EPHOH-15-001
KIAS-P15002
KOBE-TH-15-01
KUNS-2535
OU-HET-848
WU-HEP-15-01

Classification of three-generation models
on magnetized orbifolds

Tomo-hiro Abe, aaa E-mail<EMAIL_ADDRESS>Yukihiro Fujimoto, bbb
E-mail<EMAIL_ADDRESS>Tatsuo Kobayashi, ccc E-mail:
<EMAIL_ADDRESS>
Takashi Miura, ddd E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>eeeMoved to Software Systems Laboratories,
FUJITSU LABORATORIES LTD. 1-1,
spaces spaces sKamikodanaka 4-chome, Nakahara-ku Kawasaki, 211-8588 Japan
Kenji Nishiwaki, fff E-mail<EMAIL_ADDRESS>
Makoto Sakamoto, ggg E-mail<EMAIL_ADDRESS>and Yoshiyuki Tatsuta hhh
E-mail<EMAIL_ADDRESS>
aDepartment of Physics, Kyoto University, Kyoto 606-8502, Japan
bDepartment of Physics, Osaka University, Toyonaka 560-0043, Japan
cDepartment of Physics, Hokkaido University, Sapporo 060-0810, Japan
d,gDepartment of Physics, Kobe University, Kobe 657-8501, Japan
fSchool of Physics, Korea Institute for Advanced Study,
85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea
hDepartment of Physics, Waseda University, Tokyo 169-8555, Japan
2 September 2024

We classify the combinations of parameters which lead three generations of
quarks and leptons in the framework of magnetized twisted orbifolds on
$T^{2}/Z_{2}$, $T^{2}/Z_{3}$, $T^{2}/Z_{4}$ and $T^{2}/Z_{6}$ with allowing
nonzero discretized Wilson line phases and Scherk-Schwarz phases. We also
analyze two actual examples with nonzero phases leading to one-pair Higgs and
five-pair Higgses and discuss the difference from the results without nonzero
phases studied previously.

##  1 Introduction

One of the greatest achievements in particle physics is the completion of the
Standard Model (SM) as the discovery of the SM-like Higgs boson following the
works reported in [1, 2]. As is widely known, the SM is consistent with almost
all the experimental results which have ever been made, while the origin of
the SM configurations is still unknown, e.g., the origin of the
$SU(3)_{C}\times SU(2)_{W}\times U(1)_{Y}$ gauge structure and the related
quantum numbers in the matter fields, or why the number of the matter
generation is three accompanied with the large mass hierarchy.

Ten-dimensional (10D) super Yang-Mills (SYM) theory on magnetized tori [3, 4,
5, 6, 7, 8, 9] possibly is a good candidate for explaining these issues
simultaneously.111 Introducing point interactions (zero-thickness branes) in
the bulk space of a five-dimensional theory on $S^{1}$ (or a line segment) and
considering various boundary conditions of fields on them [10, 11, 12] is a
direction to attack the problems.  Here, 4D chirality and multiple matter
generations are created by the effect of magnetic fluxes in a unified gauge
group. After the introduction of the fluxes, the larger gauge group is
explicitly broken down into a small substructure, which contains the SM
counterpart. In this direction, lots of phenomenological aspects have been
explored: Yukawa couplings [7], realization of quark/lepton masses and their
mixing angles [13, 14], higher order couplings [15], flavor symmetries [16,
17, 18, 19, 20, 21, 22], massive modes [23], and others [24, 25, 26, 27, 28,
29, 30, 31, 32, 33].222 Some related works in other stringy contexts (e.g.,
intersecting D-brane model and heterotic string theory) are found in [34, 35,
36, 37, 8, 9, 38, 39, 40, 41].

In actual model building, orbifolding plays an important role, not only in
removing exotic particles, but also in deforming wave function profiles of the
SM matters, which would help us to realize the observed flavor structure of
the nature. In general under the magnetic gauge background, it is not so easy
to analyze the case more than $Z_{2}$, namely on $T^{2}$, $T^{2}/Z_{3,4,6}$
geometries, where $Z_{N}$-entangled states have very complicated forms since
mode functions on magnetized $T^{2}$ are described by theta functions.333 Note
that their geometrical aspects are discussed [42, 43, 44] within the context
of string theory. In a higher-dimensional field theory, detailed studies have
been carried out [45, 46, 47, 48, 49, 50].  However recently, general
structure of these geometries was declared in an exact and analytical way in a
“dual” description with operator formalism [51]. In that paper, the authors
treated most general cases on magnetized $T^{2}/Z_{2,3,4,6}$ with nonzero
(discretized) Wilson line phases and/or Scherk-Schwarz phases, which were
discussed in [52] (see also [53, 54, 43, 55, 56, 57, 58, 59, 60, 61]). Note
that shifted orbifold can be considered on a magnetized torus [62].

Some works toward phenomenologically realistic model building have been done
through this direction based on $U(N)$ gauge theory in [63, 64, 13, 32, 14],
where only the $Z_{2}$ orbifolding with trivial Wilson line phases and Scherk-
Schwarz phases is considered. See also Refs [27, 28] for discussions based on
$E_{6,7,8}$ groups. Now, because of the advent of [51], all the technical
difficulties for considering the generalized case have been resolved and it
would be phenomenologically meaningful that we start to pursue the situation
with nontrivial boundary conditions under the $Z_{2}$ or higher $Z_{3,4,6}$.
As a first step, in this paper, we classify all the possibilities of the
models with suitable three-generation matter structure of the quarks and the
leptons within $U(N)$ gauge group being the original gauge structure, and show
an example in the $Z_{2}$ case with nontrivial boundary conditions after
discussing the effect via nonzero Wilson line phases and/or Scherk-Schwarz
phases.

This paper is organized as follows. In section 2, we give a brief review on
the description of 10D SYM theory on $(T^{2})^{3}$ with or without
orbifoldings $Z_{2,3,4,6}$, respectively, including magnetic fluxes,
nontrivial Wilson line phases and Scherk-Schwarz phases. In section 3, we
exhaust all the possible configurations of parameters in all the $Z_{2}$,
$Z_{3}$, $Z_{4}$ and $Z_{6}$ cases with nontrivial boundary conditions. In
section 4, after we examine how much mass hierarchy in the quark sector is
deformed by introducing nonzero Wilson line phases and/or Scherk-Schwarz
phases, we propose an example for realizing (semi-)realistic quark mass
patterns and the mixing structure described by the Cabibbo-Kobayashi-Maskawa
(CKM) matrix. Section 5 is devoted to summarizing issues and discussing future
prospects. In Appendix A, we show the analytical forms describing a state on
the magnetized $T^{2}/Z_{2,3,4,6}$ orbifolds obtained in Ref. [51]. In
Appendix B, we show our results for $T^{2}/Z_{2,3,4,6}$ by writing down all
the independent combinations of parameters explicitly.

##  2 10D SYM on generalized magnetized orbifolds

Here, we review the basics of the 10D super Yang-Mills theory on generalized
magnetized orbifolds of $T^{2}/Z_{2},T^{2}/Z_{3},T^{2}/Z_{4},T^{2}/Z_{6}$ with
nonzero Wilson line phases and/or Scherk-Schwarz phases.

###  2.1 $U(N)$ gauge theory on $(T^{2})^{3}$

Firstly, we focus on the 10D magnetized SYM theory without orbifolding [34,
65, 64], with adopting notations in Refs. [52, 51],

$\displaystyle
S=\int_{M^{4}}d^{4}x\int_{(T^{2})^{3}}d^{6}z\left\\{-\frac{1}{4}\text{tr}\left(F_{MN}F^{MN}\right)+\frac{1}{2}\text{tr}\left(\overline{\lambda}\Gamma^{M}iD_{M}\lambda\right)\right\\},$
(2.1)

which is defined on a product of 4D Minkowski space and three factorizable
2-tori, $M^{4}\times(T^{2})^{3}$. The capital roman indices $M,N$ run over
$\mu(=0,1,2,3),z_{1},\overline{z_{1}},z_{2},\overline{z_{2}},z_{3},\overline{z_{3}}$,
where the $i$-th ($i=1,2,3$) 2-torus is described by the complex coordinates
$z_{i}=y_{2i+2}+iy_{2i+3}$ and its complex conjugation
$\overline{z_{i}}=y_{2i+2}-iy_{2i+3}$ made by the two Cartesian coordinate
representing the extra directions, $y_{2i+2}$ and $y_{2i+3}$. We take each
torus modulus parameter $\tau_{i}\ (\subset\mathbb{C})$ as
$\text{Im}\tau_{i}>0$ for convenience. We use the short-hand notation $d^{6}z$
meaning $\Pi_{i=1}^{3}dz_{i}d\overline{z_{i}}$. On $T^{2}_{i}$, the coordinate
$z_{i}$ is identified as $z_{i}\sim z_{i}+1\sim z_{i}+\tau_{i}$. The bulk
Lagrangian holds $\mathcal{N}=1$ supersymmetry in 10D and consists of the 10D
vector fields $A_{M}$, which are found in the covariant derivative $D_{M}$ and
the field strength $F_{MN}$, and the gaugino fields $\lambda$ described by 10D
Majorana-Weyl spinors. The explicit forms of $D_{M}\lambda$ and $F_{MN}$ are
given by

$\displaystyle D_{M}\lambda$ $\displaystyle=\partial_{M}\lambda-
ig\left[A_{M},\lambda\right],$ (2.2) $\displaystyle F_{MN}$
$\displaystyle=\partial_{M}A_{N}-\partial_{N}A_{M}-ig\left[A_{M},A_{N}\right],$
(2.3)

with the 10D gauge coupling $g$.

The gaugino fields and the 10D vector fields are Kaluza-Klein (KK) decomposed
as

$\displaystyle\lambda(x,\\{z_{i},\overline{z_{i}}\\})$
$\displaystyle=\sum_{l,m,n}\chi_{l,m,n}(x)\otimes\psi^{(1)}_{l}(z_{1},\overline{z_{1}})\otimes\psi^{(2)}_{m}(z_{2},\overline{z_{2}})\otimes\psi^{(3)}_{n}(z_{3},\overline{z_{3}}),$
(2.4) $\displaystyle A_{M}(x,\\{z_{i},\overline{z_{i}}\\})$
$\displaystyle=\sum_{l,m,n}\varphi_{l,m,n;M}(x)\otimes\phi^{(1)}_{l,M}(z_{1},\overline{z_{1}})\otimes\phi^{(2)}_{m,M}(z_{2},\overline{z_{2}})\otimes\phi^{(3)}_{n,M}(z_{3},\overline{z_{3}}),$
(2.5)

where $l,m,n$ are KK indices and $\psi^{(i)}_{l}$ is a 2D spinor describing
the $l$-th KK mode on the $i$-th $T^{2}$, whose exact form is
$\psi^{(i)}_{l}=\left(\psi^{(i)}_{l,+},\psi^{(i)}_{l,-}\right)^{\text{T}}$ and
the corresponding 2D chirality ($+$ or $-$) is denoted by $s_{i}$. We adopt
the gamma matrices $\tilde{\Gamma}^{m}$ (identified by the Cartesian
coordinates) corresponding to the $i$-th torus as

$\displaystyle\tilde{\Gamma}^{2i+2}=i\sigma_{1},\quad\tilde{\Gamma}^{2i+3}=i\sigma_{2},$
(2.6)

where $\sigma_{1,2}$ are Pauli matrices. In the following part, we only focus
on the zero modes $(l=m=n=0)$ and omit the KK indices hereafter.

We introduce factorizable Abelian magnetic fluxes on the three $T^{2}$ through
the classical vector potential of $A_{M}$ in the following forms

$\displaystyle A^{(b)}(\\{z_{i},\overline{z_{i}}\\})$
$\displaystyle=\sum_{i=1}^{3}\frac{\pi}{{g}\text{Im}\tau_{i}}\begin{pmatrix}M_{1}^{(i)}\text{Im}\left[(\overline{z_{i}}+\overline{C^{(i)}_{1}})dz_{i}\right]\mathbf{1}_{N_{1}\times
N_{1}}&&0\\\ &\ddots&\\\
0&&M_{n}^{(i)}\text{Im}\left[(\overline{z_{i}}+\overline{C^{(i)}_{n}})dz_{i}\right]\mathbf{1}_{N_{n}\times
N_{n}}\end{pmatrix}$
$\displaystyle=\sum_{i=1}^{3}\frac{1}{4i\text{Im}\tau_{i}}\left(\overline{{\bm{B}}^{(i)}}dz_{i}-{\bm{B}}^{(i)}d\overline{z_{i}}\right)$
$\displaystyle=\sum_{i=1}^{3}\left(A_{z_{i}}^{(b)}(\overline{z_{i}})dz_{i}+A_{\overline{z_{i}}}^{(b)}(z_{i})d\overline{z_{i}}\right),$
(2.7)

where $C^{(i)}_{j}\ (j=1,\cdots,n)$ represent the corresponding Wilson line
phases on $T^{2}_{i}$, and $M^{(i)}_{j}\ (j=1,\cdots,n)$ should be integers
because of Dirac’s quantization condition on $T^{2}_{i}$. On $T^{2}_{i}$, no
restriction is there on possible choices of $C^{(i)}_{j}$. Under this
background, the original gauge group $U(N)$ explicitly breaks down as
$U(N)\to\Pi_{a=1}^{n}U(N_{a})$ with $N=\sum_{a=1}^{n}N_{a}$. We can derive the
following relations,

$\displaystyle
A^{(b)}{(\\{z_{i}+\delta_{ij},\overline{z_{i}}+\delta_{ij}\\})}$
$\displaystyle=A^{(b)}(\\{z_{i},\overline{z_{i}}\\})+{d_{j}}\xi_{1}(\\{z_{i}\\}),\quad{(j=1,2,3)}$
(2.8) $\displaystyle
A^{(b)}{(\\{z_{i}+\delta_{ij}\tau_{j},\overline{z_{i}}+\delta_{ij}\overline{\tau_{j}}\\})}$
$\displaystyle=A^{(b)}(\\{z_{i},\overline{z_{i}}\\})+{d_{j}}\xi_{\tau}(\\{z_{i}\\}),\quad{(j=1,2,3)}$
(2.9) $\displaystyle\xi_{1}(\\{z_{i}\\})=\sum_{i=1}^{3}\xi_{1_{i}}(z_{i})$
$\displaystyle=\sum_{i=1}^{3}\frac{1}{2\text{Im}\tau_{i}}\text{Im}[{\bm{B}}^{(i)}],$
(2.10)
$\displaystyle\xi_{\tau}(\\{z_{i}\\})=\sum_{i=1}^{3}\xi_{\tau_{i}}(z_{i})$
$\displaystyle=\sum_{i=1}^{3}\frac{1}{2\text{Im}\tau_{i}}\text{Im}[\overline{\tau_{i}}{\bm{B}}^{(i)}],$
(2.11)

where $d_{j}$ plays as an exterior derivative on $T^{2}_{j}$. Here, the
Lagrangian density in Eq. (2.1) should be single-valued under every torus
identification $z_{i}\sim z_{i}+1\sim z_{i}+\tau_{i}\ (i=1,2,3)$, and then in
the gaugino fields $\lambda(x,\\{z_{i},\overline{z_{i}}\\})$, the following
pseudo-periodic boundary conditions should be arranged,

$\displaystyle\lambda{(x,\\{z_{i}+\delta_{ij},\overline{z_{i}}+\delta_{ij}\\})}$
$\displaystyle={U_{1_{j}}(z_{j})}\lambda(x,\\{z_{i},\overline{z_{i}}\\}){U_{1_{j}}(z_{j})^{\dagger}},\quad{(j=1,2,3)}$
(2.12)
$\displaystyle\lambda{(x,\\{z_{i}+\delta_{ij}\tau_{j},\overline{z_{i}}+\delta_{ij}\overline{\tau_{j}}\\})}$
$\displaystyle={U_{\tau_{j}}(z_{j})}\lambda(x,\\{z_{i},\overline{z_{i}}\\}){U_{\tau_{j}}(z_{j})^{\dagger}},\quad{(j=1,2,3)}$
(2.13)

with

$\displaystyle U_{1_{i}}(z_{i}):=e^{i{g}\xi_{1_{i}}(z_{i})+2\pi
i{\bm{\alpha}}^{(i)}},\quad
U_{\tau_{i}}(z_{i}):=e^{i{g}\xi_{\tau_{i}}(z_{i})+2\pi i{\bm{\beta}}^{(i)}},$
(2.14)
$\displaystyle{\bm{\alpha}}^{(i)}:=\begin{pmatrix}\alpha^{(i)}_{1}\mathbf{1}_{N_{1}\times
N_{1}}&&0\\\ &\ddots&\\\ 0&&\alpha^{(i)}_{n}\mathbf{1}_{N_{n}\times
N_{n}}\end{pmatrix},\quad{\bm{\beta}}^{(i)}:=\begin{pmatrix}\beta^{(i)}_{1}\mathbf{1}_{N_{1}\times
N_{1}}&&0\\\ &\ddots&\\\ 0&&\beta^{(i)}_{n}\mathbf{1}_{N_{n}\times
N_{n}}\end{pmatrix},$ (2.15)

where $\alpha^{(i)}_{j}$ and $\beta^{(i)}_{j}$ $(j=1,\cdots,n)$ describe
Scherk-Schwarz phases and can take any real numbers.

Now, we consider the specific case of $U(N)\to U(N_{a})\times U(N_{b})$ for
our concrete understanding on the system. In this case, the 1-form potential
in Eq. (2.7) takes the shape,

$\displaystyle A^{(b)}(\\{z_{i},\overline{z_{i}}\\})$
$\displaystyle=\sum_{i=1}^{3}\frac{\pi}{{g}\text{Im}\tau_{i}}\begin{pmatrix}M_{a}^{(i)}\text{Im}\left[(\overline{z_{i}}+\overline{C^{(i)}_{a}})dz_{i}\right]\mathbf{1}_{N_{a}\times
N_{a}}&0\\\
0&M_{b}^{(i)}\text{Im}\left[(\overline{z_{i}}+\overline{C^{(i)}_{b}})dz_{i}\right]\mathbf{1}_{N_{b}\times
N_{b}}\end{pmatrix},$ (2.16)

and the gaugino fields $\lambda$ and their $i$-th torus part are decomposed as
follows:

$\displaystyle\lambda(x,\\{z_{i},\overline{z_{i}}\\})=\begin{pmatrix}\lambda^{aa}(x,\\{z_{i},\overline{z_{i}}\\})&\lambda^{ab}(x,\\{z_{i},\overline{z_{i}}\\})\\\
\lambda^{ba}(x,\\{z_{i},\overline{z_{i}}\\})&\lambda^{bb}(x,\\{z_{i},\overline{z_{i}}\\})\end{pmatrix},\quad\psi^{(i)}(z_{i},\overline{z_{i}})=\begin{pmatrix}\psi^{(i)aa}(z_{i},\overline{z_{i}})&\psi^{(i)ab}(z_{i},\overline{z_{i}})\\\
\psi^{(i)ba}(z_{i},\overline{z_{i}})&\psi^{(i)bb}(z_{i},\overline{z_{i}})\end{pmatrix}.$
(2.17)

The fields $\lambda^{aa}$ and $\lambda^{bb}$ play as the gaugino fields under
the unbroken gauge group $U(N_{a})\times U(N_{b})$, while $\lambda^{ab}$ and
$\lambda^{ba}$ correspond to the bi-fundamental matter fields as
$(N_{a},\overline{N_{b}})$ and $(\overline{N_{a}},N_{b})$, respectively. We
obtain the zero-mode equations for these gaugino fields on the $i$-th $T^{2}$
with the 2D chirality ($+$ or $-$) as

$\displaystyle\begin{pmatrix}\partial_{\overline{z_{i}}}\psi^{(i)aa}_{+}&\left[\partial_{\overline{z_{i}}}+\frac{\pi}{2\text{Im}\tau_{i}}\left(M^{(i)}_{ab}z_{i}+C^{\prime(i)}_{ab}\right)\right]\psi^{(i)ab}_{+}\\\
\left[\partial_{\overline{z_{i}}}+\frac{\pi}{2\text{Im}\tau_{i}}\left(M^{(i)}_{ba}z_{i}+C^{\prime(i)}_{ba}\right)\right]\psi^{(i)ba}_{+}&\partial_{\overline{z_{i}}}\psi^{(i)bb}_{+}\end{pmatrix}$
$\displaystyle=0,$ (2.18)
$\displaystyle\begin{pmatrix}\partial_{z_{i}}\psi^{(i)aa}_{-}&\left[\partial_{z_{i}}-\frac{\pi}{2\text{Im}\tau_{i}}\left(M^{(i)}_{ab}\overline{z_{i}}+\overline{C^{\prime(i)}_{ab}}\right)\right]\psi^{(i)ab}_{-}\\\
\left[\partial_{z_{i}}-\frac{\pi}{2\text{Im}\tau_{i}}\left(M^{(i)}_{ba}\overline{z_{i}}+\overline{C^{\prime(i)}_{ba}}\right)\right]\psi^{(i)ba}_{-}&\partial_{z_{i}}\psi^{(i)bb}_{-}\end{pmatrix}$
$\displaystyle=0,$ (2.19)

with the short-hand notations $M^{(i)}_{ab}:=M^{(i)}_{a}-M^{(i)}_{b}$ and
$C^{\prime(i)}_{ab}:=M^{(i)}_{a}C^{(i)}_{a}-M^{(i)}_{b}C^{(i)}_{b}$. The
block-diagonal parts of the gaugino fields, $\psi^{(i)aa}$ and $\psi^{(i)bb}$,
do not feel magnetic flux and Wilson line phase, which is apparent from Eqs.
(2.18) and (2.19). On the other hand, in the block-off-diagonal parts,
$\psi^{(i)ab}$ and $\psi^{(i)ba}$, magnetic fluxes and Wilson line phases
affect properties of the matter fields. From the information in Eqs. (2.14)
and (2.15), the effective boundary conditions of the fields are easily written
down,

$\displaystyle\psi^{(i)ab}_{s_{i}}{(z_{i}+1,\overline{z_{i}}+1)}$
$\displaystyle=e^{i\frac{\pi
s_{i}}{\text{Im}\tau_{i}}\text{Im}\left[M^{(i)}_{ab}z_{i}+C^{\prime(i)}_{ab}\right]+2\pi
i\alpha^{(i)}_{ab}}\psi^{(i)ab}_{s_{i}}{(z_{i},\overline{z_{i}})},$
$\displaystyle\psi^{(i)ba}_{s_{i}}{(z_{i}+1,\overline{z_{i}}+1)}$
$\displaystyle=e^{i\frac{\pi
s_{i}}{\text{Im}\tau_{i}}\text{Im}\left[M^{(i)}_{ba}z_{i}+C^{\prime(i)}_{ba}\right]+2\pi
i\alpha^{(i)}_{ba}}\psi^{(i)ba}_{s_{i}}{(z_{i},\overline{z_{i}})},$
$\displaystyle\psi^{(i)aa}_{s_{i}}{(z_{i}+1,\overline{z_{i}}+1)}$
$\displaystyle=\psi^{(i)aa}_{s_{i}}{(z_{i},\overline{z_{i}})},$
$\displaystyle\psi^{(i)bb}_{s_{i}}{(z_{i}+1,\overline{z_{i}}+1)}$
$\displaystyle=\psi^{(i)bb}_{s_{i}}{(z_{i},\overline{z_{i}})},$ (2.20)
$\displaystyle\psi^{(i)ab}_{s_{i}}{(z_{i}+\tau_{i},\overline{z_{i}}+\overline{\tau_{i}})}$
$\displaystyle=e^{i\frac{\pi
s_{i}}{\text{Im}\tau_{i}}\text{Im}\left[\overline{\tau_{i}}\left(M^{(i)}_{ab}z_{i}+C^{\prime(i)}_{ab}\right)\right]+2\pi
i\beta^{(i)}_{ab}}\psi^{(i)ab}_{s_{i}}{(z_{i},\overline{z_{i}})},$
$\displaystyle\psi^{(i)ba}_{s_{i}}{(z_{i}+\tau_{i},\overline{z_{i}}+\overline{\tau_{i}})}$
$\displaystyle=e^{i\frac{\pi
s_{i}}{\text{Im}\tau_{i}}\text{Im}\left[\overline{\tau_{i}}\left(M^{(i)}_{ba}z_{i}+C^{\prime(i)}_{ba}\right)\right]+2\pi
i\beta^{(i)}_{ba}}\psi^{(i)ba}_{s_{i}}{(z_{i},\overline{z_{i}})},$
$\displaystyle\psi^{(i)aa}_{s_{i}}{(z_{i}+\tau_{i},\overline{z_{i}}+\overline{\tau_{i}})}$
$\displaystyle=\psi^{(i)aa}_{s_{i}}{(z_{i},\overline{z_{i}})},$
$\displaystyle\psi^{(i)bb}_{s_{i}}{(z_{i}+\tau_{i},\overline{z_{i}}+\overline{\tau_{i}})}$
$\displaystyle=\psi^{(i)bb}_{s_{i}}{(z_{i},\overline{z_{i}})},$ (2.21)

with the short-hand notations,
$\alpha^{(i)}_{ab}:=\alpha^{(i)}_{a}-\alpha^{(i)}_{b}$ and
$\beta^{(i)}_{ab}:=\beta^{(i)}_{a}-\beta^{(i)}_{b}$. We remind that $s_{i}$
shows the corresponding 2D chirality.

