# Higgs potential in gauge-Higgs unification with a flat extra dimension

Mitsuru Kakizaki<EMAIL_ADDRESS>Department of Physics,
University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan  Shin Suzuki
<EMAIL_ADDRESS>Department of Physics, University of Toyama,
3190 Gofuku, Toyama 930-8555, Japan

(July 28, 2021)

###### Abstract

We analyze the structure of the Higgs potential in gauge-Higgs unification
with a flat extra dimension. As a concrete model, we first consider the cases
where the Standard Model Higgs doublet is embedded into a higher-dimensional
$SU(3)_{w}$ gauge multiplet with five-dimensional Lorentz symmetry relaxed. In
this $SU(3)_{w}$ model with Lorentz violation, the deviation of the resulting
triple Higgs boson coupling from its SM prediction is shown to be less than
$10\%$ when the compactification scale is larger than the experimental lower
bound, which is around 5 TeV. Next, we examine the Higgs potential in other
similar gauge-Higgs unification models with a flat extra dimension. It is
pointed out that even in such models the shape of the Higgs potential around
the vacuum quickly approaches that of the minimal Higgs potential with one
Higgs doublet as the compactification scale of the extra dimension increases.
An observable deviation of the triple Higgs boson coupling at planned lepton
colliders will necessitate significant extensions of such gauge-Higgs
unification models.

††preprint: UT-HET-134

##  I Introduction

The discovery of the Higgs boson $h$ with a mass of $m_{h}=125~{}\mathrm{GeV}$
Aad  _et al._ (2012); Chatrchyan  _et al._ (2012) and the measurements of its
couplings Aad  _et al._ (2016); Aaboud  _et al._ (2018); Sirunyan  _et al._
(2019) at the CERN Large Hadron Collider (LHC) have established the mechanism
of the spontaneous electroweak symmetry breaking, on which the Standard Model
(SM) is based. Nevertheless, there still remain unknowns regarding the Higgs
sector. These include the guiding principle, the shape of the Higgs potential,
dynamics behind the electroweak symmetry breaking (EWSB), etc. Besides, new
physics beyond the SM is necessary for accounting for neutrino oscillations,
baryon asymmetry of the Universe, the existence of dark matter and cosmic
inflation. As listed above, the SM has both theoretical and phenomenological
puzzles, and many ideas about these issues have been proposed.

One of the notorious problems the SM bears is the so-called hierarchy problem.
Since the Higgs doublet is an elementary scalar field in the SM, quantum
corrections to the mass squared of the Higgs doublet is quadratically
divergent. In order to obtain the observed mass of the Higgs boson, an
unnaturally huge cancellation between the bare mass squared and the quantum
corrections is required. This fine-tuning problem can be evaded if one invokes
yet-to-be-discovered paradigms, such as supersymmetry (SUSY) Wess and Zumino
(1974); Salam and Strathdee (1974), compositeness Kaplan and Georgi (1984) and
gauge-Higgs unification (GHU) Manton (1979); Fairlie (1979a, b); Hosotani
(1983a, b, 1989). In SUSY models, the quadratic divergence from the SM
particles are canceled out by that from the corresponding superpartners. In
composite models, the SM Higgs boson is composed of a pair of fermions, and
thus chiral symmetry can avoid the problem arising from an elementary scalar.
In GHU models, the SM Higgs doublet is embedded into some extra-dimensional
component of a vector multiplet. Due to the higher-dimensional gauge symmetry,
the mass squared of the Higgs boson is protected against quadratic divergence
Hatanaka  _et al._ (1998); Haba  _et al._ (2008). In more detail, the Higgs
potential is flat at the tree level due to the higher-dimensional gauge
symmetry and induced by the quantum corrections at the loop level. Although
the higher-dimensional gauge theory is nonrenormalizable, the finiteness of
the Higgs potential has been validated at the two loop level in some GHU
models Maru and Yamashita (2006); Hosotani  _et al._ (2007). The Higgs
potential at higher-loop levels has also been investigated Hisano  _et al._
(2020).

Although the concept of GHU is elegant, it is difficult to obtain the measured
masses of the top quark and Higgs boson. Challenges in solving this problem
lead to a rich phenomenology in GHU Scrucca  _et al._ (2003); Cacciapaglia
_et al._ (2006). In this paper, we focus on the Higgs potential in GHU models
and investigate the triple Higgs boson coupling. The triple Higgs boson
coupling in the framework of GHU have been discussed in a flat five-
dimensional (5D) orbifold model with large bulk fermion representations Adachi
and Maru (2018), and in warped models Funatsu  _et al._ (2020). In contrast to
these earlier works, we employ a flat orbifold model with 5D Lorentz symmetry
relaxed in order to obtain realistic values of the masses of the top quark and
Higgs boson. As an example of broken Lorentz symmetry in the fifth dimension,
we consider the $SU(3)_{w}$ model studied in Refs. Panico  _et al._ (2006),
compute the triple Higgs boson coupling and compare the predicted value with
the expected precision at future lepton colliders, such as the International
Linear Collider (ILC) Fujii  _et al._ (2019) and the Compact LInear Collider
(CLIC) Roloff  _et al._ (2020). We show that the characteristics of the
effective potential and triple Higgs boson coupling are applicable to a wide
class of GHU models with a flat orbifold.

This paper is organized as follows. In Sec. II, we briefly review the
$SU(3)_{w}$ GHU model on a flat $S^{1}/Z_{2}$ orbifold with 5D Lorentz
symmetry relaxed, and describe the matter Lagrangian and mass spectrum. The
calculation of the resulting effective Higgs potential is demonstrated in Sec.
III. Phenomenological constraints on this model are given in Sec. IV. Then, we
show the prediction of the triple Higgs boson coupling in Sec. V. Section VI
is devoted to a summary.

