NITEP 137

Gauge Symmetry Breaking in Flux Compactification
with Wilson-line Scalar Condensate

Kento Akamatsua, Takuya Hirosea and Nobuhito Marua,b,

aDepartment of Physics, Osaka Metropolitan University,
Osaka 558-8585, Japan

bNambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP),
Osaka Metropolitan University, Osaka 558-8585, Japan

We discuss the gauge symmetry breaking of six dimensional theories in flux
compactification with a magnetic flux background and a constant vacuum
expectation value (VEV) for the scalar fields, which are zero modes of extra
spatial components of the gauge field. Although the effective potential for
the scalar fields are known not to be generated classically and radiatively in
a magnetic flux background only, the one-loop effective potential is shown to
be generated by the effects of the non-zero constant VEV. As illustrations, we
calculate the one-loop effective potential in SU(2) and SU(3) Yang-Mills
theories. In both cases, we expect that the potential minimum is located at
non-zero VEV and the gauge symmetry breaking takes place.

##  1 Introduction

Although the Standard Model (SM) has a successful theory, it still has some
problems. Many attractive scenarios based on the higher dimensional theory
have been proposed as the physics beyond the SM. In particular, flux
compactification, which has been studied in string theory [1, 2], has many
attractive aspects: explanation of the generation number of the SM fermions
[3, 4] and computation of Yukawa coupling [5, 6, 7, 8].

Recently, it has been considered that the quantum corrections to the masses of
zero-mode of the scalar field induced from extra components of higher
dimensional gauge field (called as Wilson-line (WL) scalar field) are
cancelled [9, 10, 11, 12, 13] and are finite [14]. The reason why the quantum
corrections are cancelled is that the shift symmetry from translation in extra
spaces forbids the mass term of scalar field since the zero-mode of the scalar
field can be identified with Nambu-Goldstone (NG) boson of spontaneously
broken translational symmetry (or with pseudo-NG boson in [14]). This
cancellation mechanism may be applied to the hierarchy problem in the SM which
is the problem that the quantum corrections to the mass of Higgs field are
sensitive to the square of the ultraviolet cutoff scale of the theory. If we
regard the Higgs field as the WL scalar field, which is an idea of gauge-Higgs
unification [15, 16, 17, 18], the quantum corrections to the mass of Higgs
field is cancelled as mentioned above. In the gauge-Higgs unification, the
finite Higgs mass is generated by the quantum corrections [18, 23, 19, 20, 21]
controlled by the compactification scale. If the compactification scale is
increased by the absence of the new physics discovery, the fine-tuning problem
in the Higgs mass parameter is reintroduced. In flux compactification,
however, if the translational symmetry in extra spaces is explicitly broken
around TeV scale independent of the compactification scale, the light Higgs
boson mass is radiatively generated. This logic is also applied to the
potential of the WL scalar field.

In this paper, we investigate the gauge symmetry breaking in a higher
dimensional theory in flux compactification with a magnetic flux background
and a constant WL scalar vacuum expectation value (VEV). First, we consider a
six dimensional SU(2) Yang-Mills theory compactified on a torus with a
magnetic flux and a constant VEV. Calculating the KK mass spectrum in the
presence of both flux background and the constant VEV, we obtain the one-loop
effective potential for the WL scalar field. Although the effective potential
in the flux background only is not radiatively generated, the effective
potential in both the flux background and the constant VEV is generated at
one-loop and the potential minimum at nonvanishing constant VEV is expected.
This concludes that gauge symmetry SU(2) is completely broken.

Next, we consider a six dimensional SU(3) Yang-Mills theory compactified on a
torus with a magnetic flux and a constant VEV. In this case, we consider two
types of configuration where the flux background and the constant VEV are
developed. One is that the flux background and the constant VEV are in the
eighth and the first components of SU(3). The other is that the flux
background and the constant VEV are in the eighth and the sixth components of
SU(3). Calculating the one-loop effective potential similarly as is done in
SU(2) Yang-Mills theory, we find (expect) that the potential is minimized at
nonvanishing constant VEV in the former (latter) case. In the former (latter)
case, the gauge symmetry breaking SU(3) $\to$ U(1) $\times$ U(1)(SU(3) $\to$
U(1)) is found, respectively.

This paper is organized as follows. We give a setup of a six-dimensional SU(2)
Yang-Mills theory with magnetic flux compactification and introduce the
constant VEV in section 2. We furthermore consider a six-dimensional SU(3)
Yang-Mills theory with magnetic flux compactification in section 3, where two
types of configuration for the flux background and the constant VEV to be
taken. In both sections, the one-loop effective potential for the WL scalar
field is calculated and the gauge symmetry breaking is discussed. In the last
section, we devote our summary. In Appendix A, the calculation of the KK mass
spectrum at the second order perturbation is summarized.

##  2 SU(2) Yang-Mills theory

We consider a six-dimensional SU(2) Yang-Mills theory with two nontrivial
backgrounds: a constant magnetic flux background and an ordinary constant
vacuum expectation value.

###  2.1 Set up

Six-dimensional spacetime is $M^{4}\times T^{2}$, where $M^{4}$ is a Minkowski
spacetime and $T^{2}$ is a two-dimensional square torus. The Lagrangian of
SU(2) Yang-Mills theory in six dimensions is

$\displaystyle\mathcal{L}_{6}$ $\displaystyle=-\frac{1}{4}F^{a}_{MN}F^{aMN}$
$\displaystyle=-\frac{1}{4}F^{a}_{\mu\nu}F^{a\mu\nu}-\frac{1}{2}F^{a}_{\mu
5}F^{a\mu 5}-\frac{1}{2}F^{a}_{\mu 6}F^{a\mu 6}-\frac{1}{2}F^{a}_{56}F^{a56},$
(1)

where the field strength tensor and the covariant derivative are defined by

$\displaystyle F^{a}_{MN}$
$\displaystyle=\partial_{M}A^{a}_{N}-\partial_{N}A^{a}_{M}-ig[A_{M},A_{N}]^{a},$
(2) $\displaystyle D_{M}A^{a}_{N}$
$\displaystyle=D_{M}^{ac}A_{N}^{c}=(\delta^{ac}\partial_{M}+g\epsilon^{abc}A^{b}_{M})A^{c}_{N}$
$\displaystyle=\partial_{M}A^{a}_{N}-ig[A_{M},A_{N}]^{a}.$ (3)

The spacetime indices are $M,N=0,1,2,3,5,6,\mu,\nu=0,1,2,3,m,n=5,6$, and the
gauge indices are $a,b,c=1,2,3$. The metric convention
$\eta_{MN}=\text{diag}(-1,+1,\cdots,+1)$ is employed. $\epsilon^{abc}$ is a
totally anti-symmetric tensor of SU(2).

We discuss how the two backgrounds are introduced in our model. First, the
constant magnetic flux, is given by the VEV of the fifth and the sixth
component of the gauge fields, $A^{3}_{5,6}$, which must satisfy their
classical equation of motion:

$\displaystyle D^{m}\braket{F^{a}_{mn}}=0.$ (4)

Second, the ordinary constant background is generated by the quantum
correction in the sixth component of the gauge field $A^{1}_{6}$, for
simplicity 111In general, the constant background can be introduced by the
fifth component of the gauge field $A^{1}_{5}$. . In this section, we choose a
solution

$\displaystyle\braket{A^{1}_{6}}=v,~{}~{}~{}\braket{A^{3}_{5}}=-\frac{1}{2}fx_{6},~{}~{}~{}\braket{A^{3}_{6}}=\frac{1}{2}fx_{5}~{}~{}\text{and}~{}~{}\braket{A^{1,2}_{5}}=\braket{A^{2}_{6}}=0.$
(5)

$\braket{A^{3}_{5,6}}$ introduce a magnetic field parametrized by a constant
$f$, namely $\braket{F^{3}_{56}}=f$. Note that the flux background
spontaneously breaks a translational invariance on the torus. The flux
background breaks the gauge symmetry, which is broken to U(1) in this case.
The flux is also associated with the degeneracy:

$\displaystyle\frac{g}{2\pi}\int_{T^{2}}dx_{5}dx_{6}\braket{F^{3}_{56}}=\frac{g}{2\pi}L^{2}f=N\in\mathbb{Z},$
(6)

where $L^{2}$ is an area of the torus. For simplicity, we set $L=1$ from now
on. It is useful to define $\partial$ and the scalar fields $\phi^{a}$ as

$\displaystyle\partial\equiv\partial_{z}=\partial_{5}-i\partial_{6},~{}~{}~{}z\equiv\frac{1}{2}(x_{5}+ix_{6}),~{}~{}~{}\phi^{a}=\frac{1}{\sqrt{2}}(A^{a}_{6}+iA^{a}_{5}).$
(7)

In these complex coordinates, the VEVs of $\phi^{1,3}$ are given by

$\displaystyle\braket{\phi^{1}}=\frac{v}{\sqrt{2}},\qquad\braket{\phi^{3}}=\frac{f\bar{z}}{\sqrt{2}},$
(8)

and we expand $\phi^{a}$ around the backgrounds:

$\displaystyle\phi^{a}=\braket{\phi^{a}}+\varphi^{a},$ (9)

where $\varphi^{a}$ are quantum fluctuations.

The Lagrangian (2.1) can be rewritten by using the new coordinates (7) as
follows:

$\displaystyle\mathcal{L}_{6}=$
$\displaystyle-\frac{1}{4}F^{a}_{\mu\nu}F^{a\mu\nu}-\partial_{\mu}\bar{\phi}^{a}\partial^{\mu}\phi^{a}-\frac{1}{2}DA^{a}_{\mu}\overline{D}A^{a\mu}$
$\displaystyle-\frac{i}{\sqrt{2}}(\partial_{\mu}\phi^{a}\bar{\partial}A^{a\mu}-\partial_{\mu}\bar{\phi}^{a}\partial
A^{a\mu})$
$\displaystyle+ig\left(\partial_{\mu}\phi^{a}[A^{\mu},\bar{\phi}]^{a}+\partial^{\mu}\bar{\phi}^{a}[A_{\mu},\phi]^{a}\right)$
$\displaystyle-\frac{1}{4}\left(D\bar{\phi}^{a}+\overline{D}\phi^{a}+\sqrt{2}g[\phi,\bar{\phi}]^{a}\right)^{2},$
(10)

where

$\displaystyle\begin{cases}D\Phi^{a}\equiv(D_{5}-iD_{6})\Phi^{a}=\partial\Phi^{a}-\sqrt{2}g[\phi,\Phi]^{a},\\\\[5.69054pt]
\overline{D}\Phi^{a}\equiv(D_{5}+iD_{6})\Phi^{a}=\bar{\partial}\Phi^{a}+\sqrt{2}g[\bar{\phi},\Phi]^{a},\end{cases}$
(11)

which express the covariant derivatives with respect to the complex
coordinates in the torus. $\Phi^{a}$ denote arbitrary fields in the adjoint
representation. We can remove the mixing terms between the gauge and the
scalar fields in the second line of eq. (2.1) by introducing the gauge-fixing
terms with a gauge parameter $\xi$:

$\displaystyle\mathcal{L}_{g-f}\equiv$
$\displaystyle-\frac{1}{2\xi}(D_{\mu}A^{a\mu}+\xi\mathcal{D}_{m}A^{am})^{2}$
$\displaystyle=$
$\displaystyle-\frac{1}{2\xi}D_{\mu}A^{a\mu}D_{\nu}A^{a\nu}-\frac{g}{\sqrt{2}}\left(\partial\bar{\phi}^{a}[A_{\mu},A^{\mu}]^{a}-\bar{\partial}\phi^{a}[A_{\mu},A^{\mu}]^{a}\right)$
$\displaystyle+\frac{\xi}{4}(\mathcal{D}\bar{\phi}^{a}-\overline{\mathcal{D}}\phi^{a})^{2}+\frac{i}{\sqrt{2}}(\partial_{\mu}\phi^{a}\bar{\partial}A^{a\mu}-\partial_{\mu}\bar{\phi}^{a}\partial
A^{a\mu}).$ (12)

The new covariant derivatives $\mathcal{D},\overline{\mathcal{D}}$ are defined
by replacing $\phi^{a},\bar{\phi}^{a}$ in $D,\overline{D}$ with the VEVs
$\braket{\phi^{a}},\braket{\bar{\phi}^{a}}$, respectively. Due to the gauge-
fixing, we need to introduce the Faddeev-Popov ghost fields and their
Lagrangian is

$\displaystyle\mathcal{L}_{ghost}=-\bar{c}^{a}(D_{\mu}D^{\mu}+\xi
D_{m}\mathcal{D}^{m})c^{a}.$ (13)

Then, the total Lagrangian is given by

$\displaystyle\mathcal{L}_{total}=$
$\displaystyle-\frac{1}{4}F^{a}_{\mu\nu}F^{a\mu\nu}-\frac{1}{2\xi}D_{\mu}A^{a\mu}D_{\nu}A^{a\nu}-\partial_{\mu}\bar{\phi}^{a}\partial^{\mu}\phi^{a}-\frac{1}{2}DA^{a}_{\mu}\overline{D}A^{a\mu}$
$\displaystyle-\frac{g}{\sqrt{2}}\left(\partial\bar{\phi}^{a}[A_{\mu},A^{\mu}]^{a}-\bar{\partial}\phi^{a}[A_{\mu},A^{\mu}]^{a}\right)+\frac{\xi}{4}(\mathcal{D}\bar{\phi}^{a}-\overline{\mathcal{D}}\phi^{a})^{2}$
$\displaystyle+ig\left(\partial_{\mu}\phi^{a}[A^{\mu},\bar{\phi}]^{a}+\partial^{\mu}\bar{\phi}^{a}[A_{\mu},\phi]^{a}\right)$
$\displaystyle-\frac{1}{4}\left(D\bar{\phi}^{a}+\overline{D}\phi^{a}+\sqrt{2}g[\phi,\bar{\phi}]^{a}\right)^{2}-\bar{c}^{a}(D_{\mu}D^{\mu}+\xi
D_{m}\mathcal{D}^{m})c^{a}.$ (14)

For simplicity, we choose the Feynman gauge $\xi=1$ throughout this paper.