When $M^{(i)}_{ab}>0$, the fields $\psi^{(i)ab}_{+}$ and $\psi^{(i)ba}_{-}$
contain $|M^{(i)}_{ab}|$ normalizable zero modes, while the others
$\psi^{(i)ba}_{+}$ and $\psi^{(i)ab}_{-}$ have no corresponding one. On the
other hand in $M^{(i)}_{ab}<0$, $|M^{(i)}_{ab}|$ normalizable zero modes are
generated from each of $\psi^{(i)ba}_{+}$ and $\psi^{(i)ab}_{-}$, whereas
there is nothing from $\psi^{(i)ab}_{+}$ and $\psi^{(i)ba}_{-}$. In the case
of $M^{(i)}_{ab}=0$, like $\psi^{(i)aa}_{s_{i}}$ or $\psi^{(i)bb}_{s_{i}}$,
only one non-localized mode is generated from each of the all sectors and no
phenomenological interests occur. When $M^{(i)}_{ab}>0$, which is equal to
$M^{(i)}_{ba}<0$, the wave functions of $\psi^{(i)ab}_{+}$ and
$\psi^{(i)ba}_{-}$ take the following forms:

$\displaystyle\psi^{(i)ab}_{+}(z_{i})=\sum_{I=0}^{|M^{(i)}_{ab}|-1}\begin{pmatrix}\Theta^{(I+\alpha^{(i)}_{ab},\beta^{(i)}_{ab})}_{M^{(i)}_{ab},a^{(i)}_{ab}}(z_{i},\tau_{i})\\\
0\end{pmatrix},\quad\psi^{(i)ba}_{-}(\overline{z_{i}})=\sum_{I=0}^{|M^{(i)}_{ba}|-1}\begin{pmatrix}0\\\
\Theta^{(I+\alpha^{(i)}_{ba},\beta^{(i)}_{ba})}_{M^{(i)}_{ba},\overline{a^{(i)}_{ba}}}(\overline{z_{i}},\overline{\tau_{i}})\end{pmatrix},$
(2.22)
$\displaystyle\Theta^{(I+\alpha^{(i)}_{ab},\beta^{(i)}_{ab})}_{M^{(i)}_{ab},a^{(i)}_{ab}}(z_{i},\tau_{i})$
$\displaystyle=\mathcal{N}_{|M^{(i)}_{ab}|}e^{i\pi
M^{(i)}_{ab}(z_{i}+a^{(i)}_{ab}){\mathrm{Im}(z_{i}+a^{(i)}_{ab})\over\mathrm{Im}\tau_{i}}}\cdot\vartheta\left[\begin{array}[]{c}{I+\alpha^{(i)}_{ab}\over
M^{(i)}_{ab}}\\\
-\beta^{(i)}_{ab}\end{array}\right](M^{(i)}_{ab}(z_{i}+a^{(i)}_{ab}),M^{(i)}_{ab}\tau_{i}),$
(2.25)
$\displaystyle\Theta^{(I+\alpha^{(i)}_{ba},\beta^{(i)}_{ba})}_{M^{(i)}_{ba},\overline{a^{(i)}_{ba}}}(\overline{z_{i}},\overline{\tau_{i}})$
$\displaystyle=\mathcal{N}_{|M^{(i)}_{ba}|}e^{i\pi
M^{(i)}_{ba}(\overline{z_{i}}+\overline{a^{(i)}_{ba}}){\mathrm{Im}(\overline{z_{i}}+\overline{a^{(i)}_{ba}})\over\mathrm{Im}\overline{\tau_{i}}}}\cdot\vartheta\left[\begin{array}[]{c}{I+\alpha^{(i)}_{ba}\over
M^{(i)}_{ba}}\\\
-\beta^{(i)}_{ba}\end{array}\right](M^{(i)}_{ba}(\overline{z_{i}}+\overline{a^{(i)}_{ba}}),M^{(i)}_{ba}\overline{\tau_{i}}),$
(2.28)

with the effective Wilson line phase parameter
$a^{(i)}_{ab}:=C^{\prime(i)}_{ab}/M^{(i)}_{ab}$. Here, $I\
(=0,\cdots,|M^{(i)}_{ab}|-1)$ discriminates the $|M^{(i)}_{ab}|$-degenerated
zero-mode states. The $\vartheta$ function is defined by

$\displaystyle\vartheta\left[\begin{array}[]{c}a\\\
b\end{array}\right](c\nu,c\tau)=\sum_{l=-\infty}^{\infty}e^{i\pi(a+l)^{2}c\tau}e^{2\pi
i(a+l)(c\nu+b)},$ (2.31)

with the properties

$\displaystyle\vartheta\left[\begin{array}[]{c}a\\\
b\end{array}\right](c(\nu+n),c\tau)=e^{2\pi
iacn}\vartheta\left[\begin{array}[]{c}a\\\ b\end{array}\right](c\nu,c\tau),$
(2.36) $\displaystyle\vartheta\left[\begin{array}[]{c}a\\\
b\end{array}\right](c(\nu+n\tau),c\tau)=e^{-i\pi cn^{2}\tau-2\pi
in(c\nu+b)}\vartheta\left[\begin{array}[]{c}a\\\
b\end{array}\right](c\nu,c\tau),$ (2.41)
$\displaystyle\vartheta\left[\begin{array}[]{c}a+m\\\
b+n\end{array}\right](c\nu,c\tau)=e^{2\pi
ian}\vartheta\left[\begin{array}[]{c}a\\\ b\end{array}\right](c\nu,c\tau),$
(2.46) $\displaystyle\vartheta\left[\begin{array}[]{c}a\\\
b\end{array}\right](c\nu,c\tau)=\vartheta\left[\begin{array}[]{c}a\\\
0\end{array}\right](c\nu+b,c\tau),$ (2.51)

where $a$ and $b$ are real numbers, $c$, $m$ and $n$ are integers, and $\nu$
and $\tau$ are complex numbers with $\mathrm{Im}\tau>0$. The following
orthonormality condition determines the normalization factor
$\mathcal{N}_{|M^{(i)}_{ab}|}$,

$\displaystyle\int_{T^{2}_{i}}d^{2}z_{i}\left(\Theta^{(I+\alpha^{(i)}_{ab},\beta^{(i)}_{ab})}_{M^{(i)}_{ab},a^{(i)}_{ab}}(z_{i},\tau_{i})\right)^{\ast}\left(\Theta^{(J+\alpha^{(i)}_{ab},\beta^{(i)}_{ab})}_{M^{(i)}_{ab},a^{(i)}_{ab}}(z_{i},\tau_{i})\right)$
$\displaystyle=\delta_{I,J}\quad(M^{(i)}_{ab}>0),$
$\displaystyle\int_{T^{2}_{i}}d^{2}z_{i}\left(\Theta^{(I+\alpha^{(i)}_{ba},\beta^{(i)}_{ba})}_{M^{(i)}_{ba},a^{(i)}_{ba}}(\overline{z_{i}},\overline{\tau_{i}})\right)^{\ast}\left(\Theta^{(J+\alpha^{(i)}_{ba},\beta^{(i)}_{ba})}_{M^{(i)}_{ba},a^{(i)}_{ba}}(\overline{z_{i}},\overline{\tau_{i}})\right)$
$\displaystyle=\delta_{I,J}\quad(M^{(i)}_{ba}<0),$ (2.52)

with $d^{2}z_{i}:=dz_{i}d\overline{z_{i}}$. We note that the total number of
zero modes of $\psi^{(i)ab}_{+}$ and $\psi^{(i)ba}_{-}$ (when
$M^{(i)}_{ab}>0$) is given as $\Pi_{i=1}^{3}|M_{ab}^{(i)}|$, where we should
replace $|M_{ab}^{(i)}|$ by one when $M_{ab}^{(i)}=0$. In the opposite case
with $M^{(i)}_{ab}<0$, situations are similar. Here, we can derive an
important relationship easily (in the case of $M^{(i)}_{ab}>0$),

$\displaystyle\left(\Theta^{(I+\alpha^{(i)}_{ab},\beta^{(i)}_{ab})}_{M^{(i)}_{ab},a^{(i)}_{ab}}(z_{i},\tau_{i})\right)^{\ast}=\Theta^{(-I+\alpha^{(i)}_{ba},\beta^{(i)}_{ba})}_{M^{(i)}_{ba},\overline{a^{(i)}_{ba}}}(\overline{z_{i}},\overline{\tau_{i}}),$
(2.53)

where the index $I$ is identified under the condition, $\text{mod}\
|M^{(i)}_{ab}|$, and we can always redefine $-I$ as
$I^{\prime}(=0,\cdots,|M^{(i)}_{ab}|-1)$.

In the $U(N)$ SYM on the magnetized tori, we can realize the following gauge
symmetry breaking by the flux as a typical example, $U(8)\to{U(4)_{PSC}}\times
U(2)_{L}\times U(2)_{R}$, where $U(4)_{PSC}$ is the Pati-Salam color gauge
group, and $U(2)_{L}$ and $U(2)_{R}$ are the left- and right-electroweak gauge
groups, respectively. Here, we can find all the Minimal Supersymmetric
Standard Model (MSSM) fields in this case. Precisely speaking, the requirement
is that the SM gauge group $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ should
be intact under the existence of the flux. Thereby, other possibilities of
symmetry breaking via flux, e.g., $U(8)\to U(3)_{C}\times U(1)_{1}\times
U(2)_{L}\times U(2)_{R}$ or $U(8)\to U(3)_{C}\times U(1)_{1}\times
U(2)_{L}\times U(1)_{2}\times U(1)_{3}$ are also reasonable.

On flux background, zero-mode profiles are not only split, also localized
around points different from each other. Then, we can expect that hierarchical
values in Yukawa couplings are created via overlap integrals in the Yukawa
sector of this model. The concrete form of the Yukawa couplings are as
follows:

$\displaystyle Y_{\mathcal{I},\mathcal{J},\mathcal{K}}$
$\displaystyle=c\,\lambda^{(1)}_{I_{1},J_{1},K_{1}}\lambda^{(2)}_{I_{2},J_{2},K_{2}}\lambda^{(3)}_{I_{3},J_{3},K_{3}},$
(2.54) $\displaystyle\lambda^{(i)}_{I_{i},J_{i},K_{i}}$
$\displaystyle=\int_{T^{2}_{i}}d^{2}z_{i}\,\Theta^{(I_{i}+\alpha^{(i)}_{I},\beta^{(i)}_{I})}_{M^{(i)}_{I},a^{(i)}_{I}}(z_{i},\tau_{i})\Theta^{(J_{i}+\alpha^{(i)}_{J},\beta^{(i)}_{J})}_{M^{(i)}_{J},a^{(i)}_{J}}(z_{i},\tau_{i})\left(\Theta^{(K_{i}+\alpha^{(i)}_{K},\beta^{(i)}_{K})}_{M^{(i)}_{K},a^{(i)}_{K}}(z_{i},\tau_{i})\right)^{\ast},$
(2.55)

where $c$ is a constant factor via gauge structure and
$\mathcal{I}=(I_{1},I_{2},I_{3}),\,\mathcal{J}=(J_{1},J_{2},J_{3}),\,\mathcal{K}=(K_{1},K_{2},K_{3})$,
respectively. In a suitable symmetry breaking like the above examples, we can
find conditions on the parameters,

$\displaystyle M^{(i)}_{I}+M^{(i)}_{J}$ $\displaystyle=M^{(i)}_{K},$ (2.56)
$\displaystyle\alpha^{(i)}_{I}+\alpha^{(i)}_{J}$
$\displaystyle=\alpha^{(i)}_{K},$ (2.57)
$\displaystyle\beta^{(i)}_{I}+\beta^{(i)}_{J}$
$\displaystyle=\beta^{(i)}_{K},$ (2.58) $\displaystyle
M^{(i)}_{I}a^{(i)}_{I}+M^{(i)}_{J}a^{(i)}_{J}$
$\displaystyle=M^{(i)}_{K}a^{(i)}_{K},$ (2.59)

where we implicitly use the rule in Eq. (2.56) and the relation in Eq. (2.53)
when we write down the actual form in Eq. (2.55). After utilizing the last
property in Eq. (2.51) and the following formula about the theta function,

$\displaystyle\vartheta\left[\begin{array}[]{c}r/N_{1}\\\
0\end{array}\right](z_{1},N_{1}\tau)\times\vartheta\left[\begin{array}[]{c}s/N_{2}\\\
0\end{array}\right](z_{2},N_{2}\tau)\quad(r,s\in\mathbb{R};N_{1},N_{2}\in\mathbb{Z};z_{1},z_{2},\tau\in\mathbb{C})$
(2.64) $\displaystyle=\sum_{m\in
Z_{N_{1}+N_{2}}}\vartheta\left[\begin{array}[]{c}\frac{r+s+N_{1}m}{N_{1}+N_{2}}\\\
0\end{array}\right](z_{1}+z_{2},\tau(N_{1}+N_{2}))$ (2.67)
$\displaystyle\quad\times\vartheta\left[\begin{array}[]{c}\frac{N_{2}r-N_{1}s+N_{1}N_{2}m}{N_{1}N_{2}(N_{1}+N_{2})}\\\
0\end{array}\right](z_{1}N_{2}-z_{2}N_{1},\tau N_{1}N_{2}(N_{1}+N_{2})),$
(2.70)

with reminding the constraints in Eqs. (2.56)–(2.59), we can derive the
analytical result of the Yukawa coupling in Eq. (2.55) as

$\displaystyle\lambda^{(i)}_{I_{i},J_{i},K_{i}}$
$\displaystyle=\frac{\mathcal{N}_{M^{(i)}_{I}}\mathcal{N}_{M^{(i)}_{J}}}{\mathcal{N}_{M^{(i)}_{K}}}e^{\frac{i\pi}{\text{Im}\tau_{i}}\left[a^{(i)}_{I}\text{Im}\left(M^{(i)}_{I}a^{(i)}_{I}\right)+a^{(i)}_{J}\text{Im}\left(M^{(i)}_{J}a^{(i)}_{J}\right)-a^{(i)}_{K}\text{Im}\left(M^{(i)}_{K}a^{(i)}_{K}\right)\right]}$
$\displaystyle\quad\times\sum_{m\in
Z_{M^{(i)}_{K}}}\vartheta\left[\begin{array}[]{c}\frac{M^{(i)}_{J}\left(I_{i}+\alpha^{(i)}_{I}\right)-M^{(i)}_{I}\left(J_{i}+\alpha^{(i)}_{J}\right)+mM^{(i)}_{I}M^{(i)}_{J}}{M^{(i)}_{I}M^{(i)}_{J}M^{(i)}_{K}}\\\
0\end{array}\right]{(X,Y)}$ (2.73)
$\displaystyle\quad\times\delta_{I_{i}+\alpha^{(i)}_{I}+J_{i}+\alpha^{(i)}_{J}+mM^{(i)}_{I},\,K_{i}+\alpha^{(i)}_{K}+\ell
M^{(i)}_{K}},$ (2.74)

with
$X:=M^{(i)}_{I}\beta^{(i)}_{J}-M^{(i)}_{J}\beta^{(i)}_{I}+M^{(i)}_{I}M^{(i)}_{J}\left(a^{(i)}_{I}-a^{(i)}_{J}\right)$,
$Y:=\tau_{i}M^{(i)}_{I}M^{(i)}_{J}M^{(i)}_{K}$ and possible choices of
integers $\ell$.444 In another word, we consider the Kronecker’s delta with
the condition “$\text{mod}\ |M^{(i)}_{K}|$”.

At the end of this subsection, we mention an efficient technique discussed in
[52, 51]. In the Abelian magnetic flux, we can set either the (complex) Wilson
line phase $C^{(i)}_{j}$, or the Scherk-Schwarz phases $\alpha^{(i)}_{j}$ and
$\beta^{(i)}_{j}$ as zero (in each $j$ individually) without loss of
generality. This fact could make the following analysis simplified.

###  2.2 $U(N)$ gauge theory on orbifolded $(T^{2})^{3}$

In this subsection, we examine the $U(N)$ SYM theory on magnetized
$(T^{2})^{3}$ with orbifolding. In our configuration where the six extra
dimensions are factorized as three 2-tori, apparently, we mainly focus on one
of the tori $T^{2}_{i}$ for a general discussion.

On $T^{2}_{i}$, possible twisted orbifolding is to impose the covariance on
the fields under the rotation with the angle $\omega$, $z_{i}\to\omega z_{i}$,
where $\omega$ is $e^{2\pi i/N}$ with $N=2,3,4,6$. In other words, $Z_{2}$,
$Z_{3}$, $Z_{4}$ and $Z_{6}$ (twisted) orbifoldings are realizable on
$T^{2}_{i}$. In non-Abelian gauge theories, a nontrivial gauge structure part
$P$ appears in the $Z_{N}$ manipulation as,

$\displaystyle A_{\mu}{(x,\\{\omega
z_{i},\overline{\omega}\overline{z_{i}}\\})}$
$\displaystyle=PA_{\mu}(x,\\{z_{i},\overline{z_{i}}\\})P^{-1},$ (2.75)
$\displaystyle A_{z_{i}}{(x,\\{\omega
z_{i},\overline{\omega}\overline{z_{i}}\\})}$
$\displaystyle=\overline{\omega}PA_{z_{i}}(x,\\{z_{i},\overline{z_{i}}\\})P^{-1},$
(2.76) $\displaystyle A_{\overline{z_{i}}}{(x,\\{\omega
z_{i},\overline{\omega}\overline{z_{i}}\\})}$ $\displaystyle=\omega
PA_{\overline{z_{i}}}(x,\\{z_{i},\overline{z_{i}}\\})P^{-1},$ (2.77)
$\displaystyle\lambda_{s_{i}=+}{(x,\\{\omega
z_{i},\overline{\omega}\overline{z_{i}}\\})}$
$\displaystyle=P\lambda_{s_{i}=+}(x,\\{z_{i},\overline{z_{i}}\\})P^{-1},$
(2.78) $\displaystyle\lambda_{s_{i}=-}{(x,\\{\omega
z_{i},\overline{\omega}\overline{z_{i}}\\})}$ $\displaystyle=\omega
P\lambda_{s_{i}=-}(x,\\{z_{i},\overline{z_{i}}\\})P^{-1},$ (2.79)

where $P$ should satisfy the conditions, $P\in U(N)$ and
$P^{N}={\mathbf{1}_{N\times N}}$. Here, to prevent an additional explicit
gauge symmetry breaking via the orbifoldings, we should take the following
form in $P$,

$\displaystyle P=\begin{pmatrix}\eta_{1}{\mathbf{1}_{N_{1}\times N_{1}}}&&0\\\
&\ddots&\\\ 0&&\eta_{n}{\mathbf{1}_{N_{n}\times N_{n}}}\end{pmatrix},$ (2.80)

with $\eta_{j}=\\{1,\omega,\cdots,\omega^{N-1}\\}\ (j=1,\cdots,n)$. Within the
concrete example of $U(N)\to U(N_{a})\times U(N_{b})$ discussed in the
previous subsection, $\psi^{(i)aa}_{+}$ and $\psi^{(i)bb}_{+}$ have trivial
$Z_{N}$ parity irrespective of the values of $\eta_{a}$ and $\eta_{b}$, while
$\psi^{(i)ab}_{+}$ and $\psi^{(i)ba}_{+}$ can contain nontrivial values of
$\eta_{a}\overline{\eta_{b}}$, $\overline{\eta_{a}}\eta_{b}$, respectively.
The conditions for the 2D gauginos with negative chirality are evaluated with
ease by use of the relation in Eq. (2.79).

Now, the gauge structure under the orbifoldings is declared, and thus we can
concentrate on an actual case with the $Z_{N}$ parity $\eta^{(i)}$, the (2D)
positive chirality and the state-discriminating index $I$ on $T^{2}_{i}$ as,

$\displaystyle\psi^{(i)}_{+,\eta^{(i)}}(z_{i})=\sum_{I=0}^{|M^{(i)}|-1}\psi^{(i),I}_{+,\eta^{(i)}}(z_{i}),\quad\psi^{(i),I}_{+,\eta^{(i)}}(z_{i})=\begin{pmatrix}\widetilde{\Theta}^{(I_{i}+\alpha^{(i)},\beta^{(i)})}_{M^{(i)},a^{(i)};\eta^{(i)}}(z_{i},\tau_{i})\\\
0\end{pmatrix},$ (2.81)

where we assume that $M^{(i)}$ is a positive integer. Note that the
correspondence to the negative chirality case is basically straightforward by
the replacements, $z_{i}\to\overline{z_{i}}$, $\tau_{i}\to\overline{\tau_{i}}$
and $a^{(i)}\to\overline{a^{(i)}}$.

Constructing the concrete form of
$\widetilde{\Theta}^{(I_{i}+\alpha^{(i)},\beta^{(i)})}_{M^{(i)},a^{(i)};\eta^{(i)}}(z_{i},\tau_{i})$
itself can be done straightforwardly just following the general recipe as

$\displaystyle\widetilde{\Theta}^{(I_{i}+\alpha^{(i)},\beta^{(i)})}_{M^{(i)},a^{(i)};\eta^{(i)}}(z_{i},\tau_{i})=\frac{1}{N}\sum_{x=0}^{N-1}\left(\overline{\eta^{(i)}}\right)^{x}\Theta^{(I_{i}+\alpha^{(i)},\beta^{(i)})}_{M^{(i)},a^{(i)}}(\omega^{x}z_{i},\tau_{i}),$
(2.82)

where, apparently, this state has the eigenvalue $\eta^{(i)}$ under
$z_{i}\to\omega z_{i}$. Since the function $\Theta(z_{i})$ forms a complete
set, then in principle, we can express the $Z_{N}$-rotated one as a linear
combination of $\Theta(z_{i})$. If we derive the coefficients of such a linear
combination analytically, we can construct the $T^{2}_{i}/Z_{N}$ mode
functions in an exact manner.555 Within the direct calculations with theta
functions, it is very difficult to fix analytical forms of the coefficients.
Fortunately, the method developed in Ref. [51] based on operator formalism
resolves the technical difficulty. By considering a “dual” two-dimensional
quantum mechanical system, we can estimate all the coefficients in an
analytical way.  Here, following the discussions in [51], we take the
following basis,

$\displaystyle a^{(i)}=0,$ (2.83)

for simplicity. Within taking the Abelian magnetic flux, we can always select
this choice without any loss of generality [52, 51].

By use of the results in [51], the $Z_{N}$-transformed states (in the basis
(2.83)) can be expressed as

$\displaystyle\Theta^{(I_{i}+\alpha^{(i)},\beta^{(i)})}_{M^{(i)},0}(\omega^{x}z_{i},\tau_{i})=\sum_{J_{i}=0}^{|M^{(i)}|-1}C_{I_{i}J_{i}}^{(\omega^{x})}\Theta^{(J_{i}+\alpha^{(i)},\beta^{(i)})}_{M^{(i)},0}(z_{i},\tau_{i}),$
(2.84)

with the expansion coefficients

$\displaystyle
C_{I_{i}J_{i}}^{(\omega^{x})}=\int_{T^{2}_{i}}d^{2}z_{i}\left(\Theta^{(J_{i}+\alpha^{(i)},\beta^{(i)})}_{M^{(i)},0}(z_{i},\tau_{i})\right)^{\ast}\Theta^{(I_{i}+\alpha^{(i)},\beta^{(i)})}_{M^{(i)},0}(\omega^{x}z_{i},\tau_{i}),$
(2.85)

where explicit forms of $C_{I_{i}J_{i}}^{(\omega^{x})}$ are summarized in
Appendix A. Using this information, the relation in (2.82) is written,

$\displaystyle\widetilde{\Theta}^{(I_{i}+\alpha^{(i)},\beta^{(i)})}_{M^{(i)},0;\eta^{(i)}}(z_{i},\tau_{i})$
$\displaystyle=\sum_{J_{i}=0}^{|M^{(i)}|-1}\left(\frac{1}{N}\sum_{x=0}^{N-1}\left(\overline{\eta^{(i)}}\right)^{x}C_{I_{i}J_{i}}^{(\omega^{x})}\right)\Theta^{(J_{i}+\alpha^{(i)},\beta^{(i)})}_{M^{(i)},0}(z_{i},\tau_{i})$
$\displaystyle=:\sum_{J_{i}=0}^{|M^{(i)}|-1}M_{I_{i}J_{i}}^{(Z_{N};\eta^{(i)})}\Theta^{(J_{i}+\alpha^{(i)},\beta^{(i)})}_{M^{(i)},0}(z_{i},\tau_{i}),$
(2.86)

where the number of independent physical states can be calculated as

$\displaystyle(\text{$\\#$ of physical
states})=\text{Rank}\left[M_{I_{i}J_{i}}^{(Z_{N};\eta^{(i)})}\right].$ (2.87)

In general after orbifolding, the number of independent physical states should
be reduced as
$\text{Rank}\left[M_{I_{i}J_{i}}^{(Z_{N};\eta^{(i)})}\right]<|M^{(i)}|$, but
exact numbers are only available after concrete calculation. This procedure is
possible since operator formalism tells us the analytical forms of
$C_{I_{i}J_{i}}^{(\omega^{x})}$ [51]. The detailed information on the numbers
of the survived physical states and the forms of
$C_{I_{i}J_{i}}^{(\omega^{x})}$ are summarized in Appendix A.

