#  Bottom quark and tau lepton masses in a toy ${\rm SU}(6)$ model

Ning Chen${}^{\,\heartsuit}$ , Ying-nan Mao$\,{}^{\star}$ ,
Zhaolong Teng${}^{\,\clubsuit}$ 0000-0002-0032-9012 0000-0001-8063-8968
0000-0002-7141-2331

(${}^{\heartsuit}\,{}^{\clubsuit}$School of Physics, Nankai University,
Tianjin, 300071, China
⋆ Department of Physics, School of Science, Wuhan University of Technology,
Wuhan, 430070, Hubei, China
)

###### Abstract

We study a toy ${\rm SU}(6)$ model with the symmetry breaking pattern of the
extended $331$ symmetry of ${\rm SU}(3)_{c}\otimes{\rm SU}(3)_{W}\otimes{\rm
U}(1)_{X}$. A “fermion-Higgs mismatching” symmetry breaking pattern is
proposed for more realistic model building. Within such symmetry breaking
pattern, only one Higgs doublet develops vacuum expectation value for the
spontaneous electroweak symmetry breaking, and gives tree-level top quark
mass. A natural VEV splittings in the $331$ breaking Higgs fields gives tree-
level masses to both bottom quark and tau lepton. The $125\,{\rm GeV}$ SM-like
Higgs boson discovered at the LHC can have Yukawa couplings to bottom quark
and tau lepton as in the SM prediction, and this suggests the $331$ symmetry
breaking scale to be $\sim{\cal O}(10)\,{\rm TeV}$.

_Emails:_
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###### Contents

  1. 1 Introduction
  2. 2 One-generational ${\rm SU}(6)$
    1. 2.1 The symmetry breaking pattern
    2. 2.2 An example: ${\rm SU}(8)$ with three generations
  3. 3 The Higgs sector of the ${\rm SU}(6)$
    1. 3.1 The Higgs potential
    2. 3.2 The charged and CP-odd Higgs bosons
    3. 3.3 The CP-even Higgs bosons
    4. 3.4 Summary of the Higgs spectrum
  4. 4 Bottom quark and tau lepton masses in the ${\rm SU}(6)$
    1. 4.1 The Yukawa couplings
    2. 4.2 The bottom quark mass
    3. 4.3 The tau lepton mass
    4. 4.4 The possible radiative mechanism
  5. 5 Conclusions
  6. A The gauge symmetry breaking in the $331$ model
    1. A.1 The $331$ gauge bosons
    2. A.2 The gauge couplings of fermions
    3. A.3 The mass matrices of the Higgs bosons
    4. A.4 The Yukawa couplings of fermions

##  1 Introduction

Grand Unified Theories (GUTs) [1, 2] were proposed to unify all fundamental
interactions and elementary particles described by the Standard Model (SM) at
the electroweak (EW) scale. Meanwhile, a unified description of the
generational structure as well as the SM fermion mass hierarchies have not
been realized in terms of the ${\rm SU}(5)$ or ${\rm SO}(10)$ GUTs. This is
largely due to the fact that three generations of SM fermions are accommodated
in the ${\rm SU}(5)$ or ${\rm SO}(10)$ GUTs by simple repetition of one
anomaly-free fermion generation. Consequently, the symmetry breaking patterns
do not provide any source for the observed SM fermion mass hierarchies. It was
pointed out and discussed in Refs. [3, 4, 5, 6, 7] that multiple fermion
generations, such as $n_{g}=3$ for the SM case, can be embedded non-trivially
in GUT groups of ${\rm SU}(7)$ and beyond 111Such a scenario is different from
the flavors with certain flavor symmetries, and is named as “flavors without
flavor symmetries”, see Refs. [6, 7] for the recent discussions on the fermion
mass generation from non-minimal GUTs.. Therefore, it is natural to conjecture
that the SM fermion mass hierarchies may originate from the intermediate
symmetry breaking scale of some non-minimal GUT with ${\rm SU}(N\geq 7)$ [4,
5]. Historically, the embedding of the SM generations as well as fermion mass
hierarchies were studied in the context of technicolor and extended
technicolor models [8, 9, 10, 11, 12], where the symmetry breakings are due to
the fermion bi-linear condensates. Given the discovery of a single $125\,{\rm
GeV}$ SM-like Higgs boson at the Large Hadron Collider (LHC) [13, 14] until
now, it is pragmatic to revisit the flavor issue in the framework of GUTs,
where the spontaneous symmetry breakings are achieved by the Higgs mechanism.