###  2.2 The mass of the gauge fields $A^{a}_{\mu}$

We will discuss mass eigenstates and eigenvalues of the fields
$A^{a}_{\mu},\varphi^{a},c^{a}$. In this subsection, we find mass eigenvalues
and eigenstates of the gauge fields $A^{a}_{\mu}$. The mass term of gauge
field corresponds to the background part of
$-DA^{a}_{\mu}\overline{D}A^{a\mu}/2$:

$\displaystyle\mathcal{L}_{AA}=-\frac{1}{2}\mathcal{D}A^{a}_{\mu}\overline{\mathcal{D}}A^{a\mu}=-\frac{1}{2}A^{a}_{\mu}\left[-\mathcal{D}\overline{\mathcal{D}}\hskip
1.42262pt\right]A^{a\mu}.$ (15)

We would like to regard the background covariant derivatives
$\mathcal{D},\overline{\mathcal{D}}$ as creation and annihilation operators,
respectively. In a matrix form, they are expressed as

$\displaystyle\mathcal{D}=\left[\begin{array}[]{ccc}\partial&igf\bar{z}&0\\\
-igf\bar{z}&\partial&igv\\\
0&-igv&\partial\end{array}\right],\qquad\overline{\mathcal{D}}=\left[\begin{array}[]{ccc}\bar{\partial}&-igfz&0\\\
igfz&\bar{\partial}&-igv\\\ 0&igv&\bar{\partial}\end{array}\right].$ (22)

Diagonalizing them, we obtain

$\displaystyle\begin{cases}\mathcal{D}_{diag}=\text{diag}\left(\partial,\partial+g\sqrt{f^{2}\bar{z}^{2}+v^{2}},\partial-g\sqrt{f^{2}\bar{z}^{2}+v^{2}}\right),\\\\[5.69054pt]
\overline{\mathcal{D}}_{diag}=\text{diag}\left(\bar{\partial},\bar{\partial}-g\sqrt{f^{2}z^{2}+v^{2}},\bar{\partial}+g\sqrt{f^{2}z^{2}+v^{2}}\right).\end{cases}$
(23)

Their commutation relations is

$\displaystyle\left[\overline{\mathcal{D}}_{diag},\mathcal{D}_{diag}\right]^{ac}=gf^{2}\left(\frac{z}{\sqrt{f^{2}z^{2}+v^{2}}}+\frac{\bar{z}}{\sqrt{f^{2}\bar{z}^{2}+v^{2}}}\right)\left[\begin{array}[]{ccc}0&0&0\\\
0&1&0\\\ 0&0&-1\end{array}\right],$ (27)

which depends on extra space coordinates. Therefore,
$\mathcal{D},\overline{\mathcal{D}}$ cannot be identified with creation and
annihilation operators.

Since Kaluza-Klein (KK) mass spectrum cannot be exactly solved by using the
creation and annihilation operators, we would like to find them perturbatively
by the expansion in $v$. In this expansion, $vL\ll 1$ or $v\ll 1$ in the
present case is assumed. In other words, we consider the case where the
compactification scale is much larger than the constant VEV $v$. From (22), we
define the unperturbed parts $\mathcal{D}_{3},\overline{\mathcal{D}}_{3}$ and
the perturbed part $V$ as

$\displaystyle\mathcal{D}_{3}\equiv\left[\begin{array}[]{ccc}\partial&igf\bar{z}&0\\\
-igf\bar{z}&\partial&0\\\
0&0&\partial\end{array}\right],~{}~{}~{}\overline{\mathcal{D}}_{3}\equiv\left[\begin{array}[]{ccc}\bar{\partial}&-igfz&0\\\
igfz&\bar{\partial}&0\\\ 0&0&\bar{\partial}\end{array}\right]$ (34)

and

$\displaystyle V\equiv\left[\begin{array}[]{ccc}0&0&0\\\ 0&0&igv\\\
0&-igv&0\end{array}\right],$ (38)

respectively. In these notations, the covariant derivatives can be expressed
as $\mathcal{D}=\mathcal{D}_{3}+V$ and
$\overline{\mathcal{D}}=\overline{\mathcal{D}}_{3}+\overline{V}$.
$\mathcal{D}_{3}$ and $\overline{\mathcal{D}}_{3}$ can be identified with
creation and annihilation operators, which are diagonalized as follows.

$\displaystyle\begin{cases}\mathcal{D}_{3,diag}=\text{diag}(\partial-
gf\bar{z},\partial+gf\bar{z},\partial),\\\\[5.69054pt]
\overline{\mathcal{D}}_{3,diag}=\text{diag}(\bar{\partial}+gfz,\bar{\partial}-gfz,\bar{\partial}).\end{cases}$
(39)

Their diagonalizing unitary matrix $U_{3}$, which satisfies
$U_{3}^{-1}\mathcal{D}_{3}U_{3}=\mathcal{D}_{3,diag}$ and
$U_{3}^{-1}\overline{\mathcal{D}}_{3}U_{3}=\overline{\mathcal{D}}_{3,diag}$,
is

$\displaystyle U_{3}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{ccc}1&i&0\\\
i&1&0\\\ 0&0&\sqrt{2}\end{array}\right].$ (43)

The commutation relation between $\mathcal{D}_{3,diag}$ and
$\overline{\mathcal{D}}_{3,diag}$ is

$\displaystyle\left[i\overline{\mathcal{D}}_{3,diag},i\mathcal{D}_{3,diag}\right]^{ac}=2gf\left[\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\ 0&0&0\end{array}\right].$ (47)

The creation and annihilation operators are defined as

$\displaystyle
a=\frac{i}{\sqrt{\alpha_{2}}}\overline{\mathcal{D}}_{3,diag},\qquad
a^{\dagger}=\frac{i}{\sqrt{\alpha_{2}}}\mathcal{D}_{3,diag},$ (48)

where $\alpha_{2}\equiv 2gf$. The components of the creation and annihilation
operators are summarized as follows:

$\displaystyle\begin{cases}\displaystyle
a_{1}\equiv\frac{i}{\sqrt{\alpha_{2}}}(\bar{\partial}+gfz),\\\\[11.38109pt]
\displaystyle
a_{2}\equiv\frac{i}{\sqrt{\alpha_{2}}}(\bar{\partial}-gfz),\\\\[11.38109pt]
\displaystyle
a_{3}\equiv\frac{i}{\sqrt{\alpha_{2}}}\bar{\partial},\end{cases}~{}~{}\text{and}~{}~{}~{}~{}\begin{cases}\displaystyle
a^{\dagger}_{1}\equiv\frac{i}{\sqrt{\alpha_{2}}}(\partial-
gf\bar{z}),\\\\[11.38109pt] \displaystyle
a^{\dagger}_{2}\equiv\frac{i}{\sqrt{\alpha_{2}}}(\partial+gf\bar{z}),\\\\[11.38109pt]
\displaystyle
a^{\dagger}_{3}\equiv\frac{i}{\sqrt{\alpha_{2}}}\partial.\end{cases}$ (49)

We note that $a_{3}$ and $a^{\dagger}_{3}$ have no flux effects and play no
role of creation and annihilation operators. $a^{\dagger}_{1}$ and $a_{2}$ are
creation operators and $a_{1}$ and $a^{\dagger}_{2}$ are annihilation
operators: the roles of creation and annihilation operators for $a_{2}$ and
$a^{\dagger}_{2}$ are inverted due to the commutation relation for the
$2$-direction $[a_{2},a^{\dagger}_{2}]=-1$. The ground state mode functions
are determined by

$\displaystyle a_{1}\xi_{0,j}=0,\qquad a^{\dagger}_{2}\bar{\xi}_{0,j}=0,$ (50)

where $\xi_{0,j}$ are the functions of $z$ (in detail, see [5]) and $j$ labels
the degeneracy of the ground state: $j=0,\cdots,|N|-1$. Higher mode functions
$\xi_{n,j}$ are constructed similar to the harmonic oscillator case [7]:

$\displaystyle\xi_{n_{1},j}=\frac{1}{\sqrt{n_{1}!}}(a^{\dagger}_{1})^{n_{1}}\xi_{0,j},~{}~{}\bar{\xi}_{n_{2},j}=\frac{1}{\sqrt{n_{2}!}}(a_{2})^{n_{2}}\bar{\xi}_{0,j}~{}~{}(n_{1,2}\in\mathbb{Z},~{}n_{1,2}\geq
0).$ (51)

The functions $\xi_{n,j}$ satisfy the orthogonal conditions

$\displaystyle\int_{T^{2}}d^{2}x~{}\bar{\xi}_{n,j}~{}\xi_{n^{\prime},j^{\prime}}=\delta_{n,n^{\prime}}\delta_{j,j^{\prime}}.$
(52)

To be operated by $a$ or $a^{\dagger}$, we should define the states with the
gauge indices:

$\displaystyle\psi^{1}_{n_{1},j}\equiv\left[\begin{array}[]{c}\xi_{n_{1},j}\\\
0\\\
0\end{array}\right],\qquad\psi^{2}_{n_{2},j}\equiv\left[\begin{array}[]{c}0\\\
\bar{\xi}_{n_{2},j}\\\ 0\end{array}\right].$ (59)

Moreover, using the periodic boundary condition of torus, we define the
eigenstates for the 3-direction:

$\displaystyle\psi^{3}_{l,m}\equiv\left[\begin{array}[]{c}0\\\ 0\\\
\lambda_{l,m}\end{array}\right]~{}~{}~{}(l,m\in\mathbb{Z}),$ (63)

where $\lambda_{l,m}$ are the functions of $x_{5}$ and $x_{6}$:

$\displaystyle\lambda_{l,m}\equiv\lambda_{l,m}(x_{5},x_{6})=\exp[\hskip
0.85358pt2\pi i(lx_{5}+mx_{6})].$ (64)

These functions $\lambda_{l,m}$ also satisfy the orthogonal conditions

$\displaystyle\int_{T^{2}}d^{2}x~{}\bar{\lambda}_{l,m}\lambda_{l^{\prime},m^{\prime}}=\delta_{l,l^{\prime}}\delta_{m,m^{\prime}}.$
(65)

The eigenstates satisfy the following relations

$\displaystyle\begin{cases}a\psi^{1}_{n_{1},j}=\sqrt{n_{1}}\psi^{1}_{n_{1}-1,j},\\\\[5.69054pt]
a\psi^{2}_{n_{2},j}=\sqrt{n_{2}+1}\psi^{2}_{n_{2}+1,j},\\\\[5.69054pt]
\displaystyle a\psi^{3}_{l,m}\hskip
3.1298pt=-\sqrt{\frac{4\pi^{2}}{\alpha_{2}}}(l+im)\psi^{3}_{l,m},\end{cases}~{}~{}\text{and}~{}~{}~{}\begin{cases}a^{\dagger}\psi^{1}_{n_{1},j}=\sqrt{n_{1}+1}\psi^{1}_{n_{1}+1,j},\\\\[5.69054pt]
a^{\dagger}\psi^{2}_{n_{1},j}=\sqrt{n_{2}}\psi^{2}_{n_{2}-1,j},\\\\[5.69054pt]
\displaystyle a^{\dagger}\psi^{3}_{l,m}\hskip
3.1298pt=-\sqrt{\frac{4\pi^{2}}{\alpha_{2}}}(l-im)\psi^{3}_{l,m}.\end{cases}$
(66)

For convenience we unify the labels of the eigenstates:

$\displaystyle\psi^{1}_{\\{n_{1}\\}}\equiv\psi^{1}_{n_{1},j},~{}~{}\psi^{2}_{\\{n_{2}\\}}\equiv\psi^{2}_{n_{2},j},\quad\psi^{3}_{\\{n_{3}\\}}\equiv\psi^{3}_{l,m}.$
(67)

Their products of eigenstates in the different directions are zero.

$\displaystyle\left(\psi^{a}_{\\{n_{a}\\}}\right)^{\dagger}\psi^{c}_{\\{n_{c}\\}}=0~{}~{}(a\neq
c).$ (68)

Then we define the unperturbed Hamiltonian $H_{0}$ and the perturbation
$V_{1},V_{2}$:

$\displaystyle-\mathcal{D}\overline{\mathcal{D}}=$
$\displaystyle~{}U_{3}\left[-\mathcal{D}_{3,diag}\overline{\mathcal{D}}_{3,diag}+\mathcal{D}_{3,diag}U_{3}^{-1}VU_{3}\right.$
$\displaystyle\hskip
28.45274pt\left.-U_{3}^{-1}VU_{3}\overline{\mathcal{D}}_{3,diag}+(U_{3}^{-1}VU_{3})^{2}\right]U_{3}^{-1}$
$\displaystyle\equiv$
$\displaystyle~{}U_{3}\left[H_{0}+V_{1}+V_{1}^{\dagger}+V_{2}\right]U_{3}^{-1}$
$\displaystyle\equiv$ $\displaystyle~{}U_{3}HU_{3}^{-1}.$ (69)

From the previous discussion, $H_{0}$ is expressed as

$\displaystyle H_{0}=\left[\begin{array}[]{ccc}\alpha_{2}n_{1}&0&0\\\
0&\alpha_{2}(n_{2}+1)&0\\\ 0&0&4\pi^{2}(l^{2}+m^{2})\end{array}\right],$ (73)

and its eigenstates of the gauge fields are defined by

$\displaystyle
A^{a^{\prime}}_{\mu}U_{3}^{a^{\prime}a}\equiv\widetilde{A}^{a}_{\mu}\equiv\sum_{\\{n_{a}\\}}\widetilde{A}^{a}_{\mu,\\{n_{a}\\}}\psi^{a}_{\\{n_{a}\\}},\qquad\left(U^{-1}_{3}\right)^{aa^{\prime}}A^{a^{\prime}\mu}\equiv\widetilde{A}^{a\mu}\equiv\sum_{\\{n_{a}\\}}\widetilde{A}^{a\mu}_{\\{n_{a}\\}}\psi^{a}_{\\{n_{a}\\}}.$
(74)

The perturbation $V_{1}$ and $V_{2}$ are

$\displaystyle
V_{1}=\frac{gv}{\sqrt{2}}\left[\begin{array}[]{ccc}0&0&\partial-gf\bar{z}\\\
0&0&i(\partial+gf\bar{z})\\\ \partial&-i\partial&0\end{array}\right],\qquad
V_{2}=\frac{g^{2}v^{2}}{2}\left[\begin{array}[]{ccc}1&-i&0\\\ i&1&0\\\
0&0&2\end{array}\right],$ (81)

respectively. As was seen from the unperturbed Hamiltonian (73), we find that
there are degeneracy: $\psi^{1}_{0,j}$ and $\psi^{3}_{l,m}$,
$\psi^{1}_{n+1,j}$ and $\psi^{2}_{n,j}$. We thus should be careful to
calculate their energies in perturbation.