Also, operator formalism helps us when we calculate the kinetic terms on
$T^{2}_{i}/Z_{N}$ described as

$\displaystyle\mathcal{K}_{I_{i}J_{i}}^{(Z_{N};\eta^{(i)})}=\int_{T^{2}_{i}}d^{2}z_{i}\left(\widetilde{\Theta}^{(I_{i}+\alpha^{(i)},\beta^{(i)})}_{M^{(i)},0;\eta^{(i)}}(z_{i},\tau_{i})\right)^{\ast}\widetilde{\Theta}^{(J_{i}+\alpha^{(i)},\beta^{(i)})}_{M^{(i)},0;\eta^{(i)}}(z_{i},\tau_{i}).$
(2.88)

After some calculations in operators and states, we can reach the simple
correspondence,

$\displaystyle\mathcal{K}_{I_{i}J_{i}}^{(Z_{N};\eta^{(i)})}=M_{J_{i}I_{i}}^{(\eta^{(i)})}.$
(2.89)

Here, we should mention that the kinetic terms
$\mathcal{K}_{I_{i}J_{i}}^{(Z_{N};\eta^{(i)})}$ are in general non-diagonal,
and thereby we should diagonalize them to know the correct forms with suitable
normalization. This process itself is, at least in principle, doable for any
larger $|M^{(i)}|$ through the Gram-Schmidt process for orthonormalization in
linear algebra. After the unitary transformation with the corresponding
diagonalizing matrix $U^{(Z_{N};\eta^{(i)})}$,
$\mathcal{K}_{I_{i}J_{i}}^{(Z_{N};\eta^{(i)})}$ should be performed as

$\displaystyle\mathcal{K}^{(Z_{N};\eta^{(i)})}\to\left(U^{(Z_{N};\eta^{(i)})}\right)^{\dagger}\mathcal{K}^{(Z_{N};\eta^{(i)})}U^{(Z_{N};\eta^{(i)})}=\text{diag}(\underbrace{1,\cdots,1}_{{\text{Rank}\left[\mathcal{K}^{(Z_{N};\eta^{(i)})}\right]}},0,\cdots,0).$
(2.90)

The row index of the $|M^{(i)}|$-by-$|M^{(i)}|$ diagonalizing matrix
$U^{(Z_{N};\eta^{(i)})}$ means a discriminator for the states on the original
geometry $T^{2}_{i}$, whereas the column index identifies the physical
eigenstates, where this index is meaningful only up to
$\text{Rank}\left[\mathcal{K}^{(Z_{N};\eta^{(i)})}\right]-1$ from zero.
Therefore in the physical eigenstates after considering the correct
normalization in the kinetic terms, $T^{2}_{i}/Z_{N}$ mode function should be

$\displaystyle\widetilde{\Theta}^{(I_{i}+\alpha^{(i)},\beta^{(i)})}_{M^{(i)},0;\eta^{(i)}}(z_{i},\tau_{i})\to\sum_{I_{i}=0}^{|M^{(i)}|-1}\widetilde{\Theta}^{(I_{i}+\alpha^{(i)},\beta^{(i)})}_{M^{(i)},0;\eta^{(i)}}(z_{i},\tau_{i})\left(U^{(Z_{N};\eta^{(i)})}\right)_{I_{i}I^{\prime}_{i}},$
(2.91)

where $I^{\prime}_{i}$ is the index of physical eigenstates.

Based on the knowledge, we briefly discuss the way of calculating Yukawa
couplings under the $Z_{N}$ symmetry. The $T^{2}_{i}$ part is formulated (with
the basis fixing in Eq. (2.83)) in the “ $T^{2}_{i}$ ” eigenbasis as

$\displaystyle\widetilde{\lambda}^{(i)}_{I_{i},J_{i},K_{i}}$
$\displaystyle=\int_{T^{2}_{i}}d^{2}z_{i}\,\widetilde{\Theta}^{(I_{i}+\alpha^{(i)}_{I},\beta^{(i)}_{I})}_{M^{(i)}_{I},0;\eta_{I}^{(i)}}(z_{i},\tau_{i})\widetilde{\Theta}^{(J_{i}+\alpha^{(i)}_{J},\beta^{(i)}_{J})}_{M^{(i)}_{J},0;\eta_{J}^{(i)}}(z_{i},\tau_{i})\left(\widetilde{\Theta}^{(K_{i}+\alpha^{(i)}_{K},\beta^{(i)}_{K})}_{M^{(i)}_{K},0;\eta_{K}^{(i)}}(z_{i},\tau_{i})\right)^{\ast},$
(2.92)

where we find the condition on the $Z_{N}$ parities (via the invariance of the
system),

$\displaystyle\eta_{I}^{(i)}\eta_{J}^{(i)}\overline{\eta_{K}^{(i)}}=1.$ (2.93)

Note that a complex-conjugated state holds the corresponding complex-
conjugated $Z_{N}$ parity, which is clearly understandable from Eq. (2.82).
Now, evaluating $\widetilde{\lambda}^{(i)}_{I_{i},J_{i},K_{i}}$ turns out to
be straightforward with the help of Eqs. (2.86), (2.89), (2.90) and (2.91).

##  3 Possibilities of three generations in $U(N)$ models

In this section, we classify all the reasonable possibilities of three-
generation models under the $Z_{2}$, $Z_{3}$, $Z_{4}$ and $Z_{6}$
orbifoldings. When we start from 10D $U(N)$ SYM theory, the basic pattern of
the gauge symmetry breaking under the magnetic flux is $U(N)\to U(N_{a})\times
U(N_{b})\times U(N_{c})$ with $N=N_{a}+N_{b}+N_{c}$, where the corresponding
1-form potential is

$\displaystyle
A^{(b)}(\\{z_{i},\overline{z_{i}}\\})=\sum_{i=1}^{3}\frac{\pi}{q\text{Im}\tau_{i}}\times$
$\displaystyle\quad\text{diag}\left(M_{a}^{(i)}\text{Im}\left[(\overline{z_{i}}+\overline{C^{(i)}_{a}})dz_{i}\right]\mathbf{1}_{N_{a}\times
N_{a}},M_{b}^{(i)}\text{Im}\left[(\overline{z_{i}}+\overline{C^{(i)}_{b}})dz_{i}\right]\mathbf{1}_{N_{b}\times
N_{b}},M_{c}^{(i)}\text{Im}\left[(\overline{z_{i}}+\overline{C^{(i)}_{c}})dz_{i}\right]\mathbf{1}_{N_{c}\times
N_{c}}\right).$ (3.1)

Following the notation in the previous example of $U(N)\to U(N_{a})\times
U(N_{b})$, we find six types of bi-fundamental matter fields,
$\lambda^{ab},\lambda^{bc},\lambda^{ca},\lambda^{ba},\lambda^{cb},\lambda^{ac}$,
whose gauge properties are $(N_{a},\overline{N_{b}})$,
$(N_{b},\overline{N_{c}})$, $(\overline{N_{a}},N_{c})$,
$(\overline{N_{a}},N_{b})$, $(\overline{N_{b}},N_{c})$,
$(N_{a},\overline{N_{c}})$, respectively. When we adopt the choice, $N_{a}=4$,
$N_{b}=2$, $N_{c}=2$, $U(4)_{PSC}\times U(2)_{L}\times U(2)_{R}$ gauge groups
are realized from the $U(8)$ group up to $U(1)$ factors, where the subscripts
$PSC$, $L$ and $R$ denote the Pati-Salam color, left- and right-electroweak
gauge groups, respectively.666 Some of the combinations of the $U(1)$ part
would be anomalous. Then they could be massive and decoupled via the Green-
Schwarz mechanism.  In such a situation, when the actual chirality of the
gaugino is left (negative), $\lambda^{ab}$ corresponds to the left-handed
quarks and leptons, and $\lambda^{ca}$ accords with (charge-conjugated) right-
handed quarks and leptons, respectively. When the magnetic fluxes are suitably
assigned, the situation with three-generations is materialized. Besides,
$\lambda^{bc}$ plays as up-type and down-type Higgsinos. After we assume that
(4D $\mathcal{N}=1$) supersymmetry is preserved at least locally at the $ab$,
$bc$ and $ca$ sectors, the corresponding Higgses via extra-dimensional
components of the 10D vector fields are still massless under the fluxes and
the number of the fields are the same with Higgsino fields. Also, no tachyonic
mode is expected at the tree level. Here, in general, multiple Higgs fields
appear from the $bc$ sector. Interestingly, when $\lambda^{ab}$,
$\lambda^{bc}$ and $\lambda^{ca}$ have zero modes, $\lambda^{ba}$,
$\lambda^{cb}$ and $\lambda^{ac}$ cannot contain any zero mode and thus no
exotic particle arises from these fermionic sectors. In the case of the actual
chirality being right (positive), we should flip the roles of the two
categories.

As a phenomenological point of view, we can consider the following additional
breakdowns originating from flux, $U(4)_{PSC}\to U(3)_{C}\times U(1)_{1}$ and
$U(2)_{R}\to U(1)_{2}\times U(1)_{3}$ (up to U(1) factors), where $U(3)_{C}$
is the color gauge group (up to a $U(1)$ factor). Under the latter breaking,
the up-type and down-type Higgsino/Higgs sectors can feel different magnetic
fluxes individually. Consequently, the numbers of the two types of the fields
become not the same.

###  3.1 Strategy in classification

In the $U(N)$ model, the effective Yukawa couplings after the integration
among the six extra dimensions are symbolically written as

$\displaystyle
Y_{ijk}={c}\int_{(T^{2})^{3}}d^{6}z\psi_{L_{i}}(z)\psi_{R_{j}}(z)\phi_{H_{k}}(z),$
(3.2)

where $c$ is a constant factor. $\psi_{L_{i}}(z)$, $\psi_{R_{j}}(z)$ and
$\phi_{H_{k}}(z)$ denote zero mode wave functions of the left-, right-handed
matter fields and the Higgs fields, and their generations are indicated by the
indices $i$, $j$ and $k$, respectively. When the left- and right-handed
matters are localized among different tori, the Yukawa couplings are
factorized like

$\displaystyle Y_{ijk}=a_{ik}b_{jk}.$ (3.3)

The rank of this type of matrices (about $i$ and $j$) is one and no room would
be there for generating realistic flavor structure. We can reach the same
conclusion about the Higgs fields. Thereby, realizing the Yukawa matrix
holding rank three requires the situation that all the three types of the
fields are localized on the same torus like

$\displaystyle Y_{ijk}=a^{(1)}_{ijk}a^{(2)}a^{(3)}.$ (3.4)

In this case, the contributions from $a^{(2)}$ and $a^{(3)}$ only affect the
overall normalization, and the structure of the Yukawa matrix is governed only
by $a^{(1)}$. Thus, we can mainly focus on the torus determining the matter
generations.

Based on the knowledge which we have obtained through the discussions before,
we can easily understand the constraints on the parameters describing each
sector in the $U(N)$ theory,

$\displaystyle M^{(i)}_{ab}+M^{(i)}_{bc}+M^{(i)}_{ca}$ $\displaystyle=0,$
$\displaystyle\alpha^{(i)}_{ab}+\alpha^{(i)}_{bc}+\alpha^{(i)}_{ca}$
$\displaystyle=0,$
$\displaystyle\beta^{(i)}_{ab}+\beta^{(i)}_{bc}+\beta^{(i)}_{ca}$
$\displaystyle=0,$ $\displaystyle
M^{(i)}_{ab}a^{(i)}_{ab}+M^{(i)}_{bc}a^{(i)}_{bc}+M^{(i)}_{ca}a^{(i)}_{ca}$
$\displaystyle=0,$
$\displaystyle\eta^{(i)}_{ab}\eta^{(i)}_{bc}\eta^{(i)}_{ca}$
$\displaystyle=1,$ (3.5)

where the above parameters are defined by the fundamental ones like
$M_{ab}^{(i)}=M^{(i)}_{a}-M^{(i)}_{b}$ except the $Z_{N}$ parities. The
$Z_{N}$ parities are described as

$\displaystyle\eta_{ab}^{(i)}=\eta_{a}^{(i)}\overline{\eta_{b}^{(i)}},\quad\eta_{bc}^{(i)}=\eta_{b}^{(i)}\overline{\eta^{(i)}_{c}},\quad\eta_{ca}^{(i)}=\eta_{c}^{(i)}\overline{\eta_{a}^{(i)}}.$
(3.6)

When we remember the property under complex conjugation in Eq. (2.53), we can
reach the form in Eq. (2.55). As discussed in [52, 51], it is convenient to
choose a specific basis where either the Wilson line phases or the Scherk-
Schwarz phases are zero. In the case of Abelian magnetic flux, this is
possible without loss of generality. The way of the correspondence in the
parameters in between the two bases where the (complex) Wilson line phases are
zero $\\{\alpha,\beta,Ma=0\\}$ and the Scherk-Schwarz phases are zero
$\\{\tilde{\alpha}=\tilde{\beta}=0,M\tilde{a}\\}$ is as follows with the torus
modulus parameter $\tau$ [52]:

$\displaystyle M\tilde{a}=\alpha\tau-\beta.$ (3.7)

In the following analysis, like the discussions in the previous section, we
adopt the basis where the Scherk-Schwarz phases are nonzero, and
correspondingly the Wilson line phases are zero. The interpretation into the
basis with nonzero Wilson line phases and Scherk-Schwarz phases being zero is
straightforward, where we just obey the relation in Eq. (3.7).

In the following part, we search for all the phenomenological possibilities
with three generations in the quarks and the leptons. Here, we adopt the
strategy that we scan over all the combinations of the effective parameters
describing the $ab$, $bc$ and $ca$ sectors under the constraints in Eq. (3.5).
Note that the $ba$, $cb$ and $ac$ sectors are completely fixed after we
determine the configurations of the above three parts. As we discussed before,
the number of generations is exactly calculable with the help of operator
formalism and actual numbers are summarized in Appendix A. We use this
information for the selection cut in realized matter generations. Since our
attention is on the Yukawa structure schematically depicted in Eq. (3.4), it
is enough to focus on a 2-torus of the geometry. Thus hereafter in this
section, we drop the index $i$ for identifying the tori.

Before going to scanning, we focus on how many configurations we should
consider to exhaust possibilities under the requirements in Eq. (3.5).
Apparently from (3.5), the information on the $bc$ sector is totally
determined after we set the parameters of the other two sectors. Here, we look
at the condition on the magnetic fluxes. The first line of Eq. (3.5) tells us
that at least one of the signs of the three fluxes should be different from
the others and it is enough that we focus on the two possibilities in the
signs of the fluxes of the matter sectors, $M_{ab}$ and $M_{ca}$ as,777 There
is another possibility of $M_{ab}>0,\ M_{bc}<0$, but this case is physically
the same as $M_{ab}<0,\ M_{bc}>0$.

$\displaystyle M_{ab}<0,\ {M_{ca}}<0;\quad M_{ab}<0,\ {M_{ca}}>0.$ (3.8)

Besides, after ignoring the difference coming from the combinatorics, we can
introduce the additional condition,

$\displaystyle{|M_{ab}|\leq|{M_{ca}}|}.$ (3.9)

Consequently, we only take into account of the magnetic fluxes being possible
under the conditions in (3.8) and (3.9).

We also comment on the other parameters. After the orbifolding, the Wilson
line phases and the Scherk-Schwarz phases no longer take arbitrary values [52,
51]. At this moment in time, combinatorics degrees of freedom are fixed by our
preconditions for analysis in Eqs. (3.8) and (3.9), and then we should scan
over all the possibilities in the other parameters. The possible values of the
Wilson line phases and the Scherk-Schwarz phases on $T^{2}/Z_{N}$ in the
specific basis are depicted in Appendix A.

###  3.2 Results

Now, we are ready to evaluate the numbers of allowed possibilities in
effective parameter configurations of the $Z_{2}$, $Z_{3}$, $Z_{4}$ and
$Z_{6}$ cases. Summaries of our analysis are found in Tables 1 ($Z_{2}$), 2
($Z_{3}$), 3 ($Z_{4}$) and 4 ($Z_{6}$), where “General cases” and “Trivial
BC’s only” means the cases with and without nontrivial Scherk-Schwarz phases,
respectively. Here, we only show the numbers of allowed parameter
configurations in $M_{ab},M_{ca}<0$ and $M_{ab}<0,\,M_{ca}>0$ separately.
Corresponding numbers of the Higgs pairs ($N_{H}$) are also shown. If an
allowed configuration says $M_{bc}=0$, where one non-localized Higgs pair
appears without magnetic background and this situation would not be
interesting in the phenomenological point of view, we discriminate this case
as “$1_{\text{trivial}}$” from the one Higgs cases with magnetic flux “$1$”.

We mention about the $Z_{2}$ result. The “Trivial BC’s only” case in $Z_{2}$
was already analyzed in Ref. [64] and our result is totally consistent with
them. The complete information on $Z_{2,3,4,6}$ configurations is summarized
in Appendix B. Brief comments are also given here. After allowing nonzero
Wilson line phases, the configurations with seven Higgs pairs arise in the
$Z_{2}$ case as shown in Table 1. In the cases of $Z_{3}$, $Z_{4}$ and $Z_{6}$
summarized in Tables 2, 3 and 4, we find the situations where two or four
Higgs pairs are realized, while those with nine pairs never occur.

General cases | Trivial BC’s only
---|---
$M_{ab},M_{ca}<0$ | $M_{ab}<0,\,M_{ca}>0$ | $M_{ab},M_{ca}<0$ | $M_{ab}<0,\,M_{ca}>0$
$41\,(N_{H}=5)$ | $16\,(N_{H}=1_{\text{trivial}})$ | $5\,(N_{H}=5)$ | $4\,(N_{H}=1_{\text{trivial}})$
$56\,(N_{H}=6)$ | $65\,(N_{H}=1)$ | $2\,(N_{H}=6)$ | $5\,(N_{H}=1)$
$30\,(N_{H}=7)$ |  |  |
$8\,(N_{H}=8)$ |  | $2\,(N_{H}=8)$ |
$1\,(N_{H}=9)$ |  | $1\,(N_{H}=9)$ |
$136+81=217$ in total |  $10+9=19$ in total
Table 1: Result in $Z_{2}$ case. “General cases” and “Trivial BC’s only” means the cases with and without nontrivial Scherk-Schwarz phases, respectively. Corresponding numbers of the Higgs pairs ($N_{H}$) are also shown. The case indicated by $1_{\text{trivial}}$ means the one Higgs pair appears under non-magnetized background in $bc$ sector.  General cases | Trivial BC’s only
---|---
$M_{ab},M_{ca}<0$ | $M_{ab}<0,\,M_{ca}>0$ | $M_{ab},M_{ca}<0$ | $M_{ab}<0,\,M_{ca}>0$
$11\,(N_{H}=4)$ | $17\,(N_{H}=1_{\text{trivial}})$ | $1\,(N_{H}=4)$ | $9\,(N_{H}=1_{\text{trivial}})$
$83\,(N_{H}=5)$ | $142\,(N_{H}=1)$ | $6\,(N_{H}=5)$ | $27\,(N_{H}=1)$
$190\,(N_{H}=6)$ | $21\,(N_{H}=2)$ | $7\,(N_{H}=6)$ |
$83\,(N_{H}=7)$ |  | $6\,(N_{H}=7)$ |
$11\,(N_{H}=8)$ |  | $1\,(N_{H}=8)$ |
$378+180=558$ in total |  $21+36=57$ in total
Table 2: Result in $Z_{3}$ case. The convention is the same with in Table 1. General cases | Trivial BC’s only
---|---
$M_{ab},M_{ca}<0$ | $M_{ab}<0,\,M_{ca}>0$ | $M_{ab},M_{ca}<0$ | $M_{ab}<0,\,M_{ca}>0$
$9\,(N_{H}=4)$ | $24\,(N_{H}=1_{\text{trivial}})$ | $3\,(N_{H}=4)$ | $12\,(N_{H}=1_{\text{trivial}})$
$128\,(N_{H}=5)$ | $228\,(N_{H}=1)$ | $37\,(N_{H}=5)$ | $60\,(N_{H}=1)$
$254\,(N_{H}=6)$ | $18\,(N_{H}=2)$ | $59\,(N_{H}=6)$ | $6\,(N_{H}=2)$
$120\,(N_{H}=7)$ |  | $27\,(N_{H}=7)$ |
$17\,(N_{H}=8)$ |  | $10\,(N_{H}=8)$ |
$528+270=798$ in total |  $136+78=214$ in total
Table 3: Result in $Z_{4}$ case. The convention is the same with in Table 1. General cases | Trivial BC’s only
---|---
$M_{ab},M_{ca}<0$ | $M_{ab}<0,\,M_{ca}>0$ | $M_{ab},M_{ca}<0$ | $M_{ab}<0,\,M_{ca}>0$
$14\,(N_{H}=4)$ | $24\,(N_{H}=1_{\text{trivial}})$ | $4\,(N_{H}=4)$ | $12\,(N_{H}=1_{\text{trivial}})$
$156\,(N_{H}=5)$ | $282\,(N_{H}=1)$ | $45\,(N_{H}=5)$ | $73\,(N_{H}=1)$
$326\,(N_{H}=6)$ | $27\,(N_{H}=2)$ | $76\,(N_{H}=6)$ | $8\,(N_{H}=2)$
$150\,(N_{H}=7)$ |  | $36\,(N_{H}=7)$ |
$20\,(N_{H}=8)$ |  | $10\,(N_{H}=8)$ |
$666+333=999$ in total |  $171+93=264$ in total
Table 4: Result in $Z_{6}$ case. The convention is the same with in Table 1.

##  4 Examples in $Z_{2}$ with nontrivial twisting

Based on the results which we have achieved in the previous section, we can
examine how much nontrivial boundary conditions by the Scherk-Schwarz phases
(or the Wilson line phases) affect the flavor structure in some specific
models with the $Z_{2}$ orbifolding. Here, we adopt the basis with the Wilson
line phases being zero and only debate with the torus where all the matter
fields are assumed to be localized. As we already discussed, the above
assumption is justified by the phenomenological requirement for flavors. Here,
we rewrite the expressions of the Yukawa couplings $\lambda_{I,J,K}$ on
$T^{2}$, and $\widetilde{\lambda}_{I,J,K}$ on $T^{2}/Z_{N}$, in Eqs. (2.55)
and (2.92), respectively,

$\displaystyle\lambda_{I,J,K}$
$\displaystyle=\int_{T^{2}}d^{2}z\,\Theta^{(I+\alpha_{I},\beta_{I})}_{M_{I},0}(z,\tau)\Theta^{(J+\alpha_{J},\beta_{J})}_{M_{J},0}(z,\tau)\left(\Theta^{(K+\alpha_{K},\beta_{K})}_{M_{K},0}(z,\tau)\right)^{\ast},$
(4.1) $\displaystyle\widetilde{\lambda}_{I,J,K}$
$\displaystyle=\int_{T^{2}}d^{2}z\,\widetilde{\Theta}^{(I+\alpha_{I},\beta_{I})}_{M_{I},0;\eta_{I}}(z,\tau)\widetilde{\Theta}^{(J+\alpha_{J},\beta_{J})}_{M_{J},0;\eta_{J}}(z,\tau)\left(\widetilde{\Theta}^{(K+\alpha_{K},\beta_{K})}_{M_{K},0;\eta_{K}}(z,\tau)\right)^{\ast},$
(4.2)

with dropping the torus index $i$ for convenience. From Eq. (2.74), the
analytical form of the Yukawa on $T^{2}$ without nonzero Wilson line phases is
represented as

$\displaystyle\lambda_{I,J,K}$
$\displaystyle=\frac{\mathcal{N}_{M_{I}}\mathcal{N}_{M_{J}}}{\mathcal{N}_{M_{K}}}\sum_{m\in
Z_{M_{K}}}\vartheta\left[\begin{array}[]{c}\frac{M_{J}\left(I+\alpha_{I}\right)-M_{I}\left(J+\alpha_{J}\right)+mM_{I}M_{J}}{M_{I}M_{J}M_{K}}\\\
0\end{array}\right](M_{I}\beta_{J}-M_{J}\beta_{I},\tau M_{I}M_{J}M_{K})$ (4.5)
$\displaystyle\quad\times\delta_{I+\alpha_{I}+J+\alpha_{J}+mM_{I},\,K+\alpha_{K}+\ell
M_{K}}.$ (4.6)

In the “kinetic” eigenbasis, the Yukawa couplings are expressed as

$\displaystyle\widetilde{\lambda}^{\prime}_{I^{\prime},J^{\prime},K^{\prime}}=\sum_{I=0}^{|M_{I}|-1}\sum_{J=0}^{|M_{J}|-1}\sum_{K=0}^{|M_{K}|-1}\widetilde{\lambda}_{I,J,K}\left(U^{Z_{2};\eta_{I}}\right)_{I,I^{\prime}}\left(U^{Z_{2};{\eta_{J}}}\right)_{J,J^{\prime}}\left({U^{Z_{2};{\eta_{K}}}}\right)_{K,K^{\prime}}^{\ast}$
(4.7)

where the indices for identifying kinetic eigenstates, $I^{\prime}$,
$J^{\prime}$, $K^{\prime}$, have
$\text{Rank}\left[M^{(Z_{2};\eta_{I})}\right]$,
$\text{Rank}\left[M^{(Z_{2};\eta_{J})}\right]$,
$\text{Rank}\left[M^{(Z_{2};\eta_{K})}\right]$ numbers of nonzero
configurations, respectively. In general, the mixing effect through
$U^{Z_{N};\eta_{I}}$ contributes to the physics. However in the $Z_{2}$ case,
the kinetic terms are diagonal in the $Z_{2}$-orbifolded basis (irrespective
of values of the Scherk-Schwarz phases) and we can set the diagonalizing
matrix when we calculate Yukawa couplings as

$\displaystyle\left(U^{Z_{N};\eta_{I}}\right)_{J,J^{\prime}}\to\delta_{J,J^{\prime}}.$
(4.8)

The reason behind this simplicity is that the structure of orbifolded
functions is uninvolved, where we estimate the number of survived physical
modes analytically and explicitly as in Appendix A.888 When $T^{2}/Z_{3,4,6}$,
we can also analytically evaluate the numbers in operator analysis, but an
explicit formula is not obtained yet.  In the $Z_{3,4,6}$ orbifoldings,
nontrivial effects via $U^{Z_{N};\eta_{I}}$ might be expected.

