ with unpolarized electron and positron beams are
shown up to $\sqrt{s}=1$ TeV. There is almost no difference between the cross
sections in the SM and the GHU model. In both the SM and the GHU model, the
main contribution comes from the $Zh$ process below $\sqrt{s}\simeq 500$ GeV,
while the main contribution comes from the VBF process above $\sqrt{s}\simeq
500$ GeV, where the contributions from the $Zh$ and VBF processes depend on
the initial polarizations of the electron and the positron.

Figure 10: The deviations from the SM in the GHU model with six parameter sets
$A_{\pm}$, $B_{\pm}$, $C_{\pm}$ for total cross sections
$\Delta_{\sigma}^{eeh}(P_{e^{-}},P_{e^{+}})$ are shown for
$\sqrt{s}=[200,1000]\,\mbox{GeV}$ and $(P_{e^{-}},P_{e^{+}})=(-0.8,+0.3)$ and
$(+0.8,-0.3)$, respectively. Other information is the same as in Figure 6.

In Figure 10, the $\sqrt{s}$ dependence of the deviation from the SM in the
GHU model with six parameter sets $A_{\pm}$, $B_{\pm}$, $C_{\pm}$ for the
total cross sections $\Delta_{\sigma}^{eeh}(P_{e^{-}},P_{e^{+}})$ are shown.
As the same as in Figure 6, the left and right figures show the center-of-mass
energies $\sqrt{s}=[200,1000]\,\mbox{GeV}$ and
$(P_{e^{-}},P_{e^{+}})=(-0.8,+0.3)$ and $(+0.8,-0.3)$, respectively. The
$1\sigma$ statistical errors are estimated in the SM at each $\sqrt{s}$ with
$1$ ab-1 by using the decay mode of the Higgs boson $h$ to two photons
$\gamma\gamma$, where the branching ratio for the SM Higgs boson with
$m_{H}=125$ GeV is given by $\mbox{Br}(h\to\gamma\gamma)=2.27(1\pm
0.021)\times 10^{-3}$ [16].

Figure 11: The left-right asymmetry of the $e^{-}e^{-}\to e^{-}e^{+}h$ process
and the deviation of the left-right asymmetry
$\Delta_{A_{LR}}^{eeh}(P_{e^{-}},P_{e^{+}})$ from the SM are shown. Other
information is the same as in Figure 7.

In Figure 11, the $\sqrt{s}$ dependence of the left-right asymmetry
$A_{LR}^{eeh}(P_{e^{-}},P_{e^{+}})$ and the deviation from the SM of the left-
right asymmetry $\Delta_{A_{LR}}^{eeh}(P_{e^{-}},P_{e^{+}})$ with
$(P_{e^{-}},P_{e^{+}})=(-0.8,+0.3)$ are shown, where
$A_{LR}^{eeh}(P_{e^{-}},P_{e^{+}})$ and
$\Delta_{A_{LR}}^{eeh}(P_{e^{-}},P_{e^{+}})$ are given in Eqs. (A.56) and
(A.58). Although the deviation from the SM of the left-right asymmetry is
quite large, but the number of events are small due to the small cross-
section. Therefore, it it very difficult to confirm the deviation from the SM
by using the left-right asymmetry of the $e^{-}e^{+}\to e^{-}e^{+}h$ process.

##  5 Summary and discussions

In this paper, we analyzed the three main single Higgs boson production
processes $e^{-}e^{+}\to Zh$, $e^{-}e^{+}\to\nu\bar{\nu}h$, and $e^{-}e^{+}\to
e^{-}e^{+}h$. For the Higgs strahlung process $e^{-}e^{+}\to Zh$, we showed
that for a parameter region satisfying the current experimental constraints
($m_{\rm KK}\geq 13$ TeV, $\theta_{H}\leq 0.10$), a maximum deviation from the
SM is about up to 6% for $\sqrt{s}=250$ GeV and that from the SM is about up
to 20% for $\sqrt{s}=500$ GeV, depending on the initial polarizations of the
electron and the positron. The deviation from the SM in the GHU model is
monotonically increasing with respect to $\sqrt{s}\lesssim 1$ TeV. By
observing the decay mode of the $Z$ boson to the muon pair for $\sqrt{s}=250$
GeV, $L_{\rm int}=1$ ab-1, and $(P_{e^{-}},P_{e^{+}})=(\mp 0.8,\pm 0.3)$, the
deviation from the SM is larger than 3$\sigma$ for $m_{\rm KK}\gtrsim 15$ TeV.
By observing the decay mode of the $Z$ boson to the muon pair for
$\sqrt{s}=500$ GeV, $L_{\rm int}=2$ ab-1, and $(P_{e^{-}},P_{e^{+}})=(\mp
0.8,\pm 0.3)$, the deviation from the SM is larger than 3$\sigma$ for $m_{\rm
KK}\gtrsim 20$ TeV. We also showed that the sign of the bulk mass of the
electron multiplet in the GHU model can be determined by examining the
deviation of the left-right symmetry of the $e^{-}e^{+}\to Zh$ process from
the SM. The $e^{-}e^{+}\to\nu\bar{\nu}h$ process gives the main contribution
to the single Higgs boson production at sufficiently high $\sqrt{s}$, but in
the GHU model a sufficiently large deviation from the SM can only be observed
at energy scales close to the $Z^{\prime}$ boson mass.

Although we focuses on single Higgs boson production processes in this paper,
double Higgs boson production processes are important to determine the self-
coupling constant of the Higgs boson because we need to confirm whether the
currently observed Higgs boson fully satisfies the properties of the Higgs
boson predicted by the SM or not. In the GHU model, it is shown in Ref. [32]
that the coupling constants of the Higgs boson to the $W$ and $Z$ bosons are
very close to those in the SM, while the cubic and quartic self-coupling
constants of the Higgs boson in the GHU model are deviated from those in the
SM for about 8% and 30%, respectively. According to Ref. [110], the total
cross section of the main double Higgs boson production $e^{-}e^{+}\to Zhh$ at
$\sqrt{s}=500$ GeV is about $0.1$ fb, where the main contribution comes from
the $e^{-}e^{+}\to Zh\to Zhh$ process. Even if almost all the decay modes of
the Higgs boson could be used in the analysis, the 1 $\sigma$ statistical
error of the $e^{-}e^{+}\to Zhh$ process is about $5$% by the ILC experiment
with $\sqrt{s}=500$ GeV and $L_{\rm int}=4$ ab-1. Therefore, it is difficult
to check the deviation from the SM in the GHU model at the ILC experiment. The
planned experiments at higher $\sqrt{s}$, such as the CLIC experiment, may
confirm the deviation of the self-coupling constant of the Higgs boson from
the SM by using a double Higgs boson production process
$e^{-}e^{+}\to\nu_{e}\bar{\nu}_{e}hh$ because the cross section of the process
becomes larger when $\sqrt{s}$ becomes larger, where the main contribution
comes from the
$e^{-}e^{+}\to\nu_{e}\bar{\nu}_{e}W^{+}W^{-}\to\nu_{e}\bar{\nu}_{e}h\to\nu_{e}\bar{\nu}_{e}hh$
process.

We comment on electroweak precision measurements. In this paper, we focus on
the search for the GHU model by using the single Higgs boson production
processes. By comparing the exploration region of the KK mass in the GHU model
by using the single Higgs boson production processes with that by using the
fermion pair production processes analyzed in Ref. [33], we find that fermion
pair production processes can be explored to higher KK mass scale. Therefore,
the first sign of the new physics in the ILC experiment is expected to be
found not in the Higgs boson production process but in the fermion pair
production processes because the cross section in the electron-positron
collider experiments is larger for the fermion pair production processes than
for the single Higgs boson production processes. This analysis of the Higgs
boson production process provides one important piece of information to
distinguish the GHU model from other new physics models.

## Acknowledgments

The authors would like to thank Yutaka Hosotani for useful discussions and
comments. One of the authors (N.Y.) would like to thank Cheng-Wei Chiang for
valuable comments. This work was supported in part by European Regional
Development Fund-Project Engineering Applications of Microworld Physics (No.
CZ.02.1.01/0.0/0.0/16_019/0000766) (Y.O.), and the Ministry of Science and
Technology of Taiwan under Grant No. MOST-111-2811-M-002-047-MY2 (N.Y.).

##  Appendix A Cross section

In this section, we give the formulas necessary to calculate observables of
the $e^{-}e^{+}\to Zh$ and $e^{-}e^{+}\to f\bar{f}h$ processes, where
$f\bar{f}$ stands for a pair of fermions such as the electron and the positron
($e^{-}e^{+}$).

###  A.1 Cross section of $e^{-}e^{+}\to Zh$

In the SM, the contribution to the cross section of $e^{-}e^{+}\to Zh$ comes
from the $e^{-}e^{+}\to Z{}\to Zh$ process at tree level. In the GHU model,
additional contributions come from not only the $e^{-}e^{+}\to Z^{\prime}\to
Zh$ process but also the interference between the $e^{-}e^{+}\to Z\to Zh$
process and the $e^{-}e^{+}\to Z^{\prime}\to Zh$ process.