For $\psi^{1}_{0,j}$ and $\psi^{3}_{l,m}$, the first-order perturbation energy
from $V_{1}+V^{\dagger}_{1}+V_{2}$, $E_{A,0}^{(1)}$ can be easily obtained

$\displaystyle
E^{(1)}_{A,n_{1}=0}=g^{2}v^{2}/2,~{}~{}~{}E^{(1)}_{A,3}=g^{2}v^{2}.$ (82)

Note that we have to solve secular equation for $\psi^{1}_{0,j}$ and
$\psi^{3}_{l=0,m=0}$ because there exist the degeneracy and the perturbation
of $\psi^{3}_{l\neq 0,m\neq 0}$ can be obtained by $V_{2}$. The second-order
perturbation energy $E_{A,0}^{(2)}$ for $\psi^{1}_{0,j}$ is shown in appendix
A.

For $\psi^{1}_{n+1,j},\psi^{2}_{n,j}~{}(n\geq 0)$ and $\psi^{3}_{l,m}$, the
first-order perturbation energy from $V_{1}+V^{\dagger}_{1}+V_{2}$,
$E_{A}^{(1)}$ can be easily obtained

$\displaystyle
E_{A,1^{\prime}}^{(1)}=0,~{}~{}~{}E_{A,2^{\prime}}^{(1)}=g^{2}v^{2},$ (83)

where the mode functions in new direction $1^{\prime}$ and $2^{\prime}$ are
defined as

$\displaystyle\psi^{1^{\prime}}_{n+1,j}\equiv\left(i\psi^{1}_{n+1,j}+\psi^{2}_{n,j}\right)/\sqrt{2},~{}~{}~{}\psi^{2^{\prime}}_{n+1,j}\equiv\left(\psi^{1}_{n+1,j}+i\psi^{2}_{n,j}\right)/\sqrt{2}.$
(84)

The second-order perturbation energy $E_{A,1^{\prime}}^{(2)}$,
$E_{A,2^{\prime}}$ and $E_{A,3}^{(2)}$ are shown in appendix A.

Thus, we summarize the mass of the gauge fields as

$\displaystyle\begin{cases}~{}m^{2}_{A,n_{1}=0}=\displaystyle\frac{g^{2}v^{2}}{2}+E_{A,0}^{(2)},\\\\[17.07164pt]
~{}m^{2}_{A,1^{\prime}}\hskip
13.37277pt=\displaystyle\alpha_{2}(n+1)+E_{A,1^{\prime}}^{(2)},\\\\[17.07164pt]
~{}m^{2}_{A,2^{\prime}}\hskip
13.37277pt=\displaystyle\alpha_{2}(n+1)+g^{2}v^{2}+E_{A,2^{\prime}}^{(2)},\\\\[17.07164pt]
~{}m^{2}_{A,3}\hskip 16.07578pt=\displaystyle
4\pi^{2}(l^{2}+m^{2})+g^{2}v^{2}+E_{A,3}^{(2)},\end{cases}$ (85)

and we find that all of the gauge fields have nonzero mass if $v\neq 0$.
Therefore, we conclude that the SU(2) gauge symmetry is completely broken. The
fact that non-zero VEV $v$ is realized will be seen in the potential analysis.

###  2.3 The mass of the scalar fields $\varphi^{a}$

The terms relevant to the scalar mass are

$\displaystyle\mathcal{L}$
$\displaystyle\supset-\frac{1}{4}\left\\{(\mathcal{D}\bar{\varphi}+\overline{\mathcal{D}}\varphi)^{2}-2\sqrt{2}g(\partial\braket{\bar{\varphi}^{a}}+\bar{\partial}\braket{\varphi^{a}})[\varphi,\bar{\varphi}]^{a}\right\\}+\frac{1}{4}(\mathcal{D}\bar{\varphi}-\overline{\mathcal{D}}\varphi)^{2}$
$\displaystyle=-\overline{\widetilde{\varphi}}\left(H+gf\hskip
1.70717pt\text{diag}(1,-1,0)\right)\widetilde{\varphi},$ (86)

where

$\displaystyle\overline{\varphi}^{a^{\prime}}U_{3}^{a^{\prime}a}\equiv\overline{\widetilde{\varphi}}^{a}\equiv\sum_{\\{n_{a}\\}}\overline{\widetilde{\varphi}}^{a}_{\\{n_{a}\\}}\psi^{a}_{\\{n_{a}\\}},\qquad\left(U^{-1}_{3}\right)^{aa^{\prime}}\varphi^{a^{\prime}}\equiv\widetilde{\varphi}^{a}\equiv\sum_{\\{n_{a}\\}}\widetilde{\varphi}^{a}_{\\{n_{a}\\}}\psi^{a}_{\\{n_{a}\\}}.$
(87)

Since the energy eigenvalues for $\psi^{1}_{n,j}$ and $\psi^{2}_{n,j}$ are
degenerate, we must solve the secular equation. We find the first-order
perturbation energy from $V_{1}+V^{\dagger}_{1}+V_{2}$ as

$\displaystyle
E^{(1)}_{\varphi,1^{\prime\prime}}=0,~{}~{}~{}E^{(1)}_{\varphi,2^{\prime\prime}}=g^{2}v^{2},$
(88)

where the mode functions in new direction $1^{\prime\prime}$ and
$2^{\prime\prime}$ are defined as

$\displaystyle\psi^{1^{\prime\prime}}_{n,j}\equiv\left(i\psi^{1}_{n,j}+\psi^{2}_{n,j}\right)/\sqrt{2},~{}~{}~{}\psi^{2^{\prime\prime}}_{n,j}\equiv\left(\psi^{1}_{n,j}+i\psi^{2}_{n,j}\right)/\sqrt{2}.$
(89)

The second-order perturbation energy $E_{\varphi,1^{\prime\prime}}^{(2)}$,
$E_{\varphi,2^{\prime\prime}}^{(2)}$ and $E_{\varphi,3}^{(2)}$ are shown in
appendix A.

Thus, the mass of the scalar fields are obtained as

$\displaystyle\begin{cases}~{}\displaystyle
m^{2}_{\varphi,1^{\prime\prime}}\hskip
3.1298pt=\alpha_{2}\left(n+\frac{1}{2}\right)+E_{\varphi,1^{\prime\prime}}^{(2)},\\\\[11.38109pt]
~{}\displaystyle m^{2}_{\varphi,2^{\prime\prime}}\hskip
3.1298pt=\alpha_{2}\left(n+\frac{1}{2}\right)+g^{2}v^{2}+E_{\varphi,2^{\prime\prime}}^{(2)},\\\\[11.38109pt]
~{}\displaystyle m^{2}_{\varphi,3}\hskip
7.96677pt=4\pi^{2}(l^{2}+m^{2})+g^{2}v^{2}+E_{\varphi,3}^{(2)}.\end{cases}$
(90)

###  2.4 The mass of the ghost fields $c^{a}$

The terms relevant to the ghost mass is

$\displaystyle\mathcal{L}$
$\displaystyle\supset-\bar{c}^{a}(\mathcal{D}_{m}\mathcal{D}^{m})^{ab}c^{b}~{}~{}(m=5,6),$
(91)

where

$\displaystyle\mathcal{D}_{m}\mathcal{D}^{m}$
$\displaystyle=-\left(-\mathcal{D}\overline{\mathcal{D}}\right)-\frac{1}{2}\left[\mathcal{D},\overline{\mathcal{D}}\right]$
$\displaystyle=-U_{3}\left[H+\frac{1}{2}\left(\left[\mathcal{D}_{3,diag},\overline{\mathcal{D}}_{3,diag}\right]-V_{1}-V_{1}^{\dagger}+V_{3}+V_{3}^{\dagger}\right)\right]U_{3}^{-1}$
$\displaystyle=-U_{3}\left[H_{0}+V_{2}+gf\text{diag}(1,-1,0)+V_{4}+V_{4}^{\dagger}\right]U_{3}^{-1}.$
(92)

$V_{3}$ and $V_{4}$ are defined as

$\displaystyle V_{3}\equiv
U_{3}^{-1}VU_{3}\mathcal{D}_{3,diag}~{}~{}\text{and}~{}~{}V_{4}\equiv\frac{1}{2}(V_{1}+V_{3}).$
(93)

Note that the first three terms in the equation (2.4) are the same as those of
the scalar fields. As in previous section, we solve the secular equation and
find the first-order perturbation energy from $V_{2}+V_{4}+V^{\dagger}_{4}$ as

$\displaystyle
E^{(1)}_{c,1^{\prime\prime}}=0,~{}~{}~{}E^{(1)}_{c,2^{\prime\prime}}=g^{2}v^{2}.$
(94)

The second-order perturbation energy $E_{c,1^{\prime\prime}}^{(2)}$,
$E_{c,2^{\prime\prime}}^{(2)}$ and $E_{c,3}^{(2)}$ are summarized in appendix
A.

Thus, the mass of the ghost fields are obtained as

$\displaystyle\begin{cases}~{}\displaystyle m^{2}_{c,1^{\prime\prime}}\hskip
3.1298pt=\alpha_{2}\left(n+\frac{1}{2}\right)+E_{c,1^{\prime\prime}}^{(2)},\\\\[11.38109pt]
~{}\displaystyle m^{2}_{c,2^{\prime\prime}}\hskip
3.1298pt=\alpha_{2}\left(n+\frac{1}{2}\right)+g^{2}v^{2}+E_{c,2^{\prime\prime}}^{(2)},\\\\[11.38109pt]
~{}\displaystyle m^{2}_{c,3}\hskip
7.96677pt=4\pi^{2}(l^{2}+m^{2})+g^{2}v^{2}+E_{c,3}^{(2)}.\end{cases}$ (95)

Note that the mass of ghost fields are the same as that of scalar mass at the
first order in Feynman gauge $\xi=1$. This fact greatly simplifies the
potential analysis as will be discussed later.

###  2.5 The analysis of the effective potential

Since the potential of the constant WL scalar fields is not generated at tree
level, we have to calculate the one-loop effective potential by use of KK mass
spectrum obtained in the previous subsections. We will show that the constant
WL scalar VEV can be nonzero and the gauge symmetry is broken.

In order to calculate the one-loop effective potential as general as possible,
we parametrize the KK mass spectrum as follows.

$\displaystyle m^{2}_{0}\equiv
Ag^{2}v^{2},~{}~{}~{}m^{2}_{n}\equiv\alpha(n+x)+Bg^{2}v^{2},~{}~{}~{}m^{2}_{l,m}\equiv
4\pi^{2}(l^{2}+m^{2})+Cg^{2}v^{2},$ (96)

where $x$ is 0, $1$ or $1/2$ depending on the fields under consideration. We
suppose that $A,B$ and $C$ are positive constant. $\alpha$ means $\alpha_{2}$
or $\alpha_{3}$, which will be defined in next section. Then, the typical
forms of the one-loop effective potential can be written as

$\displaystyle\nu_{0}(A)$
$\displaystyle\equiv\int\frac{d^{4}p}{(2\pi)^{4}}\ln\left[p^{2}+Ag^{2}v^{2}\right],$
(97) $\displaystyle\nu_{n}(\alpha,B;x)$
$\displaystyle\equiv\sum_{n=0}^{\infty}\int\frac{d^{4}p}{(2\pi)^{4}}\ln\left[p^{2}+\alpha(n+x)+Bg^{2}v^{2}\right],$
(98) $\displaystyle\nu_{l,m}(C)$
$\displaystyle\equiv\sum_{l=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}\int\frac{d^{4}p}{(2\pi)^{4}}\ln\left[p^{2}+4\pi^{2}(l^{2}+m^{2})+Cg^{2}v^{2}\right].$
(99)

Note that the second order perturbations are neglected in these potentials.
For an obvious reason, $\nu_{0}(A)$, $\nu_{n}(\alpha,B;x)$ and $\nu_{l,m}(C)$
will be referred as one-loop effective potential of zero-mode type, with-flux
type and without-flux type, respectively.

####  2.5.1 The zero-mode type $\nu_{0}$

First, we consider the effective potential of zero-mode type $\nu_{0}$.

$\displaystyle\nu_{0}(A)$
$\displaystyle=-\int\frac{d^{4}p}{(2\pi)^{4}}\int_{0}^{\infty}\frac{dt}{t}e^{-\left(p^{2}+Ag^{2}v^{2}\right)t}$
$\displaystyle=-\frac{1}{16\pi^{2}}\int_{0}^{\infty}\frac{dt}{t^{3}}e^{-Ag^{2}v^{2}t},$

where Schwinger’s proper time integral is introduced in the first line and the
momentum integral is performed in the second line. Obviously, this integral
diverges at $t=0$, but we can extract a finite value from it. We will propose
the idea later and the regularized effective potential of zero-mode type
$\nu_{reg,0}$ is found as

$\displaystyle\nu_{reg,0}(A)=\frac{(Ag^{2}v^{2})^{2}}{576\pi^{2}}\left(\frac{9\hskip
1.42262pt\zeta(3)}{\pi^{2}}-1\right),$ (100)

where $\zeta(3)$ is the Apéry’s constant, $\zeta(3)=1.20205\cdots$ and
$9\hskip 1.42262pt\zeta(3)/\pi^{2}-1=0.0961381\cdots$.