###  4.1 Numerical analysis in one-pair Higgs model

From the result in the previous section on the $Z_{2}$ general case, we see
the number of the possibilities with one-pair Higgs is strikingly enlarged as
$65$ from $5$. Note that the modulus parameter $\tau$ of the torus does not
receive any restriction from the $Z_{2}$ orbifolding, in contrast to the other
three cases. As discussed in [64], when the greater value the modulus
parameter $\tau$ takes, the more significant hierarchy could be realized.
However, in the simple five cases without nonzero Scherk-Schwarz phases, we
can check that no sizable hierarchy is generated even when we take a large
value. For example, even when we adopt $\tau=10\,i$, only up to
$\mathcal{O}(10^{2})$-order hierarchy can be generated.

Through the following example, we will see how the nonzero phases improve the
magnitude in the hierarchy. Here, we consider the configuration,

$\displaystyle(M_{bc},M_{ca},M_{ab})=(-2,-4,+6),$
$\displaystyle(\alpha_{bc},\alpha_{ca},\alpha_{ab})=(0,0,0),$
$\displaystyle(\beta_{bc},\beta_{ca},\beta_{ab})=(1/2,0,1/2),$
$\displaystyle(\eta_{bc},\eta_{ca},\eta_{ab})=(1,1,1),$
$\displaystyle(M_{bc^{\prime}},M_{c^{\prime}a},M_{ab})=(-1,-5,+6),$
$\displaystyle(\alpha_{bc^{\prime}},\alpha_{c^{\prime}a},\alpha_{ab})=(0,0,0),$
$\displaystyle(\beta_{bc^{\prime}},\beta_{c^{\prime}a},\beta_{ab})=(1/2,0,1/2),$
$\displaystyle(\eta_{bc^{\prime}},\eta_{c^{\prime}a},\eta_{ab})=(1,1,1),$
(4.9)

where $U(2)_{R}$ breaks down to $U(1)_{R1}$ and $U(1)_{R2}$ identified by the
indices $c$ and $c^{\prime}$. Now, the left-handed quarks, right-handed
up/down-type quarks and up/down-type Higgs bosons live in the sectors, $ab$,
$bc/bc^{\prime}$ and $ca/c^{\prime}a$, respectively. This case belongs to the
category “$M_{ab}<0,M_{ca}>0$” in the previous classification and the
correspondence to Eq. (4.6) is straightforward, e.g., $|M_{bc}|\to M_{I},\
|M_{ca}|\to M_{J},\ |M_{ab}|\to M_{K}$. By following the formula in Eq.
(2.74), each element of the up/down Yukawa couplings are calculated
analytically by use of the forms $\eta_{N}^{(u)}/\eta_{N}^{(d)}$,

$\displaystyle\eta_{N}^{(u)}:=\vartheta\left[\begin{array}[]{c}N/M_{u}\\\
0\end{array}\right](-2,M_{u}\tau),\quad\eta_{N}^{(d)}:=\vartheta\left[\begin{array}[]{c}N/M_{d}\\\
0\end{array}\right](-5/2,M_{d}\tau),$ (4.14)

where they are functions of $N$ and $M_{u}$ and $M_{d}$ are defined as
$M_{u}:=|M_{ab}M_{bc}M_{ca}|=48$ and
$M_{d}:=|M_{ab}M_{bc^{\prime}}M_{c^{\prime}a}|=30$. $\eta_{N}^{(u)}$ and
$\eta_{N}^{(d)}$ are equivalent to $\eta_{N+M_{u}}^{(u)}$ and
$\eta_{N+M_{d}}^{(d)}$, respectively. When we change $N$, the values of the
functions alter exponentially, whose maxima and minima are around $N=0$ and
$N=M_{u,d}/2$, respectively. Apart from the trivial Scherk-Schwarz case, the
“$c\nu$” index of the theta function in $\eta_{N}^{(u,d)}$ can take nonzero
values. The correspondence to the $c\nu=0$ expression is easily obtainable
when we utilize the first relation in Eq. (2.51) as

$\displaystyle\eta_{N}^{(u)}=e^{2\pi
i\frac{N}{M_{u}}(-2)}\vartheta\left[\begin{array}[]{c}N/M_{u}\\\
0\end{array}\right](0,M_{u}\tau),\quad\eta_{N}^{(d)}=e^{2\pi
i\frac{N}{M_{d}}\left(-\frac{5}{2}\right)}\vartheta\left[\begin{array}[]{c}N/M_{d}\\\
0\end{array}\right](0,M_{d}\tau),$ (4.19)

where there are only $\mathcal{O}(1)$ differences between $\eta^{(u,d)}_{N}$
with nonzero $c\nu$ and with $c\nu=0$.

On the other hand, allowed values of $N$ with nonzero Scherk-Schwarz phases
are changed since such phases modify the selection rule described with
Kronecker’s delta in Eq. (4.6). When we pick up only the most significant term
in each matrix element with ignoring $\mathcal{O}(1)$ factor, the results are
as follows,

$\displaystyle\widetilde{\lambda^{\prime}}^{(u)}_{I^{\prime},J^{\prime}}\sim\begin{pmatrix}\eta_{0}^{(u)}&0&\eta_{12}^{(u)}\\\
0&\eta_{2}^{(u)}&0\\\
\eta_{8}^{(u)}&0&\eta_{4}^{(u)}\end{pmatrix},\quad\widetilde{\lambda^{\prime}}^{(d)}_{I^{\prime},J^{\prime}}\sim\begin{pmatrix}\eta_{0}^{(d)}&\eta_{6}^{(d)}&\eta_{12}^{(d)}\\\
\eta_{5}^{(d)}&\eta_{1}^{(d)}&\eta_{23}^{(d)}\\\
\eta_{10}^{(d)}&\eta_{4}^{(d)}&\eta_{2}^{(d)}\end{pmatrix},$ (4.20)

where we omit to show the index about the Higgs because only one Higgs pair is
there. Now, we show the resultant values in Table 5,999 In the numerical
calculation, we include all the sub-leading terms which we omit to represent
in Eq. (4.20). In this calculation, note that the ratio of top and bottom
masses is determined only by the ratio of VEVs, so-called $\tan{\beta}$ in the
MSSM since we assume a MSSM-like Higgs sector.  where we notice that the
desirable amount of hierarchy is suitably realized in the mass values.101010
Here, we assume that the Yukawa couplings are proportional to the mass values,
like in the SM.  However, if we try to generate the realistic CKM mixing, a
more complicated mixing pattern would be required.

| Obtained values | Actual values
---|---|---
$(m_{u},m_{c},m_{t})/m_{t}$ | $(9.9\times 10^{-6},2.8\times 10^{-2},1)$ | $(1.5\times 10^{-5},7.5\times 10^{-3},1)$
$(m_{d},m_{s},m_{b})/m_{b}$ | $(5.0\times 10^{-3},1.6\times 10^{-2},1)$ | $(1.2\times 10^{-3},2.3\times 10^{-2},1)$
$|V_{\text{CKM}}|$ | $\begin{pmatrix}0.99&3.1\times 10^{-8}&0\\\ 3.1\times 10^{-8}&0.99&2.6\times 10^{-13}\\\ 0&2.6\times 10^{-13}&0.99\end{pmatrix}$ | $\begin{pmatrix}0.97&0.23&0.0035\\\ 0.23&0.97&0.041\\\ 0.0087&0.040&1.0\end{pmatrix}$
Table 5: The mass ratios of the quarks and the absolute values of the CKM
matrix elements when we adopt the parameters in Eq. (4.9) with one Higgs pair.
We use the experimental values in Ref. [66].

###  4.2 Gaussian Froggatt-Nielsen mechanism with discrete Scherk-Schwarz
phases

Here, we would like to search for a configuration with realistic observed
values by following the strategy discussed in [14], named Gaussian Froggatt-
Nielsen mechanism. As pointed out in [14], especially when $M_{u,d}$ are as
large as $\mathcal{O}(10^{2})$ or more, the forms of $\eta_{N}^{(u,d)}$ are
approximately described by the following simple forms with good precision,

$\displaystyle\eta_{N}^{(u,d)}\sim
e^{-\frac{\text{Im}[\tau]\pi}{M_{u,d}}N^{2}},$ (4.21)

where the real part of $\tau$ only contributes to this as an $\mathcal{O}(1)$
phase factor. Also, we would remember that the nonzeroness in “$c\nu$” of the
theta function does not lead to a sizable effect, only up to $\mathcal{O}(1)$
as in Eq. (4.19). Under some assumptions in multiple Higgs VEVs, the Yukawa
couplings would be symbolically expressed as

$\displaystyle Y_{I^{\prime},J^{\prime}}\sim
e^{-c(a_{I^{\prime}}+b_{J^{\prime}})^{2}},$ (4.22)

where $c$ is a common constant factor and $a_{I^{\prime}}$ and
$b_{J^{\prime}}$ are determined only by the magnitudes of the magnetic fluxes
on the left- and right-handed fermion sectors, respectively. It seems to be
similar to another familiar one named Froggatt-Nielsen form [67] as

$\displaystyle Y_{I^{\prime},J^{\prime}}\sim
e^{-c(a_{I^{\prime}}+b_{J^{\prime}})},$ (4.23)

where in this case, a linear form appears and $a_{I^{\prime}}$ and
$b_{J^{\prime}}$ correspond to quantum numbers named Froggatt-Nielsen charges.
When we realize the following shape,

$\displaystyle(Y)_{I^{\prime},J^{\prime}}\ll(Y)_{M^{\prime},N^{\prime}},$
(4.24)

for $I^{\prime}\leq M^{\prime}$ and $J^{\prime}\leq N^{\prime}$, we would
regenerate a desirable form seen in the CKM matrix.

In the following part, we analyze the flavor structure through Gaussian
Froggatt-Nielsen mechanism with discrete Scherk-Schwarz phases in the
parameters,

$\displaystyle(M_{ab},M_{ca},M_{bc})=(-5,-4,+9),$
$\displaystyle(\alpha_{ab},\alpha_{ca},\alpha_{bc})=(0,0,0),$
$\displaystyle(\beta_{ab},\beta_{ca},\beta_{bc})=(1/2,0,1/2),$
$\displaystyle(\eta_{ab},\eta_{ca},\eta_{bc})=(1,1,1),$
$\displaystyle(M_{ab},M_{c^{\prime}a},M_{bc^{\prime}})=(-5,-7,+12),$
$\displaystyle(\alpha_{ab},\alpha_{c^{\prime}a},\alpha_{bc^{\prime}})=(0,0,0),$
$\displaystyle(\beta_{ab},\beta_{c^{\prime}a},\beta_{bc^{\prime}})=(1/2,1/2,0),$
$\displaystyle(\eta_{ab},\eta_{c^{\prime}a},\eta_{bc^{\prime}})=(1,-1,-1),$
(4.25)

where the notation is the same with that in the previous analysis. This
configuration is categorized in the case, “$M_{ab},M_{ca}<0$”, where multiple
Higgs fields tend to be obtained. The correspondence to Eq. (4.6) is
straightforward, e.g., $|M_{ab}|\to M_{I},\ |M_{ca}|\to M_{J},\ |M_{bc}|\to
M_{K}$.

Our choice predicts five pairs of up- and down-type Higgs bosons, where enough
such possibilities are found in the $Z_{2}$ parameter landscape. With
neglecting $\mathcal{O}(1)$ phase factors and dropping sub-leading terms, we
can focus on the following part,

$\displaystyle\eta_{N}^{(u)}:=\vartheta\left[\begin{array}[]{c}N/M_{u}\\\
0\end{array}\right](-2,M_{u}\tau),\quad\eta_{N}^{(d)}:=\vartheta\left[\begin{array}[]{c}N/M_{d}\\\
0\end{array}\right](-1,M_{d}\tau),$ (4.30)
$\displaystyle\widetilde{\lambda^{\prime}}^{(u)}_{I^{\prime},J^{\prime},K^{\prime}=3}\sim\begin{pmatrix}\eta_{60}^{(u)}&\eta_{15}^{(u)}&\eta_{30}^{(u)}\\\
\eta_{24}^{(u)}&\eta_{21}^{(u)}&\eta_{6}^{(u)}\\\
\eta_{12}^{(u)}&\eta_{3}^{(u)}&\eta_{42}^{(u)}\end{pmatrix},\quad\widetilde{\lambda^{\prime}}^{(u)}_{I^{\prime},J^{\prime},K^{\prime}=4}\sim\begin{pmatrix}\eta_{20}^{(u)}&\eta_{25}^{(u)}&\eta_{70}^{(u)}\\\
\eta_{16}^{(u)}&\eta_{11}^{(u)}&\eta_{34}^{(u)}\\\
\eta_{52}^{(u)}&\eta_{7}^{(u)}&\eta_{2}^{(u)}\end{pmatrix},$ (4.31)
$\displaystyle\widetilde{\lambda^{\prime}}^{(d)}_{I^{\prime},J^{\prime},K^{\prime}=3}\sim\begin{pmatrix}\eta_{40}^{(d)}&\eta_{80}^{(d)}&\eta_{20}^{(d)}\\\
\eta_{68}^{(d)}&\eta_{32}^{(d)}&\eta_{8}^{(d)}\\\
\eta_{16}^{(d)}&\eta_{4}^{(d)}&\eta_{64}^{(d)}\end{pmatrix},\quad\widetilde{\lambda^{\prime}}^{(d)}_{I^{\prime},J^{\prime},K^{\prime}=4}\sim\begin{pmatrix}\eta_{145}^{(d)}&\eta_{25}^{(d)}&\eta_{85}^{(d)}\\\
\eta_{23}^{(d)}&\eta_{73}^{(d)}&\eta_{13}^{(d)}\\\
\eta_{61}^{(d)}&\eta_{11}^{(d)}&\eta_{1}^{(d)}\end{pmatrix},$ (4.32)

with $M_{u}=180$ and $M_{d}=420$. When we assume that the two Higgs pairs,
$H_{u3(d3)}$ and $H_{u4(d4)}$, contain VEVs and the other do not, the
effective Yukawa couplings $Y^{(u,d)}$ are described with the following
parameters $\rho_{u,d}$,

$\displaystyle\rho_{u}=\frac{\langle H_{u3}\rangle}{\langle
H_{u4}\rangle},\quad\rho_{d}=\frac{\langle H_{d3}\rangle}{\langle
H_{d4}\rangle},$ (4.33) $\displaystyle
Y^{(u)}\sim\begin{pmatrix}\eta_{20}^{(u)}&\rho_{u}\eta_{15}^{(u)}&\rho_{u}\eta_{30}^{(u)}\\\
\eta_{16}^{(u)}&\eta_{11}^{(u)}&\rho_{u}\eta_{6}^{(u)}\\\
\rho_{u}\eta_{12}^{(u)}&\rho_{u}\eta_{3}^{(u)}&\eta_{2}^{(u)}\end{pmatrix},\quad
Y^{(d)}\sim\begin{pmatrix}\rho_{d}\eta_{40}^{(d)}&\eta_{25}^{(d)}&\rho_{d}\eta_{20}^{(d)}\\\
\eta_{23}^{(d)}&\rho_{d}\eta_{32}^{(d)}&\rho_{d}\eta_{8}^{(d)}\\\
\rho_{d}\eta_{16}^{(d)}&\rho_{d}\eta_{4}^{(d)}&\eta_{1}^{(d)}\end{pmatrix}.$
(4.34)

The choice in the two ratios in the Higgs VEVs, $\rho_{u}=0.03$ and
$\rho_{d}=0.49$, leads to the (semi-)realistic observed values summarized in
Table 6.111111 In the numerical calculation, we include all the sub-leading
terms which we omit to represent in Eq. (4.34). Also like the previous
calculation, we do not touch the ratio between $m_{t}$ and $m_{b}$ described
by the ratio of the VEVs.

| Obtained values | Actual values
---|---|---
$(m_{u},m_{c},m_{t})/m_{t}$ | $(3.0\times 10^{-5},3.6\times 10^{-2},1)$ | $(1.5\times 10^{-5},7.5\times 10^{-3},1)$
$(m_{d},m_{s},m_{b})/m_{b}$ | $(2.9\times 10^{-4},8.9\times 10^{-3},1)$ | $(1.2\times 10^{-3},2.3\times 10^{-2},1)$
$|V_{\text{CKM}}|$ | $\begin{pmatrix}0.97&0.23&0.011\\\ 0.23&0.97&0.026\\\ 0.005&0.027&1.00\end{pmatrix}$ | $\begin{pmatrix}0.97&0.23&0.0035\\\ 0.23&0.97&0.041\\\ 0.0087&0.040&1.0\end{pmatrix}$
Table 6: The mass ratios of the quarks and the absolute values of the CKM
matrix elements when we adopt the parameters in Eq. (4.25) with five Higgs
pairs. We use the experimental values in Ref. [66].

##  5 Summary and Discussion

In this paper, based on the analytical formulation discussed in Ref. [51], we
have classified all the individual possibilities of flavor models generating
suitable three generations in the quarks and the leptons on generalized
$T^{2}$-based magnetized orbifolds, namely $T^{2}/Z_{2}$, $T^{2}/Z_{3}$,
$T^{2}/Z_{4}$, $T^{2}/Z_{6}$, with nontrivial Wilson line phases and/or
Scherk-Schwarz phases. Also, we examined the effects caused by the nontrivial
phases in mass hierarchy in a configuration with one Higgs pair, and
constructed a concrete model with the phases being nonzero values in the case
of five Higgs pairs, accompanied by a realization of the SM quark structure,
that is the observed mass values and mixing angles, with rather good
precision.

Now, there are lots of possible suitable candidates with three generations in
the choice of fluxes and related parameters, and thus to search for suitable
configurations to describe the nature on $T^{2}/Z_{3,4,6}$ could be an
interesting direction. Here, nontrivial contributions via diagonalizing
matrices of kinetic terms would be nontrivial and situations might be
different from in the $Z_{2}$. Revealing this point is also an important work
in the model building on magnetized background. Also, like in [13, 32], full
model embedding to 10D SYM theory is a meaningful task.

## Acknowledgments

Y.T. would like to thank Hiroyuki Abe for useful comments. This work was
supported in part by scientific grants from the Ministry of Education,
Culture, Sports, Science, and Technology under Grants No. 25$\cdot$3825
(Y.F.), No. 25400252 (T.K.) and No. 20540274 (M.S.).

## Appendix

##  Appendix A Information on $T^{2}/Z_{N}$ orbifold

In this Appendix, we summarize the information on the numbers of the survived
physical states and the forms of $C_{I_{i}J_{i}}^{(\omega^{x})}$ in
$T^{2}/Z_{N}$ orbifolds $(N=2,3,4,6)$ discussed in [51]. Hereafter in this
section, we omit the subscript $i$ and use small roman letters for showing the
indices identifying the states on $T^{2}$ (without orbifolding) for
simplicity. Also, we replace $\alpha$ and $\beta$ as $\alpha_{1}$ and
$\alpha_{2}$, respectively. Note that we evaluate the numbers of generations
in the $Z_{3,4,6}$ cases within the range required in the discussion in
section 3. Note that all the following formulas are for the situation where
the value of the magnetic flux is positive. The correspondence to the negative
case is understandable, e.g., through Eq. (2.53).

###  A.1 $T^{2}/Z_{2}$

On the $T^{2}/Z_{2}$ ($\omega=e^{\pi i}=-1$), allowed values of Scherk-Schwarz
phases are following four patterns,

$(\alpha_{1},\alpha_{2})=\left(0,0\right),\left(\frac{1}{2},0\right),\left(0,\frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{2}\right)\,.$
(A.1)

The analytic form of the expansion coefficient is

$\displaystyle C_{jk}^{(\omega^{1})}=e^{-2\pi
i\cdot\frac{2\alpha_{2}}{M}(j+\alpha_{1})}{\delta_{-2\alpha_{1}-j,k}}\,.$
(A.2)

We can find easily the formula of the number of the physical states by
analyzing the matrix

$M_{jk}^{(Z_{2};\eta)}=\frac{1}{2}\left(\delta_{j,k}+\eta\,C_{jk}^{(\omega^{1})}\right)\quad(\eta=\pm
1)\,.$ (A.3)

####  $(\alpha_{1},\alpha_{2})=(0,0)$ case

In this pattern, the number of independent physical states is given by

$\begin{split}(\textrm{{\\#} of identical physical states for
$\eta=+1$})&=\left[\frac{M}{2}\right]+1,\\\ (\textrm{{\\#} of identical
physical states for $\eta=-1$})&=\left[\frac{M-1}{2}\right],\end{split}$ (A.4)

where $\left[x\right]$ is floor function.

This case has been already studied. The above equations obtained by the
operator formalism analysis reproduce the same result as the earlier study,
Table 1 in [64].

####  $(\alpha_{1},\alpha_{2})=(\frac{1}{2},0)$ case

In this pattern, the number of independent physical states is given by

$\begin{split}(\textrm{{\\#} of identical physical states for
$\eta=+1$})&=\left[\frac{M+1}{2}\right],\\\ (\textrm{{\\#} of identical
physical states for $\eta=-1$})&=\left[\frac{M}{2}\right].\end{split}$ (A.5)

####  $(\alpha_{1},\alpha_{2})=(0,\frac{1}{2})$ case

In this pattern, the number of independent physical states is given by

$\begin{split}(\textrm{{\\#} of identical physical states for
$\eta=+1$})&=\left[\frac{M+1}{2}\right],\\\ (\textrm{{\\#} of identical
physical states for $\eta=-1$})&=\left[\frac{M}{2}\right].\end{split}$ (A.6)

####  $(\alpha_{1},\alpha_{2})=(\frac{1}{2},\frac{1}{2})$ case

In this pattern, the number of independent physical states is given by

$\begin{split}(\textrm{{\\#} of identical physical states for
$\eta=+1$})&=\left[\frac{M}{2}\right],\\\ (\textrm{{\\#} of identical physical
states for $\eta=-1$})&=\left[\frac{M+1}{2}\right].\end{split}$ (A.7)

###  A.2 $T^{2}/Z_{3}$

On the $T^{2}/Z_{3}$ ($\omega=e^{2\pi i/3}$), allowed values of Scherk-Schwarz
phases are

$\alpha:=\alpha_{1}=\alpha_{2}=\left\\{\begin{aligned}
&0,\frac{1}{3},\frac{2}{3}\quad\,({\rm for}~{}M={\rm even})\,,\\\
&\frac{1}{6},\frac{3}{6},\frac{5}{6}\quad({\rm for}~{}M={\rm
odd})\,.\end{aligned}\right.$ (A.8)

After analyzing the matrix

$M_{jk}^{(Z_{3};\eta)}=\frac{1}{3}\sum_{x=0}^{2}\bar{\eta}^{x}C_{jk}^{(\omega^{x})}\quad(\eta=1,\omega,\bar{\omega})\,,$
(A.9)

described by the elements

$\displaystyle
C^{(\omega)}_{jk}=\frac{1}{\sqrt{M}}e^{-i\frac{\pi}{12}+i\frac{3\pi\alpha^{2}}{M}}e^{i\frac{\pi}{M}k(k+6\alpha)+2\pi
i\frac{j\cdot k}{M}},$ $\displaystyle
C^{(\omega^{2})}_{jk}=\frac{1}{\sqrt{M}}e^{i\frac{\pi}{12}-i\frac{3\pi\alpha^{2}}{M}-i\frac{\pi}{M}j(j+6\alpha)}e^{-2\pi
i\frac{j\cdot k}{M}},$ (A.10)

we obtain the results shown in Tables 7, 8, 9, 10.