Besides of addressing the flavor puzzle, it was also pointed out that the non-
minimal GUTs can automatically give rise to the global Peccei-Quinn (PQ)
symmetry [16] for the strong CP problem 222The first example was based on the
rank-$2$ ${\rm SU}(9)$ theory [15], where the fermion content of
$[5\times\mathbf{\overline{9_{F}}}]\oplus\mathbf{36_{F}}$ enjoys the ${\rm
SU}(5)\otimes{\rm U}(1)$ global symmetry.. This is due to the emergent global
symmetry of ${\rm SU}(N)\otimes{\rm U}(1)$ in the rank-$2$ anti-symmetric
${\rm SU}(N+4)$ gauge theories (with $N\geq 2$), which was first pointed out
by Dimopoulos, Raby, and Susskind [17]. In this regard, the longstanding
flavor puzzle as well as the PQ quality problem [18, 19, 20] may be
simultaneously addressed within the non-minimal GUTs 333Recently, it was
realized [21] that the non-minimal GUTs with $n_{g}=3$ can generally lead to
sufficiently large dimensional PQ-breaking operators for the later problem..

Before the ambitious goal of understanding the known SM fermion mass
hierarchies in realistic non-minimal GUTs, it will be useful to ask whether
the minimal version of this class already had some general properties in
producing the SM fermion masses. Among various non-minimal GUTs with ${\rm
SU}(N\geq 7)$, indeed, an extended gauge symmetry of
$\mathcal{G}_{331}\equiv{\rm SU}(3)_{c}\otimes{\rm SU}(3)_{W}\otimes{\rm
U}(1)_{X}$ above the EW scale is usually predicted 444A specific example will
be given for the ${\rm SU}(9)$ GUT in Sec. 2.2.. This class of models are
collectively known as the $331$ model, and were previously studied in Refs.
[22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. This motivates us to consider the
${\rm SU}(6)$ as a one-generational toy model 555The ${\rm SU}(6)$ turns out
to be a toy model for the third generation in particular., which can be
spontaneously broken to $\mathcal{G}_{331}$ by its adjoint Higgs field of
$\mathbf{35_{H}}$. An advantage of considering the one-generational ${\rm
SU}(6)$ instead of the 331 model is that one can uniquely define the electric
charges for both fermions and gauge bosons in the spectrum. Meanwhile, the
previous studies based on the 331 model itself often allowed different charge
quantization schemes [24, 37, 39, 44, 48], which could potentially lead to
fermions with exotic electric charges.

After the GUT symmetry breaking, there can be three ${\rm SU}(3)_{W}$ anti-
fundamental Higgs fields in the $331$ model. In the previous studies, only one
of them developed a vacuum expectation value (VEV) of $V_{331}$ for the
symmetry breaking of ${\rm SU}(3)_{W}\otimes{\rm U}(1)_{X}\to{\rm
SU}(2)_{W}\otimes{\rm U}(1)_{Y}$, while two others developed VEVs of $v_{\rm
EW}\simeq 246\,\rm GeV$ for the electroweak symmetry breaking (EWSB).
According to the Yukawa couplings, one can identify a type-II two-Higgs-
doublet model (2HDM) at the EW scale for the $331$ model. By extending to
larger non-minimal GUTs for $n_{g}=3$, such as ${\rm SU}(9)$ as our example,
the conventional symmetry breaking pattern in the $331$ model predicts more
than two EW Higgs doublets. This is certainly very problematic given that the
direct searches for the second Higgs doublet at the Large Hadron Collider
(LHC) give null results so far. Motivated by the general features in the Higgs
sector of the non-minimal GUT, we study the alternative symmetry breaking
pattern with only one EW Higgs doublet coming from the $\mathbf{15_{H}}$ of
the ${\rm SU}(6)$. An immediate question is how do bottom quark and tau lepton
acquire masses given the vanishing tree-level Yukawa couplings. It turns out
their masses can only be obtained when two ${\rm SU}(3)_{W}$ anti-fundamental
Higgs fields from $\mathbf{\overline{6_{H}}}^{\rho=1\,,2}$ develop VEVs both
for the $331$ and EW symmetry breaking directions. A natural mass splitting
between the top quark and the $(b\,,\tau)$ in the third generation can be
achieved with $\mathcal{O}(1)$ Yukawa couplings. The corresponding $331$
breaking scale is found to be $V_{331}\sim\mathcal{O}(10)\,\rm TeV$ from the
Yukawa couplings of the SM-like Higgs boson with the $(b\,,\tau)$.
Historically, a universal $\mathcal{O}(1)$ Yukawa coupling was motivated by
observing the natural top quark mass at the EW scale, and this was generalized
as the anarchical fermion mass scenario in the studies of the neutrino masses
[51, 52]. We also wish to remind the readers, the whole discussions are based
on the $331$ model due to the minimal one-generational ${\rm SU}(6)$ symmetry
breaking. Aside from the SM fermion masses, we do not address some general
questions of gauge coupling unification or proton lifetime predictions.
Neither do we determine whether a supersymmetric extension to the current
model is necessary, with the belief that this would better be studied in more
realistic models with $n_{g}=3$. Some related discussions can be found in
Refs. [41, 43, 46].