####  A.1.1 Amplitude of $e^{-}e^{+}\to Zh$

The s-channel production process of $Zh$ via vector bosons
$V_{a}=Z,Z^{\prime}$ is given by

$\displaystyle
e_{X}^{-}(p_{e^{-}})e_{\overline{X}}^{+}(p_{e^{+}})\longrightarrow
V_{a}(q=p_{e^{-}}+p_{e^{+}}=p_{Z}+p_{h})\longrightarrow Z(p_{Z})h(p_{h}),$
(A.1)

where $X=L,R$, $\overline{L}=R$, $\overline{R}=L$. The amplitude of the
$e^{-}e^{+}\to Zh$ process is given by

$\displaystyle{\cal M}^{X}$
$\displaystyle=J_{\mu}^{X}\epsilon^{\mu*}(p_{Z})Q_{X},$ (A.2)

where $\epsilon^{\mu}$ is a polarization vector of the finial state of the $Z$
boson, $J^{X}_{\mu}$ is the fermion current defined as

$\displaystyle J^{X}_{\mu}:=\bar{v}(p_{e^{+}})\gamma_{\mu}P_{X}u(p_{e^{-}}),$
(A.3)

and

$\displaystyle
Q_{X}:=\sum_{a}\frac{g_{V_{a}e}^{X}g_{V_{a}Zh}}{(q^{2}-m_{V_{a}}^{2})+im_{V_{a}}\Gamma_{V_{a}}},$
(A.4)

where $X=L,R$; $P_{R/L}=(1\pm\gamma_{5})/2$; $g_{V_{a}e}^{L}$ and
$g_{V_{a}e}^{R}$ are the coupling constants of the left- and right-handed
electron (the right- and left-handed positron) to the gauge boson $V_{a}$,
respectively; $g_{V_{a}Zh}$ is the coupling constant of the gauge bosons
$V_{a}$ and $Z$ to the Higgs boson $h$; $m_{V_{a}}$ and $\Gamma_{V_{a}}$ are
the mass and the total decay width of $V_{a}$.

####  A.1.2 Kinematics of $e^{-}e^{+}\to Zh$

We use the following four moments for the initial state of the electron
($e^{-}$) and the positron ($e^{+}$), and the final state of the $Z$ boson
($Z$) and the Higgs boson ($h$) in the center-of-mass system:

$\displaystyle p_{e^{-}}=\frac{\sqrt{s}}{2}(1,0,0,+1),\ \
p_{e^{+}}=\frac{\sqrt{s}}{2}(1,0,0,-1),\ \ $ $\displaystyle
p_{Z}=\left(E_{Z},+\kappa\sin\theta,0,+\kappa\cos\theta\right),\ \
p_{h}=\left(E_{h},-\kappa\sin\theta,0,-\kappa\cos\theta\right),$
$\displaystyle q=(\sqrt{s},0,0,0),$ (A.5)

where we neglect the mass of electron; $m_{Z}$ and $m_{h}$ stand for the
masses of the $Z$ boson and the Higgs boson, respectively;

$\displaystyle\kappa:=\frac{\sqrt{[s-(m_{Z}+m_{h})^{2}][s-(m_{Z}-m_{h})^{2}]}}{2\sqrt{s}}=\frac{\lambda^{1/2}(s,m_{Z}^{2},m_{h}^{2})}{2\sqrt{s}},$
$\displaystyle\lambda(\alpha,\beta,\gamma):=\alpha^{2}+\beta^{2}+\gamma^{2}-2\alpha\beta-2\alpha\gamma-2\beta\gamma=(\alpha-\beta-\gamma)^{2}-4\beta\gamma$
(A.6)

From the above, we find

$\displaystyle(p_{e^{-}}\cdot
p_{Z})=\frac{s}{2}\left(E_{Z}-p\cos\theta\right),\ \ (p_{e^{+}}\cdot
p_{Z})=\frac{s}{2}\left(E_{Z}+p\cos\theta\right).$ (A.7)

The Mandelstam variables are given by

$\displaystyle s=(p_{e^{-}}+p_{e^{+}})^{2}=(p_{Z}+p_{h})^{2}=q^{2},\ \
t=(p_{e^{-}}-p_{Z})^{2}=(p_{e^{+}}-p_{h})^{2}=m_{Z}^{2}-\sqrt{s}(E_{Z}-\kappa\cos\theta),$
$\displaystyle
u=(p_{e^{-}}-p_{h})^{2}=(p_{e^{+}}-p_{Z})^{2}=m_{Z}^{2}-\sqrt{s}(E_{Z}+\kappa\cos\theta),\
\ s+t+u=m_{Z}^{2}+m_{h}^{2}.$ (A.8)

####  A.1.3 Cross section of $e^{-}e^{+}\to Zh$

For the $e_{X}^{-}e_{\overline{X}}^{+}\to Zh$ process, when only vector bosons
are involved in the intermediate states, the cross section of the initial
states of of the polarized electron and positron is given by

$\displaystyle\frac{d\sigma^{Zh}}{d\cos\theta}(P_{e^{-}},P_{e^{+}},\cos\theta)$
$\displaystyle=\frac{1}{4}\biggl{\\{}(1-P_{e^{-}})(1+P_{e^{+}})\frac{d\sigma_{L}^{Zh}}{d\cos\theta}(\cos\theta)+(1+P_{e^{-}})(1-P_{e^{+}})\frac{d\sigma_{R}^{Zh}}{d\cos\theta}(\cos\theta)\biggr{\\}},$
(A.9)

where $P_{e^{-}}$ and $P_{e^{+}}$ are the initial polarizations of the
electron and the positron, and $\sigma_{L}^{Zh}$ and $\sigma_{R}^{Zh}$ are
cross sections of $e^{-}_{L}e^{+}_{R}\to Zh$ and $e^{-}_{R}e^{+}_{L}\to Zh$,
respectively:

$\displaystyle\frac{d\sigma_{X}^{Zh}}{d\cos\theta}(\cos\theta):=\frac{d\sigma}{d\cos\theta}(e^{-}_{X}e^{+}_{\overline{X}}\to
Zh)$ $\displaystyle=\frac{\kappa}{16\pi s\sqrt{s}}|{\cal M}^{X}|^{2},$ (A.10)

where the formula for the differential cross section in the SM is given in
e.g., Refs. [111, 16]. The amplitude of the $e_{X}^{-}e_{\overline{X}}^{+}\to
Zh$ process can be obtained by using the standard method of calculation given
in e.g., Ref. [111] as

$\displaystyle\left|{\cal M}^{X}\right|^{2}$
$\displaystyle=|Q_{X}|^{2}s\frac{2m_{Z}^{2}+\kappa^{2}\sin^{2}\theta}{m_{Z}^{2}},$
(A.11)

where $Q_{X=L,R}$ are defined in Eq. (A.4). Therefore, we find the
differential cross section of the $e^{-}(P_{e^{-}})e^{+}(P_{e^{+}})\to Zh$
process with the initial polarizations $P_{e^{\pm}}$ as

$\displaystyle\frac{d\sigma^{Zh}}{d\cos\theta}(P_{e^{-}},P_{e^{+}},\cos\theta)$
$\displaystyle=\frac{1}{4}\left((1-P_{e^{-}})(1+P_{e^{+}})|Q_{L}|^{2}+(1+P_{e^{-}})(1-P_{e^{+}})|Q_{R}|^{2}\right)\frac{\kappa}{16\pi\sqrt{s}}\frac{2m_{Z}^{2}+\kappa^{2}\sin^{2}\theta}{m_{Z}^{2}}.$
(A.12)

The total cross section of the $e^{-}e^{+}\to Zh$ process with the initial
polarizations can be defined by integrating the differential cross section in
Eq. (A.9) with the angle $\theta$

$\displaystyle\sigma^{Zh}(P_{e^{-}},P_{e^{+}}):=\int_{-1}^{1}\frac{d\sigma^{Zh}}{d\cos\theta}(P_{e^{-}},P_{e^{+}},\cos\theta)d\cos\theta,$
(A.13)

where the minimum and maximal values of $\cos\theta$ are determined by each
detector and we cannot use date near $\cos\theta\simeq\pm 1$. By using Eq.
(A.12), the total cross section of $e^{-}e^{+}\to Zh$ is given by

$\displaystyle\sigma^{Zh}(P_{e^{-}},P_{e^{+}})=\frac{1}{4}(1-P_{e^{-}})(1+P_{e^{+}})\sigma_{L}^{Zh}+\frac{1}{4}(1+P_{e^{-}})(1-P_{e^{+}})\sigma_{R}^{Zh},$
(A.14)

where

$\displaystyle\sigma_{X}^{Zh}$
$\displaystyle:=\int_{-1}^{1}\frac{d\sigma_{X}^{Zh}}{d\cos\theta}(\cos\theta)d\cos\theta=|Q_{X}|^{2}\frac{\kappa}{12\pi\sqrt{s}}\frac{3m_{Z}^{2}+\kappa^{2}}{m_{Z}^{2}},$
(A.15)

where $X=L,R$; $Q_{X}$ is given in Eq. (A.4).