####  2.5.2 The with-flux type $\nu_{n}$

Next, we consider the effective potential of with-flux type $\nu_{n}$.

$\displaystyle\nu_{n}(\alpha,B;x)$
$\displaystyle=-\sum_{n}\int\frac{d^{4}p}{(2\pi)^{4}}\int_{0}^{\infty}\frac{dt}{t}e^{-\left(p^{2}+\alpha(n+x)+Bg^{2}v^{2}\right)t}$
$\displaystyle=-\frac{1}{16\pi^{2}}\int_{0}^{\infty}\frac{dt}{t^{3}}\frac{e^{-\alpha(x+Bg^{2}v^{2}/\alpha-1)t}}{e^{\alpha
t}-1}$
$\displaystyle=-\frac{\alpha^{2}}{16\pi^{2}}\int_{0}^{\infty}\frac{dy}{y^{1-(-2)}}\frac{e^{-(x+Bg^{2}v^{2}/\alpha-1)y}}{e^{y}-1}.$
(101)

To calculate this, we try to apply the integral representation of the Hurwitz
$\zeta$ function [24]

$\displaystyle\zeta(s,a)=$ $\displaystyle\hskip
1.13809pt\frac{a^{-s}}{2}+\frac{a^{1-s}}{s-1}+\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}\hskip
0.56905pta^{1-s-2k}(s)_{2k-1}$
$\displaystyle+\frac{1}{\Gamma(s)}\int_{0}^{\infty}dy\hskip
0.56905pt\frac{e^{-ay}}{y^{1-s}}\left(\frac{1}{e^{y}-1}-\frac{1}{y}+\frac{1}{2}-\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}\hskip
0.56905pty^{2k-1}\right),$ (102)

where $n$ is a non-negative integer, $B_{n}$ is the Bernoulli number and
$(s)_{n}$ is the Pochhammer symbol. The equation (2.5.2) is satisfied with the
conditions

$\displaystyle{\rm Re}~{}s>-(2n+1)~{}~{}(n\in\mathbb{Z},~{}n\geq
0),~{}~{}s\neq 1,~{}~{}{\rm Re}~{}a>0.$ (103)

Comparing the integral in (2.5.2) with the expression (2.5.2), we need to
consider a case where $s=-2$. Therefore, it is enough to take $n=1$ as
follows:

$\displaystyle\zeta(s,a)=$ $\displaystyle\hskip
1.13809pt\frac{a^{-s}}{2}+\frac{a^{1-s}}{s-1}+\frac{sa^{-s-1}}{12}$
$\displaystyle+\frac{1}{\Gamma(s)}\int_{0}^{\infty}dy\hskip
2.84526pt\frac{e^{-ay}}{y^{1-s}}\left(\frac{1}{e^{y}-1}-\frac{1}{y}+\frac{1}{2}-\frac{y}{12}\right),$
(104)

where $(s)_{1}=s$ and $B_{2}=1/6$ are used. The integral which includes a term
of $1/(e^{y}-1)$ corresponds to the with-flux type $\nu_{n}$. On the other
hand, we know the form of $\zeta(-2,a)$ with elementary functions:

$\displaystyle\zeta(-2,a)=-\frac{1}{3}\sum_{k=0}^{3}{}_{3}\mathrm{C}_{k}B_{3-k}a^{k}=\frac{a^{-(-2)}}{2}+\frac{a^{1-(-2)}}{-2-1}+\frac{(-2)a^{-(-2)-1}}{12},$
(105)

and we find this to be equivalent to the first line of the equation (2.5.2).

When we take $s=\epsilon-2~{}(\epsilon=(4-d)/2\ll 1)$, the equation (105) can
be understood as $\mathcal{O}(\epsilon^{0})$ terms of the both sides of the
equation (2.5.2). Therefore, the integral in the second line of (2.5.2) is
understood to be $\mathcal{O}(\epsilon)$. Extracting the
$\mathcal{O}(\epsilon)$ terms from the equation (2.5.2), we obtain

$\displaystyle\int_{0}^{\infty}dy\hskip
2.84526pt\frac{e^{-ay}}{y^{3}}\left(\frac{1}{e^{y}-1}-\frac{1}{y}+\frac{1}{2}-\frac{y}{12}\right)$
$\displaystyle=\frac{1}{2}\left[\zeta^{(1,0)}(-2,a)-\frac{a}{12}(2\ln
a+1)+\frac{a^{2}}{2}\ln a-\frac{a^{3}}{9}(3\ln a-1)\right],$ (106)

where we used the expansion of $\zeta(\epsilon-2,a)$ in $\epsilon$:

$\displaystyle\zeta(\epsilon-2,a)=\zeta(-2,a)+\epsilon\zeta^{(1,0)}(-2,a)+\mathcal{O}(\epsilon^{2}).$
(107)

Although each integral of the equation (2.5.2) diverges at $y=0$, the right
hand side of (2.5.2) is finite. In other words, we can interpret that the last
three terms of the left hand side in (2.5.2) work as the regulators which
extract finite quantity from the divergent integral and we are able to
evaluate the potential of with-flux type $\nu_{n}$. Then, we obtain the
regularized one-loop effective potential of with-flux type $\nu_{reg,n}$

$\displaystyle\nu_{reg,n}(\alpha,B;x)$
$\displaystyle=-\frac{\alpha^{2}}{32\pi^{2}}\left[\hskip
1.42262pt\zeta^{(1,0)}\left(-2,x+\frac{Bg^{2}v^{2}}{\alpha}-1\right)\right.$
$\displaystyle\hskip
59.75095pt-\frac{1}{12}\left(x+\frac{Bg^{2}v^{2}}{\alpha}-1\right)\left\\{2\ln\left(x+\frac{Bg^{2}v^{2}}{\alpha}-1\right)+1\right\\}$
$\displaystyle\hskip
59.75095pt+\frac{1}{2}\left(x+\frac{Bg^{2}v^{2}}{\alpha}-1\right)^{2}\ln\left(x+\frac{Bg^{2}v^{2}}{\alpha}-1\right)$
$\displaystyle\hskip
59.75095pt\left.-\frac{1}{9}\left(x+\frac{Bg^{2}v^{2}}{\alpha}-1\right)^{3}\left\\{3\ln\left(x+\frac{Bg^{2}v^{2}}{\alpha}-1\right)-1\right\\}\right].$
(108)

Note that the quantity $a=x+Bg^{2}v^{2}/\alpha-1$ of the equation (2.5.2) is
not always positive: if $a$ is zero or negative, it does not satisfy the
condition (103). This situation occur in the case $x=0$ or 1/2. In that case,
we separate the term of $n=0$:

$\displaystyle\nu_{reg,n}(\alpha,B;x<1)=$
$\displaystyle-\int\frac{d^{4}p}{(2\pi)^{4}}\int_{0}^{\infty}\frac{dt}{t}e^{-\left(p^{2}+\alpha
x+Bg^{2}v^{2}\right)t}$
$\displaystyle-\sum_{n=1}^{\infty}\int\frac{d^{4}p}{(2\pi)^{4}}\int_{0}^{\infty}\frac{dt}{t}e^{-\left(p^{2}+\alpha(n+x)+Bg^{2}v^{2}\right)t}$
$\displaystyle=$
$\displaystyle-\frac{\alpha^{2}}{16\pi^{2}}\int_{0}^{\infty}\frac{dy}{y^{3}}e^{-(x+Bg^{2}v^{2}/\alpha)y}-\frac{\alpha^{2}}{16\pi^{2}}\int_{0}^{\infty}\frac{dy}{y^{3}}\frac{e^{-(x+Bg^{2}v^{2}/\alpha)y}}{e^{y}-1}.$
(109)

Although the first term in (109) is similarly divergent at $y=0$, it can be
finite by using (2.5.2) as

$\displaystyle-\frac{1}{16\pi^{2}}\int_{0}^{\infty}\frac{dt}{t^{3}}e^{-(\alpha
x+Bg^{2}v^{2})t}\rightarrow\frac{\alpha^{2}}{576\pi^{2}}\left(x+\frac{Bg^{2}v^{2}}{\alpha}\right)^{2}\left(\frac{9\hskip
1.42262pt\zeta(3)}{\pi^{2}}-1\right).$

In this calculation, the terms except for the third term of the left hand side
in (2.5.2) work as the regulators. Thus, the regularized one-loop effective
potential of with-flux type $\nu_{reg,n}(\alpha,B;x<1)$ can be obtained

$\displaystyle\nu_{reg,n}(\alpha,B;x<1)$
$\displaystyle=-\frac{\alpha^{2}}{32\pi^{2}}\left[\hskip
1.42262pt\zeta^{(1,0)}\left(-2,x+\frac{Bg^{2}v^{2}}{\alpha}\right)\right.$
$\displaystyle\hskip
59.75095pt-\frac{1}{12}\left(x+\frac{Bg^{2}v^{2}}{\alpha}\right)\left\\{2\ln\left(x+\frac{Bg^{2}v^{2}}{\alpha}\right)+1\right\\}$
$\displaystyle\hskip
59.75095pt+\frac{1}{18}\left(x+\frac{Bg^{2}v^{2}}{\alpha}\right)^{2}\left\\{9\ln\left(x+\frac{Bg^{2}v^{2}}{\alpha}\right)-\frac{9\hskip
1.42262pt\zeta(3)}{\pi^{2}}+1\right\\}$ $\displaystyle\hskip
59.75095pt\left.-\frac{1}{9}\left(x+\frac{Bg^{2}v^{2}}{\alpha}\right)^{3}\left\\{3\ln\left(x+\frac{Bg^{2}v^{2}}{\alpha}\right)-1\right\\}\right].$
(110)

####  2.5.3 The without-flux type $\nu_{l,m}$

Finally, we consider the one-loop effective potential of without-flux type
$\nu_{l,m}$.

$\displaystyle\nu_{l,m}(C)$
$\displaystyle=-\sum_{l=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}\frac{dt}{t}\int\frac{d^{4}p}{(2\pi)^{4}}e^{-\left[p^{2}+4\pi^{2}(l^{2}+m^{2})+Cg^{2}v^{2}\right]t}$
$\displaystyle=-\frac{1}{16\pi^{2}}\sum_{l,m}\int_{0}^{\infty}\frac{dt}{t^{3}}e^{-\left[4\pi^{2}(l^{2}+m^{2})+Cg^{2}v^{2}\right]t}$
$\displaystyle=-\frac{1}{16\pi^{2}}\int_{0}^{\infty}duue^{-Cg^{2}v^{2}/u}\sum_{l,m}e^{-4\pi^{2}(l^{2}+m^{2})/u}.$

Using the Poisson resummation formula,

$\displaystyle\sum_{l=-\infty}^{\infty}\exp\left[-\frac{4\pi^{2}(l+v)^{2}}{u}\right]=\sqrt{\frac{u}{4\pi}}\sum_{r=-\infty}^{\infty}e^{-ur^{2}/4\pm
2\pi irv}$ (111)

with $v=0$, we obtain

$\displaystyle\nu_{l,m}(C)=$
$\displaystyle-\frac{1}{64\pi^{3}}\int_{0}^{\infty}duu^{2}e^{-Cg^{2}v^{2}/u}\sum_{r=-\infty}^{\infty}\sum_{s=-\infty}^{\infty}e^{-u(r^{2}+s^{2})/4}$
$\displaystyle=$ $\displaystyle-\frac{1}{64\pi^{3}}\sum_{r,s\neq
0}\left(\frac{4}{r^{2}+s^{2}}\right)^{3}\int_{0}^{\infty}duu^{2}e^{-u}\exp\left[-\frac{Cg^{2}v^{2}(r^{2}+s^{2})}{4u}\right]$
$\displaystyle+\frac{1}{64\pi^{3}}\int_{0}^{\infty}dt\hskip
1.42262pt\frac{e^{-Cg^{2}v^{2}t}}{t^{3}}\left(-\frac{1}{t}\right),$ (112)

where we note that a change of variable $t=(r^{2}+s^{2})/u$ is performed in
the second line of (2.5.3) except for $r=s=0$ mode. For the first term of the
equation (2.5.3), we consider applying the modified Bessel function of second
kind,

$\displaystyle K_{\nu}(z)$
$\displaystyle=\frac{1}{2}\left(\frac{z}{2}\right)^{\nu}\int_{0}^{\infty}dtt^{-\nu-1}\exp\left(-t-\frac{z^{2}}{4t}\right)$
$\displaystyle=\frac{1}{2}\left(\frac{z}{2}\right)^{-\nu}\int_{0}^{\infty}duu^{\nu-1}\exp\left(-\frac{z^{2}}{4u}-u\right),$
(113)

where a change of variable $u=z^{2}/4t$ is performed in the second line. This
is satisfied with the conditions ${\rm Re}~{}\nu>-1/2,~{}|\arg z|<\pi/4$.
Thus, the first term of the equation (2.5.3) becomes

$\displaystyle-\frac{1}{64\pi^{3}}\sum_{r,s\neq
0}\left(\frac{4}{r^{2}+s^{2}}\right)^{3}\int_{0}^{\infty}duu^{2}e^{-u}\exp\left[-\frac{Cg^{2}v^{2}(r^{2}+s^{2})}{4u}\right]$
$\displaystyle=-\frac{C^{3/2}g^{3}v^{3}}{4\pi^{3}}\sum_{r,s\neq
0}\left(\frac{1}{r^{2}+s^{2}}\right)^{3/2}K_{3}\left(gv\sqrt{C(r^{2}+s^{2})}\right).$

The second term of the equation (2.5.3) is regularized by using the equation
(2.5.2):

$\displaystyle\frac{1}{64\pi^{3}}\int_{0}^{\infty}dt\hskip
1.42262pt\frac{e^{-Cg^{2}v^{2}t}}{t^{3}}\left(-\frac{1}{t}\right)\rightarrow\frac{(Cg^{2}v^{2})^{3}}{4608\pi^{3}}\left(\frac{9\hskip
1.42262pt\zeta(3)}{\pi^{2}}-1\right).$

Therefore, the regularized one-loop effective potential of without-flux type
is obtained

$\displaystyle\nu_{reg,l,m}(C)=$
$\displaystyle-\frac{(Cg^{2}v^{2})^{3/2}}{4\pi^{3}}\sum_{r,s\neq
0}\left(\frac{1}{r^{2}+s^{2}}\right)^{3/2}K_{3}\left(gv\sqrt{C(r^{2}+s^{2})}\right)$
$\displaystyle+\frac{(Cg^{2}v^{2})^{3}}{4608\pi^{3}}\left(\frac{9\hskip
1.42262pt\zeta(3)}{\pi^{2}}-1\right).$ (114)

Figure 1: A picture of $\nu_{reg,0}(1)$, $\nu_{reg,n}(\alpha_{2},1;0)$,
$\nu_{reg,n}(\alpha_{2},1;1)$ and $\nu_{reg,l,m}(1)$.