The rest of the paper is organized as follows. In Sec. 2, we motivate the
possible symmetry breaking pattern from two independent aspects in the toy
${\rm SU}(6)$ model, which leads to only one Higgs doublet for the spontaneous
EWSB. In Sec. 3, we describe the Higgs sector in the ${\rm SU}(6)$ GUT, with
the emphasis on the reasonable mass generations to the bottom quark and tau
lepton through the Yukawa couplings. In Sec. 4, we describe the bottom quark
and tau lepton masses in the toy ${\rm SU}(6)$ model based on the reasonable
symmetry breaking pattern as well as the VEV assignment. Some comments will be
made for the necessary condition of the radiative mass generation in the
current context. We summarize our results and make discussions in Sec. 5. An
appendix A is provided to summarize the gauge sector as well as the fermion
Yukawa couplings of the $331$ model. All Lie group calculations in this work
are carried out by LieART [53, 54].

##  2 One-generational ${\rm SU}(6)$

The minimal anomaly-free ${\rm SU}(6)$ GUT contains the left-handed fermions
of

$\displaystyle\\{f_{L}\\}_{{\rm SU}(6)}$ $\displaystyle=$
$\displaystyle\mathbf{\overline{6_{F}}}^{\rho}\oplus\mathbf{15_{F}}\,,\quad\rho=1\,,2\,.$
(1)

The fermion sector enjoys a global symmetry of

$\displaystyle{\cal G}_{\rm flavor}$ $\displaystyle=$ $\displaystyle{\rm
SU}(2)_{F}\otimes{\rm U}(1)_{\rm PQ}\,,$ (2)

according to Ref. [17]. The most general Yukawa couplings that are invariant
under the gauge symmetry are expressed as 666Other Yukawa couplings of
$\epsilon_{\rho\sigma}\mathbf{\overline{6_{F}}}^{\rho}\mathbf{\overline{6_{F}}}^{\sigma}(\mathbf{15_{H}}+\mathbf{21_{H}})+H.c.$
are also possible. These terms are only relevant to neutrino masses, and we
will neglect them in the current discussions.

$\displaystyle-\mathcal{L}_{Y}$ $\displaystyle=$
$\displaystyle(Y_{\mathcal{D}})_{\rho\sigma}\mathbf{15_{F}}\mathbf{\overline{6_{F}}}^{\rho}\mathbf{\overline{6_{H}}}^{\sigma}+Y_{\mathcal{U}}\mathbf{15_{F}}\mathbf{15_{F}}\mathbf{15_{H}}+H.c.\,,$
(3)

where we allow the explicit ${\rm SU}(2)_{F}$-breaking term in the Yukawa
couplings, so that $(Y_{\mathcal{D}})_{\rho\sigma}\neq
Y_{\mathcal{D}}\delta_{\rho\sigma}$.

Below, we motivate our Higgs VEV assignments for the viable symmetry breaking
from three different aspects, which include

  * $\mathfrak{A}$

the null results in searching for a second Higgs doublet at the LHC,

  * $\mathfrak{B}$

the extension to the non-minimal GUTs with $n_{g}=3$, e.g., the ${\rm SU}(8)$
GUT [6, 7],

  * $\mathfrak{C}$

the natural mass generation of the bottom quark and tau lepton with Yukawa
couplings of $\sim\mathcal{O}(1)$.