The statistical error of the cross section $\sigma^{Zh}(P_{e^{-}},P_{e^{+}})$
is given by

$\displaystyle\Delta\sigma^{Zh}(P_{e^{-}},P_{e^{+}})=\frac{\sigma^{Zh}(P_{e^{-}},P_{e^{+}})}{\sqrt{N^{Zh}}},\
\ N^{Zh}=L_{\rm int}\cdot\sigma^{Zh}(P_{e^{-}},P_{e^{+}}),$ (A.16)

where $L_{\rm int}$ is an integrated luminosity, and $N^{Zh}$ is the number of
events for the $e^{-}e^{+}\to Zh$ process. Note that the $Z$ boson and the
Higgs boson cannot observed directly, so we need to choose the decay modes of
the $Z$ boson and/or the Higgs boson, and then the available number of events
must be $N^{Zh}$ multiplied by the branching ratio of each selected decay
mode. The amount of the deviation of the cross section of the $e^{-}e^{+}\to
Zh$ process from the SM in the GHU model is given by

$\displaystyle\Delta_{\sigma}^{Zh}(P_{e^{-}},P_{e^{+}}):=\frac{\left[\sigma^{Zh}(P_{e^{-}},P_{e^{+}})\right]_{\rm
GHU}}{\left[\sigma^{Zh}(P_{e^{-}},P_{e^{+}})\right]_{\rm SM}}-1,$ (A.17)

where $\left[\sigma^{Zh}(P_{e^{-}},P_{e^{+}})\right]_{\rm GHU}$ and
$\left[\sigma^{Zh}(P_{e^{-}},P_{e^{+}})\right]_{\rm SM}$ stand for the cross
sections of the $e^{-}e^{+}\to Zh$ process in the SM and the GHU model,
respectively. The same notation is used for other cases in the followings.

####  A.1.4 Left-right asymmetry of $e^{-}e^{+}\to Zh$

As the same in fermion pair productions, we can define a left-right asymmetry
[93, 86, 87, 33, 36] of the $e^{-}e^{+}\to Zh$ process as

$\displaystyle
A_{LR}^{Zh}:=\frac{\sigma_{L}^{Zh}-\sigma_{R}^{Zh}}{\sigma_{L}^{Zh}+\sigma_{R}^{Zh}}=\frac{|Q_{L}|^{2}-|Q_{R}|^{2}}{|Q_{L}|^{2}+|Q_{R}|^{2}},$
(A.18)

where $\sigma_{X}^{Zh}$ and $Q_{X}$ ($X=L,R$) are given in Eq. (A.15) and Eq.
(A.4), respectively.

An observable left-right asymmetry is defined as

$\displaystyle
A_{LR}^{Zh}(P_{e^{-}},P_{e^{+}}):=\frac{\sigma^{Zh}(P_{e^{-}},P_{e^{+}})-\sigma^{Zh}(-P_{e^{-}},-P_{e^{+}})}{\sigma^{Zh}(P_{e^{-}},P_{e^{+}})+\sigma^{Zh}(-P_{e^{-}},-P_{e^{+}})}$
(A.19)

for $P_{e^{-}}<0$ and $|P_{e^{-}}|>|P_{e^{+}}|$. From Eq. (A.12), we find

$\displaystyle A_{LR}^{Zh}(P_{e^{-}},P_{e^{+}},\cos\theta)$
$\displaystyle=-\frac{P_{e^{-}}-P_{e^{+}}}{1-P_{e^{-}}P_{e^{+}}}\frac{|Q_{L}|^{2}-|Q_{R}|^{2}}{|Q_{L}|^{2}+|Q_{R}|^{2}}.$
(A.20)

The relation between the theoretically defined left-right asymmetry
$A_{LR}^{Zh}$ given in Eq. (A.18) and the observable left-right asymmetry
$A_{LR}^{Zh}(P_{e^{-}},P_{e^{+}})$ given in Eq. (A.20) is given by

$\displaystyle A_{LR}^{Zh}=\frac{1}{-P_{\rm
eff}}A_{LR}^{Zh}(P_{e^{-}},P_{e^{+}}),\ \ P_{\rm
eff}:=\frac{P_{e^{-}}-P_{e^{+}}}{1-P_{e^{-}}P_{e^{+}}}.$ (A.21)

The statistical error of the left-right asymmetry is given by

$\displaystyle\Delta A_{LR}^{Zh}$
$\displaystyle=2\frac{\sqrt{N_{L}^{Zh}N_{R}^{Zh}}\left(\sqrt{N_{L}^{Zh}}+\sqrt{N_{R}^{Zh}}\right)}{(N_{L}^{Zh}+N_{R}^{Zh})^{2}},$
(A.22)

where $N_{L}^{Zh}=L_{\rm int}\,\sigma^{Zh}(P_{e^{-}},P_{e^{+}})$ and
$N_{R}^{Zh}=L_{\rm int}\,\sigma^{Zh}(-P_{e^{-}},-P_{e^{+}})$ are the numbers
of the events for $P_{e^{-}}<0$ and $|P_{e^{-}}|>|P_{e^{+}}|$. The amount of
the deviation from the SM in Eq. (A.18) is characterized by

$\displaystyle\Delta_{A_{LR}}^{Zh}$
$\displaystyle:=\frac{\left[A_{LR}^{Zh}\right]_{\rm
GHU}}{\left[A_{LR}^{Zh}\right]_{\rm SM}}-1.$ (A.23)

###  A.2 Cross section of $e^{-}e^{+}\to f\bar{f}h$

The cross sections of the $e^{-}e^{+}\to\nu\bar{\nu}h$ process via the
$W^{\pm}$ bosons and of the $e^{-}e^{+}\to e^{-}e^{+}h$ process via the $Z$
bosons are estimated in Refs. [112, 113, 114]. When we consider the cross
section of the $e^{-}e^{+}\to f\bar{f}h$ processes
$(f\bar{f}=\nu\bar{\nu},e^{-}e^{+})$ in the SM, we need to take into account
not only the vector boson fusion process $e^{-}e^{+}\to f\bar{f}VV\to
f\bar{f}h$ but also the $Zh$ process $e^{-}e^{+}\to Z\to Zh\to f\bar{f}h$. In
the GHU model, additional contributions come from the $W^{\prime}$ and
$Z^{\prime}$ bosons.

####  A.2.1 Amplitude of $e^{-}e^{+}\to f\bar{f}VV\to f\bar{f}h$

The Higgs boson production processes via vector gauge bosons
$V_{a}=W^{(\prime)\pm}$ or $Z^{(\prime)}$ are given by

$\displaystyle
e_{X}^{-}(p_{1})e_{\overline{Y}}^{+}(p_{2})\longrightarrow\nu_{e}(p_{3})\bar{\nu}_{e}(p_{4})W^{(\prime)-}(q_{1}=p_{1}-p_{3})W^{(\prime)+}(q_{2}=p_{2}-p_{4})\longrightarrow\nu_{e}(p_{3})\bar{\nu}_{e}(p_{4})h(p_{h})\,$
$\displaystyle e_{X}^{-}(p_{1})e_{\overline{Y}}^{+}(p_{2})\longrightarrow
e^{-}(p_{3})e^{+}(p_{4})Z^{(\prime)}(q_{1}=p_{1}-p_{3})Z^{(\prime)}(q_{2}=p_{2}-p_{4})\longrightarrow
e^{-}(p_{3})e^{+}(p_{4})h(p_{h}),$ (A.24)

where $X,Y=L,R$; $\overline{L}=R$, $\overline{R}=L$; $p_{1}$ and $p_{2}$ are
the momenta of the initial states of electron and positron; $p_{3}$ and
$p_{4}$ are the momenta of the final states of fermions; and $p_{h}$ is the
momenta of the Higgs boson. The amplitude is given by

$\displaystyle{\cal M}_{VVh}^{XY}$
$\displaystyle=\sum_{V_{a},V_{b}=\\{W^{(\prime)}\\}\ \mbox{or}\
\\{Z^{(\prime)}\\}}g_{V_{a}V_{b}h}g_{V_{a}e}^{X}g_{V_{b}e}^{Y}g^{\mu\nu}{J_{\mu\nu}^{XY}}\frac{1}{(q_{1}^{2}-m_{V_{a}}^{2})+im_{V_{a}}\Gamma_{V_{a}}}\frac{1}{(q_{2}^{2}-m_{V_{b}}^{2})+im_{V_{b}}\Gamma_{V_{b}}},$
(A.25)

where we defined

$\displaystyle
J_{\mu\nu}^{XY}:=\bar{u}(p_{3})\gamma_{\mu}P_{X}u(p_{1})\bar{v}(p_{2})\gamma_{\nu}P_{Y}v(p_{4}),$
(A.26)

and $P_{R/L}=(1\pm\gamma_{5})/2$. We neglected electron and neutrino masses.