The several types of the one-loop potentials
$\nu_{reg,0}(1),\nu_{reg,n}(\alpha_{2},1;0),\nu_{reg,n}(\alpha_{2},1;1)$ and
$\nu_{reg,l,m}(1)$ are shown in Figure 1.

###  2.6 The one-loop effective potential of SU(2) Yang-Mills theory

We now apply the above results obtained in the previous subsections to SU(2)
Yang-Mills theory. The effective potentials can be calculated by using the
masses of the gauge fields $A^{a}_{\mu}$, the scalar fields $\varphi^{a}$ and
the ghost fields $c^{a}$,

$\displaystyle V_{A}$
$\displaystyle=\frac{3}{2}\frac{1}{(2\pi)^{2}}\Big{(}\nu_{0}(1/2)+\nu_{l,m}(1)+\nu_{n}(\alpha_{2},1;1)\Big{)},$
(115) $\displaystyle V_{\varphi}$
$\displaystyle=\frac{1}{2}\frac{1}{(2\pi)^{2}}\Big{(}\nu_{l,m}(1)+\nu_{n}(\alpha_{3},1;1/2)\Big{)},$
(116) $\displaystyle V_{c}$
$\displaystyle=-\frac{1}{2}\frac{1}{(2\pi)^{2}}\Big{(}\nu_{l,m}(1)+\nu_{n}(\alpha_{3},1;1/2)\Big{)}=-V_{\varphi}.$
(117)

Since these effective potentials are divergent, we extract the finite value
from them. By using (100), (108) and (114), the regularized effective
potential is expressed as

$\displaystyle V_{reg,A}$
$\displaystyle=\frac{3}{2}\frac{1}{(2\pi)^{2}}\Big{(}\nu_{reg,0}(1/2)+\nu_{reg,l,m}(1)+\nu_{reg,n}(\alpha_{2},1;1)\Big{)},$
(118) $\displaystyle V_{reg,\varphi}$
$\displaystyle=\frac{1}{2}\frac{1}{(2\pi)^{2}}\Big{(}\nu_{reg,l,m}(1)+\nu_{reg,n}(\alpha_{3},1;1/2)\Big{)},$
(119) $\displaystyle V_{reg,c}$
$\displaystyle=-\frac{1}{2}\frac{1}{(2\pi)^{2}}\Big{(}\nu_{reg,l,m}(1)+\nu_{reg,n}(\alpha_{3},1;1/2)\Big{)}=-V_{reg,\varphi}.$
(120)

Since $V_{\varphi}$ and $V_{c}$ cancel each other because of the same the KK
mass spectrum in Feynman gauge, we have only to consider $V_{reg,A}$ as the
total effective potential $V\equiv V_{reg,A}+V_{reg,\varphi}+V_{reg,c}$ 222If
you do not choose Feynman gauge $\xi=1$, the cancellation of $V_{reg,c}$ and
$V_{reg,\varphi}$ does not occur. See [12] for this implications. . In detail,
the total effective potential can be expressed as

$\displaystyle V=$ $\displaystyle\hskip 2.84526ptV_{reg,A}$ $\displaystyle=$
$\displaystyle\hskip
2.84526pt\frac{(g^{2}v^{2})^{2}}{6144\pi^{4}}\left(\frac{9\zeta(3)}{\pi}-1\right)$
$\displaystyle-\frac{3(g^{2}v^{2})^{3/2}}{32\pi^{5}}\sum_{r,s\neq
0}\left(\frac{1}{r^{2}+s^{2}}\right)^{3/2}K_{3}(gv\sqrt{r^{2}+s^{2}})+\frac{(g^{2}v^{2})^{3}}{12288\pi^{5}}\left(\frac{9\zeta(3)}{\pi}-1\right)$
$\displaystyle-\frac{3N^{2}}{16\pi^{2}}\left[\hskip
1.42262pt\zeta^{(1,0)}\left(-2,\frac{g^{2}v^{2}}{4\pi
N}\right)\right.-\frac{1}{12}\left(\frac{g^{2}v^{2}}{4\pi
N}\right)\left\\{2\ln\left(\frac{g^{2}v^{2}}{4\pi N}\right)+1\right\\}$
$\displaystyle\hskip 48.36958pt+\frac{1}{2}\left(\frac{g^{2}v^{2}}{4\pi
N}\right)^{2}\ln\left(\frac{g^{2}v^{2}}{4\pi
N}\right)\left.-\frac{1}{9}\left(\frac{g^{2}v^{2}}{4\pi
N}\right)^{3}\left\\{3\ln\left(\frac{g^{2}v^{2}}{4\pi
N}\right)-1\right\\}\right].$ (121)

Figure 2: One-loop effective potentials of SU(2)Yang-Mills theory. $N=3$ is
taken for simplicity. The shaded region represents a range where our
perturbation analysis breaks down.

The one-loop effective potentials with $N=3$ are shown in figure 2. The
cancellation of the potentials between the scalar (blue line) and the ghost
(green line) loop contributions can be explicitly verified. As can be seen
from the total potential (red line) in figure 2, we can conclude that the
origin of the potential is at least not a minimum and the minimum of the
effective potential is expected to locate at a nonzero constant VEV $v\neq 0$
and the gauge symmetry SU(2) is completely broken.

Here we should note that we cannot determine the location of the minimum since
the location is expected to be beyond the range which our perturbation is
valid as is shown in figure 2. Even if the value of the minimum is not
determined, if we assume that the potential is bounded below, the pattern of
gauge symmetry breaking can be confirmed from the gauge boson spectrum (85).

##  3 SU(3) Yang-Mills theory

In a context of gauge-Higgs unification, SU(3) gauge symmetry is minimal to
realize the zero mode of WL scalar as an SU(2) Higgs doublet. In our case, if
SU(3) is broken to SU(2) $\times$ U(1) by the VEV of the constant magnetic
flux, the SU(2) Higgs doublet would appear and the electroweak symmetry
breaking can be discussed. As a first step toward the discussion, we now
consider a six-dimensional SU(3) Yang-Mills theory. We consider two cases of
directions where we introduce the VEVs

$\displaystyle(1)~{}\braket{\phi^{1}}=\frac{v}{\sqrt{2}}~{}~{}\text{and}~{}~{}\braket{\phi^{8}}=\frac{f\bar{z}}{\sqrt{2}},$
(122)
$\displaystyle(2)~{}\braket{\phi^{6}}=\frac{v}{\sqrt{2}}~{}~{}\text{and}~{}~{}\braket{\phi^{8}}=\frac{f\bar{z}}{\sqrt{2}},$
(123)

where the superscripts of $\phi$ denote the gauge indices of SU(3) and the
structure constant is changed to that of SU(3) accordingly. Note that the flux
background breaks the SU(3) gauge symmetry, which is broken SU(2)$\times$
U(1). The case (1) is not included in SU(2) theory, because two kinds of VEV
are both taken in the components of the unbroken symmetry. Other cases of
developing the VEVs are reduced to the above two cases by the gauge rotations.

###  3.1 Case (1)

In this subsection, let us consider the case (1) where the background
covariant derivatives $\mathcal{D},\overline{\mathcal{D}}$ are expressed as

$\displaystyle\mathcal{D}^{ac}=\left[\begin{array}[]{cccccccc}\partial&0&0&0&0&0&0&0\\\
0&\partial&igv&0&0&0&0&0\\\ 0&-igv&\partial&0&0&0&0&0\\\
0&0&0&\partial&\frac{\sqrt{3}}{2}igf\bar{z}&0&\frac{1}{2}igv&0\\\
0&0&0&-\frac{\sqrt{3}}{2}igf\bar{z}&\partial&-\frac{1}{2}igv&0&0\\\
0&0&0&0&\frac{1}{2}igv&\partial&\frac{\sqrt{3}}{2}igf\bar{z}&0\\\
0&0&0&-\frac{1}{2}igv&0&-\frac{\sqrt{3}}{2}igf\bar{z}&\partial&0\\\
0&0&0&0&0&0&0&\partial\end{array}\right],$ (132)
$\displaystyle\overline{\mathcal{D}}^{ac}=\left[\begin{array}[]{cccccccc}\bar{\partial}&0&0&0&0&0&0&0\\\
0&\bar{\partial}&-igv&0&0&0&0&0\\\ 0&igv&\bar{\partial}&0&0&0&0&0\\\
0&0&0&\bar{\partial}&-\frac{\sqrt{3}}{2}igfz&0&-\frac{1}{2}igv&0\\\
0&0&0&\frac{\sqrt{3}}{2}igfz&\bar{\partial}&\frac{1}{2}igv&0&0\\\
0&0&0&0&-\frac{1}{2}igv&\bar{\partial}&-\frac{\sqrt{3}}{2}igfz&0\\\
0&0&0&\frac{1}{2}igv&0&\frac{\sqrt{3}}{2}igfz&\bar{\partial}&0\\\
0&0&0&0&0&0&0&\bar{\partial}\end{array}\right].$ (141)

Diagonalizing them, we obtain

$\displaystyle\begin{cases}\mathcal{D}_{diag}=\text{diag}\left(\partial,\partial-
gv,\partial+gv,\partial-\frac{1}{2}gv-\frac{\sqrt{3}}{2}gf\bar{z},\partial-\frac{1}{2}gv+\frac{\sqrt{3}}{2}gf\bar{z},\right.\\\\[5.69054pt]
\hskip
71.13188pt\left.\partial+\frac{1}{2}gv-\frac{\sqrt{3}}{2}gf\bar{z},\partial+\frac{1}{2}gv+\frac{\sqrt{3}}{2}gf\bar{z},\partial\right),\\\\[14.22636pt]
\overline{\mathcal{D}}_{diag}=\text{diag}\left(\bar{\partial},\bar{\partial}+gv,\bar{\partial}-gv,\bar{\partial}+\frac{1}{2}gv+\frac{\sqrt{3}}{2}gfz,\bar{\partial}+\frac{1}{2}gv-\frac{\sqrt{3}}{2}gfz,\right.\\\\[5.69054pt]
\hskip
71.13188pt\left.\bar{\partial}-\frac{1}{2}gv+\frac{\sqrt{3}}{2}gfz,\bar{\partial}-\frac{1}{2}gv-\frac{\sqrt{3}}{2}gfz,\bar{\partial}\right).\end{cases}$
(142)

Their diagonalizing unitary matrix $U$ is given by

$\displaystyle U=\frac{1}{2}\left[\begin{array}[]{cccccccc}2&0&0&0&0&0&0&0\\\
0&\sqrt{2}&\sqrt{2}i&0&0&0&0&0\\\ 0&\sqrt{2}i&\sqrt{2}&0&0&0&0&0\\\
0&0&0&1&i&-1&i&0\\\ 0&0&0&i&1&-i&1&0\\\ 0&0&0&1&-i&1&i&0\\\
0&0&0&i&-1&i&1&0\\\ 0&0&0&0&0&0&0&2\end{array}\right].$ (151)

The commutation relation of $\mathcal{D}$ and $\overline{\mathcal{D}}$ become
just a constant matrix:

$\displaystyle\left[i\overline{\mathcal{D}}_{diag},i\mathcal{D}_{diag}\right]^{ac}=\sqrt{3}gf\text{diag}(0,0,0,1,-1,1,-1,0),$
(152)

where the creation and annihilation operators can be given as

$\displaystyle
a=\frac{i}{\sqrt{\alpha_{3}}}\overline{\mathcal{D}}_{diag},\qquad
a^{\dagger}=\frac{i}{\sqrt{\alpha_{3}}}\mathcal{D}_{diag}~{}~{}(\alpha_{3}\equiv\sqrt{3}gf).$
(153)

Defining further

$\displaystyle a_{4,6}$
$\displaystyle=\frac{i}{\sqrt{\alpha_{3}}}\left(\bar{\partial}+\frac{\sqrt{3}}{2}gfz\right),~{}~{}~{}a^{\dagger}_{4,6}=\frac{i}{\sqrt{\alpha_{3}}}\left(\partial-\frac{\sqrt{3}}{2}gf\bar{z}\right),$
(154) $\displaystyle a_{5,7}$
$\displaystyle=\frac{i}{\sqrt{\alpha_{3}}}\left(\bar{\partial}-\frac{\sqrt{3}}{2}gfz\right),~{}~{}~{}a^{\dagger}_{5,7}=\frac{i}{\sqrt{\alpha_{3}}}\left(\partial+\frac{\sqrt{3}}{2}gf\bar{z}\right)$
(155)

in the matrix form of the creation and annihilation operator, the diagonalized
part of $\mathcal{D}_{diag}$ and $\overline{\mathcal{D}}_{diag}$ can be
expressed as

$\displaystyle\begin{cases}\displaystyle(i\overline{\mathcal{D}}_{diag})^{44}=\sqrt{\alpha_{3}}a_{4,6}+\frac{1}{2}igv,\\\\[11.38109pt]
\displaystyle(i\overline{\mathcal{D}}_{diag})^{55}=\sqrt{\alpha_{3}}a_{5,7}+\frac{1}{2}igv,\\\\[11.38109pt]
\displaystyle(i\overline{\mathcal{D}}_{diag})^{66}=\sqrt{\alpha_{3}}a_{4,6}-\frac{1}{2}igv,\\\\[11.38109pt]
\displaystyle(i\overline{\mathcal{D}}_{diag})^{77}=\sqrt{\alpha_{3}}a_{5,7}-\frac{1}{2}igv,\end{cases}~{}~{}\text{and}~{}~{}~{}\begin{cases}\displaystyle(i\mathcal{D}_{diag})^{44}=\sqrt{\alpha_{3}}a^{\dagger}_{4,6}-\frac{1}{2}igv,\\\\[11.38109pt]
\displaystyle(i\mathcal{D}_{diag})^{55}=\sqrt{\alpha_{3}}a^{\dagger}_{5,7}-\frac{1}{2}igv,\\\\[11.38109pt]
\displaystyle(i\mathcal{D}_{diag})^{66}=\sqrt{\alpha_{3}}a^{\dagger}_{4,6}+\frac{1}{2}igv,\\\\[11.38109pt]
\displaystyle(i\mathcal{D}_{diag})^{77}=\sqrt{\alpha_{3}}a^{\dagger}_{5,7}+\frac{1}{2}igv.\end{cases}$
(156)

Note that other components are just spatial derivatives and do not play any
role of creation and annihilation operators.