$|M|$ | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22
---|---|---|---|---|---|---|---|---|---|---|---
$\eta$ | 1 | 1 | 1 | 3 | 3 | 3 | 5 | 5 | 5 | 7 | 7 | 7
$\omega$ | 0 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 8
$\bar{\omega}$ | 1 | 1 | 1 | 3 | 3 | 3 | 5 | 5 | 5 | 7 | 7
Table 7: The numbers of linearly independent zero-mode eigenstates with $Z_{3}$ eigenvalue $\eta$ for $M={\rm even}$ and $(\alpha_{1},\alpha_{2})=(0,0)$ on $T^{2}/Z_{3}$. $|M|$ | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22
---|---|---|---|---|---|---|---|---|---|---|---
$\eta$ | 1 | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 6 | 6 | 7 | 8
$\omega$ | 1 | 1 | 2 | 3 | 3 | 4 | 5 | 5 | 6 | 7 | 7
$\bar{\omega}$ | 0 | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 6 | 6 | 7
Table 8: The numbers of linearly independent zero-mode eigenstates with $Z_{3}$ eigenvalue $\eta$ for $M={\rm even}$ and $(\alpha_{1},\alpha_{2})=(\frac{1}{3},\frac{1}{3}),(\frac{2}{3},\frac{2}{3})$ on $T^{2}/Z_{3}$. $|M|$ | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21
---|---|---|---|---|---|---|---|---|---|---|---
$\eta$ | 1 | 1 | 1 | 2 | 3 | 3 | 4 | 5 | 5 | 6 | 7 | 7
$\omega$ | 0 | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 6 | 6 | 7
$\bar{\omega}$ | 0 | 1 | 1 | 2 | 3 | 3 | 4 | 5 | 5 | 6 | 7
Table 9: The numbers of linearly independent zero-mode eigenstates with $Z_{3}$ eigenvalue $\eta$ for $M={\rm odd}$ and $(\alpha_{1},\alpha_{2})=(\frac{1}{6},\frac{1}{6}),(\frac{5}{6},\frac{5}{6})$ on $T^{2}/Z_{3}$. $|M|$ | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21
---|---|---|---|---|---|---|---|---|---|---|---
$\eta$ | 1 | 0 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 8
$\omega$ | 1 | 1 | 1 | 3 | 3 | 3 | 5 | 5 | 5 | 7 | 7
$\bar{\omega}$ | 0 | 0 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6
Table 10: The numbers of linearly independent zero-mode eigenstates with
$Z_{3}$ eigenvalue $\eta$ for $M={\rm odd}$ and
$(\alpha_{1},\alpha_{2})=(\frac{3}{6},\frac{3}{6})$ on $T^{2}/Z_{3}$.

###  A.3 $T^{2}/Z_{4}$

On the $T^{2}/Z_{4}$ ($\omega=e^{\pi i/2}=i$), allowed values of Scherk-
Schwarz phases are

$\alpha:=\alpha_{1}=\alpha_{2}=0,\frac{1}{2}\,.$ (A.11)

After analyzing the matrix

$M_{jk}^{(Z_{4};\eta)}=\frac{1}{4}\sum_{x=0}^{3}\bar{\eta}^{x}C_{jk}^{(\omega^{x})}\quad(\eta=\pm
1,\pm i)\,,$ (A.12)

described by the elements

$\displaystyle C_{jk}^{(\omega)}=\frac{1}{\sqrt{M}}e^{2\pi
i\frac{\alpha^{2}}{M}}e^{2\pi i\frac{j\cdot k}{M}+2\pi i\frac{2\alpha}{M}k},$
$\displaystyle C_{jk}^{(\omega^{2})}=e^{-2\pi
i\frac{2\alpha}{M}(\alpha+j)}\delta_{{-2\alpha-j,k}},$ $\displaystyle
C_{jk}^{(\omega^{3})}=\frac{1}{\sqrt{M}}e^{-2\pi i\frac{\alpha^{2}}{M}-2\pi
i\frac{2\alpha}{M}j}e^{-2\pi i\frac{j\cdot k}{M}},$ (A.13)

we obtain the results summarized in Tables 11 and 12.

$|M|$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\eta$ | $+1$ | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4
$+i$ | 0 | 0 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4
$-1$ | 0 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4
$-i$ | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3
$|M|$ | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30
$\eta$ | $+1$ | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 8 | 8 | 8
$+i$ | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7
$-1$ | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 8
$-i$ | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 7 | 7
Table 11: The numbers of linearly independent zero-mode eigenstates with $Z_{4}$ eigenvalue $\eta$ for $(\alpha_{1},\alpha_{2})=(0,0)$ on $T^{2}/Z_{4}$. $|M|$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\eta$ | $+1$ | 0 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4
$+i$ | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4
$-1$ | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3
$-i$ | 0 | 0 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4
$|M|$ | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30
$\eta$ | $+1$ | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 8
$+i$ | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 8 | 8
$-1$ | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 7 | 7 | 7
$-i$ | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7
Table 12: The numbers of linearly independent zero-mode eigenstates with
$Z_{4}$ eigenvalue $\eta$ for
$(\alpha_{1},\alpha_{2})=(\frac{1}{2},\frac{1}{2})$ on $T^{2}/Z_{3}$.

###  A.4 $T^{2}/Z_{6}$

On the $T^{2}/Z_{6}$ ($\omega=e^{\pi i/3}$), allowed values of Scherk-Schwarz
phases are

$\alpha:=\alpha_{1}=\alpha_{2}=\left\\{\begin{aligned} &0\quad\,({\rm
for}~{}M={\rm even})\\\ &\frac{1}{2}\quad({\rm for}~{}M={\rm
odd})\,.\end{aligned}\right.$ (A.14)

After analyzing the matrix

$M_{jk}^{(Z_{6};\eta)}=\frac{1}{6}\sum_{x=0}^{5}\bar{\eta}^{x}C_{jk}^{(\omega^{x})}\quad(\eta=1,\omega,\omega^{2},\omega^{3},\omega^{4},\omega^{5})\,,$
(A.15)

described by the elements

$\displaystyle
C_{jk}^{(\omega)}=\frac{1}{\sqrt{M}}e^{i\frac{\pi}{12}+i\frac{\pi}{M}\alpha^{2}}e^{-i\frac{\pi}{M}k^{2}+2\pi
i\frac{\alpha}{M}k+2\pi i\frac{j\cdot k}{M}},$ $\displaystyle
C_{jk}^{(\omega^{2})}=\frac{1}{\sqrt{M}}e^{-i\frac{\pi}{12}+i\frac{3\pi\alpha^{2}}{M}+i\frac{\pi}{M}j^{2}+2\pi
i\frac{\alpha}{M}j}e^{i\frac{4\pi\alpha}{M}k+2\pi i\frac{j\cdot k}{M}},$
$\displaystyle
C_{jk}^{(\omega^{3})}=e^{-i\frac{4\pi\alpha^{2}}{M}-i\frac{4\pi\alpha}{M}j}\delta_{{-2\alpha-j,k}},$
$\displaystyle
C_{jk}^{(\omega^{4})}=\frac{1}{\sqrt{M}}e^{i\frac{\pi}{12}-i\frac{3\pi\alpha^{2}}{M}-i\frac{4\pi\alpha}{M}j}e^{-i\frac{\pi}{M}k^{2}-2\pi
i\frac{\alpha}{M}k-2\pi i\frac{j\cdot k}{M}},$ $\displaystyle
C_{jk}^{(\omega^{5})}=\frac{1}{\sqrt{M}}e^{-i\frac{\pi}{12}-i\frac{\pi}{M}\alpha^{2}{+}i\frac{\pi}{M}j^{2}-2\pi
i\frac{\alpha}{M}j}e^{-2\pi i\frac{j\cdot k}{M}},$ (A.16)

we obtain the results in Tables 13 and 14.

$|M|$ | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22
---|---|---|---|---|---|---|---|---|---|---|---
$\eta$ | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4
$\omega$ | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4
$\omega^{2}$ | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4
$\omega^{3}$ | 0 | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3
$\omega^{4}$ | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4
$\omega^{5}$ | 0 | 0 | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3
$|M|$ | 24 | 26 | 28 | 30 | 32 | 34 | 36 | 38 | 40 | 42 | 44
$\eta$ | 1 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8
$\omega$ | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7 | 7
$\omega^{2}$ | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7 | 7 | 8
$\omega^{3}$ | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7
$\omega^{4}$ | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7 | 7
$\omega^{5}$ | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7
Table 13: The numbers of linearly independent zero-mode eigenstates with $Z_{6}$ eigenvalue $\eta$ for $M={\rm even}$ and $(\alpha_{1},\alpha_{2})=(0,0)$ on $T^{2}/Z_{6}$. $|M|$ | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21
---|---|---|---|---|---|---|---|---|---|---|---
$\eta$ | 1 | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4
$\omega$ | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4
$\omega^{2}$ | 0 | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3
$\omega^{3}$ | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4
$\omega^{4}$ | 0 | 0 | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3
$\omega^{5}$ | 0 | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3
$|M|$ | 23 | 25 | 27 | 29 | 31 | 33 | 35 | 37 | 39 | 41 | 43
$\eta$ | 1 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7 | 7
$\omega$ | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7 | 7 | 8
$\omega^{2}$ | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7
$\omega^{3}$ | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7 | 7
$\omega^{4}$ | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7
$\omega^{5}$ | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7
Table 14: The numbers of linearly independent zero-mode eigenstates with
$Z_{6}$ eigenvalue $\eta$ for $M={\rm odd}$ and
$(\alpha_{1},\alpha_{2})=(\frac{1}{2},\frac{1}{2})$ on $T^{2}/Z_{6}$.

##  Appendix B Possible parameter configurations in $Z_{2,3,4,6}$

In this part, we show the explicit information on the possible configurations
in the cases of $Z_{2,3,4,6}$ with nontrivial boundary conditions. In the
analysis, we adopt the basis where the Scherk-Schwarz phases are nonzero, and
correspondingly the Wilson line phases are zero. The interpretation into the
basis with nonzero Wilson line phases and Scherk-Schwarz phases being zero is
straightforward, where we just obey the relation in Eq. (3.7).

Results of the two cases mentioned in Eq. (3.8), $M_{ab}<0,\ M_{bc}<0$, and,
$M_{ab}<0,\ M_{bc}>0$, are separately stored in tables B.1 and B.1 ($Z_{2}$),
B.2 and B.2 ($Z_{3}$), B.3 and B.3 ($Z_{4}$) and B.4 and B.4 ($Z_{6}$),
respectively. Here, we skip to represent the parameters of $bc$
(Higgsino/Higgs) sector since they are not independent and automatically fixed
by the configurations of the other two sectors through the constraints in Eq.
(3.5). If an allowed configuration says $M_{bc}=0$, where one non-localized
Higgs pair appears without magnetic background and this situation would be not
interesting in the phenomenological point of view, we discriminate this case
as “1” from the one Higgs cases with magnetic flux “$1$”. A $Z_{N}$ parity
$\eta$ is represented as a function of $i$ in this form, $\eta=\omega^{i}$
($\omega=e^{2\pi i/N}$).

Note that the “Trivial BC’s only” case in $Z_{2}$ was already analyzed in Ref.
[64] and our result is completely consistent with the previous ones. In the
$Z_{3,4,6}$ orbifoldings, the two values of the Scherk-Schwarz phases should
be the same as $\alpha=\beta$.

###  B.1 $T^{2}/Z_{2}$

Table 15: Possible parameter configurations on $T^{2}/Z_{2}$ with
$M_{ab}<0,\,M_{ca}<0$. $T^{2}/Z_{2}$ with $M_{ab}<0,\,M_{ca}<0$
---
$ab$-sector |  $ca$-sector |  $bc$-sector
$M_{ab}$ | $\omega^{i}_{ab}$ | $\alpha_{ab}$ | $\beta_{ab}$ | $M_{ca}$ | $\omega^{i}_{ca}$ | $\alpha_{ca}$ | $\beta_{ca}$ | # of Higgs
$-4$ | $0$ | $0$ | $0$ | $-4$ | $0$ | $0$ | $0$ | $5$
$-4$ | $0$ | $0$ | $0$ | $-5$ | $0$ | $0$ | $0$ | $5$
$-4$ | $0$ | $0$ | $0$ | $-5$ | $0$ | $0$ | $1/2$ | $5$
$-4$ | $0$ | $0$ | $0$ | $-5$ | $0$ | $1/2$ | $0$ | $5$
$-4$ | $0$ | $0$ | $0$ | $-5$ | $1$ | $1/2$ | $1/2$ | $5$
$-5$ | $0$ | $0$ | $0$ | $-5$ | $0$ | $0$ | $0$ | $6$
$-5$ | $0$ | $0$ | $0$ | $-5$ | $0$ | $0$ | $1/2$ | $5$
$-5$ | $0$ | $0$ | $0$ | $-5$ | $0$ | $1/2$ | $0$ | $5$
$-5$ | $0$ | $0$ | $1/2$ | $-5$ | $0$ | $0$ | $1/2$ | $6$
$-5$ | $0$ | $0$ | $1/2$ | $-5$ | $0$ | $1/2$ | $0$ | $5$
$-5$ | $0$ | $1/2$ | $0$ | $-5$ | $0$ | $1/2$ | $0$ | $6$
$-5$ | $0$ | $0$ | $0$ | $-5$ | $1$ | $1/2$ | $1/2$ | $5$
$-5$ | $0$ | $0$ | $1/2$ | $-5$ | $1$ | $1/2$ | $1/2$ | $5$
$-5$ | $0$ | $1/2$ | $0$ | $-5$ | $1$ | $1/2$ | $1/2$ | $5$
$-5$ | $1$ | $1/2$ | $1/2$ | $-5$ | $1$ | $1/2$ | $1/2$ | $6$
$-4$ | $0$ | $0$ | $0$ | $-6$ | $0$ | $0$ | $1/2$ | $5$
$-4$ | $0$ | $0$ | $0$ | $-6$ | $0$ | $1/2$ | $0$ | $5$
$-4$ | $0$ | $0$ | $0$ | $-6$ | $0$ | $1/2$ | $1/2$ | $5$
$-4$ | $0$ | $0$ | $0$ | $-6$ | $1$ | $0$ | $1/2$ | $5$
$-4$ | $0$ | $0$ | $0$ | $-6$ | $1$ | $1/2$ | $0$ | $5$
$-4$ | $0$ | $0$ | $0$ | $-6$ | $1$ | $1/2$ | $1/2$ | $5$
$-5$ | $0$ | $0$ | $0$ | $-6$ | $0$ | $0$ | $1/2$ | $6$
$-5$ | $0$ | $0$ | $0$ | $-6$ | $0$ | $1/2$ | $0$ | $6$
$-5$ | $0$ | $0$ | $0$ | $-6$ | $0$ | $1/2$ | $1/2$ | $5$
$-5$ | $0$ | $0$ | $1/2$ | $-6$ | $0$ | $0$ | $1/2$ | $6$
$-5$ | $0$ | $0$ | $1/2$ | $-6$ | $0$ | $1/2$ | $0$ | $5$
$-5$ | $0$ | $0$ | $1/2$ | $-6$ | $0$ | $1/2$ | $1/2$ | $6$
$-5$ | $0$ | $1/2$ | $0$ | $-6$ | $0$ | $0$ | $1/2$ | $5$
$-5$ | $0$ | $1/2$ | $0$ | $-6$ | $0$ | $1/2$ | $0$ | $6$
$-5$ | $0$ | $1/2$ | $0$ | $-6$ | $0$ | $1/2$ | $1/2$ | $6$
$-5$ | $0$ | $0$ | $0$ | $-6$ | $1$ | $0$ | $1/2$ | $5$
$-5$ | $0$ | $0$ | $0$ | $-6$ | $1$ | $1/2$ | $0$ | $5$
$-5$ | $0$ | $0$ | $0$ | $-6$ | $1$ | $1/2$ | $1/2$ | $6$
$-5$ | $0$ | $0$ | $1/2$ | $-6$ | $1$ | $0$ | $1/2$ | $5$
$-5$ | $0$ | $0$ | $1/2$ | $-6$ | $1$ | $1/2$ | $0$ | $6$
$-5$ | $0$ | $0$ | $1/2$ | $-6$ | $1$ | $1/2$ | $1/2$ | $5$
$-5$ | $0$ | $1/2$ | $0$ | $-6$ | $1$ | $0$ | $1/2$ | $6$
$-5$ | $0$ | $1/2$ | $0$ | $-6$ | $1$ | $1/2$ | $0$ | $5$
$-5$ | $0$ | $1/2$ | $0$ | $-6$ | $1$ | $1/2$ | $1/2$ | $5$
$-5$ | $1$ | $1/2$ | $1/2$ | $-6$ | $0$ | $0$ | $1/2$ | $5$
$-5$ | $1$ | $1/2$ | $1/2$ | $-6$ | $0$ | $1/2$ | $0$ | $5$
$-5$ | $1$ | $1/2$ | $1/2$ | $-6$ | $0$ | $1/2$ | $1/2$ | $5$
$-5$ | $1$ | $1/2$ | $1/2$ | $-6$ | $1$ | $0$ | $1/2$ | $6$
$-5$ | $1$ | $1/2$ | $1/2$ | $-6$ | $1$ | $1/2$ | $0$ | $6$
$-5$ | $1$ | $1/2$ | $1/2$ | $-6$ | $1$ | $1/2$ | $1/2$ | $6$
$-6$ | $0$ | $0$ | $1/2$ | $-6$ | $0$ | $0$ | $1/2$ | $7$
$-6$ | $0$ | $0$ | $1/2$ | $-6$ | $0$ | $1/2$ | $0$ | $6$
$-6$ | $0$ | $0$ | $1/2$ | $-6$ | $0$ | $1/2$ | $1/2$ | $6$
$-6$ | $0$ | $1/2$ | $0$ | $-6$ | $0$ | $1/2$ | $0$ | $7$
$-6$ | $0$ | $1/2$ | $0$ | $-6$ | $0$ | $1/2$ | $1/2$ | $6$
$-6$ | $0$ | $1/2$ | $1/2$ | $-6$ | $0$ | $1/2$ | $1/2$ | $7$
$-6$ | $0$ | $0$ | $1/2$ | $-6$ | $1$ | $0$ | $1/2$ | $5$
$-6$ | $0$ | $0$ | $1/2$ | $-6$ | $1$ | $1/2$ | $0$ | $6$
$-6$ | $0$ | $0$ | $1/2$ | $-6$ | $1$ | $1/2$ | $1/2$ | $6$
$-6$ | $0$ | $1/2$ | $0$ | $-6$ | $1$ | $0$ | $1/2$ | $6$
$-6$ | $0$ | $1/2$ | $0$ | $-6$ | $1$ | $1/2$ | $0$ | $5$
$-6$ | $0$ | $1/2$ | $0$ | $-6$ | $1$ | $1/2$ | $1/2$ | $6$
$-6$ | $0$ | $1/2$ | $1/2$ | $-6$ | $1$ | $0$ | $1/2$ | $6$
$-6$ | $0$ | $1/2$ | $1/2$ | $-6$ | $1$ | $1/2$ | $0$ | $6$
$-6$ | $0$ | $1/2$ | $1/2$ | $-6$ | $1$ | $1/2$ | $1/2$ | $5$
$-6$ | $1$ | $0$ | $1/2$ | $-6$ | $1$ | $0$ | $1/2$ | $7$
$-6$ | $1$ | $0$ | $1/2$ | $-6$ | $1$ | $1/2$ | $0$ | $6$
$-6$ | $1$ | $0$ | $1/2$ | $-6$ | $1$ | $1/2$ | $1/2$ | $6$
$-6$ | $1$ | $1/2$ | $0$ | $-6$ | $1$ | $1/2$ | $0$ | $7$
$-6$ | $1$ | $1/2$ | $0$ | $-6$ | $1$ | $1/2$ | $1/2$ | $6$
$-6$ | $1$ | $1/2$ | $1/2$ | $-6$ | $1$ | $1/2$ | $1/2$ | $7$
$-4$ | $0$ | $0$ | $0$ | $-7$ | $0$ | $1/2$ | $1/2$ | $5$
$-4$ | $0$ | $0$ | $0$ | $-7$ | $1$ | $0$ | $0$ | $5$
$-4$ | $0$ | $0$ | $0$ | $-7$ | $1$ | $0$ | $1/2$ | $5$
$-4$ | $0$ | $0$ | $0$ | $-7$ | $1$ | $1/2$ | $0$ | $5$
$-5$ | $0$ | $0$ | $0$ | $-7$ | $0$ | $1/2$ | $1/2$ | $6$
$-5$ | $0$ | $0$ | $1/2$ | $-7$ | $0$ | $1/2$ | $1/2$ | $6$
$-5$ | $0$ | $1/2$ | $0$ | $-7$ | $0$ | $1/2$ | $1/2$ | $6$
$-5$ | $0$ | $0$ | $0$ | $-7$ | $1$ | $0$ | $0$ | $5$
$-5$ | $0$ | $0$ | $0$ | $-7$ | $1$ | $0$ | $1/2$ | $6$
$-5$ | $0$ | $0$ | $0$ | $-7$ | $1$ | $1/2$ | $0$ | $6$
$-5$ | $0$ | $0$ | $1/2$ | $-7$ | $1$ | $0$ | $0$ | $6$
$-5$ | $0$ | $0$ | $1/2$ | $-7$ | $1$ | $0$ | $1/2$ | $5$
$-5$ | $0$ | $0$ | $1/2$ | $-7$ | $1$ | $1/2$ | $0$ | $6$
$-5$ | $0$ | $1/2$ | $0$ | $-7$ | $1$ | $0$ | $0$ | $6$
$-5$ | $0$ | $1/2$ | $0$ | $-7$ | $1$ | $0$ | $1/2$ | $6$
$-5$ | $0$ | $1/2$ | $0$ | $-7$ | $1$ | $1/2$ | $0$ | $5$
$-5$ | $1$ | $1/2$ | $1/2$ | $-7$ | $0$ | $1/2$ | $1/2$ | $5$
$-5$ | $1$ | $1/2$ | $1/2$ | $-7$ | $1$ | $0$ | $0$ | $6$
$-5$ | $1$ | $1/2$ | $1/2$ | $-7$ | $1$ | $0$ | $1/2$ | $6$
$-5$ | $1$ | $1/2$ | $1/2$ | $-7$ | $1$ | $1/2$ | $0$ | $6$
$-6$ | $0$ | $0$ | $1/2$ | $-7$ | $0$ | $1/2$ | $1/2$ | $7$
$-6$ | $0$ | $1/2$ | $0$ | $-7$ | $0$ | $1/2$ | $1/2$ | $7$
$-6$ | $0$ | $1/2$ | $1/2$ | $-7$ | $0$ | $1/2$ | $1/2$ | $7$
$-6$ | $0$ | $0$ | $1/2$ | $-7$ | $1$ | $0$ | $0$ | $6$
$-6$ | $0$ | $0$ | $1/2$ | $-7$ | $1$ | $0$ | $1/2$ | $6$
$-6$ | $0$ | $0$ | $1/2$ | $-7$ | $1$ | $1/2$ | $0$ | $7$
$-6$ | $0$ | $1/2$ | $0$ | $-7$ | $1$ | $0$ | $0$ | $6$
$-6$ | $0$ | $1/2$ | $0$ | $-7$ | $1$ | $0$ | $1/2$ | $7$
$-6$ | $0$ | $1/2$ | $0$ | $-7$ | $1$ | $1/2$ | $0$ | $6$
$-6$ | $0$ | $1/2$ | $1/2$ | $-7$ | $1$ | $0$ | $0$ | $7$
$-6$ | $0$ | $1/2$ | $1/2$ | $-7$ | $1$ | $0$ | $1/2$ | $6$
$-6$ | $0$ | $1/2$ | $1/2$ | $-7$ | $1$ | $1/2$ | $0$ | $6$
$-6$ | $1$ | $0$ | $1/2$ | $-7$ | $0$ | $1/2$ | $1/2$ | $6$
$-6$ | $1$ | $1/2$ | $0$ | $-7$ | $0$ | $1/2$ | $1/2$ | $6$
$-6$ | $1$ | $1/2$ | $1/2$ | $-7$ | $0$ | $1/2$ | $1/2$ | $6$
$-6$ | $1$ | $0$ | $1/2$ | $-7$ | $1$ | $0$ | $0$ | $7$
$-6$ | $1$ | $0$ | $1/2$ | $-7$ | $1$ | $0$ | $1/2$ | $7$
$-6$ | $1$ | $0$ | $1/2$ | $-7$ | $1$ | $1/2$ | $0$ | $6$
$-6$ | $1$ | $1/2$ | $0$ | $-7$ | $1$ | $0$ | $0$ | $7$
$-6$ | $1$ | $1/2$ | $0$ | $-7$ | $1$ | $0$ | $1/2$ | $6$
$-6$ | $1$ | $1/2$ | $0$ | $-7$ | $1$ | $1/2$ | $0$ | $7$
$-6$ | $1$ | $1/2$ | $1/2$ | $-7$ | $1$ | $0$ | $0$ | $6$
$-6$ | $1$ | $1/2$ | $1/2$ | $-7$ | $1$ | $0$ | $1/2$ | $7$
$-6$ | $1$ | $1/2$ | $1/2$ | $-7$ | $1$ | $1/2$ | $0$ | $7$
$-7$ | $0$ | $1/2$ | $1/2$ | $-7$ | $0$ | $1/2$ | $1/2$ | $8$
$-7$ | $0$ | $1/2$ | $1/2$ | $-7$ | $1$ | $0$ | $0$ | $7$
$-7$ | $0$ | $1/2$ | $1/2$ | $-7$ | $1$ | $0$ | $1/2$ | $7$
$-7$ | $0$ | $1/2$ | $1/2$ | $-7$ | $1$ | $1/2$ | $0$ | $7$
$-7$ | $1$ | $0$ | $0$ | $-7$ | $1$ | $0$ | $0$ | $8$
$-7$ | $1$ | $0$ | $0$ | $-7$ | $1$ | $0$ | $1/2$ | $7$
$-7$ | $1$ | $0$ | $0$ | $-7$ | $1$ | $1/2$ | $0$ | $7$
$-7$ | $1$ | $0$ | $1/2$ | $-7$ | $1$ | $0$ | $1/2$ | $8$
$-7$ | $1$ | $0$ | $1/2$ | $-7$ | $1$ | $1/2$ | $0$ | $7$
$-7$ | $1$ | $1/2$ | $0$ | $-7$ | $1$ | $1/2$ | $0$ | $8$
$-4$ | $0$ | $0$ | $0$ | $-8$ | $1$ | $0$ | $0$ | $5$
$-5$ | $0$ | $0$ | $0$ | $-8$ | $1$ | $0$ | $0$ | $6$
$-5$ | $0$ | $0$ | $1/2$ | $-8$ | $1$ | $0$ | $0$ | $6$
$-5$ | $0$ | $1/2$ | $0$ | $-8$ | $1$ | $0$ | $0$ | $6$
$-5$ | $1$ | $1/2$ | $1/2$ | $-8$ | $1$ | $0$ | $0$ | $6$
$-6$ | $0$ | $0$ | $1/2$ | $-8$ | $1$ | $0$ | $0$ | $7$
$-6$ | $0$ | $1/2$ | $0$ | $-8$ | $1$ | $0$ | $0$ | $7$
$-6$ | $0$ | $1/2$ | $1/2$ | $-8$ | $1$ | $0$ | $0$ | $7$
$-6$ | $1$ | $0$ | $1/2$ | $-8$ | $1$ | $0$ | $0$ | $7$
$-6$ | $1$ | $1/2$ | $0$ | $-8$ | $1$ | $0$ | $0$ | $7$
$-6$ | $1$ | $1/2$ | $1/2$ | $-8$ | $1$ | $0$ | $0$ | $7$
$-7$ | $0$ | $1/2$ | $1/2$ | $-8$ | $1$ | $0$ | $0$ | $8$
$-7$ | $1$ | $0$ | $0$ | $-8$ | $1$ | $0$ | $0$ | $8$
$-7$ | $1$ | $0$ | $1/2$ | $-8$ | $1$ | $0$ | $0$ | $8$
$-7$ | $1$ | $1/2$ | $0$ | $-8$ | $1$ | $0$ | $0$ | $8$
$-8$ | $1$ | $0$ | $0$ | $-8$ | $1$ | $0$ | $0$ | $9$