##  II Model

Here, we briefly outline the 5D GHU model proposed in Ref. Panico  _et al._
(2006). In this model, the extra dimension is compactified on a flat
$S^{1}/Z_{2}$ orbifold with the $S^{1}$ compactification radius $R$, and
Lorentz symmetry in the fifth dimension is assumed to be broken in order to
obtain realistic masses of the top quark and Higgs boson, as discussed later.
The electroweak gauge group is extended to $SU(3)_{w}\times U(1)^{\prime}$.
The 5D $SU(3)_{w}$ and $U(1)^{\prime}$ gauge coupling constants, $g_{5}$ and
$g_{5}^{\prime}$, are related to the corresponding four-dimensional (4D) gauge
coupling constants, $g$ and $g^{\prime}$, through

$\displaystyle g_{5}\equiv g\sqrt{2\pi R},\quad g_{5}^{\prime}\equiv
g^{\prime}\sqrt{2\pi R}.$ (1)

The extended electroweak gauge group $SU(3)_{w}\times U(1)^{\prime}$ is broken
down to $SU(2)_{L}\times U(1)_{w}\times U(1)^{\prime}$ by the nontrivial
$Z_{2}$ orbifold boundary condition whose projection matrix is expressed as

$\displaystyle P=\begin{pmatrix}-1&0&0\\\ 0&-1&0\\\ 0&0&+1\\\ \end{pmatrix},$
(2)

in the $SU(3)_{w}$ space. After the orbifold compactification, there appear
massless 4D bosonic fields from the $SU(2)_{L}\times U(1)_{w}$ components of
the $SU(3)_{w}$ gauge fields $A_{\mu}^{a}$, $U(1)^{\prime}$ gauge field
$A_{\mu}^{\prime}$, and doublet scalar component $A_{5}^{a}$, which is charged
under $U(1)^{\prime}$ and plays the role of the Higgs doublet. The vacuum
expectation value (VEV) of $A_{5}^{a}$ induces an additional spontaneous
symmetry breaking to $U(1)_{\rm EM}$. It is convenient to parametrize the VEV
of $A_{5}^{a}$ using a parameter $\alpha$ as

$\displaystyle\langle A_{5}^{a}\rangle=\frac{2\alpha}{g_{5}R}\delta^{a7},$ (3)

where $\delta^{ab}$ is the Kronecker delta in the $SU(3)_{w}$ adjoint
representation. The EWSB in this model is detailed in Sec. III.

Notice that the extra $U(1)^{\prime}$ gauge group and its gauge coupling
constant $g^{\prime}$ are necessary for modifying the Weinberg angle
$\theta_{W}$. If the $SU(2)_{L}\times U(1)_{Y}$ gauge group is a subgroup of
the $SU(3)_{w}$, an unacceptably large Weinberg angle of $\theta_{W}=\pi/3$ is
predicted. Thanks to the existence of the $U(1)^{\prime}$ gauge group, one can
set the hypercharge generator $Y$ to the linear combination of the $U(1)_{w}$
and $U(1)^{\prime}$ generators as $Y=t_{8}/\sqrt{3}+t^{\prime}$. Then, the
hypercharge gauge field $A_{Y}$ and its orthogonal gauge field $A_{X}$ are
given by

$\displaystyle
A_{Y}=\frac{g^{\prime}A^{8}+\sqrt{3}gA^{\prime}}{\sqrt{3g^{2}+g^{\prime
2}}},\quad
A_{X}=\frac{\sqrt{3}gA^{8}+g^{\prime}A^{\prime}}{\sqrt{3g^{2}+g^{\prime 2}}}.$
(4)

The $U(1)_{Y}$ gauge coupling constant $g_{Y}$ is given by a combination of
the $SU(3)_{w}$ and $U(1)^{\prime}$ coupling constants as
$g_{Y}=\sqrt{3}gg^{\prime}/\sqrt{3g^{2}+g^{\prime 2}}$. The Weinberg angle is
modified to

$\displaystyle\sin^{2}\theta_{W}=\frac{g_{Y}^{2}}{g^{2}+g_{Y}^{2}}=\frac{3}{4+3g^{2}/g^{\prime
2}}\,.$ (5)

By adjusting the value of $g^{\prime}$, we can obtain the measured Weinberg
angle.

###  II.1 Matter Lagrangian

Let us turn to the matter sector of this $SU(3)_{w}$ GHU model, where both
massive bulk fermions and massless brane-localized fermions are introduced. By
adjusting mixing terms between the bulk and brane fermions, one can reproduce
the observed hierarchy of the fermion masses. Each brane fermion can be
localized at either of the two fixed points, $y=0$ or $y=\pi R$ of the
$S^{1}/Z_{2}$ orbifold. In our analysis, we locate the left-handed and right-
handed brane fermions at the separate fixed points in order to reconcile the
masses of the top quark and Higgs boson with the observed values. This choice
corresponds to the $\delta=1$ set-up in Ref. Panico  _et al._ (2006). In our
analysis, we focus on the third generation quarks because radiative
contributions to the Higgs potential from the first and second generation
quarks as well as those from leptons are relatively small. We introduce bulk
fermions that are coupled to the brane top (bottom) quark. These bulk fermions
are assigned to a symmetric (fundamental) representation of $SU(3)_{w}$ and
periodic on $S^{1}$. In addition, in order to adjust the shape of the Higgs
potential, one needs another set of bulk fermions that are symmetric
representations of $SU(3)_{w}$ and antiperiodic on $S^{1}$.