####  3.1.1 The mass of the gauge fields $A^{a}_{\mu}$

In the same way as the calculations in subsection 2.2, we obtain the gauge
mass matrix

$\displaystyle-\mathcal{D}_{diag}\overline{\mathcal{D}}_{diag}\equiv
m^{2}_{A},$ (157)

where the components are

$\displaystyle\begin{cases}~{}(m^{2}_{A})^{11}=4\pi^{2}(l_{1}^{2}+m_{1}^{2}),&\displaystyle~{}(m^{2}_{A})^{22}=4\pi^{2}\left\\{l_{2}^{2}+\left(m_{2}-\frac{gv}{2\pi}\right)^{2}\right\\},\\\\[11.38109pt]
\displaystyle~{}(m^{2}_{A})^{33}=4\pi^{2}\left\\{l_{2}^{3}+\left(m_{3}+\frac{gv}{2\pi}\right)^{2}\right\\},&\displaystyle~{}(m^{2}_{A})^{44}=\alpha_{3}n_{4}+\frac{1}{4}g^{2}v^{2},\\\\[11.38109pt]
\displaystyle~{}(m^{2}_{A})^{55}=\alpha_{3}(n_{5}+1)+\frac{1}{4}g^{2}v^{2},&\displaystyle~{}(m^{2}_{A})^{66}=\alpha_{3}n_{6}+\frac{1}{4}g^{2}v^{2},\\\\[11.38109pt]
\displaystyle~{}(m^{2}_{A})^{77}=\alpha_{3}(n_{7}+1)+\frac{1}{4}g^{2}v^{2},&~{}(m^{2}_{A})^{88}=4\pi^{2}(l_{8}^{2}+m_{8}^{2}).\end{cases}$
(158)

From the massless modes in (158), we find that the SU(2)$\times$ U(1) gauge
symmetry is broken into U(1) $\times$ U(1) if $v\neq 0$.

####  3.1.2 The mass of the scalar fields $\varphi^{a}$

The scalar masses are calculated from the terms

$\displaystyle
2\left(-\mathcal{D}_{diag}\overline{\mathcal{D}}_{diag}\right)^{ac}-4igf(U^{-1})^{a^{\prime}a}f^{8a^{\prime}c^{\prime}}U^{c^{\prime}c}\equiv
m^{2}_{\varphi},$ (159)

and the resulting KK mass spectrums are found as

$\displaystyle\begin{cases}~{}(m^{2}_{\varphi})^{11}=4\pi^{2}(l_{1}^{2}+m_{1}^{2}),&~{}(m^{2}_{\varphi})^{22}=\displaystyle
4\pi^{2}\left\\{l_{2}^{2}+\left(m_{2}-\frac{gv}{2\pi}\right)^{2}\right\\},\\\\[11.38109pt]
~{}(m^{2}_{\varphi})^{33}=\displaystyle
4\pi^{2}\left\\{l_{2}^{3}+\left(m_{3}+\frac{gv}{2\pi}\right)^{2}\right\\},&~{}(m^{2}_{\varphi})^{44}\displaystyle=\alpha_{3}\left(n_{4}+\frac{1}{2}\right)+\frac{1}{4}g^{2}v^{2},\\\\[11.38109pt]
~{}(m^{2}_{\varphi})^{55}=\displaystyle\alpha_{3}\left(n_{5}+\frac{1}{2}\right)+\frac{1}{4}g^{2}v^{2},&~{}(m^{2}_{\varphi})^{66}=\displaystyle\alpha_{3}\left(n_{6}+\frac{1}{2}\right)+\frac{1}{4}g^{2}v^{2},\\\\[11.38109pt]
~{}(m^{2}_{\varphi})^{77}=\displaystyle\alpha_{3}\left(n_{7}+\frac{1}{2}\right)+\frac{1}{4}g^{2}v^{2},&~{}(m^{2}_{\varphi})^{88}=4\pi^{2}(l_{8}^{2}+m_{8}^{2}).\end{cases}.$
(160)

####  3.1.3 The mass of the ghost fields $c^{a}$

The ghost masses are calculated from the terms

$\displaystyle-\mathcal{D}_{diag}\overline{\mathcal{D}}_{diag}+\frac{\alpha_{3}}{2}[a,a^{\dagger}]\equiv
m^{2}_{c},$ (161)

and the resulting KK spectrum are obtained

$\displaystyle\begin{cases}\displaystyle~{}(m^{2}_{c})^{11}=4\pi^{2}(l^{2}_{1}+m^{2}_{1}),&\displaystyle~{}(m^{2}_{c})^{22}=4\pi^{2}\left\\{l^{2}_{2}+\left(m_{2}-\frac{gv}{2\pi}\right)^{2}\right\\},\\\\[11.38109pt]
\displaystyle~{}(m^{2}_{c})^{33}=4\pi^{2}\left\\{l^{2}_{3}+\left(m_{3}+\frac{gv}{2\pi}\right)^{2}\right\\},&\displaystyle~{}(m^{2}_{c})^{44}=\alpha_{3}\left(n_{4}+\frac{1}{2}\right)+\frac{1}{4}g^{2}v^{2},\\\\[11.38109pt]
\displaystyle~{}(m^{2}_{c})^{55}=\alpha_{3}\left(n_{5}+\frac{1}{2}\right)+\frac{1}{4}g^{2}v^{2},&\displaystyle~{}(m^{2}_{c})^{66}=\alpha_{3}\left(n_{6}+\frac{1}{2}\right)+\frac{1}{4}g^{2}v^{2},\\\\[11.38109pt]
\displaystyle~{}(m^{2}_{c})^{77}=\alpha_{3}\left(n_{7}+\frac{1}{2}\right)+\frac{1}{4}g^{2}v^{2},&\displaystyle~{}(m^{2}_{c})^{88}=4\pi^{2}(l^{2}_{8}+m^{2}_{8}).\end{cases}$
(162)

####  3.1.4 The potential calculation

The effective potential can be calculated by using the masses of the gauge
fields $A^{a}_{\mu}$, the scalar fields $\varphi^{a}$, the ghost fields
$c^{a}$ and (98) and (99):

$\displaystyle V_{A}$
$\displaystyle=\frac{3}{2}\frac{1}{(2\pi)^{2}}\Big{(}2\nu_{l,m}(0)+\nu_{+}(gv/2\pi)+\nu_{-}(gv/2\pi)+2\nu_{n}(\alpha_{3},1/4;0)+2\nu_{n}(\alpha_{3},1/4;1)\Big{)},$
(163) $\displaystyle V_{\varphi}$
$\displaystyle=\frac{1}{2}\frac{1}{(2\pi)^{2}}\Big{(}2\nu_{l,m}(0)+\nu_{+}(gv/2\pi)+\nu_{-}(gv/2\pi)+4\nu_{n}(\alpha_{3},1/4;1/2)\Big{)},$
(164) $\displaystyle V_{c}$
$\displaystyle=-\frac{1}{2}\frac{1}{(2\pi)^{2}}\Big{(}2\nu_{l,m}(0)+\nu_{+}(gv/2\pi)+\nu_{-}(gv/2\pi)+4\nu_{n}(\alpha_{3},1/4;1/2)\Big{)}=-V_{\varphi},$
(165)

where $\nu_{\pm}(V)$ is defined as

$\displaystyle\nu_{\pm}(V)$
$\displaystyle\equiv\sum_{l=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}\int\frac{d^{4}p}{(2\pi)^{4}}\ln\left[p^{2}+4\pi^{2}\left\\{l^{2}+\left(m\pm
V\right)^{2}\right\\}\right].$ (166)

The terms which do not contain the constant VEV $v$, $\nu_{l,m}(0)$, are
irrelevant to determine the potential minimum.

We consider here new one-loop effective potentials of without-flux type
$\nu_{\pm}$.

$\displaystyle\nu_{\pm}(V)$
$\displaystyle=-\sum_{l,m}\int_{0}^{\infty}\frac{dt}{t}\int\frac{d^{4}p}{(2\pi)^{4}}\exp\left({-\left[p^{2}+4\pi^{2}\left\\{l^{2}+\left(m\pm
V\right)\right\\}t\right]}\right)$
$\displaystyle=-\frac{1}{16\pi^{2}}\sum_{l,m}\int_{0}^{\infty}\frac{dt}{t^{3}}\exp\left(-4\pi^{2}\left\\{l^{2}+\left(m\pm
V\right)\right\\}t\right)$
$\displaystyle=-\frac{1}{16\pi^{2}}\int_{0}^{\infty}duu\sum_{l,m}\exp\left(-4\pi^{2}\left\\{l^{2}+\left(m\pm
V\right)^{2}\right\\}/u\right).$

Using the Poisson resummation formula (111), $\nu_{\pm}$ become

$\displaystyle\nu_{\pm}(V)=-\frac{1}{64\pi^{3}}\sum_{r=-\infty}^{\infty}\sum_{s=-\infty}^{\infty}e^{\pm
2\pi iVs}\int_{0}^{\infty}duu^{2}e^{-u(r^{2}+s^{2})/4}.$

Since the term with $s=0$ has no dependence on $v$, it may be removed and we
obtain the regularized one-loop effective potentials of without-flux type
$\nu_{reg,\pm}$

$\displaystyle\nu_{reg,\pm}(V)$
$\displaystyle=-\frac{1}{64\pi^{3}}\sum_{r}\sum_{s\neq 0}e^{\pm 2\pi
iVs}\int_{0}^{\infty}duu^{2}e^{-u(r^{2}+s^{2})/4}$
$\displaystyle=-\frac{4}{\pi^{3}}\sum_{r=-\infty}^{\infty}\sum_{s=1}^{\infty}\frac{1}{(r^{2}+s^{2})^{3}}\cos\left(2\pi
Vs\right).$ (167)

Note that this periodic potential often appears in gauge-Higgs unification
[22, 23].

By using (108), (110), (114) and (167), each regularized effective potential
is expressed as

$\displaystyle V_{reg,A}$
$\displaystyle=\frac{3}{2}\frac{1}{(2\pi)^{2}}\Big{(}2\nu_{reg,l,m}(0)+2\nu_{reg,\pm}(gv/2\pi)+2\nu_{reg,n}(\alpha_{3},1/4;0)+2\nu_{reg,n}(\alpha_{3},1/4;1)\Big{)},$
(168) $\displaystyle V_{reg,\varphi}$
$\displaystyle=\frac{1}{2}\frac{1}{(2\pi)^{2}}\Big{(}2\nu_{reg,l,m}(0)+2\nu_{reg,\pm}(gv/2\pi)+4\nu_{reg,n}(\alpha_{3},1/4;1/2)\Big{)},$
(169) $\displaystyle V_{reg,c}$
$\displaystyle=-\frac{1}{2}\frac{1}{(2\pi)^{2}}\Big{(}2\nu_{reg,l,m}(0)+2\nu_{reg,\pm}(gv/2\pi)+4\nu_{reg,n}(\alpha_{3},1/4;1/2)\Big{)}=-V_{reg,\varphi}.$
(170)

Because of $V_{reg,c}=-V_{reg,\varphi}$, we consider $V_{reg,A}$ as the total
effective potential $V\equiv V_{reg,A}+V_{reg,\varphi}+V_{reg,c}$. In detail,
the total one-loop effective potential can be expressed as

$\displaystyle V=V_{reg,A}$
$\displaystyle\supset-\frac{3}{\pi^{5}}\sum_{r=-\infty}^{\infty}\sum_{s=1}^{\infty}\frac{1}{(r^{2}+s^{2})^{3}}\cos\left(gvs\right)$
$\displaystyle\quad-\frac{9N^{2}}{16\pi^{2}}\left[\hskip
1.42262pt\zeta^{(1,0)}\left(-2,\frac{g^{2}v^{2}}{8\sqrt{3}\pi
N}\right)\right.-\frac{1}{12}\left(\frac{g^{2}v^{2}}{8\sqrt{3}\pi
N}\right)\left(2\ln\frac{g^{2}v^{2}}{8\sqrt{3}\pi N}+1\right)$
$\displaystyle\hskip
65.44142pt+\frac{1}{36}\left(\frac{g^{2}v^{2}}{8\sqrt{3}\pi
N}\right)^{2}\left\\{18\ln\frac{g^{2}v^{2}}{8\sqrt{3}\pi
N}-\frac{9\zeta(3)}{2\pi^{2}}+1\right\\}$ $\displaystyle\hskip
65.44142pt\left.-\frac{1}{9}\left(\frac{g^{2}v^{2}}{8\sqrt{3}\pi
N}\right)^{3}\left(3\ln\frac{g^{2}v^{2}}{8\sqrt{3}\pi N}-1\right)\right].$
(171)

In (171), $2\nu_{reg,l,m}(0)$ term was dropped because it is irrelevant to
finding the potential minimum.