After some tedious calculations, we obtain

$\displaystyle\sum_{\rm spin}\left|{\cal M}_{VVh}^{XY}\right|^{2}$
$\displaystyle=16\sum_{V_{a},V_{b},V_{c},V_{d}=\\{W^{(\prime)}\\}\ \mbox{or}\
\\{Z^{(\prime)}\\}}g_{V_{a}V_{b}h}g_{V_{c}V_{d}h}^{*}\times\left\\{\begin{array}[]{cc}g_{V_{a}e}^{X}g_{V_{b}e}^{X}g_{V_{c}e}^{X*}g_{V_{d}e}^{X*}(p_{1}\cdot
p_{2})(p_{3}\cdot p_{4})&\mbox{for}\ X=Y\\\
g_{V_{a}e}^{X}g_{V_{b}e}^{Y}g_{V_{c}e}^{X*}g_{V_{d}e}^{Y*}(p_{1}\cdot
p_{4})(p_{2}\cdot p_{3})&\mbox{for}\ X\not=Y\\\ \end{array}\right\\}$ (A.29)
$\displaystyle\ \
\times\frac{1}{(q_{1}^{2}-m_{V_{a}}^{2})+im_{V_{a}}\Gamma_{V_{a}}}\frac{1}{(q_{2}^{2}-m_{V_{b}}^{2})+im_{V_{b}}\Gamma_{V_{b}}}\frac{1}{(q_{1}^{2}-m_{V_{c}}^{2})-im_{V_{c}}\Gamma_{V_{c}}}\frac{1}{(q_{2}^{2}-m_{V_{d}}^{2})-im_{V_{d}}\Gamma_{V_{d}}}.$
(A.30)

####  A.2.2 Amplitude of $e^{-}e^{+}\to Z^{(\prime)}\to Z^{(\prime)}h\to
f\bar{f}h$

The Higgs boson production processes via neutral gauge bosons
$V_{a}=Z,Z^{\prime}$ are given by

$\displaystyle e_{X}^{-}(p_{1})e_{\overline{Y}}^{+}(p_{2})\longrightarrow
Z^{(\prime)}(q_{1}^{\prime}=p_{1}+p_{2})\longrightarrow
Z^{(\prime)}(q_{2}^{\prime}=p_{3}+p_{4})h(p_{h})\longrightarrow\nu(p_{3})\bar{\nu}(p_{4})h(p_{h}),$
$\displaystyle e_{X}^{-}(p_{1})e_{\overline{Y}}^{+}(p_{2})\longrightarrow
Z^{(\prime)}(q_{1}^{\prime}=p_{1}+p_{2})\longrightarrow
Z^{(\prime)}(q_{2}^{\prime}=p_{3}+p_{4})h(p_{h})\longrightarrow
e^{-}(p_{3})e^{+}(p_{4})h(p_{h}),$ (A.31)

The amplitude is given by

$\displaystyle{\cal M}_{Zh}^{XY}$
$\displaystyle=\sum_{V_{a},V_{b}=\\{Z^{(\prime)}\\}}g_{V_{a}V_{b}h}g_{V_{a}e}^{X}g_{V_{b}e}^{Y}g^{\mu\nu}{J_{\mu\nu}^{\prime
XY}}\frac{1}{(q_{1}^{\prime
2}-m_{V_{a}}^{2})+im_{V_{a}}\Gamma_{V_{a}}}\frac{1}{(q_{2}^{\prime
2}-m_{V_{b}}^{2})+im_{V_{b}}\Gamma_{V_{b}}},$ (A.32)

where we define $J_{\mu\nu}^{\prime XY}$ $(X,Y=L,R)$ as

$\displaystyle J_{\mu\nu}^{\prime
XY}=\bar{v}(p_{2})\gamma_{\mu}P_{X}u(p_{1})\bar{u}(p_{3})\gamma_{\nu}P_{Y}v(p_{4}).$
(A.33)

By the same calculation as for the square of the amplitude given in Eq.
(A.30), we find

$\displaystyle\sum_{\rm spin}\left|{\cal M}_{Zh}^{XY}\right|^{2}$
$\displaystyle=16\sum_{V_{a},V_{b},V_{c},V_{d}=\\{Z^{(\prime)}\\}}g_{V_{a}V_{b}h}g_{V_{c}V_{d}h}^{*}\times\left\\{\begin{array}[]{cc}g_{V_{a}e}^{X}g_{V_{b}e}^{X}g_{V_{c}e}^{X*}g_{V_{d}e}^{X*}(p_{1}\cdot
p_{3})(p_{2}\cdot p_{4})&\mbox{for}\ X=Y\\\
g_{V_{a}e}^{X}g_{V_{b}e}^{Y}g_{V_{c}e}^{X*}g_{V_{d}e}^{Y*}(p_{1}\cdot
p_{4})(p_{2}\cdot p_{3})&\mbox{for}\ X\not=Y\\\ \end{array}\right\\}$ (A.36)
$\displaystyle\ \ \times\frac{1}{(q_{1}^{\prime
2}-m_{V_{a}}^{2})+im_{V_{a}}\Gamma_{V_{a}}}\frac{1}{(q_{2}^{\prime
2}-m_{V_{b}}^{2})+im_{V_{b}}\Gamma_{V_{b}}}\frac{1}{(q_{1}^{\prime
2}-m_{V_{c}}^{2})-im_{V_{c}}\Gamma_{V_{c}}}\frac{1}{(q_{2}^{\prime
2}-m_{V_{d}}^{2})-im_{V_{d}}\Gamma_{V_{d}}},$ (A.37)

where when the fermion pair in the final state is $e^{-}e^{+}$,
$g_{V_{b}e}^{L/R}$ and $g_{V_{d}e}^{L/R}$ are gauge coupling constants of the
left- or right-handed electron, while when the fermion pair in the final state
is $\nu_{e}\bar{\nu}_{e}$, $g_{V_{b}e}^{L/R}$ and $g_{V_{d}e}^{L/R}$ are gauge
coupling constants of the left-handed or right-handed neutrinos The
$g_{V_{a}e}^{L/R}$ and $g_{V_{c}e}^{L/R}$ are always the gauge coupling
constants of the left-handed or right-handed electron.

####  A.2.3 Interference of $e^{-}e^{+}\to f\bar{f}VV\to f\bar{f}h$ and
$e^{-}e^{+}\to Z^{(\prime)}\to Z^{(\prime)}h\to f\bar{f}h$

The square of the amplitude of the $e_{X}^{+}e_{Y}^{-}\to f\bar{f}h$ process
is given by

$\displaystyle\left|{\cal M}^{XY}\right|^{2}=\left|{\cal M}_{VVh}^{XY}+{\cal
M}_{Zh}^{XY}\right|^{2}=\left|{\cal M}_{VVh}^{XY}\right|^{2}+\left|{\cal
M}_{Zh}^{XY}\right|^{2}+2\mbox{Re}\left({\cal M}_{VVh}^{XY}{\cal
M}_{Zh}^{XY*}\right).$ (A.38)

The last term stands for the interference term between $e^{-}e^{+}\to
f\bar{f}VV\to f\bar{f}h$ and $e^{-}e^{+}\to Z^{(\prime)}\to Z^{(\prime)}h\to
f\bar{f}h$. This term also need to be calculated.

As the same as Eqs. (A.25) and (A.32), after some tedious calculations, we
find

$\displaystyle 2\mbox{Re}\left[\sum_{\rm spin}{\cal M}_{VVh}{\cal
M}_{Zh}^{*}\right]=$
$\displaystyle-16\mbox{Re}\Bigg{[}\sum_{V_{a},V_{b}=\\{W^{(\prime)}\\}\
\mbox{or}\
\\{Z^{(\prime)}\\}}\sum_{V_{c},V_{d}=\\{Z^{(\prime)}\\}}g_{V_{a}V_{b}h}g_{V_{c}^{*}V_{d}h}^{*}$
$\displaystyle\times\left\\{\begin{array}[]{cl}g_{V_{a}e}^{X}g_{V_{b}e}^{X}g_{V_{c}e}^{X*}g_{V_{d}e}^{X*}A+ig_{V_{a}e}^{X}g_{V_{b}e}^{X}g_{V_{c}e}^{X*}g_{V_{d}e}^{X*}B&\mbox{for}\
X=Y=L\\\
g_{V_{a}e}^{X}g_{V_{b}e}^{X}g_{V_{c}e}^{X*}g_{V_{d}e}^{X*}A-ig_{V_{a}e}^{X}g_{V_{b}e}^{X}g_{V_{c}e}^{X*}g_{V_{d}e}^{X*}B&\mbox{for}\
X=Y=R\\\ 0&\mbox{for}\ X\not=Y\\\ \end{array}\right\\}$ (A.42)
$\displaystyle\times\frac{1}{(q_{1}^{2}-m_{V_{a}}^{2})+im_{V_{a}}\Gamma_{V_{a}}}\frac{1}{(q_{2}^{2}-m_{V_{b}}^{2})+im_{V_{b}}\Gamma_{V_{b}}}\frac{1}{(q_{1}^{\prime
2}-m_{V_{c}}^{2})-im_{V_{c}}\Gamma_{V_{c}}}\frac{1}{(q_{2}^{\prime
2}-m_{V_{d}}^{2})-im_{V_{d}}\Gamma_{V_{d}}}\Bigg{]},$ (A.43)

where

$\displaystyle A:=\left(p_{1}\cdot p_{4}\right)\left(p_{2}\cdot
p_{3}\right)+\left(p_{1}\cdot p_{3}\right)\left(p_{2}\cdot
p_{4}\right)-\left(p_{1}\cdot p_{2}\right)\left(p_{3}\cdot p_{4}\right),\ \
B:=p_{1}^{\alpha}p_{2}^{\beta}p_{3}^{\gamma}p_{4}^{\delta}\epsilon_{\gamma\beta\alpha\delta}.$
(A.44)

####  A.2.4 Kinematics of $e^{-}e^{+}\to f\bar{f}h$

In the center-of-mass system of the initial electron and positron, the initial
energies are $E_{1}=E_{2}=\sqrt{\hat{s}}/2$, where $\hat{s}$ is the center-of-
mass energy for the initial electron and positron. When we take the same
convention in Ref. [113], we define $\eta$ and $\zeta$ as