Table 16: Possible parameter configurations on $T^{2}/Z_{2}$ with
$M_{ab}<0,\,M_{ca}>0$. $T^{2}/Z_{2}$ with $M_{ab}<0,\,M_{ca}>0$
---
$ab$-sector |  $ca$-sector |  $bc$-sector
$M_{ab}$ | $\omega^{i}_{ab}$ | $\alpha_{ab}$ | $\beta_{ab}$ | $M_{ca}$ | $\omega^{i}_{ca}$ | $\alpha_{ca}$ | $\beta_{ca}$ | # of Higgs
$-4$ | $0$ | $0$ | $0$ | $4$ | $0$ | $0$ | $0$ | 1
$-4$ | $0$ | $0$ | $0$ | $5$ | $0$ | $0$ | $0$ | $1$
$-4$ | $0$ | $0$ | $0$ | $5$ | $0$ | $0$ | $1/2$ | $1$
$-4$ | $0$ | $0$ | $0$ | $5$ | $0$ | $1/2$ | $0$ | $1$
$-4$ | $0$ | $0$ | $0$ | $5$ | $1$ | $1/2$ | $1/2$ | $1$
$-5$ | $0$ | $0$ | $0$ | $5$ | $0$ | $0$ | $0$ | 1
$-5$ | $0$ | $0$ | $1/2$ | $5$ | $0$ | $0$ | $1/2$ | 1
$-5$ | $0$ | $1/2$ | $0$ | $5$ | $0$ | $1/2$ | $0$ | 1
$-5$ | $1$ | $1/2$ | $1/2$ | $5$ | $1$ | $1/2$ | $1/2$ | 1
$-4$ | $0$ | $0$ | $0$ | $6$ | $0$ | $0$ | $1/2$ | $1$
$-4$ | $0$ | $0$ | $0$ | $6$ | $0$ | $1/2$ | $0$ | $1$
$-4$ | $0$ | $0$ | $0$ | $6$ | $0$ | $1/2$ | $1/2$ | $1$
$-4$ | $0$ | $0$ | $0$ | $6$ | $1$ | $0$ | $1/2$ | $1$
$-4$ | $0$ | $0$ | $0$ | $6$ | $1$ | $1/2$ | $0$ | $1$
$-4$ | $0$ | $0$ | $0$ | $6$ | $1$ | $1/2$ | $1/2$ | $1$
$-5$ | $0$ | $0$ | $0$ | $6$ | $0$ | $0$ | $1/2$ | $1$
$-5$ | $0$ | $0$ | $0$ | $6$ | $0$ | $1/2$ | $0$ | $1$
$-5$ | $0$ | $0$ | $1/2$ | $6$ | $0$ | $0$ | $1/2$ | $1$
$-5$ | $0$ | $0$ | $1/2$ | $6$ | $0$ | $1/2$ | $1/2$ | $1$
$-5$ | $0$ | $1/2$ | $0$ | $6$ | $0$ | $1/2$ | $0$ | $1$
$-5$ | $0$ | $1/2$ | $0$ | $6$ | $0$ | $1/2$ | $1/2$ | $1$
$-5$ | $0$ | $0$ | $0$ | $6$ | $1$ | $1/2$ | $1/2$ | $1$
$-5$ | $0$ | $0$ | $1/2$ | $6$ | $1$ | $1/2$ | $0$ | $1$
$-5$ | $0$ | $1/2$ | $0$ | $6$ | $1$ | $0$ | $1/2$ | $1$
$-5$ | $1$ | $1/2$ | $1/2$ | $6$ | $1$ | $0$ | $1/2$ | $1$
$-5$ | $1$ | $1/2$ | $1/2$ | $6$ | $1$ | $1/2$ | $0$ | $1$
$-5$ | $1$ | $1/2$ | $1/2$ | $6$ | $1$ | $1/2$ | $1/2$ | $1$
$-6$ | $0$ | $0$ | $1/2$ | $6$ | $0$ | $0$ | $1/2$ | 1
$-6$ | $0$ | $1/2$ | $0$ | $6$ | $0$ | $1/2$ | $0$ | 1
$-6$ | $0$ | $1/2$ | $1/2$ | $6$ | $0$ | $1/2$ | $1/2$ | 1
$-6$ | $1$ | $0$ | $1/2$ | $6$ | $1$ | $0$ | $1/2$ | 1
$-6$ | $1$ | $1/2$ | $0$ | $6$ | $1$ | $1/2$ | $0$ | 1
$-6$ | $1$ | $1/2$ | $1/2$ | $6$ | $1$ | $1/2$ | $1/2$ | 1
$-4$ | $0$ | $0$ | $0$ | $7$ | $0$ | $1/2$ | $1/2$ | $1$
$-4$ | $0$ | $0$ | $0$ | $7$ | $1$ | $0$ | $0$ | $1$
$-4$ | $0$ | $0$ | $0$ | $7$ | $1$ | $0$ | $1/2$ | $1$
$-4$ | $0$ | $0$ | $0$ | $7$ | $1$ | $1/2$ | $0$ | $1$
$-5$ | $0$ | $0$ | $0$ | $7$ | $0$ | $1/2$ | $1/2$ | $1$
$-5$ | $0$ | $0$ | $1/2$ | $7$ | $0$ | $1/2$ | $1/2$ | $1$
$-5$ | $0$ | $1/2$ | $0$ | $7$ | $0$ | $1/2$ | $1/2$ | $1$
$-5$ | $0$ | $0$ | $0$ | $7$ | $1$ | $0$ | $1/2$ | $1$
$-5$ | $0$ | $0$ | $0$ | $7$ | $1$ | $1/2$ | $0$ | $1$
$-5$ | $0$ | $0$ | $1/2$ | $7$ | $1$ | $0$ | $0$ | $1$
$-5$ | $0$ | $0$ | $1/2$ | $7$ | $1$ | $1/2$ | $0$ | $1$
$-5$ | $0$ | $1/2$ | $0$ | $7$ | $1$ | $0$ | $0$ | $1$
$-5$ | $0$ | $1/2$ | $0$ | $7$ | $1$ | $0$ | $1/2$ | $1$
$-5$ | $1$ | $1/2$ | $1/2$ | $7$ | $1$ | $0$ | $0$ | $1$
$-5$ | $1$ | $1/2$ | $1/2$ | $7$ | $1$ | $0$ | $1/2$ | $1$
$-5$ | $1$ | $1/2$ | $1/2$ | $7$ | $1$ | $1/2$ | $0$ | $1$
$-6$ | $0$ | $0$ | $1/2$ | $7$ | $0$ | $1/2$ | $1/2$ | $1$
$-6$ | $0$ | $1/2$ | $0$ | $7$ | $0$ | $1/2$ | $1/2$ | $1$
$-6$ | $0$ | $1/2$ | $1/2$ | $7$ | $0$ | $1/2$ | $1/2$ | $1$
$-6$ | $0$ | $0$ | $1/2$ | $7$ | $1$ | $1/2$ | $0$ | $1$
$-6$ | $0$ | $1/2$ | $0$ | $7$ | $1$ | $0$ | $1/2$ | $1$
$-6$ | $0$ | $1/2$ | $1/2$ | $7$ | $1$ | $0$ | $0$ | $1$
$-6$ | $1$ | $0$ | $1/2$ | $7$ | $1$ | $0$ | $0$ | $1$
$-6$ | $1$ | $0$ | $1/2$ | $7$ | $1$ | $0$ | $1/2$ | $1$
$-6$ | $1$ | $1/2$ | $0$ | $7$ | $1$ | $0$ | $0$ | $1$
$-6$ | $1$ | $1/2$ | $0$ | $7$ | $1$ | $1/2$ | $0$ | $1$
$-6$ | $1$ | $1/2$ | $1/2$ | $7$ | $1$ | $0$ | $1/2$ | $1$
$-6$ | $1$ | $1/2$ | $1/2$ | $7$ | $1$ | $1/2$ | $0$ | $1$
$-7$ | $0$ | $1/2$ | $1/2$ | $7$ | $0$ | $1/2$ | $1/2$ | 1
$-7$ | $1$ | $0$ | $0$ | $7$ | $1$ | $0$ | $0$ | 1
$-7$ | $1$ | $0$ | $1/2$ | $7$ | $1$ | $0$ | $1/2$ | 1
$-7$ | $1$ | $1/2$ | $0$ | $7$ | $1$ | $1/2$ | $0$ | 1
$-4$ | $0$ | $0$ | $0$ | $8$ | $1$ | $0$ | $0$ | $1$
$-5$ | $0$ | $0$ | $0$ | $8$ | $1$ | $0$ | $0$ | $1$
$-5$ | $0$ | $0$ | $1/2$ | $8$ | $1$ | $0$ | $0$ | $1$
$-5$ | $0$ | $1/2$ | $0$ | $8$ | $1$ | $0$ | $0$ | $1$
$-5$ | $1$ | $1/2$ | $1/2$ | $8$ | $1$ | $0$ | $0$ | $1$
$-6$ | $0$ | $0$ | $1/2$ | $8$ | $1$ | $0$ | $0$ | $1$
$-6$ | $0$ | $1/2$ | $0$ | $8$ | $1$ | $0$ | $0$ | $1$
$-6$ | $0$ | $1/2$ | $1/2$ | $8$ | $1$ | $0$ | $0$ | $1$
$-6$ | $1$ | $0$ | $1/2$ | $8$ | $1$ | $0$ | $0$ | $1$
$-6$ | $1$ | $1/2$ | $0$ | $8$ | $1$ | $0$ | $0$ | $1$
$-6$ | $1$ | $1/2$ | $1/2$ | $8$ | $1$ | $0$ | $0$ | $1$
$-7$ | $0$ | $1/2$ | $1/2$ | $8$ | $1$ | $0$ | $0$ | $1$
$-7$ | $1$ | $0$ | $0$ | $8$ | $1$ | $0$ | $0$ | $1$
$-7$ | $1$ | $0$ | $1/2$ | $8$ | $1$ | $0$ | $0$ | $1$
$-7$ | $1$ | $1/2$ | $0$ | $8$ | $1$ | $0$ | $0$ | $1$
$-8$ | $1$ | $0$ | $0$ | $8$ | $1$ | $0$ | $0$ | 1