Now we are in a position to specify the configuration of the above-mentioned
bulk and brane-localized fermion fields. There are three types of a pair of
colored 5D bulk fermions as follows: periodic bulk fermions
$(\Psi_{t},\tilde{\Psi}_{t})$ in the $(\bm{3},\bar{\bm{6}})$ representation;
periodic bulk fermions $(\Psi_{b},\tilde{\Psi}_{b})$ in the $(\bm{3},\bm{3})$
representation; and antiperiodic bulk fermions $(\Psi_{A},\tilde{\Psi}_{A})$
in the $(\bm{1},\bm{6})$ representation. Here, the numbers in the parentheses
denote representations under $SU(3)_{c}\times SU(3)_{w}$. The hypercharge of
$(\Psi_{A},\tilde{\Psi}_{A})$ is chosen such that
$(\Psi_{A},\tilde{\Psi}_{A})$ do not mix with the bulk or brane leptons, which
are necessary for the lepton masses. For each pair of the bulk fermions,
opposite $Z_{2}$ parities are assigned so that the mixing terms between
$\Psi_{j}$ and $\tilde{\Psi}_{j}$ ($j=t,b,A$) with 5D mass parameters $M_{j}$
are allowed. One of the characteristic features of this model is that
$SO(4,1)$ Lorentz invariance is relaxed. Parameters $k_{j}$ and
$\tilde{k}_{j}$ are introduced to parametrize such effects residing in the
kinetic terms for bulk fermions $\Psi_{j}$ and $\tilde{\Psi}_{j}$,
respectively. A left-handed $SU(2)_{L}$ doublet $Q_{L}=(t_{L},b_{L})^{\rm T}$
are localized at the $y=0$ brane, and right-handed $SU(2)_{L}$ singlets
$t_{R}$ and $b_{R}$ at the $y=\pi R$ brane, respectively. The matter contents
are summarized in Table 1. The field configuration on the $S^{1}/Z_{2}{}$
orbifold is depicted in Fig. 1. We introduce parameters $e_{1}$ and $e_{2}$
with mass dimension half in order to parametrize boundary couplings for mixing
terms between the bulk and brane fermions. From the remarks above, the 5D
matter Lagrangian is written as

$\displaystyle\mathcal{L}_{\rm matter}=$
$\displaystyle\sum_{j=t,b,A}\left\\{\bar{\Psi}_{j}\left(i\not{D}_{4}-k_{j}D_{5}\gamma^{5}\right)\Psi_{j}+\bar{\tilde{\Psi}}_{j}\left(i\not{D}_{4}-\tilde{k}_{j}D_{5}\gamma^{5}\right)\tilde{\Psi}_{j}+\left(\bar{\Psi}_{j}M_{j}\tilde{\Psi}_{j}+{\rm
h.c.}\right)\right\\}$
$\displaystyle+\delta(y-0)\left\\{\bar{Q}_{L}i\not{D}_{4}Q_{L}+\left(e^{b}_{1}\bar{Q}_{L}\psi_{b}+e^{t}_{1}\bar{Q}^{c}_{R}\psi_{t}+{\rm
h.c.}\right)\right\\}$ $\displaystyle+\delta(y-\pi
R)\left\\{\bar{t}_{R}i\not{D}_{4}t_{R}+\bar{b}_{R}i\not{D}_{4}b_{R}+\left(e^{b}_{2}\bar{b}_{R}\chi_{b}+e^{t}_{2}\bar{t}^{c}_{L}\chi_{t}+{\rm
h.c.}\right)\right\\},$ (6)

where $\not{D}_{4}\equiv\gamma^{\mu}D_{\mu}$ with $D_{\mu}$ and $D_{5}$ being
covariant derivatives, and $\psi_{t,b}$ and $\chi_{t,b}$ are the $SU(2)_{L}$
doublet and singlet components of the bulk fermions $\Psi_{t,b}$. The SM
fermions are approximately given by the brane fermion components.

Table 1: Matter contents and the quantum numbers. The hypercharge of $(\Psi_{A},\tilde{\Psi}_{A})$ is chosen such that $(\Psi_{A},\tilde{\Psi}_{A})$ do not mix with the bulk or brane leptons. The color factor is denoted by $C_{F}$. Fields | $SU(3)_{c}\times SU(3)_{w}$ | periodicity ($\eta$) | $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ |  $C_{F}$
---|---|---|---|---
$(\Psi_{t},\tilde{\Psi}_{t})$ | $({\bf 3},{\bf\bar{6}})$ | periodic (0) | $({\bf 3},{\bf 1})_{2/3}+({\bf 3},{\bf 2})_{1/6}+({\bf 3},{\bf 3})_{-1/3}$ | $3$
$(\Psi_{b},\tilde{\Psi}_{b})$ | $({\bf 3},{\bf 3})$ | periodic (0) | $({\bf 3},{\bf 1})_{-1/3}+({\bf 3},{\bf 2})_{1/6}$ | $3$
$(\Psi_{A},\tilde{\Psi}_{A})$ | $({\bf 1},{\bf 6})$ | antiperiodic (1) | $({\bf 1},{\bf 1})_{X}+({\bf 1},{\bf 2})_{X+1/2}+({\bf 1},{\bf 3})_{X+1}$ | $1$
$Q_{L}$ |  |  | $({\bf 3},{\bf 2})_{1/6}$ | $3$
$t_{R}$ |  |  | $({\bf 3},{\bf 1})_{2/3}$ | $3$
$b_{R}$ |  |  | $({\bf 3},{\bf 1})_{-1/3}$ | $3$
Figure 1: Field configuration on the $S^{1}/Z_{2}$ orbifold.

For the sake of simplicity, we take $k_{j}=\tilde{k}_{j}$, and the
counterparts in the gauge and Higgs sectors are set to the unity in the
following analysis. In place of $M_{j}$ and $e^{a}_{i}$, we introduce
dimensionless parameters $\lambda^{j}=\pi RM_{j}$ and
$\epsilon_{i}^{a}=\sqrt{\pi R/2}e^{a}_{i}$ ($a=t,b$; $i=1,2$), respectively.
We are left with 10 input parameters,
$k_{t},k_{b},k_{A},\lambda^{t},\lambda^{b},\lambda^{A},\epsilon^{t}_{1},\epsilon^{t}_{2},\epsilon^{b}_{1}$
and $\epsilon^{b}_{2}$.