###  2.1 The symmetry breaking pattern

The viable ${\rm SU}(6)$ breaking pattern is expected to be

$\displaystyle{\rm SU}(6)\xrightarrow{\Lambda_{\rm GUT}}{\cal
G}_{331}\xrightarrow{V_{331}}{\cal G}_{\rm SM}\,,$ $\displaystyle{\cal
G}_{331}={\rm SU}(3)_{c}\otimes{\rm SU}(3)_{W}\otimes{\rm
U}(1)_{X}\,,\quad{\cal G}_{\rm SM}={\rm SU}(3)_{c}\otimes{\rm
SU}(2)_{W}\otimes{\rm U}(1)_{Y}\,,$ (4)

where the GUT scale symmetry breaking is achieved by an ${\rm SU}(6)$ adjoint
Higgs field of $\mathbf{35_{H}}$. The ${\rm U}(1)_{X}$ charge for the
$\mathbf{6}\in{\rm SU}(6)$ and ${\rm U}(1)_{Y}$ charge for the
$\mathbf{3}_{W}\in{\rm SU}(3)_{W}$ are defined by

$\displaystyle X(\mathbf{6})$ $\displaystyle\equiv$ $\displaystyle{\rm
diag}(-\frac{1}{3}\,,-\frac{1}{3}\,,-\frac{1}{3}\,,+\frac{1}{3}\,,+\frac{1}{3}\,,+\frac{1}{3})\,,$
(5a) $\displaystyle Y(\mathbf{3}_{W})$ $\displaystyle\equiv$
$\displaystyle{\rm diag}(\frac{1}{6}+X\,,\frac{1}{6}+X\,,-\frac{1}{3}+X)\,.$
(5b)

The electric charge operator of the ${\rm SU}(3)_{W}$ fundamental
representation is expressed as a $3\times 3$ diagonal matrix

$\displaystyle Q(\mathbf{3}_{W})$ $\displaystyle\equiv$ $\displaystyle T_{{\rm
SU}(3)}^{3}+Y={\rm diag}(\frac{2}{3}+X\,,-\frac{1}{3}+X\,,-\frac{1}{3}+X)\,.$
(6)

with the first ${\rm SU}(3)$ Cartan generator of

$\displaystyle T_{{\rm SU}(3)}^{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\rm diag}(1\,,-1\,,0)\,.$ (7)

Accordingly, we find that Higgs fields in Eq. (3) are decomposed as follows

$\displaystyle\mathbf{\overline{6_{H}}}^{\rho}=(\mathbf{\overline{3}}\,,\mathbf{1}\,,+\frac{1}{3})_{\mathbf{H}}^{\rho}\oplus\overbrace{(\mathbf{1}\,,\mathbf{\overline{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{\rho}}^{\mathbf{\Phi}_{\mathbf{\overline{3}}\,,\rho}}\,,$
$\displaystyle\mathbf{15_{H}}=(\mathbf{\overline{3}}\,,\mathbf{1}\,,-\frac{2}{3})_{\mathbf{H}}\oplus\overbrace{(\mathbf{1}\,,\mathbf{\overline{3}}\,,+\frac{2}{3})_{\mathbf{H}}}^{\mathbf{\Phi}_{\mathbf{\overline{3}}}^{\prime}}\oplus(\mathbf{3}\,,\mathbf{3}\,,0)_{\mathbf{H}}\,,$
(8)

for the symmetry breaking pattern in Eq. (2.1). Two
$(\mathbf{1}\,,\mathbf{\overline{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{\rho}\subset\mathbf{\overline{6_{H}}}^{\rho}$
contain SM-singlet directions after the second stage symmetry breaking in Eq.
(2.1). Meanwhile, the
$(\mathbf{1}\,,\mathbf{\overline{2}}\,,+\frac{1}{2})_{\mathbf{H}}\subset(\mathbf{1}\,,\mathbf{\overline{3}}\,,+\frac{2}{3})_{\mathbf{H}}\subset\mathbf{15_{H}}$
can only develop VEV to trigger the spontaneous EWSB of ${\rm
SU}(2)_{W}\otimes{\rm U}(1)_{Y}\to{\rm U}(1)_{\rm em}$. Under the symmetry
breaking pattern in Eq. (2.1) and the charge quantization given in Eqs. (5a),
(5b), and (6), we summarize the ${\rm SU}(6)$ fermions and their names in Tab.
1. For the SM fermions marked by solid underlines, we name them by the third
generational SM fermions. This will become manifest from their mass origin
within the current context.