$\displaystyle E_{3}=(1-\eta)\frac{1}{2}\sqrt{\hat{s}},\ \
E_{4}=(1-\zeta)\frac{1}{2}\sqrt{\hat{s}},$ (A.45)

so that the energy of the Higgs boson is

$\displaystyle E_{h}=(\eta+\zeta)\frac{1}{2}\sqrt{\hat{s}}.$ (A.46)

The final state momenta lie in a plane and

$\displaystyle({\bf p_{h}})^{2}=({\bf p}_{3}+{\bf
p}_{4})^{2}=E_{h}^{2}-m_{h}^{2}.$ (A.47)

We define the angle of $\theta$ by

$\displaystyle\cos\theta=-\hat{\bf p}_{3}\cdot\hat{\bf
p}_{4}=1-2\frac{\zeta\eta-\frac{m_{h}^{2}}{\hat{s}}}{(1-\zeta)(1-\eta)}.$
(A.48)

The kinematic boundaries occur for $\cos\theta=\pm 1$, and the allowed region
must be satisfied as

$\displaystyle\zeta\eta\geq\frac{m_{h}^{2}}{\hat{s}},\ \ \zeta+\eta\leq
1+\frac{m_{h}^{2}}{\hat{s}}.$ (A.49)

To specify the final state in the center-of-mass system of the initial
electron and positron, we take the following basis:

$\displaystyle\hat{\bf p}_{1}=\hat{\bf x},\ \ \hat{\bf p}_{2}=-\hat{\bf x},\ \
\hat{\bf p}_{3}=\hat{\bf x}\cos\alpha\cos\beta+\hat{\bf y}\sin\alpha-\hat{\bf
z}\sin\beta\cos\alpha,$ $\displaystyle\hat{\bf p}_{4}=-\hat{\bf
x}\cos(\alpha-\theta)\cos\beta-\hat{\bf y}\sin(\alpha-\theta)+\hat{\bf
z}\sin\beta\cos(\alpha-\theta).$ (A.50)

The Lorentz scalars are given as

$\displaystyle(p_{1}\cdot p_{2})=\frac{1}{2}\hat{s},\ \ (p_{2}\cdot
p_{3})=\frac{1}{4}(1-\eta)(1-\cos\alpha\sin\beta)\hat{s},$
$\displaystyle(p_{1}\cdot
p_{3})=\frac{1}{4}(1-\eta)(1+\cos\alpha\sin\beta)\hat{s},\ \ (p_{3}\cdot
p_{4})=\frac{1}{4}(1-\zeta)(1-\eta)(1+\cos\theta)\hat{s},$
$\displaystyle(p_{1}\cdot
p_{4})=\frac{1}{4}(1-\zeta)(1-\cos(\alpha-\theta)\sin\beta)\hat{s},\ \
(p_{2}\cdot
p_{4})=\frac{1}{4}(1-\zeta)(1+\cos(\alpha-\theta)\sin\beta)\hat{s},$
$\displaystyle(q_{1})^{2}=-\frac{1}{2}(1-\eta)(1+\cos\alpha\sin\beta)\hat{s},\
\ (q_{2})^{2}=-\frac{1}{2}(1-\zeta)(1+\cos(\alpha-\theta)\sin\beta)\hat{s},$
$\displaystyle(q_{1}^{\prime})^{2}=\hat{s},\ \
(q_{2}^{\prime})^{2}=\frac{1}{2}(1-\zeta)(1-\eta)(1+\cos\theta)\hat{s}.$
(A.51)

####  A.2.5 Cross section of $e^{-}e^{+}\to f\bar{f}h$

The differential cross section of the $e^{-}e^{+}\to f\bar{f}h$ process is
given by

$\displaystyle d\sigma^{ffh}$
$\displaystyle=\frac{1}{16(2\pi)^{4}\hat{s}}|{\cal M}|^{2}dE_{3}dE_{4}d\alpha
d\cos\beta=\frac{1}{64(2\pi)^{4}}|{\cal M}|^{2}d\eta d\zeta d\alpha
d\cos\beta,$ (A.52)

by using the formula in e.g., Refs. [113, 16]. The total cross section with
the initial polarization of the electron and the positron is give by

$\displaystyle\sigma^{ffh}(P_{e^{-}},P_{e^{+}})$
$\displaystyle=\frac{1}{4}\bigg{\\{}(1-P_{e^{-}})(1+P_{e^{+}})\sigma_{LL}^{ffh}+(1+P_{e^{-}})(1-P_{e^{+}})\sigma_{RR}^{ffh}$
$\displaystyle\hskip
20.00003pt+(1-P_{e^{-}})(1-P_{e^{+}})\sigma_{LR}^{ffh}+(1+P_{e^{-}})(1+P_{e^{+}})\sigma_{RL}^{ffh}\bigg{\\}},$
(A.53)

where

$\displaystyle\sigma_{XY}^{ffh}$
$\displaystyle:=\frac{1}{64(2\pi)^{4}}\int_{\frac{m_{h}^{2}}{\hat{s}}}^{1}d\eta\int_{\frac{m_{h}^{2}}{\hat{s}\eta}}^{1+\frac{m_{h}^{2}}{\hat{s}}-\eta}d\zeta\int_{0}^{2\pi}d\alpha\int_{-1}^{1}d\cos\beta|{\cal
M}^{XY}|^{2},$ (A.54)

where $|{\cal M}^{XY}|^{2}(X,Y=L,R)$ are given by substituting Eqs. (A.30),
(A.37), (A.43) into Eq. (A.38). and we omitted $\sum_{\rm spin}$ to simplify
our notation.

The amount of the deviation of the cross section of the $e^{-}e^{+}\to
f\bar{f}h$ process from the SM in the GHU model is given by

$\displaystyle\Delta_{\sigma}^{ffh}(P_{e^{-}},P_{e^{+}}):=\frac{\left[\sigma^{ffh}(P_{e^{-}},P_{e^{+}})\right]_{\rm
GHU}}{\left[\sigma^{ffh}(P_{e^{-}},P_{e^{+}})\right]_{\rm SM}}-1,$ (A.55)

where $\left[\sigma^{ffh}(P_{e^{-}},P_{e^{+}})\right]_{\rm GHU}$ and
$\left[\sigma^{ffh}(P_{e^{-}},P_{e^{+}})\right]_{\rm SM}$ stand for the cross
sections of the $e^{-}e^{+}\to f\bar{f}h$ process in the SM and the GHU model,
respectively.

####  A.2.6 Left-right asymmetry of $e^{-}e^{+}\to f\bar{f}h$

Unlike the left-right asymmetry of the $e^{-}e^{+}\to Zh$ process, we cannot
define a left-right asymmetry of the $e^{-}e^{+}\to f\bar{f}h$ process that
does not depend on the initial polarization of electron and positron even if
the off-shell particles are limited to vector bosons only. Still we can define
a left-right asymmetry that depends on the initial polarization of the
electron and the positron as

$\displaystyle
A_{LR}^{ffh}(P_{e^{-}},P_{e^{+}}):=\frac{\sigma^{ffh}(P_{e^{-}},P_{e^{+}})-\sigma^{ffh}(-P_{e^{-}},-P_{e^{+}})}{\sigma^{ffh}(P_{e^{-}},P_{e^{+}})+\sigma^{ffh}(-P_{e^{-}},-P_{e^{+}})}$
(A.56)

for $P_{e^{-}}<0$ and $|P_{e^{-}}|>|P_{e^{+}}|$. By using Eq. (A.53), we find

$\displaystyle A_{LR}^{ffh}(P_{e^{-}},P_{e^{+}})$
$\displaystyle=-\frac{(P_{e^{-}}-P_{e^{+}})(\sigma_{LL}^{ffh}-\sigma_{RR}^{ffh})-(P_{e^{-}}+P_{e^{+}})(\sigma_{LR}^{ffh}-\sigma_{RL}^{ffh})}{(1-P_{e^{-}}P_{e^{+}})(\sigma_{LL}^{ffh}+\sigma_{RR}^{ffh})+(1+P_{e^{-}}P_{e^{+}})(\sigma_{LR}^{ffh}+\sigma_{RL}^{ffh})}.$
(A.57)

We can define the amount of the deviation from the SM in Eq. (A.56) as

$\displaystyle\Delta_{A_{LR}}^{ffh}(P_{e^{-}},P_{e^{+}})$
$\displaystyle:=\frac{\left[A_{LR}^{ffh}(P_{e^{-}},P_{e^{+}})\right]_{\rm
GHU}}{\left[A_{LR}^{ffh}(P_{e^{-}},P_{e^{+}})\right]_{\rm SM}}-1,$ (A.58)

where $\left[A_{LR}^{ffh}(P_{e^{-}},P_{e^{+}})\right]_{\rm GHU}$ and
${\left[A_{LR}^{ffh}(P_{e^{-}},P_{e^{+}})\right]_{\rm SM}}$ are
$A_{LR}^{ffh}(P_{e^{-}},P_{e^{+}})$ in Eq. (A.57) calculated with the
parameter sets of the SM and the GHU model.