###  II.2 Mass Spectrum

Here, we briefly describe the mass spectrum of this model. In the absence of
5D mass terms for the bulk gauge fields, the masses of their Kaluza-Klein (KK)
modes at the level $n$ scale as $m^{(n)}=n/R$. When the EWSB takes place, the
zero-mode and KK-mode masses of the gauge fields that are coupled with nonzero
$\langle A_{5}^{a}\rangle$ are shifted upward by $\alpha$. As for the
$U(1)_{X}$ gauge sector, due to quantum anomaly, the zero mode of $A_{X}$
acquire a brane-localized mass $M_{X}$, whose natural value is the cut-off
scale of the model Panico  _et al._ (2006); Scrucca  _et al._ (2003). In the
rest of this paper, we disregard $A_{X}$ in the low-energy effective theory we
discuss. For details, see Refs. Panico  _et al._ (2006); Scrucca  _et al._
(2003). In the limit where $M_{X}\to\infty$, the masses of the weak gauge
bosons are given by

$\displaystyle m_{W^{\pm}}^{(n)}(\alpha)=\frac{n+\alpha}{R},\quad
m_{Z}^{(n)}(\alpha)=\frac{n+\alpha\sec\theta_{W}}{R}.$ (7)

Therefore, we can obtain $m_{W^{\pm}}^{(0)}=m_{Z}^{(0)}\cos\theta_{W}$ for the
zero modes.

The computation of the 4D mass spectrum for the bulk fermions is more
complicated. We consider the cases where 4D fermions arising from the bulk
fermions are sufficiently heavy and the effect of the bulk-brane mixing terms
$\epsilon^{a}_{i}$ is negligible in computing the 4D masses of the bulk
fermions. In this approximation, the 4D masses of the bulk fermions take the
form of

$\displaystyle
m_{j}^{(n)}(q\alpha)=\sqrt{\left(k_{j}\frac{n+\eta/2+q\alpha}{R}\right)^{2}+\left(\frac{\lambda^{j}}{\pi
R}\right)^{2}},$ (8)

where $q$ is the charge related to the coupling with the Higgs doublet, and
determined from the $SU(3)_{w}$ representation of the bulk fermion. The
fundamental representation with a $SU(3)_{w}$ index can have $q=0,1$, and the
symmetric representation with two $SU(3)_{w}$ indices can have $q=0,1,2$. The
parameter $\eta=0\,(1)$ stands for periodic (antiperiodic) bulk fermions.
Using this formalism, there appear five sets of KK masses that depends on
$\alpha$:
$m_{t}^{(n)}(\alpha),m_{t}^{(n)}(2\alpha),m_{b}^{(n)}(\alpha),m_{A}^{(n)}(\alpha)$
and $m_{A}^{(n)}(2\alpha)$ .

The masses of the brane fermions are induced by the bulk-brane mixing terms.
By diagonalizing the full kinetic operators for the bulk and brane fermions,
we obtain the mass eigenvalues,

$\displaystyle m^{t}_{0}=\frac{k_{t}}{\sqrt{2}\pi
R}\frac{\epsilon^{t}_{1}\epsilon^{t}_{2}}{k_{t}^{2}}\mathrm{Im}f_{1}(\lambda^{t}/k_{t},2\alpha),$
$\displaystyle m^{b}_{0}=\frac{k_{b}}{\pi
R}\frac{\epsilon^{b}_{1}\epsilon^{b}_{2}}{k_{b}^{2}}\mathrm{Im}f_{1}(\lambda^{b}/k_{b},\alpha).$
(9)

After rescaling the mass eigenvalues using the normalization factors,

$\displaystyle
Z^{t}_{i}=1+\delta_{i1}\frac{(\epsilon^{b}_{1})^{2}}{k_{b}\lambda^{b}}\mathrm{Re}f_{0}(\lambda^{b}/k_{b},0)+\delta_{i2}\frac{(\epsilon^{t}_{2})^{2}}{2k_{t}\lambda^{t}}\mathrm{Re}f_{0}(\lambda^{t}/k_{t},0)+\frac{(\epsilon^{t}_{i})^{2}}{2^{\delta_{i2}}k_{t}\lambda^{t}}\mathrm{Re}f_{0}(\lambda^{t}/k_{t},2\alpha),$
$\displaystyle
Z^{b}_{i}=1+\frac{(\epsilon^{b}_{i})^{2}}{k_{b}\lambda^{b}}\mathrm{Re}f_{0}(\lambda^{b}/k_{b},\alpha)+\delta_{i1}\frac{(\epsilon^{t}_{1})^{2}}{k_{t}\lambda^{t}}\mathrm{Re}f_{0}(\lambda^{t}/k_{t},\alpha),$
(10)

with

$\displaystyle
f_{\delta}(x,\alpha)=\sum^{\infty}_{k=-\infty}\mathrm{e}^{-|2k+\delta|(x+i\pi\alpha)},$
(11)

we obtain the physical masses of the brane fermions as

$\displaystyle m^{a}=\frac{m_{0}^{a}}{\sqrt{Z_{1}^{a}Z_{2}^{a}}}.$ (12)

##  III Effective Potential

As a remnant of 5D gauge invariance, the Higgs potential is zero at the
leading order of perturbation theory. However, the Higgs potential is induced
by radiative corrections. The effective potential at the one-loop level is
given by

$\displaystyle V_{\rm
eff}(\alpha)=\sum_{I}\frac{\sigma_{I}}{2}\int\frac{d^{4}p_{E}}{(2\pi)^{4}}\ln\left\\{p_{E}^{2}+m_{I}(\alpha)^{2}\right\\},$
(13)

where $\sigma_{I}=1$ for bosons and $\sigma_{I}=-1$ for fermions, and the sum
runs over all 4D fields whose masses depend on $\alpha$. As for the 4D fields
originating from the bulk fields, we perform the integration of the Euclidean
momentum $p_{E}$ for the KK modes and decompose these contributions into an
infinite $\alpha$-independent part and finite $\alpha$-dependent part
Antoniadis  _et al._ (2001). After the integration, the one-loop contribution
from each bulk gauge field takes the form of