$\rm SU(6)$ | ${\cal G}_{331}$ | ${\cal G}_{\rm SM}$
---|---|---
$\mathbf{\overline{6_{F}}}^{1}$ | $(\mathbf{\overline{3}}\,,\mathbf{1}\,,+\frac{1}{3})_{\mathbf{F}}^{1}\leavevmode\nobreak\ :\leavevmode\nobreak\ (\mathcal{B}_{R}^{1})^{c}$ | $(\mathbf{\overline{3}}\,,\mathbf{1}\,,+\frac{1}{3})_{\mathbf{F}}^{1}\leavevmode\nobreak\ :\leavevmode\nobreak\ \underline{{b_{R}}^{c}}$
| $(\mathbf{1}\,,\mathbf{\overline{3}}\,,-\frac{1}{3})_{\mathbf{F}}^{1}\leavevmode\nobreak\ :\leavevmode\nobreak\ \mathcal{L}_{L}^{1}$ | $(\mathbf{1}\,,\mathbf{\overline{2}}\,,-\frac{1}{2})_{\mathbf{F}}^{1}\leavevmode\nobreak\ :\leavevmode\nobreak\ \underline{\ell_{L}=(\tau_{L}\,,-\nu_{L})^{T}}$
|  | $(\mathbf{1}\,,\mathbf{1}\,,0)_{\mathbf{F}}^{1}\leavevmode\nobreak\ :\leavevmode\nobreak\ \widetilde{N}_{L}^{1}$
$\mathbf{\overline{6_{F}}}^{2}$ | $(\mathbf{\overline{3}}\,,\mathbf{1}\,,+\frac{1}{3})_{\mathbf{F}}^{2}\leavevmode\nobreak\ :\leavevmode\nobreak\ (\mathcal{B}_{R}^{2})^{c}$ | $(\mathbf{\overline{3}}\,,\mathbf{1}\,,+\frac{1}{3})_{\mathbf{F}}^{2}\leavevmode\nobreak\ :\leavevmode\nobreak\ {B_{R}}^{c}$
| $(\mathbf{1}\,,\mathbf{\overline{3}}\,,-\frac{1}{3})_{\mathbf{F}}^{2}\leavevmode\nobreak\ :\leavevmode\nobreak\ \mathcal{L}_{L}^{2}$ | $(\mathbf{1}\,,\mathbf{\overline{2}}\,,-\frac{1}{2})_{\mathbf{F}}^{2}\leavevmode\nobreak\ :\leavevmode\nobreak\ (E_{L}\,,-N_{L})^{T}$
|  | $(\mathbf{1}\,,\mathbf{1}\,,0)_{\mathbf{F}}^{2}\leavevmode\nobreak\ :\leavevmode\nobreak\ \widetilde{N}_{L}^{2}$
$\mathbf{15_{F}}$ | $(\mathbf{\overline{3}}\,,\mathbf{1}\,,-\frac{2}{3})_{\mathbf{F}}\leavevmode\nobreak\ :\leavevmode\nobreak\ {t_{R}}^{c}$ | $(\mathbf{\overline{3}}\,,\mathbf{1}\,,-\frac{2}{3})_{\mathbf{F}}\leavevmode\nobreak\ :\leavevmode\nobreak\ \underline{{t_{R}}^{c}}$
| $(\mathbf{1}\,,\mathbf{\overline{3}}\,,+\frac{2}{3})_{\mathbf{F}}\leavevmode\nobreak\ :\leavevmode\nobreak\ (\mathcal{E}_{R})^{c}$ | $(\mathbf{1}\,,\mathbf{\overline{2}}\,,+\frac{1}{2})_{\mathbf{F}}\leavevmode\nobreak\ :\leavevmode\nobreak\ ({N_{R}}^{c}\,,{E_{R}}^{c})^{T}$
|  | $(\mathbf{1}\,,\mathbf{1}\,,+1)_{\mathbf{F}}\leavevmode\nobreak\ :\leavevmode\nobreak\ \underline{{\tau_{R}}^{c}}$
| $(\mathbf{3}\,,\mathbf{3}\,,0)_{\mathbf{F}}\leavevmode\nobreak\ :\leavevmode\nobreak\ \mathcal{Q}_{L}$ | $(\mathbf{3}\,,\mathbf{2}\,,+\frac{1}{6})_{\mathbf{F}}\leavevmode\nobreak\ :\leavevmode\nobreak\ \underline{q_{L}=(t_{L}\,,b_{L})^{T}}$
|  | $(\mathbf{3}\,,\mathbf{1}\,,-\frac{1}{3})_{\mathbf{F}}\leavevmode\nobreak\ :\leavevmode\nobreak\ B_{L}$
Table 1:  The $\rm SU(6)$ fermion representations under the ${\cal G}_{331}$
and the ${\cal G}_{\rm SM}$. All SM fermions are marked by solid underlines.