## References

  * [1] ATLAS Collaboration, G. Aad et al., “The ATLAS Experiment at the CERN Large Hadron Collider,” JINST 3 (2008) S08003.
  * [2] CMS Collaboration, S. Chatrchyan et al., “The CMS Experiment at the CERN LHC,” JINST 3 (2008) S08004.
  * [3] M. Cepeda et al., “Report from Working Group 2: Higgs Physics at the HL-LHC and HE-LHC,” CERN Yellow Rep. Monogr. 7 (2019) 221–584, arXiv:1902.00134 [hep-ph].
  * [4] T. Behnke, J. E. Brau, B. Foster, J. Fuster, M. Harrison, J. M. Paterson, M. Peskin, M. Stanitzki, N. Walker, and H. Yamamoto, “The International Linear Collider Technical Design Report - Volume 1: Executive Summary,” arXiv:1306.6327 [physics.acc-ph].
  * [5] H. Baer, T. Barklow, K. Fujii, Y. Gao, A. Hoang, S. Kanemura, J. List, H. E. Logan, A. Nomerotski, M. Perelstein, et al., “The International Linear Collider Technical Design Report - Volume 2: Physics,” arXiv:1306.6352 [hep-ph].
  * [6] C. Adolphsen, M. Barone, B. Barish, K. Buesser, P. Burrows, J. Carwardine, J. Clark, H. Mainaud Durand, G. Dugan, E. Elsen, et al., “The International Linear Collider Technical Design Report - Volume 3.I: Accelerator & in the Technical Design Phase,” arXiv:1306.6353 [physics.acc-ph].
  * [7] C. Adolphsen, M. Barone, B. Barish, K. Buesser, P. Burrows, J. Carwardine, J. Clark, H. Mainaud Durand, G. Dugan, E. Elsen, et al., “The International Linear Collider Technical Design Report - Volume 3.II: Accelerator Baseline Design,” arXiv:1306.6328 [physics.acc-ph].
  * [8] H. Abramowicz et al., “The International Linear Collider Technical Design Report - Volume 4: Detectors,” arXiv:1306.6329 [physics.ins-det].
  * [9] LCC Physics Working Group Collaboration, K. Fujii et al., “Tests of the Standard Model at the International Linear Collider,” arXiv:1908.11299 [hep-ex].
  * [10] ILC International Development Team Collaboration, A. Aryshev et al., “The International Linear Collider: Report to Snowmass 2021,” arXiv:2203.07622 [physics.acc-ph].
  * [11] L. Linssen, A. Miyamoto, M. Stanitzki, and H. Weerts, “Physics and Detectors at CLIC: CLIC Conceptual Design Report,” arXiv:1202.5940 [physics.ins-det].
  * [12] FCC Collaboration, A. Abada et al., “FCC-hh: The Hadron Collider: Future Circular Collider Conceptual Design Report Volume 3,” Eur. Phys. J. ST 228 no. 4, (2019) 755–1107.
  * [13] A. Blondel and P. Janot, “FCC-ee Overview: New Opportunities Create New Challenges,” Eur. Phys. J. Plus 137 no. 1, (2022) 92, arXiv:2106.13885 [hep-ex].
  * [14] S. Dasu et al., “Strategy for Understanding the Higgs Physics: The Cool Copper Collider,” arXiv:2203.07646 [hep-ex].
  * [15] CEPC Study Group Collaboration, M. Dong et al., “CEPC Conceptual Design Report: Volume 2 - Physics & Detector,” arXiv:1811.10545 [hep-ex].
  * [16] Particle Data Group Collaboration, R. L. Workman et al., “Review of Particle Physics,” PTEP 2022 (2022) 083C01.
  * [17] Y. Hosotani, “Dynamical Mass Generation by Compact Extra Dimensions,” Phys.Lett. B126 (1983) 309.
  * [18] Y. Hosotani, “Dynamics of Nonintegrable Phases and Gauge Symmetry Breaking,” Annals Phys. 190 (1989) 233.
  * [19] A. T. Davies and A. McLachlan, “Gauge Group Breaking By Wilson Loops,” Phys. Lett. B200 (1988) 305.
  * [20] A. T. Davies and A. McLachlan, “Congruency Class Effects in the Hosotani Model,” Nucl. Phys. B317 (1989) 237.
  * [21] H. Hatanaka, T. Inami, and C. S. Lim, “The Gauge Hierarchy Problem and Higher Dimensional Gauge Theories,” Mod. Phys. Lett. A13 (1998) 2601–2612, arXiv:hep-th/9805067.
  * [22] H. Hatanaka, “Matter Representations and Gauge Symmetry Breaking via Compactified Space,” Prog. Theor. Phys. 102 (1999) 407–418, arXiv:hep-th/9905100 [hep-th].
  * [23] K. Agashe, R. Contino, and A. Pomarol, “The Minimal Composite Higgs Model,” Nucl. Phys. B719 (2005) 165–187, arXiv:hep-ph/0412089 [hep-ph].
  * [24] A. D. Medina, N. R. Shah, and C. E. M. Wagner, “Gauge-Higgs Unification and Radiative Electroweak Symmetry Breaking in Warped Extra Dimensions,” Phys. Rev. D76 (2007) 095010, arXiv:0706.1281 [hep-ph].
  * [25] Y. Hosotani, K. Oda, T. Ohnuma, and Y. Sakamura, “Dynamical Electroweak Symmetry Breaking in $SO(5)\times U(1)$ Gauge-Higgs Unification with Top and Bottom Quarks,” Phys.Rev. D78 (2008) 096002, arXiv:0806.0480 [hep-ph].
  * [26] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa, and N. Yamatsu, “GUT Inspired $SO(5)\times U(1)\times SU(3)$ Gauge-Higgs Unification,” Phys. Rev. D 99 (2019) 095010, arXiv:1902.01603 [hep-ph].
  * [27] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa, and T. Shimotani, “Novel Universality and Higgs Decay H$\to\gamma\gamma$, gg in the SO(5)$\times$U(1) Gauge-Higgs Unification,” Phys. Lett. B722 (2013) 94–99, arXiv:1301.1744 [hep-ph].
  * [28] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa, and T. Shimotani, “LHC Signals of the $SO(5)\times U(1)$ Gauge-Higgs Unification,” Phys. Rev. D89 no. 9, (2014) 095019, arXiv:1404.2748 [hep-ph].
  * [29] S. Funatsu, H. Hatanaka, and Y. Hosotani, “$H\to Z\gamma$ in the Gauge-Higgs Unification,” Phys. Rev. D92 (2015) 115003, arXiv:1510.06550 [hep-ph].
  * [30] S. Funatsu, H. Hatanaka, Y. Hosotani, and Y. Orikasa, “Collider Signals of $W^{\prime}$ and $Z^{\prime}$ Bosons in the Gauge-Higgs Unification,” Phys. Rev. D 95 no. 3, (2017) 035032, arXiv:1612.03378 [hep-ph].
  * [31] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa, and N. Yamatsu, “CKM Matrix and FCNC Suppression in $SO(5)\times U(1)\times SU(3)$ Gauge-Higgs Unification,” Phys. Rev. D 101 (2020) 055016, arXiv:1909.00190 [hep-ph].
  * [32] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa, and N. Yamatsu, “Effective Potential and Universality in GUT-Inspired Gauge-Higgs Unification,” Phys. Rev. D 102 (2020) 015005, arXiv:2002.09262 [hep-ph].
  * [33] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa, and N. Yamatsu, “Fermion Pair Production at $e^{-}e^{+}$ Linear Collider Experiments in GUT Inspired Gauge-Higgs Unification,” Phys. Rev. D 102 (2020) 015029, arXiv:2006.02157 [hep-ph].
  * [34] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa, and N. Yamatsu, “Electroweak and Left-Right Phase Transitions in $SO(5)\times U(1)\times SU(3)$ Gauge-Higgs Unification,” Phys. Rev. D 104 (4, 2021) 115018, arXiv:2104.02870 [hep-ph].
  * [35] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa, and N. Yamatsu, “Signals of $W^{\prime}$ and $Z^{\prime}$ Bosons at the LHC in the $SU(3)\times SO(5)\times U(1)$ Gauge-Higgs Unification,” Phys. Rev. D 105 (2022) 055015, arXiv:2111.05624 [hep-ph].
  * [36] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa, and N. Yamatsu, “Bhabha Scattering in the Gauge-Higgs Unification,” Phys. Rev. D 106 (3, 2022) 015015, arXiv:2203.16030 [hep-ph].
  * [37] E. Senaha, “Symmetry Restoration and Breaking at Finite Temperature: An Introductory Review,” Symmetry 12 no. 5, (2020) 733.