###  B.2 $T^{2}/Z_{3}$

Table 17: Possible parameter configurations on $T^{2}/Z_{3}$ with
$M_{ab}<0,\,M_{ca}<0$. $T^{2}/Z_{3}$ with $M_{ab}<0,\,M_{ca}<0$
---
$ab$-sector |  $ca$-sector |  $bc$-sector
$M_{ab}$ | $\omega^{i}_{ab}$ | $\alpha_{ab}$ | $M_{ca}$ | $\omega^{i}_{ca}$ | $\alpha_{ca}$ | # of Higgs
$-6$ | $0$ | $0$ | $-6$ | $0$ | $0$ | $5$
$-6$ | $0$ | $0$ | $-7$ | $0$ | $1/6$ | $5$
$-6$ | $0$ | $0$ | $-7$ | $0$ | $5/6$ | $5$
$-6$ | $0$ | $0$ | $-7$ | $1$ | $1/2$ | $4$
$-7$ | $0$ | $1/6$ | $-7$ | $0$ | $1/6$ | $5$
$-7$ | $0$ | $1/6$ | $-7$ | $0$ | $5/6$ | $5$
$-7$ | $0$ | $5/6$ | $-7$ | $0$ | $5/6$ | $5$
$-7$ | $0$ | $1/6$ | $-7$ | $1$ | $1/2$ | $4$
$-7$ | $0$ | $5/6$ | $-7$ | $1$ | $1/2$ | $4$
$-7$ | $1$ | $1/2$ | $-7$ | $1$ | $1/2$ | $4$
$-6$ | $0$ | $0$ | $-8$ | $0$ | $0$ | $5$
$-6$ | $0$ | $0$ | $-8$ | $0$ | $1/3$ | $5$
$-6$ | $0$ | $0$ | $-8$ | $0$ | $2/3$ | $5$
$-6$ | $0$ | $0$ | $-8$ | $1$ | $1/3$ | $4$
$-6$ | $0$ | $0$ | $-8$ | $1$ | $2/3$ | $4$
$-6$ | $0$ | $0$ | $-8$ | $2$ | $0$ | $4$
$-7$ | $0$ | $1/6$ | $-8$ | $0$ | $0$ | $5$
$-7$ | $0$ | $1/6$ | $-8$ | $0$ | $1/3$ | $6$
$-7$ | $0$ | $1/6$ | $-8$ | $0$ | $2/3$ | $5$
$-7$ | $0$ | $5/6$ | $-8$ | $0$ | $0$ | $5$
$-7$ | $0$ | $5/6$ | $-8$ | $0$ | $1/3$ | $5$
$-7$ | $0$ | $5/6$ | $-8$ | $0$ | $2/3$ | $6$
$-7$ | $0$ | $1/6$ | $-8$ | $1$ | $1/3$ | $4$
$-7$ | $0$ | $1/6$ | $-8$ | $1$ | $2/3$ | $5$
$-7$ | $0$ | $5/6$ | $-8$ | $1$ | $1/3$ | $5$
$-7$ | $0$ | $5/6$ | $-8$ | $1$ | $2/3$ | $4$
$-7$ | $0$ | $1/6$ | $-8$ | $2$ | $0$ | $5$
$-7$ | $0$ | $5/6$ | $-8$ | $2$ | $0$ | $5$
$-7$ | $1$ | $1/2$ | $-8$ | $0$ | $0$ | $4$
$-7$ | $1$ | $1/2$ | $-8$ | $0$ | $1/3$ | $5$
$-7$ | $1$ | $1/2$ | $-8$ | $0$ | $2/3$ | $5$
$-7$ | $1$ | $1/2$ | $-8$ | $1$ | $1/3$ | $5$
$-7$ | $1$ | $1/2$ | $-8$ | $1$ | $2/3$ | $5$
$-7$ | $1$ | $1/2$ | $-8$ | $2$ | $0$ | $6$
$-8$ | $0$ | $0$ | $-8$ | $0$ | $0$ | $5$
$-8$ | $0$ | $0$ | $-8$ | $0$ | $1/3$ | $6$
$-8$ | $0$ | $0$ | $-8$ | $0$ | $2/3$ | $6$
$-8$ | $0$ | $1/3$ | $-8$ | $0$ | $1/3$ | $6$
$-8$ | $0$ | $1/3$ | $-8$ | $0$ | $2/3$ | $5$
$-8$ | $0$ | $2/3$ | $-8$ | $0$ | $2/3$ | $6$
$-8$ | $0$ | $0$ | $-8$ | $1$ | $1/3$ | $5$
$-8$ | $0$ | $0$ | $-8$ | $1$ | $2/3$ | $5$
$-8$ | $0$ | $1/3$ | $-8$ | $1$ | $1/3$ | $5$
$-8$ | $0$ | $1/3$ | $-8$ | $1$ | $2/3$ | $5$
$-8$ | $0$ | $2/3$ | $-8$ | $1$ | $1/3$ | $5$
$-8$ | $0$ | $2/3$ | $-8$ | $1$ | $2/3$ | $5$
$-8$ | $0$ | $0$ | $-8$ | $2$ | $0$ | $6$
$-8$ | $0$ | $1/3$ | $-8$ | $2$ | $0$ | $5$
$-8$ | $0$ | $2/3$ | $-8$ | $2$ | $0$ | $5$
$-8$ | $1$ | $1/3$ | $-8$ | $1$ | $1/3$ | $5$
$-8$ | $1$ | $1/3$ | $-8$ | $1$ | $2/3$ | $6$
$-8$ | $1$ | $2/3$ | $-8$ | $1$ | $2/3$ | $5$
$-8$ | $1$ | $1/3$ | $-8$ | $2$ | $0$ | $6$
$-8$ | $1$ | $2/3$ | $-8$ | $2$ | $0$ | $6$
$-8$ | $2$ | $0$ | $-8$ | $2$ | $0$ | $5$
$-6$ | $0$ | $0$ | $-9$ | $0$ | $1/6$ | $5$
$-6$ | $0$ | $0$ | $-9$ | $0$ | $5/6$ | $5$
$-6$ | $0$ | $0$ | $-9$ | $1$ | $1/6$ | $5$
$-6$ | $0$ | $0$ | $-9$ | $1$ | $1/2$ | $4$
$-6$ | $0$ | $0$ | $-9$ | $1$ | $5/6$ | $5$
$-6$ | $0$ | $0$ | $-9$ | $2$ | $1/6$ | $5$
$-6$ | $0$ | $0$ | $-9$ | $2$ | $5/6$ | $5$
$-7$ | $0$ | $1/6$ | $-9$ | $0$ | $1/6$ | $6$
$-7$ | $0$ | $1/6$ | $-9$ | $0$ | $5/6$ | $5$
$-7$ | $0$ | $5/6$ | $-9$ | $0$ | $1/6$ | $5$
$-7$ | $0$ | $5/6$ | $-9$ | $0$ | $5/6$ | $6$
$-7$ | $0$ | $1/6$ | $-9$ | $1$ | $1/6$ | $5$
$-7$ | $0$ | $1/6$ | $-9$ | $1$ | $1/2$ | $5$
$-7$ | $0$ | $1/6$ | $-9$ | $1$ | $5/6$ | $5$
$-7$ | $0$ | $5/6$ | $-9$ | $1$ | $1/6$ | $5$
$-7$ | $0$ | $5/6$ | $-9$ | $1$ | $1/2$ | $5$
$-7$ | $0$ | $5/6$ | $-9$ | $1$ | $5/6$ | $5$
$-7$ | $0$ | $1/6$ | $-9$ | $2$ | $1/6$ | $5$
$-7$ | $0$ | $1/6$ | $-9$ | $2$ | $5/6$ | $6$
$-7$ | $0$ | $5/6$ | $-9$ | $2$ | $1/6$ | $6$
$-7$ | $0$ | $5/6$ | $-9$ | $2$ | $5/6$ | $5$
$-7$ | $1$ | $1/2$ | $-9$ | $0$ | $1/6$ | $5$
$-7$ | $1$ | $1/2$ | $-9$ | $0$ | $5/6$ | $5$
$-7$ | $1$ | $1/2$ | $-9$ | $1$ | $1/6$ | $5$
$-7$ | $1$ | $1/2$ | $-9$ | $1$ | $1/2$ | $6$
$-7$ | $1$ | $1/2$ | $-9$ | $1$ | $5/6$ | $5$
$-7$ | $1$ | $1/2$ | $-9$ | $2$ | $1/6$ | $6$
$-7$ | $1$ | $1/2$ | $-9$ | $2$ | $5/6$ | $6$
$-8$ | $0$ | $0$ | $-9$ | $0$ | $1/6$ | $6$
$-8$ | $0$ | $0$ | $-9$ | $0$ | $5/6$ | $6$
$-8$ | $0$ | $1/3$ | $-9$ | $0$ | $1/6$ | $6$
$-8$ | $0$ | $1/3$ | $-9$ | $0$ | $5/6$ | $6$
$-8$ | $0$ | $2/3$ | $-9$ | $0$ | $1/6$ | $6$
$-8$ | $0$ | $2/3$ | $-9$ | $0$ | $5/6$ | $6$
$-8$ | $0$ | $0$ | $-9$ | $1$ | $1/6$ | $5$
$-8$ | $0$ | $0$ | $-9$ | $1$ | $1/2$ | $6$
$-8$ | $0$ | $0$ | $-9$ | $1$ | $5/6$ | $5$
$-8$ | $0$ | $1/3$ | $-9$ | $1$ | $1/6$ | $6$
$-8$ | $0$ | $1/3$ | $-9$ | $1$ | $1/2$ | $5$
$-8$ | $0$ | $1/3$ | $-9$ | $1$ | $5/6$ | $5$
$-8$ | $0$ | $2/3$ | $-9$ | $1$ | $1/6$ | $5$
$-8$ | $0$ | $2/3$ | $-9$ | $1$ | $1/2$ | $5$
$-8$ | $0$ | $2/3$ | $-9$ | $1$ | $5/6$ | $6$
$-8$ | $0$ | $0$ | $-9$ | $2$ | $1/6$ | $6$
$-8$ | $0$ | $0$ | $-9$ | $2$ | $5/6$ | $6$
$-8$ | $0$ | $1/3$ | $-9$ | $2$ | $1/6$ | $5$
$-8$ | $0$ | $1/3$ | $-9$ | $2$ | $5/6$ | $6$
$-8$ | $0$ | $2/3$ | $-9$ | $2$ | $1/6$ | $6$
$-8$ | $0$ | $2/3$ | $-9$ | $2$ | $5/6$ | $5$
$-8$ | $1$ | $1/3$ | $-9$ | $0$ | $1/6$ | $6$
$-8$ | $1$ | $1/3$ | $-9$ | $0$ | $5/6$ | $5$
$-8$ | $1$ | $2/3$ | $-9$ | $0$ | $1/6$ | $5$
$-8$ | $1$ | $2/3$ | $-9$ | $0$ | $5/6$ | $6$
$-8$ | $1$ | $1/3$ | $-9$ | $1$ | $1/6$ | $5$
$-8$ | $1$ | $1/3$ | $-9$ | $1$ | $1/2$ | $6$
$-8$ | $1$ | $1/3$ | $-9$ | $1$ | $5/6$ | $6$
$-8$ | $1$ | $2/3$ | $-9$ | $1$ | $1/6$ | $6$
$-8$ | $1$ | $2/3$ | $-9$ | $1$ | $1/2$ | $6$
$-8$ | $1$ | $2/3$ | $-9$ | $1$ | $5/6$ | $5$
$-8$ | $1$ | $1/3$ | $-9$ | $2$ | $1/6$ | $6$
$-8$ | $1$ | $1/3$ | $-9$ | $2$ | $5/6$ | $6$
$-8$ | $1$ | $2/3$ | $-9$ | $2$ | $1/6$ | $6$
$-8$ | $1$ | $2/3$ | $-9$ | $2$ | $5/6$ | $6$
$-8$ | $2$ | $0$ | $-9$ | $0$ | $1/6$ | $6$
$-8$ | $2$ | $0$ | $-9$ | $0$ | $5/6$ | $6$
$-8$ | $2$ | $0$ | $-9$ | $1$ | $1/6$ | $6$
$-8$ | $2$ | $0$ | $-9$ | $1$ | $1/2$ | $6$
$-8$ | $2$ | $0$ | $-9$ | $1$ | $5/6$ | $6$
$-8$ | $2$ | $0$ | $-9$ | $2$ | $1/6$ | $5$
$-8$ | $2$ | $0$ | $-9$ | $2$ | $5/6$ | $5$
$-9$ | $0$ | $1/6$ | $-9$ | $0$ | $1/6$ | $6$
$-9$ | $0$ | $1/6$ | $-9$ | $0$ | $5/6$ | $7$
$-9$ | $0$ | $5/6$ | $-9$ | $0$ | $5/6$ | $6$
$-9$ | $0$ | $1/6$ | $-9$ | $1$ | $1/6$ | $6$
$-9$ | $0$ | $1/6$ | $-9$ | $1$ | $1/2$ | $6$
$-9$ | $0$ | $1/6$ | $-9$ | $1$ | $5/6$ | $5$
$-9$ | $0$ | $5/6$ | $-9$ | $1$ | $1/6$ | $5$
$-9$ | $0$ | $5/6$ | $-9$ | $1$ | $1/2$ | $6$
$-9$ | $0$ | $5/6$ | $-9$ | $1$ | $5/6$ | $6$
$-9$ | $0$ | $1/6$ | $-9$ | $2$ | $1/6$ | $6$
$-9$ | $0$ | $1/6$ | $-9$ | $2$ | $5/6$ | $6$
$-9$ | $0$ | $5/6$ | $-9$ | $2$ | $1/6$ | $6$
$-9$ | $0$ | $5/6$ | $-9$ | $2$ | $5/6$ | $6$
$-9$ | $1$ | $1/6$ | $-9$ | $1$ | $1/6$ | $6$
$-9$ | $1$ | $1/6$ | $-9$ | $1$ | $1/2$ | $6$
$-9$ | $1$ | $1/6$ | $-9$ | $1$ | $5/6$ | $6$
$-9$ | $1$ | $1/2$ | $-9$ | $1$ | $1/2$ | $6$
$-9$ | $1$ | $1/2$ | $-9$ | $1$ | $5/6$ | $6$
$-9$ | $1$ | $5/6$ | $-9$ | $1$ | $5/6$ | $6$
$-9$ | $1$ | $1/6$ | $-9$ | $2$ | $1/6$ | $6$
$-9$ | $1$ | $1/6$ | $-9$ | $2$ | $5/6$ | $7$
$-9$ | $1$ | $1/2$ | $-9$ | $2$ | $1/6$ | $6$
$-9$ | $1$ | $1/2$ | $-9$ | $2$ | $5/6$ | $6$
$-9$ | $1$ | $5/6$ | $-9$ | $2$ | $1/6$ | $7$
$-9$ | $1$ | $5/6$ | $-9$ | $2$ | $5/6$ | $6$
$-9$ | $2$ | $1/6$ | $-9$ | $2$ | $1/6$ | $6$
$-9$ | $2$ | $1/6$ | $-9$ | $2$ | $5/6$ | $5$
$-9$ | $2$ | $5/6$ | $-9$ | $2$ | $5/6$ | $6$
$-6$ | $0$ | $0$ | $-10$ | $0$ | $0$ | $5$
$-6$ | $0$ | $0$ | $-10$ | $1$ | $1/3$ | $5$
$-6$ | $0$ | $0$ | $-10$ | $1$ | $2/3$ | $5$
$-6$ | $0$ | $0$ | $-10$ | $2$ | $0$ | $6$
$-6$ | $0$ | $0$ | $-10$ | $2$ | $1/3$ | $5$
$-6$ | $0$ | $0$ | $-10$ | $2$ | $2/3$ | $5$
$-7$ | $0$ | $1/6$ | $-10$ | $0$ | $0$ | $6$
$-7$ | $0$ | $5/6$ | $-10$ | $0$ | $0$ | $6$
$-7$ | $0$ | $1/6$ | $-10$ | $1$ | $1/3$ | $6$
$-7$ | $0$ | $1/6$ | $-10$ | $1$ | $2/3$ | $5$
$-7$ | $0$ | $5/6$ | $-10$ | $1$ | $1/3$ | $5$
$-7$ | $0$ | $5/6$ | $-10$ | $1$ | $2/3$ | $6$
$-7$ | $0$ | $1/6$ | $-10$ | $2$ | $0$ | $6$
$-7$ | $0$ | $1/6$ | $-10$ | $2$ | $1/3$ | $5$
$-7$ | $0$ | $1/6$ | $-10$ | $2$ | $2/3$ | $6$
$-7$ | $0$ | $5/6$ | $-10$ | $2$ | $0$ | $6$
$-7$ | $0$ | $5/6$ | $-10$ | $2$ | $1/3$ | $6$
$-7$ | $0$ | $5/6$ | $-10$ | $2$ | $2/3$ | $5$
$-7$ | $1$ | $1/2$ | $-10$ | $0$ | $0$ | $6$
$-7$ | $1$ | $1/2$ | $-10$ | $1$ | $1/3$ | $6$
$-7$ | $1$ | $1/2$ | $-10$ | $1$ | $2/3$ | $6$
$-7$ | $1$ | $1/2$ | $-10$ | $2$ | $0$ | $6$
$-7$ | $1$ | $1/2$ | $-10$ | $2$ | $1/3$ | $6$
$-7$ | $1$ | $1/2$ | $-10$ | $2$ | $2/3$ | $6$
$-8$ | $0$ | $0$ | $-10$ | $0$ | $0$ | $7$
$-8$ | $0$ | $1/3$ | $-10$ | $0$ | $0$ | $6$
$-8$ | $0$ | $2/3$ | $-10$ | $0$ | $0$ | $6$
$-8$ | $0$ | $0$ | $-10$ | $1$ | $1/3$ | $6$
$-8$ | $0$ | $0$ | $-10$ | $1$ | $2/3$ | $6$
$-8$ | $0$ | $1/3$ | $-10$ | $1$ | $1/3$ | $6$
$-8$ | $0$ | $1/3$ | $-10$ | $1$ | $2/3$ | $5$
$-8$ | $0$ | $2/3$ | $-10$ | $1$ | $1/3$ | $5$
$-8$ | $0$ | $2/3$ | $-10$ | $1$ | $2/3$ | $6$
$-8$ | $0$ | $0$ | $-10$ | $2$ | $0$ | $6$
$-8$ | $0$ | $0$ | $-10$ | $2$ | $1/3$ | $6$
$-8$ | $0$ | $0$ | $-10$ | $2$ | $2/3$ | $6$
$-8$ | $0$ | $1/3$ | $-10$ | $2$ | $0$ | $6$
$-8$ | $0$ | $1/3$ | $-10$ | $2$ | $1/3$ | $6$
$-8$ | $0$ | $1/3$ | $-10$ | $2$ | $2/3$ | $6$
$-8$ | $0$ | $2/3$ | $-10$ | $2$ | $0$ | $6$
$-8$ | $0$ | $2/3$ | $-10$ | $2$ | $1/3$ | $6$
$-8$ | $0$ | $2/3$ | $-10$ | $2$ | $2/3$ | $6$
$-8$ | $1$ | $1/3$ | $-10$ | $0$ | $0$ | $6$
$-8$ | $1$ | $2/3$ | $-10$ | $0$ | $0$ | $6$
$-8$ | $1$ | $1/3$ | $-10$ | $1$ | $1/3$ | $6$
$-8$ | $1$ | $1/3$ | $-10$ | $1$ | $2/3$ | $6$
$-8$ | $1$ | $2/3$ | $-10$ | $1$ | $1/3$ | $6$
$-8$ | $1$ | $2/3$ | $-10$ | $1$ | $2/3$ | $6$
$-8$ | $1$ | $1/3$ | $-10$ | $2$ | $0$ | $6$
$-8$ | $1$ | $1/3$ | $-10$ | $2$ | $1/3$ | $6$
$-8$ | $1$ | $1/3$ | $-10$ | $2$ | $2/3$ | $7$
$-8$ | $1$ | $2/3$ | $-10$ | $2$ | $0$ | $6$
$-8$ | $1$ | $2/3$ | $-10$ | $2$ | $1/3$ | $7$
$-8$ | $1$ | $2/3$ | $-10$ | $2$ | $2/3$ | $6$
$-8$ | $2$ | $0$ | $-10$ | $0$ | $0$ | $6$
$-8$ | $2$ | $0$ | $-10$ | $1$ | $1/3$ | $6$
$-8$ | $2$ | $0$ | $-10$ | $1$ | $2/3$ | $6$
$-8$ | $2$ | $0$ | $-10$ | $2$ | $0$ | $5$
$-8$ | $2$ | $0$ | $-10$ | $2$ | $1/3$ | $6$
$-8$ | $2$ | $0$ | $-10$ | $2$ | $2/3$ | $6$
$-9$ | $0$ | $1/6$ | $-10$ | $0$ | $0$ | $7$
$-9$ | $0$ | $5/6$ | $-10$ | $0$ | $0$ | $7$
$-9$ | $0$ | $1/6$ | $-10$ | $1$ | $1/3$ | $6$
$-9$ | $0$ | $1/6$ | $-10$ | $1$ | $2/3$ | $6$
$-9$ | $0$ | $5/6$ | $-10$ | $1$ | $1/3$ | $6$
$-9$ | $0$ | $5/6$ | $-10$ | $1$ | $2/3$ | $6$
$-9$ | $0$ | $1/6$ | $-10$ | $2$ | $0$ | $6$
$-9$ | $0$ | $1/6$ | $-10$ | $2$ | $1/3$ | $7$
$-9$ | $0$ | $1/6$ | $-10$ | $2$ | $2/3$ | $6$
$-9$ | $0$ | $5/6$ | $-10$ | $2$ | $0$ | $6$
$-9$ | $0$ | $5/6$ | $-10$ | $2$ | $1/3$ | $6$
$-9$ | $0$ | $5/6$ | $-10$ | $2$ | $2/3$ | $7$
$-9$ | $1$ | $1/6$ | $-10$ | $0$ | $0$ | $6$
$-9$ | $1$ | $1/2$ | $-10$ | $0$ | $0$ | $6$
$-9$ | $1$ | $5/6$ | $-10$ | $0$ | $0$ | $6$
$-9$ | $1$ | $1/6$ | $-10$ | $1$ | $1/3$ | $7$
$-9$ | $1$ | $1/6$ | $-10$ | $1$ | $2/3$ | $6$
$-9$ | $1$ | $1/2$ | $-10$ | $1$ | $1/3$ | $6$
$-9$ | $1$ | $1/2$ | $-10$ | $1$ | $2/3$ | $6$
$-9$ | $1$ | $5/6$ | $-10$ | $1$ | $1/3$ | $6$
$-9$ | $1$ | $5/6$ | $-10$ | $1$ | $2/3$ | $7$
$-9$ | $1$ | $1/6$ | $-10$ | $2$ | $0$ | $7$
$-9$ | $1$ | $1/6$ | $-10$ | $2$ | $1/3$ | $6$
$-9$ | $1$ | $1/6$ | $-10$ | $2$ | $2/3$ | $7$
$-9$ | $1$ | $1/2$ | $-10$ | $2$ | $0$ | $6$
$-9$ | $1$ | $1/2$ | $-10$ | $2$ | $1/3$ | $7$
$-9$ | $1$ | $1/2$ | $-10$ | $2$ | $2/3$ | $7$
$-9$ | $1$ | $5/6$ | $-10$ | $2$ | $0$ | $7$
$-9$ | $1$ | $5/6$ | $-10$ | $2$ | $1/3$ | $7$
$-9$ | $1$ | $5/6$ | $-10$ | $2$ | $2/3$ | $6$
$-9$ | $2$ | $1/6$ | $-10$ | $0$ | $0$ | $6$
$-9$ | $2$ | $5/6$ | $-10$ | $0$ | $0$ | $6$
$-9$ | $2$ | $1/6$ | $-10$ | $1$ | $1/3$ | $6$
$-9$ | $2$ | $1/6$ | $-10$ | $1$ | $2/3$ | $7$
$-9$ | $2$ | $5/6$ | $-10$ | $1$ | $1/3$ | $7$
$-9$ | $2$ | $5/6$ | $-10$ | $1$ | $2/3$ | $6$
$-9$ | $2$ | $1/6$ | $-10$ | $2$ | $0$ | $6$
$-9$ | $2$ | $1/6$ | $-10$ | $2$ | $1/3$ | $6$
$-9$ | $2$ | $1/6$ | $-10$ | $2$ | $2/3$ | $6$
$-9$ | $2$ | $5/6$ | $-10$ | $2$ | $0$ | $6$
$-9$ | $2$ | $5/6$ | $-10$ | $2$ | $1/3$ | $6$
$-9$ | $2$ | $5/6$ | $-10$ | $2$ | $2/3$ | $6$
$-10$ | $0$ | $0$ | $-10$ | $0$ | $0$ | $7$
$-10$ | $0$ | $0$ | $-10$ | $1$ | $1/3$ | $6$
$-10$ | $0$ | $0$ | $-10$ | $1$ | $2/3$ | $6$
$-10$ | $0$ | $0$ | $-10$ | $2$ | $0$ | $6$
$-10$ | $0$ | $0$ | $-10$ | $2$ | $1/3$ | $7$
$-10$ | $0$ | $0$ | $-10$ | $2$ | $2/3$ | $7$
$-10$ | $1$ | $1/3$ | $-10$ | $1$ | $1/3$ | $7$
$-10$ | $1$ | $1/3$ | $-10$ | $1$ | $2/3$ | $6$
$-10$ | $1$ | $2/3$ | $-10$ | $1$ | $2/3$ | $7$
$-10$ | $1$ | $1/3$ | $-10$ | $2$ | $0$ | $7$
$-10$ | $1$ | $1/3$ | $-10$ | $2$ | $1/3$ | $7$
$-10$ | $1$ | $1/3$ | $-10$ | $2$ | $2/3$ | $7$
$-10$ | $1$ | $2/3$ | $-10$ | $2$ | $0$ | $7$
$-10$ | $1$ | $2/3$ | $-10$ | $2$ | $1/3$ | $7$
$-10$ | $1$ | $2/3$ | $-10$ | $2$ | $2/3$ | $7$
$-10$ | $2$ | $0$ | $-10$ | $2$ | $0$ | $7$
$-10$ | $2$ | $0$ | $-10$ | $2$ | $1/3$ | $6$
$-10$ | $2$ | $0$ | $-10$ | $2$ | $2/3$ | $6$
$-10$ | $2$ | $1/3$ | $-10$ | $2$ | $1/3$ | $6$
$-10$ | $2$ | $1/3$ | $-10$ | $2$ | $2/3$ | $7$
$-10$ | $2$ | $2/3$ | $-10$ | $2$ | $2/3$ | $6$
$-6$ | $0$ | $0$ | $-11$ | $1$ | $1/2$ | $6$
$-6$ | $0$ | $0$ | $-11$ | $2$ | $1/6$ | $6$
$-6$ | $0$ | $0$ | $-11$ | $2$ | $5/6$ | $6$
$-7$ | $0$ | $1/6$ | $-11$ | $1$ | $1/2$ | $6$
$-7$ | $0$ | $5/6$ | $-11$ | $1$ | $1/2$ | $6$
$-7$ | $0$ | $1/6$ | $-11$ | $2$ | $1/6$ | $6$
$-7$ | $0$ | $1/6$ | $-11$ | $2$ | $5/6$ | $6$
$-7$ | $0$ | $5/6$ | $-11$ | $2$ | $1/6$ | $6$
$-7$ | $0$ | $5/6$ | $-11$ | $2$ | $5/6$ | $6$
$-7$ | $1$ | $1/2$ | $-11$ | $1$ | $1/2$ | $6$
$-7$ | $1$ | $1/2$ | $-11$ | $2$ | $1/6$ | $6$
$-7$ | $1$ | $1/2$ | $-11$ | $2$ | $5/6$ | $6$
$-8$ | $0$ | $0$ | $-11$ | $1$ | $1/2$ | $6$
$-8$ | $0$ | $1/3$ | $-11$ | $1$ | $1/2$ | $6$
$-8$ | $0$ | $2/3$ | $-11$ | $1$ | $1/2$ | $6$
$-8$ | $0$ | $0$ | $-11$ | $2$ | $1/6$ | $6$
$-8$ | $0$ | $0$ | $-11$ | $2$ | $5/6$ | $6$
$-8$ | $0$ | $1/3$ | $-11$ | $2$ | $1/6$ | $7$
$-8$ | $0$ | $1/3$ | $-11$ | $2$ | $5/6$ | $6$
$-8$ | $0$ | $2/3$ | $-11$ | $2$ | $1/6$ | $6$
$-8$ | $0$ | $2/3$ | $-11$ | $2$ | $5/6$ | $7$
$-8$ | $1$ | $1/3$ | $-11$ | $1$ | $1/2$ | $6$
$-8$ | $1$ | $2/3$ | $-11$ | $1$ | $1/2$ | $6$
$-8$ | $1$ | $1/3$ | $-11$ | $2$ | $1/6$ | $6$
$-8$ | $1$ | $1/3$ | $-11$ | $2$ | $5/6$ | $7$
$-8$ | $1$ | $2/3$ | $-11$ | $2$ | $1/6$ | $7$
$-8$ | $1$ | $2/3$ | $-11$ | $2$ | $5/6$ | $6$
$-8$ | $2$ | $0$ | $-11$ | $1$ | $1/2$ | $6$
$-8$ | $2$ | $0$ | $-11$ | $2$ | $1/6$ | $6$
$-8$ | $2$ | $0$ | $-11$ | $2$ | $5/6$ | $6$
$-9$ | $0$ | $1/6$ | $-11$ | $1$ | $1/2$ | $6$
$-9$ | $0$ | $5/6$ | $-11$ | $1$ | $1/2$ | $6$
$-9$ | $0$ | $1/6$ | $-11$ | $2$ | $1/6$ | $7$
$-9$ | $0$ | $1/6$ | $-11$ | $2$ | $5/6$ | $6$
$-9$ | $0$ | $5/6$ | $-11$ | $2$ | $1/6$ | $6$
$-9$ | $0$ | $5/6$ | $-11$ | $2$ | $5/6$ | $7$
$-9$ | $1$ | $1/6$ | $-11$ | $1$ | $1/2$ | $7$
$-9$ | $1$ | $1/2$ | $-11$ | $1$ | $1/2$ | $6$
$-9$ | $1$ | $5/6$ | $-11$ | $1$ | $1/2$ | $7$
$-9$ | $1$ | $1/6$ | $-11$ | $2$ | $1/6$ | $7$
$-9$ | $1$ | $1/6$ | $-11$ | $2$ | $5/6$ | $7$
$-9$ | $1$ | $1/2$ | $-11$ | $2$ | $1/6$ | $7$
$-9$ | $1$ | $1/2$ | $-11$ | $2$ | $5/6$ | $7$
$-9$ | $1$ | $5/6$ | $-11$ | $2$ | $1/6$ | $7$
$-9$ | $1$ | $5/6$ | $-11$ | $2$ | $5/6$ | $7$
$-9$ | $2$ | $1/6$ | $-11$ | $1$ | $1/2$ | $7$
$-9$ | $2$ | $5/6$ | $-11$ | $1$ | $1/2$ | $7$
$-9$ | $2$ | $1/6$ | $-11$ | $2$ | $1/6$ | $6$
$-9$ | $2$ | $1/6$ | $-11$ | $2$ | $5/6$ | $7$
$-9$ | $2$ | $5/6$ | $-11$ | $2$ | $1/6$ | $7$
$-9$ | $2$ | $5/6$ | $-11$ | $2$ | $5/6$ | $6$
$-10$ | $0$ | $0$ | $-11$ | $1$ | $1/2$ | $6$
$-10$ | $0$ | $0$ | $-11$ | $2$ | $1/6$ | $7$
$-10$ | $0$ | $0$ | $-11$ | $2$ | $5/6$ | $7$
$-10$ | $1$ | $1/3$ | $-11$ | $1$ | $1/2$ | $7$
$-10$ | $1$ | $2/3$ | $-11$ | $1$ | $1/2$ | $7$
$-10$ | $1$ | $1/3$ | $-11$ | $2$ | $1/6$ | $8$
$-10$ | $1$ | $1/3$ | $-11$ | $2$ | $5/6$ | $7$
$-10$ | $1$ | $2/3$ | $-11$ | $2$ | $1/6$ | $7$
$-10$ | $1$ | $2/3$ | $-11$ | $2$ | $5/6$ | $8$
$-10$ | $2$ | $0$ | $-11$ | $1$ | $1/2$ | $8$
$-10$ | $2$ | $1/3$ | $-11$ | $1$ | $1/2$ | $7$
$-10$ | $2$ | $2/3$ | $-11$ | $1$ | $1/2$ | $7$
$-10$ | $2$ | $0$ | $-11$ | $2$ | $1/6$ | $7$
$-10$ | $2$ | $0$ | $-11$ | $2$ | $5/6$ | $7$
$-10$ | $2$ | $1/3$ | $-11$ | $2$ | $1/6$ | $6$
$-10$ | $2$ | $1/3$ | $-11$ | $2$ | $5/6$ | $7$
$-10$ | $2$ | $2/3$ | $-11$ | $2$ | $1/6$ | $7$
$-10$ | $2$ | $2/3$ | $-11$ | $2$ | $5/6$ | $6$
$-11$ | $1$ | $1/2$ | $-11$ | $1$ | $1/2$ | $8$
$-11$ | $1$ | $1/2$ | $-11$ | $2$ | $1/6$ | $8$
$-11$ | $1$ | $1/2$ | $-11$ | $2$ | $5/6$ | $8$
$-11$ | $2$ | $1/6$ | $-11$ | $2$ | $1/6$ | $7$
$-11$ | $2$ | $1/6$ | $-11$ | $2$ | $5/6$ | $7$
$-11$ | $2$ | $5/6$ | $-11$ | $2$ | $5/6$ | $7$
$-6$ | $0$ | $0$ | $-12$ | $2$ | $0$ | $6$
$-7$ | $0$ | $1/6$ | $-12$ | $2$ | $0$ | $6$
$-7$ | $0$ | $5/6$ | $-12$ | $2$ | $0$ | $6$
$-7$ | $1$ | $1/2$ | $-12$ | $2$ | $0$ | $6$
$-8$ | $0$ | $0$ | $-12$ | $2$ | $0$ | $6$
$-8$ | $0$ | $1/3$ | $-12$ | $2$ | $0$ | $7$
$-8$ | $0$ | $2/3$ | $-12$ | $2$ | $0$ | $7$
$-8$ | $1$ | $1/3$ | $-12$ | $2$ | $0$ | $7$
$-8$ | $1$ | $2/3$ | $-12$ | $2$ | $0$ | $7$
$-8$ | $2$ | $0$ | $-12$ | $2$ | $0$ | $7$
$-9$ | $0$ | $1/6$ | $-12$ | $2$ | $0$ | $7$
$-9$ | $0$ | $5/6$ | $-12$ | $2$ | $0$ | $7$
$-9$ | $1$ | $1/6$ | $-12$ | $2$ | $0$ | $7$
$-9$ | $1$ | $1/2$ | $-12$ | $2$ | $0$ | $8$
$-9$ | $1$ | $5/6$ | $-12$ | $2$ | $0$ | $7$
$-9$ | $2$ | $1/6$ | $-12$ | $2$ | $0$ | $7$
$-9$ | $2$ | $5/6$ | $-12$ | $2$ | $0$ | $7$
$-10$ | $0$ | $0$ | $-12$ | $2$ | $0$ | $8$
$-10$ | $1$ | $1/3$ | $-12$ | $2$ | $0$ | $8$
$-10$ | $1$ | $2/3$ | $-12$ | $2$ | $0$ | $8$
$-10$ | $2$ | $0$ | $-12$ | $2$ | $0$ | $7$
$-10$ | $2$ | $1/3$ | $-12$ | $2$ | $0$ | $7$
$-10$ | $2$ | $2/3$ | $-12$ | $2$ | $0$ | $7$
$-11$ | $1$ | $1/2$ | $-12$ | $2$ | $0$ | $8$
$-11$ | $2$ | $1/6$ | $-12$ | $2$ | $0$ | $7$
$-11$ | $2$ | $5/6$ | $-12$ | $2$ | $0$ | $7$
$-12$ | $2$ | $0$ | $-12$ | $2$ | $0$ | $7$