In previous studies of the $331$ model [22, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37, 38, 45, 46, 47, 48, 49], there were two Higgs doublets
at the EW scale, which come from the $\mathbf{\overline{6_{H}}}^{1}$ (chosen
to be $\rho=1$ without loss of generality) and the $\mathbf{15_{H}}$,
respectively. Schematically, one expresses the VEVs of ${\rm SU}(3)$ anti-
fundamentals as follows

$\displaystyle\langle(\mathbf{1}\,,\mathbf{\overline{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{1}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}0\\\
v_{d}\\\ 0\\\
\end{array}\right)\,,\quad\langle(\mathbf{1}\,,\mathbf{\overline{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{2}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}0\\\
0\\\ V_{331}\\\ \end{array}\right)\,,$ (15)
$\displaystyle\langle(\mathbf{1}\,,\mathbf{\overline{3}}\,,+\frac{2}{3})_{\mathbf{H}}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}v_{u}\\\
0\\\ 0\\\ \end{array}\right)\,,$ (19)

with $v_{u}^{2}+v_{d}^{2}=v_{\rm EW}^{2}\approx(246\,{\rm GeV})^{2}$. In such
a scenario, the bottom quark and tau lepton masses are given by Yukawa
couplings as follows

$\displaystyle\mathbf{15_{F}}\mathbf{\overline{6_{H}}}^{1}\mathbf{\overline{6_{F}}}^{1}+H.c.$
$\displaystyle\supset$
$\displaystyle\Big{[}(\mathbf{3}\,,\mathbf{3}\,,0)_{\mathbf{F}}\otimes(\mathbf{\overline{3}}\,,\mathbf{1}\,,+\frac{1}{3})_{\mathbf{F}}^{1}$
(20) $\displaystyle\oplus$
$\displaystyle(\mathbf{1}\,,\mathbf{\overline{3}}\,,+\frac{2}{3})_{\mathbf{F}}\otimes(\mathbf{1}\,,\mathbf{\overline{3}}\,,-\frac{1}{3})_{\mathbf{F}}^{1}\Big{]}\otimes(\mathbf{1}\,,\mathbf{\overline{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{1}+H.c.$
$\displaystyle\supset$
$\displaystyle\Big{[}(\mathbf{3}\,,\mathbf{2}\,,+\frac{1}{6})_{\mathbf{F}}\otimes(\mathbf{\overline{3}}\,,\mathbf{1}\,,+\frac{1}{3})_{\mathbf{F}}^{1}$
$\displaystyle\oplus$
$\displaystyle(\mathbf{1}\,,\mathbf{1}\,,+1)_{\mathbf{F}}\otimes(\mathbf{1}\,,\mathbf{\overline{2}}\,,-\frac{1}{2})_{\mathbf{F}}^{1}\Big{]}\otimes(\mathbf{1}\,,\mathbf{\overline{2}}\,,-\frac{1}{2})_{\mathbf{H}}^{1}+H.c.\,,$

while the top quark mass is given by Yukawa coupling as follows

$\displaystyle\mathbf{15_{F}}\mathbf{15_{F}}\mathbf{15_{H}}+H.c.$
$\displaystyle\supset$
$\displaystyle(\mathbf{3}\,,\mathbf{3}\,,0)_{\mathbf{F}}\otimes(\mathbf{\overline{3}}\,,\mathbf{1}\,,-\frac{2}{3})_{\mathbf{F}}\otimes(\mathbf{1}\,,\mathbf{\overline{3}}\,,+\frac{2}{3})_{\mathbf{H}}+H.c.$
(21) $\displaystyle\supset$
$\displaystyle(\mathbf{3}\,,\mathbf{2}\,,+\frac{1}{6})_{\mathbf{F}}\otimes(\mathbf{\overline{3}}\,,\mathbf{1}\,,-\frac{2}{3})_{\mathbf{F}}\otimes(\mathbf{1}\,,\mathbf{\overline{2}}\,,+\frac{1}{2})_{\mathbf{H}}+H.c.\,.$