  * [38] Y. Hosotani and N. Yamatsu, “Gauge-Higgs Grand Unification,” Prog. Theor. Exp. Phys. 2015 (2015) 111B01, arXiv:1504.03817 [hep-ph].
  * [39] N. Yamatsu, “Gauge Coupling Unification in Gauge-Higgs Grand Unification,” Prog. Theor. Exp. Phys. 2016 (2016) 043B02, arXiv:1512.05559 [hep-ph].
  * [40] A. Furui, Y. Hosotani, and N. Yamatsu, “Toward Realistic Gauge-Higgs Grand Unification,” Prog. Theor. Exp. Phys. 2016 (2016) 093B01, arXiv:1606.07222 [hep-ph].
  * [41] Y. Hosotani and N. Yamatsu, “Electroweak Symmetry Breaking and Mass Spectra in Six-Dimensional Gauge-Higgs Grand Unification,” Prog. Theor. Exp. Phys. 2018 no. 2, (2018) 023B05, arXiv:1710.04811 [hep-ph].
  * [42] Y. Hosotani and N. Yamatsu, “Gauge-Higgs Seesaw Mechanism in 6-Dimensional Grand Unification,” Prog. Theor. Exp. Phys. 2017 no. 9, (2017) 091B01, arXiv:1706.03503 [hep-ph].
  * [43] C. Englert, D. J. Miller, and D. D. Smaranda, “Phenomenology of GUT-Inspired Gauge-Higgs Unification,” Phys. Lett. B 802 (2020) 135261, arXiv:1911.05527 [hep-ph].
  * [44] C. Englert, D. J. Miller, and D. D. Smaranda, “The Weinberg Angle and 5D RGE Effects in a SO(11) GUT Theory,” Phys. Lett. B 807 (2020) 135548, arXiv:2003.05743 [hep-ph].
  * [45] H. Georgi and S. L. Glashow, “Unity of All Elementary Particle Forces,” Phys. Rev. Lett. 32 (1974) 438–441.
  * [46] K. Inoue, A. Kakuto, and Y. Nakano, “Unification of the Lepton-Quark World by the Gauge Group SU(6),” Prog.Theor.Phys. 58 (1977) 630.
  * [47] H. Fritzsch and P. Minkowski, “Unified Interactions of Leptons and Hadrons,” Ann. Phys. 93 (1975) 193–266.
  * [48] F. Gursey, P. Ramond, and P. Sikivie, “A Universal Gauge Theory Model Based on $E_{6}$,” Phys. Lett. B60 (1976) 177.
  * [49] R. Slansky, “Group Theory for Unified Model Building,” Phys. Rept. 79 (1981) 1–128.
  * [50] N. Yamatsu, “Finite-Dimensional Lie Algebras and Their Representations for Unified Model Building,” arXiv:1511.08771 [hep-ph].
  * [51] Y. Kawamura, “Gauge Symmetry Breaking from Extra Space $S^{1}/Z_{2}$,” Prog. Theor. Phys. 103 (2000) 613–619, arXiv:hep-ph/9902423 [hep-ph].
  * [52] Y. Kawamura, “Split Multiplets, Coupling Unification and Extra Dimension,” Prog. Theor. Phys. 105 (2001) 691–696, arXiv:hep-ph/0012352.
  * [53] Y. Kawamura, “Triplet-Doublet Splitting, Proton Stability and Extra Dimension,” Prog. Theor. Phys. 105 (2001) 999–1006, arXiv:hep-ph/0012125.
  * [54] L. J. Hall and Y. Nomura, “Gauge Unification in Higher Dimensions,” Phys.Rev. D64 (2001) 055003, arXiv:hep-ph/0103125 [hep-ph].
  * [55] L. J. Hall, Y. Nomura, and D. Tucker-Smith, “Gauge Higgs Unification in Higher Dimensions,” Nucl. Phys. B 639 (2002) 307–330, arXiv:hep-ph/0107331.
  * [56] G. Burdman and Y. Nomura, “Unification of Higgs and Gauge Fields in Five-Dimensions,” Nucl. Phys. B656 (2003) 3–22, arXiv:hep-ph/0210257 [hep-ph].
  * [57] C. S. Lim and N. Maru, “Towards a Realistic Grand Gauge-Higgs Unification,” Phys.Lett. B653 (2007) 320–324, arXiv:0706.1397 [hep-ph].
  * [58] K. Kojima, K. Takenaga, and T. Yamashita, “Grand Gauge-Higgs Unification,” Phys. Rev. D84 (2011) 051701, arXiv:1103.1234 [hep-ph].
  * [59] K. Kojima, K. Takenaga, and T. Yamashita, “Gauge Symmetry Breaking Patterns in an SU(5) Grand Gauge-Higgs Unification Model,” Phys. Rev. D95 no. 1, (2017) 015021, arXiv:1608.05496 [hep-ph].
  * [60] K. Kojima, K. Takenaga, and T. Yamashita, “The Standard Model Gauge Symmetry from Higher-Rank Unified Groups in Grand Gauge-Higgs Unification Models,” JHEP 06 (2017) 018, arXiv:1704.04840 [hep-ph].
  * [61] N. Yamatsu, “Special Grand Unification,” Prog. Theor. Exp. Phys. 2017 no. 6, (2017) 061B01, arXiv:1704.08827 [hep-ph].
  * [62] N. Yamatsu, “String-Inspired Special Grand Unification,” Prog. Theor. Exp. Phys. 2017 no. 10, (2017) 101B01, arXiv:1708.02078 [hep-ph].
  * [63] N. Yamatsu, “Family Unification in Special Grand Unification,” Prog. Theor. Exp. Phys. 2018 no. 9, (2018) 091B01, arXiv:1807.10855 [hep-ph].
  * [64] N. Maru and Y. Yatagai, “Fermion Mass Hierarchy in Grand Gauge-Higgs Unification,” PTEP 2019 no. 8, (2019) 083B03, arXiv:1903.08359 [hep-ph].
  * [65] Y. Kawamura, E. Kodaira, K. Kojima, and T. Yamashita, “On Representation Matrices of Boundary Conditions in $SU(n)$ Gauge Theories Compactified on Two-Dimensional Orbifolds,” arXiv:2211.00877 [hep-th].
  * [66] ATLAS Collaboration, G. Aad et al., “Search for High-Mass Dilepton Resonances Using 139 fb-1 of $pp$ Collision Data Collected at $\sqrt{s}=$13 TeV with the ATLAS Detector,” Phys. Lett. B 796 (2019) 68–87, arXiv:1903.06248 [hep-ex].
  * [67] ATLAS Collaboration, G. Aad et al., “Search for New Non-Resonant Phenomena in High-Mass Dilepton Final States with the ATLAS Detector,” JHEP 11 (2020) 005, arXiv:2006.12946 [hep-ex]. [Erratum: JHEP 04, 142 (2021)].
  * [68] ATLAS Collaboration, G. Aad et al., “Search for a Heavy Charged Boson in Events with a Charged Lepton and Missing Transverse Momentum from $pp$ Collisions at $\sqrt{s}=13$ TeV with the ATLAS Detector,” Phys. Rev. D 100 no. 5, (2019) 052013, arXiv:1906.05609 [hep-ex].
  * [69] ATLAS Collaboration, G. Aad et al., “Search for New Resonances in Mass Distributions of Jet Pairs Using 139 fb-1 of $pp$ Collisions at $\sqrt{s}=13$ TeV with the ATLAS Detector,” JHEP 03 (2020) 145, arXiv:1910.08447 [hep-ex].
  * [70] CMS Collaboration, A. M. Sirunyan et al., “Search for High Mass Dijet Resonances with a New Background Prediction Method in Proton-Proton Collisions at $\sqrt{s}=$ 13 TeV,” JHEP 05 (2020) 033, arXiv:1911.03947 [hep-ex].
  * [71] CMS Collaboration, A. M. Sirunyan et al., “Search for Resonant and Nonresonant New Phenomena in High-Mass Dilepton Final States at $\sqrt{s}$ = 13 TeV,” JHEP 07 (2021) 208, arXiv:2103.02708 [hep-ex].
  * [72] S. Funatsu, H. Hatanaka, Y. Hosotani, and Y. Orikasa, “Distinct Signals of the Gauge-Higgs Unification in $e^{+}e^{-}$ Collider Experiments,” Phys. Lett. B775 (2017) 297–302, arXiv:1705.05282 [hep-ph].
  * [73] J. Yoon and M. E. Peskin, “Fermion Pair Production in $SO(5)\times U(1)$ Gauge-Higgs Unification Models,” arXiv:1811.07877 [hep-ph].
  * [74] J. Yoon and M. E. Peskin, “Dissection of an $SO(5)\times U(1)$ Gauge-Higgs Unification Model,” Phys. Rev. D100 no. 1, (2019) 015001, arXiv:1810.12352 [hep-ph].
  * [75] S. Funatsu, “Forward-Backward Asymmetry in the Gauge-Higgs Unification at the International Linear Collider,” Eur. Phys. J. C79 no. 10, (2019) 854, arXiv:1905.10007 [hep-ph].
  * [76] S. Bilokin, R. Pöschl, and F. Richard, “Measurement of $b$ Quark EW Couplings at ILC,” arXiv:1709.04289 [hep-ex].
  * [77] F. Richard, “Bhabha Scattering at ILC250,” arXiv:1804.02846 [hep-ex].
  * [78] A. Irles, R. Pöschl, F. Richard, and H. Yamamoto, “Complementarity between ILC250 and ILC-GigaZ,” in Linear Collider Community Meeting Lausanne, Switzerland.  2019\.  arXiv:1905.00220 [hep-ex].