Table 18: Possible parameter configurations on $T^{2}/Z_{3}$ with
$M_{ab}<0,\,M_{ca}>0$. $T^{2}/Z_{3}$ with $M_{ab}<0,\,M_{ca}>0$
---
$ab$-sector |  $ca$-sector |  $bc$-sector
$M_{ab}$ | $\omega^{i}_{ab}$ | $\alpha_{ab}$ | $M_{ca}$ | $\omega^{i}_{ca}$ | $\alpha_{ca}$ | # of Higgs
$-6$ | $0$ | $0$ | $6$ | $0$ | $0$ | 1
$-6$ | $0$ | $0$ | $7$ | $0$ | $1/6$ | $1$
$-6$ | $0$ | $0$ | $7$ | $0$ | $5/6$ | $1$
$-7$ | $0$ | $1/6$ | $7$ | $0$ | $5/6$ | 1
$-7$ | $0$ | $5/6$ | $7$ | $0$ | $1/6$ | 1
$-6$ | $0$ | $0$ | $8$ | $0$ | $0$ | $1$
$-6$ | $0$ | $0$ | $8$ | $0$ | $1/3$ | $1$
$-6$ | $0$ | $0$ | $8$ | $0$ | $2/3$ | $1$
$-7$ | $0$ | $1/6$ | $8$ | $0$ | $0$ | $1$
$-7$ | $0$ | $1/6$ | $8$ | $0$ | $2/3$ | $1$
$-7$ | $0$ | $5/6$ | $8$ | $0$ | $0$ | $1$
$-7$ | $0$ | $5/6$ | $8$ | $0$ | $1/3$ | $1$
$-8$ | $0$ | $0$ | $8$ | $0$ | $0$ | 1
$-8$ | $0$ | $1/3$ | $8$ | $0$ | $2/3$ | 1
$-8$ | $0$ | $2/3$ | $8$ | $0$ | $1/3$ | 1
$-6$ | $0$ | $0$ | $9$ | $0$ | $1/6$ | $1$
$-6$ | $0$ | $0$ | $9$ | $0$ | $5/6$ | $1$
$-6$ | $0$ | $0$ | $9$ | $1$ | $1/6$ | $1$
$-6$ | $0$ | $0$ | $9$ | $1$ | $5/6$ | $1$
$-6$ | $0$ | $0$ | $9$ | $2$ | $1/6$ | $1$
$-6$ | $0$ | $0$ | $9$ | $2$ | $5/6$ | $1$
$-7$ | $0$ | $1/6$ | $9$ | $0$ | $1/6$ | $1$
$-7$ | $0$ | $1/6$ | $9$ | $0$ | $5/6$ | $1$
$-7$ | $0$ | $5/6$ | $9$ | $0$ | $1/6$ | $1$
$-7$ | $0$ | $5/6$ | $9$ | $0$ | $5/6$ | $1$
$-7$ | $0$ | $1/6$ | $9$ | $1$ | $5/6$ | $1$
$-7$ | $0$ | $5/6$ | $9$ | $1$ | $1/6$ | $1$
$-7$ | $0$ | $1/6$ | $9$ | $2$ | $1/6$ | $1$
$-7$ | $0$ | $5/6$ | $9$ | $2$ | $5/6$ | $1$
$-7$ | $1$ | $1/2$ | $9$ | $1$ | $1/6$ | $1$
$-7$ | $1$ | $1/2$ | $9$ | $1$ | $5/6$ | $1$
$-7$ | $1$ | $1/2$ | $9$ | $2$ | $1/6$ | $1$
$-7$ | $1$ | $1/2$ | $9$ | $2$ | $5/6$ | $1$
$-8$ | $0$ | $0$ | $9$ | $0$ | $1/6$ | $1$
$-8$ | $0$ | $0$ | $9$ | $0$ | $5/6$ | $1$
$-8$ | $0$ | $1/3$ | $9$ | $0$ | $5/6$ | $1$
$-8$ | $0$ | $2/3$ | $9$ | $0$ | $1/6$ | $1$
$-8$ | $0$ | $1/3$ | $9$ | $2$ | $1/6$ | $1$
$-8$ | $0$ | $2/3$ | $9$ | $2$ | $5/6$ | $1$
$-8$ | $1$ | $1/3$ | $9$ | $1$ | $1/6$ | $1$
$-8$ | $1$ | $2/3$ | $9$ | $1$ | $5/6$ | $1$
$-8$ | $1$ | $1/3$ | $9$ | $2$ | $5/6$ | $1$
$-8$ | $1$ | $2/3$ | $9$ | $2$ | $1/6$ | $1$
$-8$ | $2$ | $0$ | $9$ | $1$ | $1/6$ | $1$
$-8$ | $2$ | $0$ | $9$ | $1$ | $5/6$ | $1$
$-9$ | $0$ | $1/6$ | $9$ | $0$ | $5/6$ | 1
$-9$ | $0$ | $5/6$ | $9$ | $0$ | $1/6$ | 1
$-9$ | $1$ | $1/6$ | $9$ | $2$ | $5/6$ | 1
$-9$ | $1$ | $5/6$ | $9$ | $2$ | $1/6$ | 1
$-9$ | $2$ | $1/6$ | $9$ | $1$ | $5/6$ | 1
$-9$ | $2$ | $5/6$ | $9$ | $1$ | $1/6$ | 1
$-6$ | $0$ | $0$ | $10$ | $0$ | $0$ | $1$
$-6$ | $0$ | $0$ | $10$ | $1$ | $1/3$ | $1$
$-6$ | $0$ | $0$ | $10$ | $1$ | $2/3$ | $1$
$-6$ | $0$ | $0$ | $10$ | $2$ | $0$ | $2$
$-6$ | $0$ | $0$ | $10$ | $2$ | $1/3$ | $1$
$-6$ | $0$ | $0$ | $10$ | $2$ | $2/3$ | $1$
$-7$ | $0$ | $1/6$ | $10$ | $0$ | $0$ | $1$
$-7$ | $0$ | $5/6$ | $10$ | $0$ | $0$ | $1$
$-7$ | $0$ | $1/6$ | $10$ | $1$ | $2/3$ | $1$
$-7$ | $0$ | $5/6$ | $10$ | $1$ | $1/3$ | $1$
$-7$ | $0$ | $1/6$ | $10$ | $2$ | $0$ | $1$
$-7$ | $0$ | $1/6$ | $10$ | $2$ | $1/3$ | $1$
$-7$ | $0$ | $1/6$ | $10$ | $2$ | $2/3$ | $1$
$-7$ | $0$ | $5/6$ | $10$ | $2$ | $0$ | $1$
$-7$ | $0$ | $5/6$ | $10$ | $2$ | $1/3$ | $1$
$-7$ | $0$ | $5/6$ | $10$ | $2$ | $2/3$ | $1$
$-7$ | $1$ | $1/2$ | $10$ | $1$ | $1/3$ | $1$
$-7$ | $1$ | $1/2$ | $10$ | $1$ | $2/3$ | $1$
$-7$ | $1$ | $1/2$ | $10$ | $2$ | $0$ | $2$
$-7$ | $1$ | $1/2$ | $10$ | $2$ | $1/3$ | $1$
$-7$ | $1$ | $1/2$ | $10$ | $2$ | $2/3$ | $1$
$-8$ | $0$ | $0$ | $10$ | $0$ | $0$ | $1$
$-8$ | $0$ | $1/3$ | $10$ | $0$ | $0$ | $1$
$-8$ | $0$ | $2/3$ | $10$ | $0$ | $0$ | $1$
$-8$ | $0$ | $1/3$ | $10$ | $1$ | $2/3$ | $1$
$-8$ | $0$ | $2/3$ | $10$ | $1$ | $1/3$ | $1$
$-8$ | $0$ | $0$ | $10$ | $2$ | $1/3$ | $1$
$-8$ | $0$ | $0$ | $10$ | $2$ | $2/3$ | $1$
$-8$ | $0$ | $1/3$ | $10$ | $2$ | $0$ | $1$
$-8$ | $0$ | $1/3$ | $10$ | $2$ | $1/3$ | $1$
$-8$ | $0$ | $2/3$ | $10$ | $2$ | $0$ | $1$
$-8$ | $0$ | $2/3$ | $10$ | $2$ | $2/3$ | $1$
$-8$ | $1$ | $1/3$ | $10$ | $1$ | $1/3$ | $1$
$-8$ | $1$ | $2/3$ | $10$ | $1$ | $2/3$ | $1$
$-8$ | $1$ | $1/3$ | $10$ | $2$ | $0$ | $1$
$-8$ | $1$ | $1/3$ | $10$ | $2$ | $1/3$ | $1$
$-8$ | $1$ | $1/3$ | $10$ | $2$ | $2/3$ | $1$
$-8$ | $1$ | $2/3$ | $10$ | $2$ | $0$ | $1$
$-8$ | $1$ | $2/3$ | $10$ | $2$ | $1/3$ | $1$
$-8$ | $1$ | $2/3$ | $10$ | $2$ | $2/3$ | $1$
$-8$ | $2$ | $0$ | $10$ | $1$ | $1/3$ | $1$
$-8$ | $2$ | $0$ | $10$ | $1$ | $2/3$ | $1$
$-8$ | $2$ | $0$ | $10$ | $2$ | $0$ | $1$
$-9$ | $0$ | $1/6$ | $10$ | $0$ | $0$ | $1$
$-9$ | $0$ | $5/6$ | $10$ | $0$ | $0$ | $1$
$-9$ | $0$ | $1/6$ | $10$ | $2$ | $1/3$ | $1$
$-9$ | $0$ | $5/6$ | $10$ | $2$ | $2/3$ | $1$
$-9$ | $1$ | $1/6$ | $10$ | $1$ | $1/3$ | $1$
$-9$ | $1$ | $5/6$ | $10$ | $1$ | $2/3$ | $1$
$-9$ | $1$ | $1/6$ | $10$ | $2$ | $0$ | $1$
$-9$ | $1$ | $1/6$ | $10$ | $2$ | $2/3$ | $1$
$-9$ | $1$ | $1/2$ | $10$ | $2$ | $1/3$ | $1$
$-9$ | $1$ | $1/2$ | $10$ | $2$ | $2/3$ | $1$
$-9$ | $1$ | $5/6$ | $10$ | $2$ | $0$ | $1$
$-9$ | $1$ | $5/6$ | $10$ | $2$ | $1/3$ | $1$
$-9$ | $2$ | $1/6$ | $10$ | $1$ | $2/3$ | $1$
$-9$ | $2$ | $5/6$ | $10$ | $1$ | $1/3$ | $1$
$-10$ | $0$ | $0$ | $10$ | $0$ | $0$ | 1
$-10$ | $1$ | $1/3$ | $10$ | $2$ | $2/3$ | 1
$-10$ | $1$ | $2/3$ | $10$ | $2$ | $1/3$ | 1
$-10$ | $2$ | $1/3$ | $10$ | $1$ | $2/3$ | 1
$-10$ | $2$ | $2/3$ | $10$ | $1$ | $1/3$ | 1
$-6$ | $0$ | $0$ | $11$ | $1$ | $1/2$ | $2$
$-6$ | $0$ | $0$ | $11$ | $2$ | $1/6$ | $2$
$-6$ | $0$ | $0$ | $11$ | $2$ | $5/6$ | $2$
$-7$ | $0$ | $1/6$ | $11$ | $1$ | $1/2$ | $1$
$-7$ | $0$ | $5/6$ | $11$ | $1$ | $1/2$ | $1$
$-7$ | $0$ | $1/6$ | $11$ | $2$ | $1/6$ | $1$
$-7$ | $0$ | $1/6$ | $11$ | $2$ | $5/6$ | $2$
$-7$ | $0$ | $5/6$ | $11$ | $2$ | $1/6$ | $2$
$-7$ | $0$ | $5/6$ | $11$ | $2$ | $5/6$ | $1$
$-7$ | $1$ | $1/2$ | $11$ | $1$ | $1/2$ | $2$
$-7$ | $1$ | $1/2$ | $11$ | $2$ | $1/6$ | $2$
$-7$ | $1$ | $1/2$ | $11$ | $2$ | $5/6$ | $2$
$-8$ | $0$ | $1/3$ | $11$ | $1$ | $1/2$ | $1$
$-8$ | $0$ | $2/3$ | $11$ | $1$ | $1/2$ | $1$
$-8$ | $0$ | $0$ | $11$ | $2$ | $1/6$ | $1$
$-8$ | $0$ | $0$ | $11$ | $2$ | $5/6$ | $1$
$-8$ | $0$ | $1/3$ | $11$ | $2$ | $1/6$ | $1$
$-8$ | $0$ | $1/3$ | $11$ | $2$ | $5/6$ | $1$
$-8$ | $0$ | $2/3$ | $11$ | $2$ | $1/6$ | $1$
$-8$ | $0$ | $2/3$ | $11$ | $2$ | $5/6$ | $1$
$-8$ | $1$ | $1/3$ | $11$ | $1$ | $1/2$ | $1$
$-8$ | $1$ | $2/3$ | $11$ | $1$ | $1/2$ | $1$
$-8$ | $1$ | $1/3$ | $11$ | $2$ | $1/6$ | $2$
$-8$ | $1$ | $1/3$ | $11$ | $2$ | $5/6$ | $1$
$-8$ | $1$ | $2/3$ | $11$ | $2$ | $1/6$ | $1$
$-8$ | $1$ | $2/3$ | $11$ | $2$ | $5/6$ | $2$
$-8$ | $2$ | $0$ | $11$ | $1$ | $1/2$ | $2$
$-8$ | $2$ | $0$ | $11$ | $2$ | $1/6$ | $1$
$-8$ | $2$ | $0$ | $11$ | $2$ | $5/6$ | $1$
$-9$ | $0$ | $1/6$ | $11$ | $2$ | $1/6$ | $1$
$-9$ | $0$ | $5/6$ | $11$ | $2$ | $5/6$ | $1$
$-9$ | $1$ | $1/6$ | $11$ | $1$ | $1/2$ | $1$
$-9$ | $1$ | $5/6$ | $11$ | $1$ | $1/2$ | $1$
$-9$ | $1$ | $1/6$ | $11$ | $2$ | $1/6$ | $1$
$-9$ | $1$ | $1/6$ | $11$ | $2$ | $5/6$ | $1$
$-9$ | $1$ | $1/2$ | $11$ | $2$ | $1/6$ | $1$
$-9$ | $1$ | $1/2$ | $11$ | $2$ | $5/6$ | $1$
$-9$ | $1$ | $5/6$ | $11$ | $2$ | $1/6$ | $1$
$-9$ | $1$ | $5/6$ | $11$ | $2$ | $5/6$ | $1$
$-9$ | $2$ | $1/6$ | $11$ | $1$ | $1/2$ | $1$
$-9$ | $2$ | $5/6$ | $11$ | $1$ | $1/2$ | $1$
$-9$ | $2$ | $1/6$ | $11$ | $2$ | $5/6$ | $1$
$-9$ | $2$ | $5/6$ | $11$ | $2$ | $1/6$ | $1$
$-10$ | $1$ | $1/3$ | $11$ | $2$ | $5/6$ | $1$
$-10$ | $1$ | $2/3$ | $11$ | $2$ | $1/6$ | $1$
$-10$ | $2$ | $1/3$ | $11$ | $1$ | $1/2$ | $1$
$-10$ | $2$ | $2/3$ | $11$ | $1$ | $1/2$ | $1$
$-6$ | $0$ | $0$ | $12$ | $2$ | $0$ | $2$
$-7$ | $0$ | $1/6$ | $12$ | $2$ | $0$ | $2$
$-7$ | $0$ | $5/6$ | $12$ | $2$ | $0$ | $2$
$-7$ | $1$ | $1/2$ | $12$ | $2$ | $0$ | $2$
$-8$ | $0$ | $0$ | $12$ | $2$ | $0$ | $2$
$-8$ | $0$ | $1/3$ | $12$ | $2$ | $0$ | $1$
$-8$ | $0$ | $2/3$ | $12$ | $2$ | $0$ | $1$
$-8$ | $1$ | $1/3$ | $12$ | $2$ | $0$ | $2$
$-8$ | $1$ | $2/3$ | $12$ | $2$ | $0$ | $2$
$-8$ | $2$ | $0$ | $12$ | $2$ | $0$ | $1$
$-9$ | $0$ | $1/6$ | $12$ | $2$ | $0$ | $1$
$-9$ | $0$ | $5/6$ | $12$ | $2$ | $0$ | $1$
$-9$ | $1$ | $1/6$ | $12$ | $2$ | $0$ | $1$
$-9$ | $1$ | $1/2$ | $12$ | $2$ | $0$ | $2$
$-9$ | $1$ | $5/6$ | $12$ | $2$ | $0$ | $1$
$-9$ | $2$ | $1/6$ | $12$ | $2$ | $0$ | $1$
$-9$ | $2$ | $5/6$ | $12$ | $2$ | $0$ | $1$
$-10$ | $1$ | $1/3$ | $12$ | $2$ | $0$ | $1$
$-10$ | $1$ | $2/3$ | $12$ | $2$ | $0$ | $1$
$-10$ | $2$ | $0$ | $12$ | $2$ | $0$ | $1$

###  B.3 $T^{2}/Z_{4}$

Table 19: Possible parameter configurations on $T^{2}/Z_{4}$ with
$M_{ab}<0,\,M_{ca}<0$. $T^{2}/Z_{4}$ with $M_{ab}<0,\,M_{ca}<0$
---
$ab$-sector |  $ca$-sector |  $bc$-sector
$M_{ab}$ | $\omega^{i}_{ab}$ | $\alpha_{ab}$ | $M_{ca}$ | $\omega^{i}_{ca}$ | $\alpha_{ca}$ | # of Higgs
$-8$ | $0$ | $0$ | $-8$ | $0$ | $0$ | $5$
$-8$ | $0$ | $0$ | $-9$ | $0$ | $0$ | $5$
$-8$ | $0$ | $0$ | $-9$ | $1$ | $1/2$ | $4$
$-9$ | $0$ | $0$ | $-9$ | $0$ | $0$ | $5$
$-9$ | $0$ | $0$ | $-9$ | $1$ | $1/2$ | $4$
$-9$ | $1$ | $1/2$ | $-9$ | $1$ | $1/2$ | $5$
$-8$ | $0$ | $0$ | $-10$ | $0$ | $0$ | $5$
$-8$ | $0$ | $0$ | $-10$ | $0$ | $1/2$ | $5$
$-8$ | $0$ | $0$ | $-10$ | $1$ | $1/2$ | $4$
$-8$ | $0$ | $0$ | $-10$ | $2$ | $0$ | $5$
$-9$ | $0$ | $0$ | $-10$ | $0$ | $0$ | $5$
$-9$ | $0$ | $0$ | $-10$ | $0$ | $1/2$ | $5$
$-9$ | $0$ | $0$ | $-10$ | $1$ | $1/2$ | $5$
$-9$ | $0$ | $0$ | $-10$ | $2$ | $0$ | $5$
$-9$ | $1$ | $1/2$ | $-10$ | $0$ | $0$ | $5$
$-9$ | $1$ | $1/2$ | $-10$ | $0$ | $1/2$ | $4$
$-9$ | $1$ | $1/2$ | $-10$ | $1$ | $1/2$ | $5$
$-9$ | $1$ | $1/2$ | $-10$ | $2$ | $0$ | $5$
$-10$ | $0$ | $0$ | $-10$ | $0$ | $0$ | $6$
$-10$ | $0$ | $0$ | $-10$ | $0$ | $1/2$ | $5$
$-10$ | $0$ | $1/2$ | $-10$ | $0$ | $1/2$ | $6$
$-10$ | $0$ | $0$ | $-10$ | $1$ | $1/2$ | $5$
$-10$ | $0$ | $1/2$ | $-10$ | $1$ | $1/2$ | $4$
$-10$ | $0$ | $0$ | $-10$ | $2$ | $0$ | $5$
$-10$ | $0$ | $1/2$ | $-10$ | $2$ | $0$ | $5$
$-10$ | $1$ | $1/2$ | $-10$ | $1$ | $1/2$ | $5$
$-10$ | $1$ | $1/2$ | $-10$ | $2$ | $0$ | $5$
$-10$ | $2$ | $0$ | $-10$ | $2$ | $0$ | $6$
$-8$ | $0$ | $0$ | $-11$ | $0$ | $0$ | $5$
$-8$ | $0$ | $0$ | $-11$ | $0$ | $1/2$ | $5$
$-8$ | $0$ | $0$ | $-11$ | $1$ | $0$ | $4$
$-8$ | $0$ | $0$ | $-11$ | $1$ | $1/2$ | $5$
$-8$ | $0$ | $0$ | $-11$ | $2$ | $0$ | $5$
$-8$ | $0$ | $0$ | $-11$ | $3$ | $1/2$ | $5$
$-9$ | $0$ | $0$ | $-11$ | $0$ | $0$ | $6$
$-9$ | $0$ | $0$ | $-11$ | $0$ | $1/2$ | $5$
$-9$ | $0$ | $0$ | $-11$ | $1$ | $0$ | $4$
$-9$ | $0$ | $0$ | $-11$ | $1$ | $1/2$ | $5$
$-9$ | $0$ | $0$ | $-11$ | $2$ | $0$ | $5$
$-9$ | $0$ | $0$ | $-11$ | $3$ | $1/2$ | $5$
$-9$ | $1$ | $1/2$ | $-11$ | $0$ | $0$ | $5$
$-9$ | $1$ | $1/2$ | $-11$ | $0$ | $1/2$ | $4$
$-9$ | $1$ | $1/2$ | $-11$ | $1$ | $0$ | $5$
$-9$ | $1$ | $1/2$ | $-11$ | $1$ | $1/2$ | $5$
$-9$ | $1$ | $1/2$ | $-11$ | $2$ | $0$ | $5$
$-9$ | $1$ | $1/2$ | $-11$ | $3$ | $1/2$ | $6$
$-10$ | $0$ | $0$ | $-11$ | $0$ | $0$ | $6$
$-10$ | $0$ | $0$ | $-11$ | $0$ | $1/2$ | $5$
$-10$ | $0$ | $1/2$ | $-11$ | $0$ | $0$ | $5$
$-10$ | $0$ | $1/2$ | $-11$ | $0$ | $1/2$ | $6$
$-10$ | $0$ | $0$ | $-11$ | $1$ | $0$ | $5$
$-10$ | $0$ | $0$ | $-11$ | $1$ | $1/2$ | $5$
$-10$ | $0$ | $1/2$ | $-11$ | $1$ | $0$ | $5$
$-10$ | $0$ | $1/2$ | $-11$ | $1$ | $1/2$ | $5$
$-10$ | $0$ | $0$ | $-11$ | $2$ | $0$ | $5$
$-10$ | $0$ | $1/2$ | $-11$ | $2$ | $0$ | $5$
$-10$ | $0$ | $0$ | $-11$ | $3$ | $1/2$ | $6$
$-10$ | $0$ | $1/2$ | $-11$ | $3$ | $1/2$ | $5$
$-10$ | $1$ | $1/2$ | $-11$ | $0$ | $0$ | $5$
$-10$ | $1$ | $1/2$ | $-11$ | $0$ | $1/2$ | $5$
$-10$ | $1$ | $1/2$ | $-11$ | $1$ | $0$ | $5$
$-10$ | $1$ | $1/2$ | $-11$ | $1$ | $1/2$ | $5$
$-10$ | $1$ | $1/2$ | $-11$ | $2$ | $0$ | $6$
$-10$ | $1$ | $1/2$ | $-11$ | $3$ | $1/2$ | $6$
$-10$ | $2$ | $0$ | $-11$ | $0$ | $0$ | $5$
$-10$ | $2$ | $0$ | $-11$ | $0$ | $1/2$ | $5$
$-10$ | $2$ | $0$ | $-11$ | $1$ | $0$ | $5$
$-10$ | $2$ | $0$ | $-11$ | $1$ | $1/2$ | $6$
$-10$ | $2$ | $0$ | $-11$ | $2$ | $0$ | $6$
$-10$ | $2$ | $0$ | $-11$ | $3$ | $1/2$ | $5$
$-11$ | $0$ | $0$ | $-11$ | $0$ | $0$ | $6$
$-11$ | $0$ | $0$ | $-11$ | $0$ | $1/2$ | $6$
$-11$ | $0$ | $1/2$ | $-11$ | $0$ | $1/2$ | $6$
$-11$ | $0$ | $0$ | $-11$ | $1$ | $0$ | $5$
$-11$ | $0$ | $0$ | $-11$ | $1$ | $1/2$ | $5$
$-11$ | $0$ | $1/2$ | $-11$ | $1$ | $0$ | $5$
$-11$ | $0$ | $1/2$ | $-11$ | $1$ | $1/2$ | $5$
$-11$ | $0$ | $0$ | $-11$ | $2$ | $0$ | $6$
$-11$ | $0$ | $1/2$ | $-11$ | $2$ | $0$ | $5$
$-11$ | $0$ | $0$ | $-11$ | $3$ | $1/2$ | $6$
$-11$ | $0$ | $1/2$ | $-11$ | $3$ | $1/2$ | $5$
$-11$ | $1$ | $0$ | $-11$ | $1$ | $0$ | $6$
$-11$ | $1$ | $0$ | $-11$ | $1$ | $1/2$ | $5$
$-11$ | $1$ | $1/2$ | $-11$ | $1$ | $1/2$ | $6$
$-11$ | $1$ | $0$ | $-11$ | $2$ | $0$ | $5$
$-11$ | $1$ | $1/2$ | $-11$ | $2$ | $0$ | $6$
$-11$ | $1$ | $0$ | $-11$ | $3$ | $1/2$ | $6$
$-11$ | $1$ | $1/2$ | $-11$ | $3$ | $1/2$ | $6$
$-11$ | $2$ | $0$ | $-11$ | $2$ | $0$ | $6$
$-11$ | $2$ | $0$ | $-11$ | $3$ | $1/2$ | $5$
$-11$ | $3$ | $1/2$ | $-11$ | $3$ | $1/2$ | $6$
$-8$ | $0$ | $0$ | $-12$ | $0$ | $1/2$ | $5$