All charged fermion masses in the third generation are due to the spontaneous
EWSB. Therefore, the Higgs sector at the EW scale is described by a type-II
2HDM 777This was also pointed out in Ref. [55], though a different symmetry
breaking pattern was considered there.. Furthermore, the other Yukawa coupling
of
$\mathbf{15_{F}}\mathbf{\overline{6_{H}}}^{2}\mathbf{\overline{6_{F}}}^{2}+H.c.$
give fermion masses of $m_{B}=m_{E}=m_{N}\sim{\cal O}(V_{331})$ [46],
according to the VEV assignment in Eq. (15). By integrating out these massive
fermions, the residual massless fermions are found to be anomaly-free under
the ${\cal G}_{\rm SM}$. Loosely speaking, one anti-fundamental fermion of
$\mathbf{\overline{6_{F}}}^{2}$ acquires mass with one Higgs field of
$(\mathbf{1}\,,\mathbf{\overline{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{2}\subset\mathbf{\overline{6_{H}}}^{2}$
developing its VEV of $\mathcal{O}(V_{331})$ 888In the current context, we do
not discuss the mass origin for neutrinos of
$\widetilde{N}_{L}^{\rho}\subset\mathbf{\overline{6_{F}}}^{\rho}$. . Thus, we
name such symmetry breaking pattern as the “fermion-Higgs matching pattern”.

However, the ongoing probes of the second Higgs doublet at the LHC lack direct
evidences for the predicted neutral and charged Higgs bosons from various
channels [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67]. In the type-II
2HDM, hierarchical Yukawa couplings of $Y_{\mathcal{U}}\gg Y_{\mathcal{D}}$
can be expected for the third-generational SM fermion masses. As will be shown
below, the suppressed $(b\,,\tau)$ masses can be realized with a more natural
Yukawa couplings of $Y_{\mathcal{U}}\sim Y_{\mathcal{D}}\sim\mathcal{O}(1)$ in
the current context. Aside from the experimental facts, it is most natural to
consider the following VEVs for the Higgs fields

$\displaystyle\langle(\mathbf{1}\,,\mathbf{\overline{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{\rho=1\,,2}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}0\\\
0\\\ V_{\rho}\\\
\end{array}\right)\,,\quad\langle(\mathbf{1}\,,\mathbf{\overline{3}}\,,+\frac{2}{3})_{\mathbf{H}}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}v_{u}\\\
0\\\ 0\\\ \end{array}\right)\,,$ (28)

purely from the group theoretical point of view. Obviously, two
$\mathcal{G}_{331}$-breaking VEVs in Eq. (28) are in the SM-singlet
components, and one expects the natural hierarchy of $V_{1}\sim
V_{2}\sim\mathcal{O}(V_{331})\gg v_{u}=v_{\rm EW}$. By taking the ${\rm
SU}(2)_{F}$-invariant Yukawa couplings of
$(Y_{\mathcal{D}})_{\rho\sigma}=Y_{\mathcal{D}}\delta_{\rho\sigma}$ in Eq. (3)
for simplicity, we have the following mass terms

$\displaystyle
Y_{\mathcal{D}}\mathbf{15_{F}}\mathbf{\overline{6_{H}}}^{\rho}\mathbf{\overline{6_{F}}}^{\rho}+H.c.\supset
Y_{\mathcal{D}}\Big{[}(\mathbf{3}\,,\mathbf{3}\,,0)_{\mathbf{F}}\otimes(\mathbf{1}\,,\mathbf{\bar{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{\rho}\otimes(\mathbf{\bar{3}}\,,\mathbf{1}\,,+\frac{1}{3})_{\mathbf{F}}^{\rho}$
(37) $\displaystyle\oplus$
$\displaystyle(\mathbf{1}\,,\mathbf{\bar{3}}\,,+\frac{2}{3})_{\mathbf{F}}\otimes(\mathbf{1}\,,\mathbf{\bar{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{\rho}\otimes(\mathbf{1}\,,\mathbf{\bar{3}}\,,-\frac{1}{3})_{\mathbf{F}}^{\rho}\Big{]}+H.c.$
$\displaystyle\Rightarrow$
$\displaystyle\frac{1}{\sqrt{2}}(\bar{b}_{L}\,,\bar{B}_{L})\cdot\left(\begin{array}[]{cc}0&0\\\
Y_{\mathcal{D}}V_{1}&Y_{\mathcal{D}}V_{2}\\\
\end{array}\right)\cdot\left(\begin{array}[]{c}b_{R}\\\ B_{R}\\\
\end{array}\right)+\frac{1}{\sqrt{2}}(\bar{\tau}_{L}\,,\bar{E}_{L})\cdot\left(\begin{array}[]{cc}0&-Y_{\mathcal{D}}V_{1}\\\
0&-Y_{\mathcal{D}}V_{2}\\\
\end{array}\right)\cdot\left(\begin{array}[]{c}\tau_{R}\\\ E_{R}\\\
\end{array}\right)$ $\displaystyle+$ $\displaystyle H.c.\,,$ (38)