$\displaystyle
V_{V}(\alpha)=-\frac{9}{64\pi^{6}R^{4}}\sum_{n=1}^{\infty}\frac{1}{n^{5}}\cos(2\pi
n\alpha).$ (14)

Then, the total contribution from the bulk gauge fields is given by

$\displaystyle V_{g}(\alpha)=2V_{V}(\alpha)+V_{V}(\alpha\sec\theta_{W}).$ (15)

Similarly, we can calculate contributions from the bulk fermion pairs to the
effective potential. Taking the effects of the 5D mass parameter $\lambda^{j}$
and the Lorentz-violating parameter $k_{j}$, the contribution from
$(\Psi_{j},\bar{\Psi}_{j})$ pair is given by

$\displaystyle
V_{\Psi_{j}}(q\alpha)=\frac{3k_{j}^{4}C_{F}}{8\pi^{6}R^{4}}\sum_{n=1}^{\infty}\frac{(\sigma_{S})^{n}}{n^{5}}\left[1+2n\frac{\lambda^{j}}{k_{j}}+\frac{4}{3}n^{2}\frac{(\lambda^{j})^{2}}{k_{j}^{2}}\right]\mathrm{e}^{-2n\lambda^{j}/k_{j}}\cos(2\pi
nq\alpha),$ (16)

where $\sigma_{S}=(-1)^{\eta}$ and $C_{F}$ stands for the color factor. Then,
the total contribution from the bulk fermion pairs is given by

$\displaystyle
V_{f}(\alpha)=V_{\Psi_{t}}(\alpha)+V_{\Psi_{t}}(2\alpha)+V_{\Psi_{b}}(\alpha)+V_{\Psi_{A}}(\alpha)+V_{\Psi_{A}}(2\alpha).$
(17)

As for the brane fermions, we need to incorporate the normalization factors of
the kinetic operators as well as the mass eigenvalues given in Eq. (12) into
the calculation of the effective potential. By changing the integral variable
from $p_{E}$ to $x=\pi Rp_{E}$, one obtains

$\displaystyle V_{t}(\alpha)=$
$\displaystyle\frac{-C_{F}}{4\pi^{6}R^{4}}\int_{0}^{\infty}dx\,x^{3}\ln\left[\prod^{2}_{i=1}\mathrm{Re}\left[1+\delta_{i1}\frac{(\epsilon^{b}_{1})^{2}}{k_{b}x^{b}}f_{0}\left(\frac{x^{b}}{k_{b}},0\right)+\delta_{i2}\frac{(\epsilon^{t}_{2})^{2}}{2k_{t}x^{t}}f_{0}\left(\frac{x^{t}}{k_{t}},0\right)\right.\right.$
$\displaystyle\left.\left.+\frac{(\epsilon^{t}_{1})^{2}}{2^{\delta_{i2}}k_{t}x^{t}}f_{0}\left(\frac{x^{t}}{k_{t}},2\alpha\right)\right]+\prod^{2}_{i=1}\mathrm{Im}\frac{(\epsilon^{t}_{i})^{2}}{2^{\delta_{i2}}k_{t}x}\left[f_{1}\left(\frac{x^{t}}{k_{t}},2\alpha\right)\right]\right],$
(18)

and

$\displaystyle V_{b}(\alpha)=$
$\displaystyle\frac{-C_{F}}{4\pi^{6}R^{4}}\int_{0}^{\infty}dx\,x^{3}\ln\left[\prod^{2}_{i=1}\mathrm{Re}\left[1+\frac{(\epsilon^{b}_{i})^{2}}{k_{b}x^{b}}f_{0}\left(\frac{x^{b}}{k_{b}},\alpha\right)+\delta_{i1}\frac{(\epsilon^{t}_{1})^{2}}{k_{t}x^{t}}f_{0}\left(\frac{x^{t}}{k_{t}},\alpha\right)\right]\right.$
$\displaystyle\left.+\prod^{2}_{i=1}\mathrm{Im}\left[\frac{(\epsilon^{b}_{i})^{2}}{k_{b}x}f_{1}\left(\frac{x^{b}}{k_{b}},\alpha\right)\right]\right],$
(19)

with

$\displaystyle x^{a}=\pi R\sqrt{p_{E}^{2}+M_{a}^{2}}.$ (20)

These contributions also turn out to be finite up to the $\alpha$-independent
constant part as in the case of the bulk fields.

Putting the above contributions together, we obtain the total one-loop
effective potential,

$\displaystyle V_{\rm
eff}(\alpha)=V_{g}(\alpha)+V_{f}(\alpha)+V_{t}(\alpha)+V_{b}(\alpha).$ (21)

The EWSB occurs when $\alpha$ develops a non-zero VEV $\alpha_{0}$. The value
of $\alpha_{0}$ is obtained by the tadpole condition,

$\displaystyle\left(\frac{gR}{2}\right)\left.\frac{\partial V_{\rm
eff}}{\partial\alpha}\right|_{\alpha=\alpha_{0}}=0.$ (22)

The mass squared of the Higgs boson is given by

$\displaystyle
m_{h}^{2}=\left(\frac{gR}{2}\right)^{2}\left.\frac{\partial^{2}V_{\rm
eff}(\alpha)}{\partial\alpha^{2}}\right|_{\alpha=\alpha_{0}},$ (23)

and the triple Higgs boson coupling by

$\displaystyle\lambda_{hhh}=\left(\frac{gR}{2}\right)^{3}\left.\frac{\partial^{3}V_{\rm
eff}(\alpha)}{\partial\alpha^{3}}\right|_{\alpha=\alpha_{0}}.$ (24)