(a) The effective potential $V$.

(b) A closeup of the effective potential.

Figure 3: A picture of the effective potential $V$ with $N=3$ and its parts
$V_{A},V_{\varphi}~{}\text{and}~{}V_{c}$. (b) is a closeup of the effective
potential (a).

The effective potential (171) of SU(3) Yang-Mills theory with $N=3$ is shown
in Figure 3. Note that $V_{\varphi}$ and $V_{c}$ are cancelled each other and
$V_{A}$ itself becomes the total potential $V$. It seems that there is a local
minimum at $gv=0$, but it is not correct. The behavior of $V$ around $gv=0$ is
shown in figure 3(b) and we find that there are two local minima at $gv\neq
0$. The reason why $V$ becomes convex upward at $gv=0$ is due to the existence
of the term $g^{2}v^{2}\ln g^{2}v^{2}$ in $V_{A}$: it becomes dominant as $gv$
gets close to zero. Furthermore, we emphasize that the logarithmic term is
originated from the potential of with-flux type. If the magnetic flux is
absent, we have no such contribution to the potential. Then, the potential is
a periodic one as seen in the gauge-Higgs unification, where the origin of the
potential becomes convex downward, which implies the origin can be a local
minimum. The effects from the potential of with-flux type are very crucial in
our analysis of gauge symmetry breaking. Thus, we find that the minimum of the
effective potential has a nonzero VEV $v\neq 0$, and the gauge symmetry SU(2)
$\times$ U(1) is broken into U(1) $\times$ U(1).

###  3.2 Case (2)

In this section, we consider the case (2).

$\displaystyle\braket{\phi^{6}}=\frac{v}{\sqrt{2}},\qquad\braket{\phi^{8}}=\frac{f\bar{z}}{\sqrt{2}}.$
(172)

In this case, the components of VEV where the constant WL scalar develops are
in the broken generators under SU(3) $\to$ SU(2) $\times$ U(1). This case is
similar to section 2. In this situation, the background covariant derivatives
$\mathcal{D}$, $\overline{\mathcal{D}}$ are expressed as

$\displaystyle\mathcal{D}^{ac}$
$\displaystyle=\left[\begin{array}[]{cccccccc}\partial&0&0&0&-\frac{1}{2}igv&0&0&0\\\
0&\partial&0&\frac{1}{2}igv&0&0&0&0\\\ 0&0&\partial&0&0&0&\frac{1}{2}igv&0\\\
0&-\frac{1}{2}igv&0&\partial&\frac{\sqrt{3}}{2}igf\bar{z}&0&0&0\\\
\frac{1}{2}igv&0&0&-\frac{\sqrt{3}}{2}igf\bar{z}&\partial&0&0&0\\\
0&0&0&0&0&\partial&\frac{\sqrt{3}}{2}igf\bar{z}&0\\\
0&0&-\frac{1}{2}igv&0&0&-\frac{\sqrt{3}}{2}igf\bar{z}&\partial&\frac{\sqrt{3}}{2}igv\\\
0&0&0&0&0&0&-\frac{\sqrt{3}}{2}igv&\partial\end{array}\right],$ (181)
$\displaystyle\overline{\mathcal{D}}^{ac}$
$\displaystyle=\left[\begin{array}[]{cccccccc}\bar{\partial}&0&0&0&\frac{1}{2}igv&0&0&0\\\
0&\bar{\partial}&0&-\frac{1}{2}igv&0&0&0&0\\\
0&0&\bar{\partial}&0&0&0&-\frac{1}{2}igv&0\\\
0&\frac{1}{2}igv&0&\bar{\partial}&-\frac{\sqrt{3}}{2}igfz&0&0&0\\\
-\frac{1}{2}igv&0&0&\frac{\sqrt{3}}{2}igfz&\bar{\partial}&0&0&0\\\
0&0&0&0&0&\bar{\partial}&-\frac{\sqrt{3}}{2}igfz&0\\\
0&0&\frac{1}{2}igv&0&0&\frac{\sqrt{3}}{2}igfz&\bar{\partial}&-\frac{\sqrt{3}}{2}igv\\\
0&0&0&0&0&0&\frac{\sqrt{3}}{2}igv&\bar{\partial}\end{array}\right].$ (190)

If we diagonalize them, the eigenvalues are the same form as (23). Therefore,
we apply the perturbation theory as we have seen in section 2.2. Defining the
unperturbed parts to be $\mathcal{D}_{8}$, $\overline{\mathcal{D}}_{8}$ and
the perturbation part $V$ such as (34) and (38) respectively, the covariant
derivatives can be represented as $\mathcal{D}=\mathcal{D}_{8}+V$ and
$\overline{\mathcal{D}}=\overline{\mathcal{D}}_{8}+\overline{V}$.
Diagonalizing $\mathcal{D}_{8}$, $\overline{\mathcal{D}}_{8}$, we obtain the
eigenvalues (142) with $v=0$. Their diagonalizing unitary matrix is given by

$\displaystyle
U_{8}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{cccccccc}\sqrt{2}&0&0&0&0&0&0&0\\\
0&\sqrt{2}&0&0&0&0&0&0\\\ 0&0&\sqrt{2}&0&0&0&0&0\\\ 0&0&0&1&i&0&0&0\\\
0&0&0&i&1&0&0&0\\\ 0&0&0&0&0&1&i&0\\\ 0&0&0&0&0&i&1&0\\\
0&0&0&0&0&0&0&\sqrt{2}\end{array}\right],$ (199)

which is different from $U$ in section 3.1. We can apply the discussion in
section 2.2 by replacing $U_{3}$ in section 2.2 with $U_{8}$. According to
(69), we define $V_{1}\equiv\mathcal{D}_{8,diag}U^{-1}_{8}VU_{8}$ and
$V_{2}\equiv(U^{-1}_{8}VU_{8})^{2}$. In particular, we focus on $V_{2}$:

$\displaystyle
V_{2}=\frac{g^{2}v^{2}}{4}\left[\begin{array}[]{cccccccc}1&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&-\sqrt{3}\\\ 0&0&0&1&0&0&0&0\\\
0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&2&-2i&0\\\ 0&0&0&0&0&2i&2&0\\\
0&0&-\sqrt{3}&0&0&0&0&3\end{array}\right].$ (208)

From (158) with $v=0$ and $V_{2}$, we find that the pair of $\psi^{3}_{l,m}$,
$\psi^{8}_{l,m}$ and $\psi^{6}_{n+1,j}$, $\psi^{7}_{n,j}$ are degenerate, and
we must solve the secular equations. As a result, the first-order perturbation
energy $E^{(1)}$ has

$\displaystyle
E^{(1)}_{3^{\prime}}=g^{2}v^{2},~{}~{}~{}E^{(1)}_{8^{\prime}}=0,~{}~{}~{}E^{(1)}_{6^{\prime}}=0,~{}~{}~{}E^{(1)}_{7^{\prime}}=g^{2}v^{2},$
(209)

where the mode functions in new direction $3^{\prime}$ and $8^{\prime}$ are
defined as

$\displaystyle\psi^{3^{\prime}}_{l,m}$
$\displaystyle=(-\psi^{3}_{l,m}+\sqrt{3}\psi^{8}_{l,m})/2,~{}~{}~{}\psi^{8^{\prime}}_{l,m}=(\sqrt{3}\psi^{3}_{l,m}+\psi^{8}_{l,m})/2,$
(210) $\displaystyle\psi^{6^{\prime}}_{n,j}$
$\displaystyle=(i\psi^{6}_{n+1,j}+\psi^{7}_{n,j})/\sqrt{2},~{}~{}~{}\psi^{7^{\prime}}_{n,j}=(\psi^{6}_{n+1,j}+i\psi^{7}_{n,j})/\sqrt{2}.$
(211)

Thus, the masses of gauge fields can be obtained as

$\displaystyle-\mathcal{D}_{diag}\overline{\mathcal{D}}_{diag}\equiv
m^{2}_{A},$ (212)
$\displaystyle\begin{cases}\displaystyle~{}(m^{2}_{A})^{11}=4\pi^{2}(l_{1}^{2}+m_{1}^{2})+\frac{1}{4}g^{2}v^{2},&\displaystyle~{}(m^{2}_{A})^{22}=4\pi^{2}(l_{2}^{2}+m_{2}^{2})+\frac{1}{4}g^{2}v^{2},\\\\[11.38109pt]
\displaystyle~{}(m^{2}_{A})^{3^{\prime}3^{\prime}}=4\pi^{2}(l_{3^{\prime}}^{2}+m^{2}_{3^{\prime}})+g^{2}v^{2},&\displaystyle~{}(m^{2}_{A})^{44}=\alpha_{3}n_{4}+\frac{1}{4}g^{2}v^{2},\\\\[11.38109pt]
\displaystyle~{}(m^{2}_{A})^{55}=\alpha_{3}(n_{5}+1)+\frac{1}{4}g^{2}v^{2},&\displaystyle~{}(m^{2}_{A})^{6^{\prime}6^{\prime}}=\frac{1}{2}g^{2}v^{2}~{}(n_{6}=0),~{}\alpha_{3}(n_{6^{\prime}}+1)~{}(n_{6^{\prime}}\geq
0),\\\\[11.38109pt]
\displaystyle~{}(m^{2}_{A})^{7^{\prime}7^{\prime}}=\alpha_{3}(n_{7^{\prime}}+1)+g^{2}v^{2},&~{}(m^{2}_{A})^{8^{\prime}8^{\prime}}=4\pi^{2}(l_{8^{\prime}}^{2}+m_{8^{\prime}}^{2}),\end{cases}$
(213)

where we ignore the second order perturbation energy.

Figure 4: A picture of the effective potential with $N=3$. The shaded region
represents a range where our perturbation analysis breaks down.

In previous discussion, the total one-loop effective potential has been
expressed by $V=V_{reg,A}$. From (213), we have

$\displaystyle V$
$\displaystyle=\frac{3}{2}\frac{1}{(2\pi)^{2}}\Big{(}2\nu_{reg,l,m}(1/4)+\nu_{reg,l,m}(1)+\nu_{reg,n}(\alpha_{3},1/4;0)$
$\displaystyle\hskip
56.9055pt+\nu_{reg,n}(\alpha_{3},1/4;1)+\nu_{reg,n}(\alpha_{3},1;1)+\nu_{reg,0}(1/2)\Big{)}.$
(214)

The effective potential of SU(3) Yang-Mills theory with $N=3$ is shown in
Figure 4. The shape of the potential in figure 4 is similar to that (the red
line) in Figure 2. The minimum of the effective potential might have a nonzero
constant VEV $v\neq 0$ and the gauge symmetry SU(2) $\times$ U(1) is broken to
U(1).

We should comment here again as was done in SU(2) case that we cannot
determine the location of the minimum since the location is expected to be
beyond the range which our perturbation is valid as is shown in figure 4. Even
if the value of the minimum is not determined, if we assume that the potential
is bounded below, the pattern of gauge symmetry breaking can be confirmed from
the gauge boson spectrum (158).

##  4 Summary

We have studied six dimensional Yang-Mills theories compactified on a torus
with a magnetic flux and a constant VEV in this paper. Before constructing
realistic models, we have discussed simple models of SU(2) and SU(3) Yang-
Mills theories to understand basic structures of the gauge symmetry breaking.

We first gave a setup of SU(2) model and derived the KK masses in terms of
perturbation theory in quantum mechanics. By using the KK masses, we
calculated the one-loop effective potential. In that computations, we focus on
the integral representation of Hurwitz $\zeta$ function and the regularized
effective potential could be obtained. From the obtained one-loop effective
potential and the mass of the gauge fields, we have seen that the SU(2) gauge
symmetry is completely broken because of the flux background and the constant
VEV.

Next, we considered an SU(3) model where two types of directions to introduce
the flux background and the constant VEV. The extension to SU(3) is necessary
for WL scalar fields to be an SU(2) doublet in the SM. In the case of section
3.1, the flux background and the constant VEV were introduced in the 8-th and
the 1-st components of SU(3) gauge symmetry, respectively. The one-loop
effective potential was calculated and it was found that the potential has a
nonzero VEV $v\neq 0$, which implies that the SU(3) gauge symmetry is broken
to U(1) $\times$ U(1) by the flux background and the constant VEV. On the
other hand, in the case of section 3.2, the flux background and the constant
VEV were introduced in the 8-th and the 6-th components of SU(3) gauge
symmetry, respectively. This case corresponds to gauge-Higgs unification in
that the constant VEV is taken in one of the component of the broken
generators of the original symmetry, SU(3)/(SU(2)$\times$ U(1)) in our model.
The one-loop effective potential was found to expect a nonzero VEV $v\neq 0$
and the SU(3) gauge symmetry is broken to U(1) by the flux background and the
constant VEV.

Although the results obtained in this paper are very interesting, it cannot be
realistic as it stands. If we identify the WL scalar field with the SM SU(2)
Higgs doublet in our SU(3) model, the gauge symmetry breaking pattern SU(3)
$\to$ SU(2) $\times$ U(1) $\to$ U(1)(or U(1)$\times$ U(1)) is not a correct
pattern of the electroweak symmetry breaking SU(2) $\times$ U(1) $\to$ U(1).
In order to realize such an electroweak symmetry breaking, it would be
interesting to take into account fermion field contributions to the one-loop
effective potential.