  * [79] A. Irles, R. Pöschl, and F. Richard, “Production and Measurement of $e^{+}e^{-}\rightarrow c\bar{c}$ Signatures at the 250 GeV ILC,” in International Workshop on Future Linear Colliders (LCWS 2019), Japan.  2019\.  arXiv:2002.05805 [hep-ex].
  * [80] K. Fujii et al., “Physics Case for the 250 GeV Stage of the International Linear Collider,” arXiv:1710.07621 [hep-ex].
  * [81] ILC Collaboration, H. Aihara et al., “The International Linear Collider. A Global Project,” arXiv:1901.09829 [hep-ex].
  * [82] P. Bambade et al., “The International Linear Collider: A Global Project,” arXiv:1903.01629 [hep-ex].
  * [83] ILC International Development Team Working Group 3 Collaboration, R. Pöschl, “Top and Heavy Quark Studies at Linear Colliders,” PoS EPS-HEP2021 (2022) 491.
  * [84] B. Schrempp, F. Schrempp, N. Wermes, and D. Zeppenfeld, “Bounds on New Contact Interactions From Future $e^{+}e^{-}$ Colliders,” Nucl. Phys. B296 (1988) 1–25.
  * [85] D. C. Kennedy, B. W. Lynn, C. J. C. Im, and R. G. Stuart, “Electroweak Cross-Sections and Asymmetries at the $Z^{0}$,” Nucl. Phys. B321 (1989) 83–107.
  * [86] SLD Collaboration, K. Abe et al., “Precise Measurement of the Left-Right Cross-Section Asymmetry in Z Boson Production by $e^{+}e^{-}$ Collisions,” Phys. Rev. Lett. 73 (1994) 25–29, arXiv:hep-ex/9404001 [hep-ex].
  * [87] SLD Collaboration, K. Abe et al., “An Improved Measurement of the Left-Right $Z^{0}$ Cross-Section Asymmetry,” Phys. Rev. Lett. 78 (1997) 2075–2079, arXiv:hep-ex/9611011 [hep-ex].
  * [88] A. Blondel, B. W. Lynn, F. M. Renard, and C. Verzegnassi, “Precision Measurements of Final State Weak Coupling From Polarized Electron–Positron Annihilation,” Nucl. Phys. B304 (1988) 438–450.
  * [89] SLD Collaboration, K. Abe et al., “Measurement of $A_{b}$ and $A_{c}$ from the Left-Right Forward-Backward Asymmetry of Leptons in Hadronic Events at the $Z^{0}$ Resonance,” Phys. Rev. Lett. 74 (1995) 2895–2899.
  * [90] SLD Collaboration, K. Abe et al., “Measurement of $A_{b}$ from the Left-Right Forward-Backward Asymmetry of $b$ Quark Production in $Z^{0}$ Decays Using a Momentum-Weighted Track-Charge Technique,” Phys. Rev. Lett. 74 (1995) 2890–2894.
  * [91] SLD Collaboration, K. Abe et al., “Measurement of the Left-Right Forward-Backward Asymmetry for Charm Quarks with $D^{*+}$ and $D^{+}$ Mesons,” Phys. Rev. Lett. 75 (1995) 3609–3613.
  * [92] SLD Collaboration, K. Abe et al., “Polarized Bhabha Scattering a Precision Measurement of the Electron Neutral Current Couplings,” Phys. Rev. Lett. 74 (1995) 2880–2884, arXiv:hep-ex/9410009 [hep-ex].
  * [93] G. Moortgat-Pick et al., “The Role of Polarized Positrons and Electrons in Revealing Fundamental Interactions at the Linear Collider,” Phys. Rept. 460 (2008) 131–243, arXiv:hep-ph/0507011 [hep-ph].
  * [94] ALEPH, DELPHI, L3, OPAL, LEP Electroweak Collaboration, S. Schael et al., “Electroweak Measurements in Electron-Positron Collisions at W-Boson-Pair Energies at LEP,” Phys. Rept. 532 (2013) 119–244, arXiv:1302.3415 [hep-ex].
  * [95] D. Bardin, Y. Dydyshka, L. Kalinovskaya, L. Rumyantsev, A. Arbuzov, R. Sadykov, and S. Bondarenko, “One-Loop Electroweak Radiative Corrections to Polarized Bhabha Scattering,” Phys. Rev. D98 no. 1, (2018) 013001, arXiv:1801.00125 [hep-ph].
  * [96] V. I. Borodulin, R. N. Rogalyov, and S. R. Slabospitskii, “CORE 3.1 (COmpendium of RElations, Version 3.1),” arXiv:1702.08246 [hep-ph].
  * [97] A. Djouadi, “The Anatomy of Electro-Weak Symmetry Breaking. I: The Higgs Boson in the Standard Model,” Phys.Rept. 457 (2008) 1–216, arXiv:hep-ph/0503172 [hep-ph].
  * [98] J. Yan, S. Watanuki, K. Fujii, A. Ishikawa, D. Jeans, J. Strube, J. Tian, and H. Yamamoto, “Measurement of the Higgs Boson Mass and $e^{+}e^{-}\to ZH$ Cross Section Using $Z\to\mu^{+}\mu^{-}$ and $Z\to e^{+}e^{-}$ at the ILC,” Phys. Rev. D 94 no. 11, (2016) 113002, arXiv:1604.07524 [hep-ex]. [Erratum: Phys.Rev.D 103, 099903 (2021)].
  * [99] G. Belanger, F. Boudjema, J. Fujimoto, T. Ishikawa, T. Kaneko, K. Kato, and Y. Shimizu, “Full One Loop Electroweak Radiative Corrections to Single Higgs Production in $e^{+}e^{-}$,” Phys. Lett. B 559 (2003) 252–262, arXiv:hep-ph/0212261.
  * [100] F. Boudjema, J. Fujimoto, T. Ishikawa, T. Kaneko, K. Kato, Y. Kurihara, Y. Shimizu, and Y. Yasui, “Electroweak Corrections to Higgs Production through ZZ Fusion at the Linear Collider,” Phys. Lett. B 600 (2004) 65–76, arXiv:hep-ph/0407065.
  * [101] A. Denner, S. Dittmaier, M. Roth, and M. M. Weber, “Electroweak Radiative Corrections to Single Higgs Boson Production in $e^{+}e^{-}$ Annihilation,” Phys. Lett. B 560 (2003) 196–203, arXiv:hep-ph/0301189.
  * [102] A. Denner, S. Dittmaier, M. Roth, and M. M. Weber, “Electroweak Radiative Corrections to $e^{+}e^{-}\to\nu\bar{\nu}H$,” Nucl. Phys. B 660 (2003) 289–321, arXiv:hep-ph/0302198.
  * [103] C.-x. Yue, S.-z. Wang, and D.-q. Yu, “Littlest Higgs Model and Associated ZH Production at High Energy $e^{+}e^{-}$ Collider,” Phys. Rev. D 68 (2003) 115004, arXiv:hep-ph/0309113.
  * [104] C.-X. Yue, W. Wang, Z.-J. Zong, and F. Zhang, “Probing Signals of the Littlest Higgs Model via the WW Fusion Processes at the High Energy $e^{+}e^{-}$ Collider,” Eur. Phys. J. C 42 (2005) 331–339, arXiv:hep-ph/0504253.
  * [105] X.-l. Wang, Y.-b. Liu, J.-h. Chen, and H. Yang, “The Correction of the Littlest Higgs Model to the Higgs Production Process $e^{+}e^{-}\to e^{+}e^{-}H$ at the ILC,” Eur. Phys. J. C 49 (2007) 593–597, arXiv:hep-ph/0607131.
  * [106] J. Han, S. Li, B. Yang, and N. Liu, “Higgs Boson Production and Decay at $e^{+}e^{-}$ Colliders as a Probe of the Left-Right Twin Higgs Model,” Nucl. Phys. B 896 (2015) 200–211, arXiv:1504.08236 [hep-ph].
  * [107] L. Randall and R. Sundrum, “A Large Mass Hierarchy from a Small Extra Dimension,” Phys. Rev. Lett. 83 (1999) 3370–3373, arXiv:hep-ph/9905221.
  * [108] ALEPH Collaboration, A. Collaboration, D. Collaboration, L. Collaboration, O. Collaboration, S. Collaboration, L. E. W. Group, S. E. Group, and S. H. F. Group, “Precision Electroweak Measurements on the $Z$ Resonance,” Phys. Rept. 427 (2006) 257, arXiv:hep-ex/0509008.
  * [109] Z.-z. Xing, H. Zhang, and S. Zhou, “Updated Values of Running Quark and Lepton Masses,” Phys. Rev. D 77 (2008) 113016, arXiv:0712.1419 [hep-ph].
  * [110] H. Abramowicz et al., “Higgs Physics at the CLIC Electron–Positron Linear Collider,” Eur. Phys. J. C 77 no. 7, (2017) 475, arXiv:1608.07538 [hep-ex].
  * [111] M. E. Peskin and D. V. Schroeder, “An Introduction to Quantum Field Theory,”. Reading, USA: Addison-Wesley (1995) 842 p.
  * [112] D. R. T. Jones and S. T. Petcov, “Heavy Higgs Bosons at LEP,” Phys. Lett. B 84 (1979) 440–444.
  * [113] R. N. Cahn, “Production of Heavy Higgs Bosons: Comparisons of Exact and Approximate Results,” Nucl. Phys. B 255 (1985) 341. [Erratum: Nucl.Phys.B 262, 744 (1985)].
  * [114] G. Altarelli, B. Mele, and F. Pitolli, “Heavy Higgs Production at Future Colliders,” Nucl. Phys. B 287 (1987) 205–224.