The compactification scale $1/R$ is determined by using the relation
$m_{W^{\pm}}^{(0)}=\alpha_{0}/R$. In order to avoid experimental constraints,
$1/R$ should be sufficiently large. In other words, $\alpha_{0}$ should be
very small. It should be emphasized that tuning of model parameters is
required for obtaining very small $\alpha_{0}$. This is because the Higgs
potential in GHU is generated only by radiative corrections and is sensitive
to model parameters. Therefore, it is nontrivial whether the Higgs field
develops a nonzero VEV. We also have to adjust the mass of the Higgs boson to
its experimental value by tuning the model parameters. Namely, the mass of the
discovered Higgs boson and the tadpole condition strongly constrain the shape
of the Higgs potential and the allowed model parameter space.

##  IV Constraints

Before discussing the triple Higgs boson coupling, we mention several
constraints on the model parameters. The experimental limits on the masses of
the top quark and the Higgs boson, and the compactification scale put severe
constraints on GHU models in generic. In the previous study performed in Ref.
Panico  _et al._ (2006), the Lorentz-violating parameters $k_{j}$ are
introduced to adjust the masses of the top quark and the Higgs boson. In Refs.
Panico  _et al._ (2006, 2007), the $\rho$ parameter and the
$Zb_{L}\bar{b}_{L}$ coupling are investigated and turned out to be strongly
affected by the KK modes of the bulk fields, such as the sine modes of
$A_{X}^{(n)}$ and the lightest modes of the bulk fermions. An improved
analysis of electroweak observables in a slightly refined $SU(3)_{w}$ GHU
model puts the lower bound on the compactification scale as $1/R\gtrsim 5$ TeV
for $m_{h}=125$ GeV Panico  _et al._ (2007). Since the works of Refs. Panico
_et al._ (2006, 2007) were done before the discovery of the Higgs boson, we
revisit the $SU(3)_{w}$ model to investigate viable parameter regions.

We perform a random scan of model parameters, and numerically evaluate the
masses of the top quark and the Higgs boson as well as the compactification
scale from the effective potential. We scan the following model parameter
regions in accordance with the previous study Panico  _et al._ (2006):

$\displaystyle\left\\{\begin{array}[]{l}1.5<k_{t}<2.5,\\\ 1.25<k_{b}<2.25,\\\
1.1\times k_{t}<k_{A}<1.5\times k_{t},\end{array}\right.\hskip
14.22636pt\left\\{\begin{array}[]{l}0.5<\lambda^{t}<1.5,\\\
5<\lambda^{b}<7,\\\ 0.75<\lambda^{A}<3.5,\end{array}\right.\hskip
14.22636pt\left\\{\begin{array}[]{l}0.75<\epsilon^{t}_{1,2}<7.5,\\\
2<\epsilon^{b}_{1,2}<7.\end{array}\right.$ (33)

Figure 2 shows the scatter plots of the predicted values of the top quark mass
$m_{t}$ and the Higgs boson mass $m_{h}$. The left (right) panel shows cases
for compactification scales larger than $800~{}\mathrm{GeV}$
($5~{}\mathrm{TeV}$). Color codes are assigned to each range of the Lorentz
violating parameter $k_{t}$: Blue points stands for $2.25<k_{t}<2.5$, green
for $2<k_{t}<2.25$, red for $1.75<k_{t}<2$, and black for $1.5<k_{t}<1.75$.
The left plot is consistent with the results presented in Ref. Panico  _et
al._ (2006). It is clear from the plots that the mass of the Higgs boson is
positively correlated with $k_{t}$. As the compactification scale $1/R$ is
increased, a finer tuning is required for obtaining the correct weak-scale
VEV. Therefore, as shown in the right panel of Fig. 2, the number of allowed
parameter sets is smaller if we impose the experimental lower bound on the
compactification scale, $1/R>5~{}\mathrm{TeV}$ Panico  _et al._ (2007). The
bottom line is that there is still enough room for reproducing the measured
values of the masses of the top and Higgs boson for $1.5<k_{t}<2$.

---
Figure 2: Scatter plots of the predicted values of the top quark mass $m_{t}$
and the Higgs boson mass $m_{h}$ for $1/R>800$ GeV (left) and for $1/R>5$ TeV
(right). Color codes are assigned to each range of the Lorentz violating
parameter $k_{t}$: Blue points stands for $2.25<k_{t}<2.5$, green for
$2<k_{t}<2.25$, red for $1.75<k_{t}<2$, and black for $1.5<k_{t}<1.75$.

##  V Triple Higgs Boson Coupling

After imposing the above-mentioned experimental constraints, we investigate to
what extent the predicted triple Higgs boson coupling $\lambda_{hhh}$ can
deviate from the corresponding SM value $\lambda_{hhh}^{\rm SM}$. The current
experimental constraint on $\lambda_{hhh}$ is still weak. Searches for Higgs
boson pair production at the LHC have put constraints on the triple Higgs
boson coupling as $-5.0<\lambda_{hhh}/\lambda_{hhh}^{\rm SM}<12.0$ for ATLAS
Aaboud  _et al._ (2018) and $-11.8<\lambda_{hhh}/\lambda_{hhh}^{\rm SM}<18.8$
for CMS Sirunyan  _et al._ (2019) at the 95% confidence level. At future
collider experiments, these bounds will be significantly improved. At the
High-Luminosity LHC, $0.52<\lambda_{hhh}/\lambda_{hhh}^{\rm SM}<1.5$ and
$0.57<\lambda_{hhh}/\lambda_{hhh}^{\rm SM}<1.5$ with and without systematic
uncertainties ($1~{}\sigma$), respectively Cepeda  _et al._ (2019). At the ILC
with a center-of-mass energy of $\sqrt{s}=1~{}\mathrm{TeV}$ and integrated
luminosity of $L=4~{}{\rm ab}^{-1}$, a precision of 10% is estimated Fujii
_et al._ (2019). At the CLIC with $\sqrt{s}=3$ TeV and $L=5~{}{\rm ab}^{-1}$,
the triple Higgs boson coupling will be measured with a relative uncertainty
of $-8\%$ to $11\%$ ($1~{}\sigma$) Roloff  _et al._ (2020).