In some models discussed in this paper, the location of the potential minimum
could not be determined in a parameter region where our perturbation is valid.
It would be desirable to obtain the value of the potential minimum in a
perturbative region by extending our analysis to the models with fermions.
These issues are left for our future work.

## Acknowledgments

This work is supported by JSPS KAKENHI Grant Number JP22J15562 (T.H.).

##  Appendix A The second-order perturbation energy

In this appendix, we represent the second-order perturbation energy. Since
$V_{2}$ has an order of $\mathcal{O}(g^{2}v^{2})$, the second-order
perturbation energy from $V_{2}$ has an order of
$\mathcal{O}((g^{2}v^{2})^{2})$ and we neglect it.

###  A.1 The mass of the gauge field

The second-order perturbation energy from $V_{1}+V^{\dagger}_{1}$ for
$\psi^{1}_{0,j}$ is

$\displaystyle E_{A,0}^{(2)}=$
$\displaystyle-\sum_{(l,m)\neq(0,0)}\frac{\left|\int_{T^{2}}d^{2}x\left(\psi^{3}_{l,m}\right)^{\dagger}(V_{1}+V^{\dagger}_{1})\psi^{1}_{0,j}\right|^{2}}{4\pi^{2}(l^{2}+m^{2})}$
$\displaystyle=$ $\displaystyle-\frac{g^{2}v^{2}}{2}\left[\hskip
1.42262pt\sum_{(l,m)\neq(0,0)}\frac{4\pi^{2}(l^{2}+m^{2})}{4\pi^{2}(l^{2}+m^{2})}\left|\int_{T^{2}}d^{2}x~{}\bar{\lambda}_{l,m}\xi_{0,j}\right|^{2}\right]$
$\displaystyle\equiv$ $\displaystyle-\frac{g^{2}v^{2}}{2}\left[\hskip
1.42262pt\sum_{(l,m)\neq(0,0)}\left|C_{l,m,0,j}\right|^{2}\right].$ (215)

We do not describe the integrals $C_{l,m,0,j}$ in detail.

The second-order perturbation energy from $V_{1}+V^{\dagger}_{1}$ for
$\psi^{1^{\prime}}_{n+1,j},\psi^{2^{\prime}}_{n+1,j},\psi^{3}_{l=0,m=0}$ and
$\psi^{3}_{l\neq 0,m\neq 0}$ are

$\displaystyle
E_{A,1^{\prime}}^{(2)}=-\sum_{l,m}\frac{\left|\int_{T^{2}}d^{2}x\left(\psi_{l,m}^{3}\right)^{\dagger}\left(V_{1}+V_{1}^{\dagger}\right)\psi_{n+1,j}^{1^{\prime}}\right|^{2}}{4\pi^{2}(l^{2}+m^{2})-\alpha_{2}(n+1)},$
(216) $\displaystyle
E_{A,2^{\prime}}^{(2)}=-\sum_{l,m}\frac{\left|\int_{T^{2}}d^{2}x\left(\psi_{l,m}^{3}\right)^{\dagger}\left(V_{1}+V_{1}^{\dagger}\right)\psi_{n+1,j}^{2^{\prime}}\right|^{2}}{4\pi^{2}(l^{2}+m^{2})-\alpha_{2}(n+1)},$
(217) $\displaystyle E_{A,3,l=0,m=0}^{(2)}=\hskip 2.84526pt$
$\displaystyle\sum_{n,j}\frac{\left|\int_{T^{2}}d^{2}x\left(\psi_{l,m}^{3}\right)^{\dagger}\left(V_{1}+V_{1}^{\dagger}\right)\psi_{n+1,j}^{1^{\prime}}\right|^{2}}{4\pi^{2}(l^{2}+m^{2})-\alpha_{2}(n+1)}$
$\displaystyle+\sum_{n,j}\frac{\left|\int_{T^{2}}d^{2}x\left(\psi_{l,m}^{3}\right)^{\dagger}\left(V_{1}+V_{1}^{\dagger}\right)\psi_{n+1,j}^{2^{\prime}}\right|^{2}}{4\pi^{2}(l^{2}+m^{2})-\alpha_{2}(n+1)}$
(218)

and

$\displaystyle E_{A,3,l\neq 0,m\neq 0}^{(2)}=\hskip 3.98337pt$
$\displaystyle\frac{g^{2}v^{2}}{2}\sum_{j}\left|C_{l,m,0,j}\right|^{2}+\sum_{n,j}\frac{\left|\int_{T^{2}}d^{2}x\left(\psi_{l,m}^{3}\right)^{\dagger}\left(V_{1}+V_{1}^{\dagger}\right)\psi_{n+1,j}^{1^{\prime}}\right|^{2}}{4\pi^{2}(l^{2}+m^{2})-\alpha_{2}(n+1)}$
$\displaystyle+\sum_{n,j}\frac{\left|\int_{T^{2}}d^{2}x\left(\psi_{l,m}^{3}\right)^{\dagger}\left(V_{1}+V_{1}^{\dagger}\right)\psi_{n+1,j}^{2^{\prime}}\right|^{2}}{4\pi^{2}(l^{2}+m^{2})-\alpha_{2}(n+1)}.$
(219)

###  A.2 The mass of the scalar field

The second-order perturbation energy from $V_{1}+V_{1}^{\dagger}$,
$E_{\varphi}^{(2)}$ are

$\displaystyle
E_{\varphi,1^{\prime\prime}}^{(2)}=-\sum_{l,m}\frac{\left|\int_{T^{2}}d^{2}x\left(\psi_{l,m}^{3}\right)^{\dagger}\left(V_{1}+V_{1}^{\dagger}\right)\psi_{n,j}^{1^{\prime\prime}}\right|^{2}}{4\pi^{2}(l^{2}+m^{2})-\alpha_{2}(n+1/2)},$
(220) $\displaystyle
E_{\varphi,2^{\prime\prime}}^{(2)}=-\sum_{l,m}\frac{\left|\int_{T^{2}}d^{2}x\left(\psi_{l,m}^{3}\right)^{\dagger}\left(V_{1}+V_{1}^{\dagger}\right)\psi_{n,j}^{2^{\prime\prime}}\right|^{2}}{4\pi^{2}(l^{2}+m^{2})-\alpha_{2}(n+1/2)}$
(221)

and

$\displaystyle E_{\varphi,3}^{(2)}=$
$\displaystyle~{}\sum_{n,j}\frac{\left|\int_{T^{2}}d^{2}x\left(\psi_{l,m}^{3}\right)^{\dagger}\left(V_{1}+V_{1}^{\dagger}\right)\psi_{n,j}^{1^{\prime\prime}}\right|^{2}}{4\pi^{2}(l^{2}+m^{2})-\alpha_{2}(n+1/2)}$
$\displaystyle+\sum_{n,j}\frac{\left|\int_{T^{2}}d^{2}x\left(\psi_{l,m}^{3}\right)^{\dagger}\left(V_{1}+V_{1}^{\dagger}\right)\psi_{n,j}^{2^{\prime\prime}}\right|^{2}}{4\pi^{2}(l^{2}+m^{2})-\alpha_{2}(n+1/2)}.$
(222)

###  A.3 The mass of the ghost field

The second-order perturbation energy by $V_{4}+V_{4}^{\dagger}$, $E_{4}^{(2)}$
are

$\displaystyle
E_{c,1^{\prime\prime}}^{(2)}=-\sum_{l,m}\frac{\left|\int_{T^{2}}d^{2}x\left(\psi_{l,m}^{3}\right)^{\dagger}\left(V_{4}+V_{4}^{\dagger}\right)\psi_{n,j}^{1^{\prime\prime}}\right|^{2}}{4\pi^{2}(l^{2}+m^{2})-\alpha_{2}(n+1/2)},$
(223) $\displaystyle
E_{c,2^{\prime\prime}}^{(2)}=-\sum_{l,m}\frac{\left|\int_{T^{2}}d^{2}x\left(\psi_{l,m}^{3}\right)^{\dagger}\left(V_{4}+V_{4}^{\dagger}\right)\psi_{n,j}^{2^{\prime\prime}}\right|^{2}}{4\pi^{2}(l^{2}+m^{2})-\alpha_{2}(n+1/2)}$
(224)

and

$\displaystyle E_{c,3}^{(2)}=$
$\displaystyle~{}\sum_{n,j}\frac{\left|\int_{T^{2}}d^{2}x\left(\psi_{l,m}^{3}\right)^{\dagger}\left(V_{4}+V_{4}^{\dagger}\right)\psi_{n,j}^{1^{\prime\prime}}\right|^{2}}{4\pi^{2}(l^{2}+m^{2})-\alpha_{2}(n+1/2)}$
$\displaystyle+\sum_{n,j}\frac{\left|\int_{T^{2}}d^{2}x\left(\psi_{l,m}^{3}\right)^{\dagger}\left(V_{4}+V_{4}^{\dagger}\right)\psi_{n,j}^{2^{\prime\prime}}\right|^{2}}{4\pi^{2}(l^{2}+m^{2})-\alpha_{2}(n+1/2)}.$
(225)

## References

  * [1] R. Blumenhagen, B. Kors, D. Lust and S. Stieberger, “Four-dimensional String Compactifications with D-Branes, Orientifolds and Fluxes,” Phys. Rept. 445, 1 (2007) [hep-th/0610327].
  * [2] L. E. Ibanez and A. M. Uranga, “String theory and particle physics: An introduction to string theory,” Cambridge Univ. Press (2012).
  * [3] E. Witten, “Some Properties of O(32) Superstrings,” Phys. Lett. 149B, 351 (1984).
  * [4] H. Abe, K. S. Choi, T. Kobayashi and H. Ohki, “Three generation magnetized orbifold models,” Nucl. Phys. B 814, 265-292 (2009) [hep-th/0812.3534].
  * [5] D. Cremades, L. E. Ibanez and F. Marchesano, “Computing Yukawa couplings from magnetized extra dimensions,” JHEP 0405 079 (2004) [hep-th/0404229].
  * [6] H. Abe, K. -S. Choi, T. Kobayashi and H. Ohki, “Higher Order Couplings in Magnetized Brane Models,” JHEP 0906 080 (2009) [hep-th/0903.3800].
  * [7] Y. Hamada and T. Kobayashi, “Massive Modes in Magnetized Brane Models,” Prog. Theor. Phys. 128 (2012) 903 [hep-th/1207.6867].
  * [8] Y. Matsumoto and Y. Sakamura, “Yukawa couplings in 6D gauge-Higgs unification on $T^{2}/Z_{N}$ with magnetic fluxes,” PTEP 2016 (2016) 5, 053B06 [hep-th/1602.01994].
  * [9] W. Buchmuller, M. Dierigl, E. Dudas and J. Schweizer, “Effective field theory for magnetic compactifications,” JHEP 1704 (2017) 052 [hep-th/1611.03798].
  * [10] D. M. Ghilencea and H. M. Lee, “Wilson lines and UV sensitivity in magnetic compactifications,” JHEP 1706 039 (2017) [hep-th/1703.10418].
  * [11] W. Buchmuller, M. Dierigl, E. Dudas, “Flux compactifications and naturalness,” JHEP 1808 151 (2018) [hep-th/1804.07497].
  * [12] T. Hirose and N. Maru, “Cancellation of One-loop Corrections to Scalar Masses in Yang-Mills Theory with Flux Compactification,” JHEP 1908 (2019) 054 [hep-th/1904.06028].
  * [13] M. Honda and T. Shibasaki, “Wilson-line Scalar as a Nambu-Goldstone Boson in Flux Compactifications and Higher-loop Corrections,” JHEP 03 (2020) 031 [hep-th/1912.04581].
  * [14] T. Hirose and N. Maru, “Nonvanishing Finite Scalar Mass in Flux Compactification,” JHEP 06 (2021) 159 [hep-th/2104.01779].
  * [15] N. S. Manton, “A New Six-Dimensional Approach to the Weinberg-Salam Model,” Nucl. Phys. B 158 (1979) 141-153.
  * [16] Y.Hosotani, “Dynamical Mass Generation by Compact Extra Dimension,” Phys. Lett. 126B (1983) 309.
  * [17] Y.Hostonani, “Dynamical of Non-integrable Phases and Gauge Symmetry Breaking,” ANNALS 190, (1989) 233-253
  * [18] H. Hatanaka, T. Inami and C.S. Lim, “The Gauge hierarchy problem and higher dimensional gauge theories,” Mod. Phys. Lett. A 13, 2601 (1998); [hep-th/9805067].
  * [19] C.S. Lim, N. Maru and K. Hasegawa, “Six dimensional gauge-Higgs unification with an extra space S**2 and the hierarchy problem,” J. Phys. Soc. Jap. 77, 074101 (2008); arXiv:hep-th/0605180.
  * [20] N. Maru and T. Yamashita, “Two-loop calculation of Higgs mass in gauge-Higgs unification: 5D massless QED compactified on S**1,” Nucl. Phys. B 754, 127 (2006); [arXiv:hep-ph/0603237].
  * [21] Y. Hosotani, N. Maru, K. Takenaga and T. Yamashita, “Two loop finiteness of Higgs mass and potential in the gauge-Higgs unification,” Prog. Theor. Phys. 118, 1053 (2007); [arXiv:0709.2844 [hep-ph]].
  * [22] Masahiro Kubo, C.S.Lim and Hiroyuki Yamashita, “The Hosotani mechanism in bulk gauge theories with an orbifold extra space $S^{1}/Z_{2}$,” Mod. Phys. Lett. A17 (2002) 2249; [hep-ph/0111327]
  * [23] I. Antoniadis, K. Benakli and M. Quiros, “Finite Higgs mass without supersymmetry,” New J. Phys. 3, 20 (2001); [arXiv:hep-th/0108005].
  * [24] E. Elizalde, “Ten physical applications of spectral zeta functions, ” second edition, Berlin, Springer, Germany (2012)