###  V.1 Numerical Result

As a measure of the triple Higgs boson coupling, we introduce the deviation
parameter defined by

$\displaystyle\Delta\lambda=\frac{\lambda_{hhh}-\lambda_{hhh}^{\rm
SM}}{\lambda_{hhh}^{\rm SM}}.$ (34)

In evaluating ${\lambda_{hhh}^{\rm SM}}$, we use the effective potential
method and include the top quark contribution at the one-loop level. Actually,
the evaluated values of the masses of the top quark and Higgs boson and other
4D parameters should be regarded as those at some high energy scale, and the
corresponding measured values are corrected using renormalization group
equations. Detailed analysis of the renormalization group equations is beyond
the scope of this paper. Taking such uncertainties into account, we employ
$152~{}{\rm GeV}<m_{t}<182~{}{\rm GeV}$ and $110~{}{\rm GeV}<m_{h}<140~{}{\rm
GeV}$ as allowed values in our parameter scan, whose range is given in Eq.
(33). In Fig. 3, we show the correlation between the deviation of the triple
Higgs boson coupling $\Delta\lambda$ and the compactification scale $1/R$. The
orange band shows the $1~{}\sigma$ accuracy expected at the ILC. From Fig. 3,
it is clear that the deviation $\Delta\lambda$ is primarily characterized by
the compactification scale $1/R$. At very large compactification scales, the
deviation of the triple Higgs boson coupling almost vanishes as

$\displaystyle\lim_{1/R\to\infty}\Delta\lambda=0.$ (35)

Compared with the current constraint on the compactification scale,
$1/R>5~{}\mathrm{TeV}$, additional extensions to this GHU model are
unavoidable if a significant deviation is observed at the future colliders.

Figure 3: Correlation between the deviation of the triple Higgs boson coupling
$\Delta\lambda$ and the compactification scale $1/R$. The orange band shows
the $1~{}\sigma$ accuracy expected at the ILC.

Let us investigate to what extent each field contributes to the total
deviation of the triple Higgs boson coupling $\Delta\lambda$. To this end, we
decompose the predicted triple Higgs boson coupling into potential terms as

$\displaystyle\lambda_{hhh}=\sum_{i}\delta\lambda_{i},$ (36)

where $\delta\lambda_{i}$ denotes the contribution from a field $i$,

$\displaystyle\quad\delta\lambda_{i}=\left.\frac{\partial^{3}V_{i}}{\partial\alpha^{3}}\right|_{\alpha=\alpha_{0}}-\left.\frac{3}{\alpha}\frac{\partial^{2}V_{i}}{\partial\alpha^{2}}\right|_{\alpha=\alpha_{0}}+\left.\frac{3}{\alpha^{2}}\frac{\partial
V_{i}}{\partial\alpha}\right|_{\alpha=\alpha_{0}}.$ (37)

It is convenient to introduce the following deviation parameter:

$\displaystyle\Delta\lambda_{i}=\frac{\delta\lambda_{i}-\lambda_{hhh}^{\rm
SM}}{\lambda_{hhh}^{\rm SM}},$ (38)

which quantifies the extent of the contribution from the field $i$. In Fig. 4,
green points shows the contributions to the deviation parameters
$\Delta\lambda_{i}$ for $\Psi_{t}(2\alpha)$ (upper left panel), $t$ (upper
right), $\Psi_{A}(2\alpha)$ (lower left), and their sum (lower right),

$\displaystyle\Delta\lambda_{\Psi_{t}(2\alpha)+t+\Psi_{A}(2\alpha)}=\frac{\delta\lambda_{\Psi_{t}(2\alpha)}+\delta\lambda_{t}+\delta\lambda_{\Psi_{A}(2\alpha)}-\lambda_{hhh}^{\rm
SM}}{\lambda_{hhh}^{\rm SM}}.$ (39)

For comparison, the total deviation parameter $\Delta\lambda$ is also
displayed (red). Particles with larger $q$ values have stronger couplings to
the Higgs field, and give dominant contributions to the deviation of the
triple Higgs boson coupling. The total deviation $\Delta\lambda$ is well
approximated by the sum of the contributions from $\Psi_{t}(2\alpha)$, $t$ and
$\Psi_{A}(2\alpha)$.

---
Figure 4: Contributions to the deviation parameters $\Delta\lambda_{i}$ for
$\Psi_{t}(2\alpha)$ (upper left panel), $t$ (upper right), $\Psi_{A}(2\alpha)$
(lower left), and their sum (lower right). For comparison, the total deviation
parameter $\Delta\lambda$ is also displayed (red).

###  V.2 Analysis

From the numerical results presented above, the deviation of the triple Higgs
boson coupling converges to zero as the compactification scale increases. This
means that the shape of our GHU Higgs potential resembles the SM Higgs
potential at around the VEV for small $\alpha_{0}$ although their tree-level
shapes are completely different. Here, we analyze the shape of the Higgs
potential at small $\alpha$.

In general, the Higgs potential in GHU with a flat extra dimension is
constructed by summing up the contributions of bulk gauge fields, bulk
fermions, and brane fermions, as discussed in Sec. III. Although the field-
dependent masses of the brane fermions are complicated functions of $\alpha$
as discussed in Sec. II.2, they are approximated by linear functions of
$\alpha$ for $\alpha\ll 1$ as is the case for the fermions in the SM.
Expanding the GHU Higgs potential with respect to $\alpha$, we obtain

$\displaystyle V_{\rm
eff}(\phi)\simeq-\left(\frac{1}{R}\right)^{2}A\phi^{2}+B\phi^{4}+C\phi^{4}\ln\frac{\phi^{2}}{v^{2}}+\left(\frac{1}{R}\right)^{-2}D\phi^{6},$
(40)

where $A$, $B$, $C$ and $D$ are dimensionless parameters that are functions of
the model parameters. From the tadpole condition,

$\displaystyle\left.\frac{\partial V_{\rm
eff}}{\partial\phi}\right|_{\phi=v}=0,$ (41)

and the condition for reproducing the mass of the discovered Higgs boson,

$\displaystyle\left.\frac{\partial^{2}V_{\rm
eff}}{\partial\phi^{2}}\right|_{\phi=v}=m_{h}^{2},$ (42)

we obtain

$\displaystyle
0=-\left(\frac{1}{R}\right)^{2}\frac{A}{2v^{2}}+B+\frac{C}{2}+\frac{3}{2}\left(\frac{1}{R}\right)^{-2}Dv^{2},$
(43)

and

$\displaystyle
m_{h}^{2}=4\left(\frac{1}{R}\right)^{2}A+8Cv^{2}+12\left(\frac{1}{R}\right)^{-2}Dv^{4},$
(44)

respectively. Namely, two of the four coefficients are not independent. Since
the compactification scale should be above a few TeV ($\alpha_{0}\ll 1$), Eqs.
(43) and (44) suggests that the parameter $A$ should be very small for
accounting for the VEV and the mass of the Higgs boson. This fact implies the
so-called little hierarchy problem. In GHU, one needs to tune bulk field
parameters to adequately suppress the parameter $A$. In our GHU model, this
suppression is accomplished by introducing the additional bulk fields
($\Psi_{A},\tilde{\Psi}_{A}$).

Eliminating the parameters $A$ and $B$ using the tadpole and mass conditions,
Eqs. (43) and (44), the triple Higgs boson coupling is written as

$\displaystyle\lambda_{hhh}=\frac{3}{v}\left[m_{h}^{2}+\frac{16}{3}Cv^{2}+16\left(\frac{1}{R}\right)^{-2}Dv^{4}\right].$
(45)

Since the term proportional to $\phi^{4}\ln\phi^{2}$ arises from loop diagrams
in which only SM particles are involved, the value of the coefficient $C$ is
the same as that in the SM. Therefore, the deviation of the triple Higgs boson
coupling from the SM is suppressed by the square of the compactification scale
as

$\displaystyle\Delta\lambda=\frac{48Dv^{3}}{\lambda_{hhh}^{\rm
SM}}\left(\frac{1}{R}\right)^{-2}.$ (46)

Namely, in the cases when the $\phi^{6}$ term and higher order ones are
negligibly small, the shape of the potential and the triple Higgs boson
coupling should be the same as the SM ones irrespective of the origin of the
potential. In deriving Eq. (46), we have not assumed specific bulk fields or
Lorentz-violating parameters. We emphasize that such a conclusion is
applicable to a wide range of GHU models that reduces to the Higgs sector with
one Higgs doublet below the compactification scale of a flat extra dimension.
This point has not been manifested in earlier works. Therefore, if a
significant deviation of the triple Higgs boson coupling is observed at
collider experiments in the future, we have to extensively modify the GHU
framework.

Finally, we mention the sign of the deviation parameter, which is controlled
by the coefficient $D$. As for the bulk fields, the deviation derived from
each contribution tends to take a larger value when $p$ and $k$ are large and
$\lambda$ is small. Its sign is given by $\sigma_{S}\sigma_{I}$, which is
determined by the periodicity on $S^{1}$ and the spin of each bulk field. In
our model, the deviation goes in the negative direction because the bulk
fermion $\Psi(2\alpha)$ with $\sigma_{S}=1$ and $\sigma_{I}=-1$ gives rise to
the largest contribution. This observation about the dependence on $R$ and the
sign accounts for our numerical results about the deviation of the triple
Higgs boson coupling presented in Fig. 3.

##  VI Summary

In sharp contrast to the Higgs potential assumed in the SM, the GHU Higgs
potential is formed exclusively from quantum corrections because the leading-
order Higgs potential vanishes as a remnant of higher-dimensional gauge
symmetry. In order to investigate phenomenological difference between the GHU
Higgs potential and SM one, we have focused on the triple Higgs boson
coupling. Taking the $SU(3)_{w}$ model with 5D Lorentz symmetry relaxed as an
example of the GHU framework with a flat extra dimension, it has been
illustrated that the deviation of the triple Higgs boson coupling from the SM
prediction quickly vanishes as the compactification scale $1/R$ increases. For
compactification scales larger than the experimental limit $1/R\gtrsim
5~{}\mathrm{TeV}$, the deviation has been found to be smaller than 10%.

We have also discussed the characteristics of the Higgs potential in GHU in
which one Higgs doublet appears below the compactification scale of a flat
extra dimension. We have expanded the Higgs potential with respect to
$\alpha$, and found that the GHU potential has the same shape as the SM one up
to higher-order terms suppressed by the compactification scale. This fact
indicates that the deviation of the triple Higgs boson coupling vanishes at
large compactification scales in such a class of GHU models. Therefore,
compared with the expected accuracy of the measurement of the triple Higgs
boson coupling, the observation of its noticeable deviation at future collider
experiments will demand us to significantly modify the GHU framework.

###### Acknowledgements.

The authors would like to thank Shuichiro Funatsu for useful discussion and
comments. The work of M. K. was supported, in part, by JSPS Grant-in-Aid for
Scientific Research KAKENHI Grant Numbers 20H00160 and 21K03571. The work of
S. S. was supported, in part, by the Sasakawa Scientific Research Grant from
The Japan Science Society.

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