with the alternative VEV assignment in Eq. (28). Clearly, both bottom quark
and tau lepton remain massless after the spontaneous breaking of the $331$
symmetry. Meanwhile, there is still only one anti-fundamental fermion of
$\mathbf{\overline{6_{F}}}^{2}$ becoming massive. In this regard, the
alternative symmetry breaking pattern achieved by both
$(\mathbf{1}\,,\mathbf{\overline{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{\rho}\subset\mathbf{\overline{6_{H}}}^{\rho}$
is also valid from the anomaly-free condition. Thus, we name the VEV
assignment in Eq. (28) as the “fermion-Higgs mismatching pattern” of symmetry
breaking. As we shall show below, this VEV assignment leads to a distinct
Higgs spectrum from the 2HDM at the EW scale.

###  2.2 An example: ${\rm SU}(8)$ with three generations

Besides of the above phenomenological consideration, a better motivation of
the current study can be made for non-minimal GUTs with multiple generations.
Let us take the ${\rm SU}(8)$ GUT as an example, which can automatically lead
to $n_{g}=3$ with the following fermion content [6, 7]

$\displaystyle\\{f_{L}\\}_{{\rm SU}(8)}$ $\displaystyle=$
$\displaystyle\left[\mathbf{\overline{8_{F}}}^{\rho}\oplus\mathbf{28_{F}}\right]\bigoplus\left[\mathbf{\overline{8_{F}}}^{\dot{\rho}}\oplus\mathbf{56_{F}}\right]\,,$
(39)

according to the rule in Ref. [3]. This setup enjoys an emergent global
symmetry of $\left[{\rm SU}(4)_{1}\otimes{\rm
U}(1)_{1}\right]\otimes\left[{\rm SU}(5)_{2}\otimes{\rm U}(1)_{2}\right]$ [8,
21], with the flavor indices of $\rho=1\,,...\,,4$ and
$\dot{\rho}=5\,,...\,,9$. To focus on the third-generational fermions, we only
consider the rank-$2$ sector in Eq. (39). ${\rm SU}(4)_{1}$-invariant Yukawa
couplings of
$\mathbf{28_{F}}\mathbf{\overline{8_{H}}}^{\rho}\mathbf{\overline{8_{F}}}^{\rho}+H.c.$
can be expected to give bottom quark and tau lepton masses, which are
analogous to
$\mathbf{15_{F}}\mathbf{\overline{6_{H}}}^{\rho}\mathbf{\overline{6_{F}}}^{\rho}+H.c.$
in the one-generational ${\rm SU}(6)$ model. Another gauge-invariant Yukawa
coupling of $\mathbf{28_{F}}\mathbf{28_{F}}\mathbf{70_{H}}+H.c.$ is expected
to give the top quark mass. A possible symmetry breaking pattern of ${\rm
SU}(8)$ can be expected as follows

$\displaystyle{\rm SU}(8)\xrightarrow{\Lambda_{\rm GUT}}{\rm
SU}(4)_{c}\otimes{\rm SU}(4)_{W}\otimes{\rm
U}(1)_{X_{0}}\xrightarrow{V_{441}}{\rm SU}(3)_{c}\otimes{\rm
SU}(4)_{W}\otimes{\rm U}(1)_{X_{1}}$ (40) $\displaystyle\xrightarrow{V_{341}}$
$\displaystyle{\rm SU}(3)_{c}\otimes{\rm SU}(3)_{W}\otimes{\rm
U}(1)_{X_{2}}\xrightarrow{V_{331}}{\rm SU}(3)_{c}\otimes{\rm
SU}(2)_{L}\otimes{\rm U}(1)_{Y}\,.$

Here, $(V_{441}\,,V_{341}\,,V_{331})$ represent three intermediate symmetry-
breaking scales above the EW scale. The corresponding ${\rm U}(1)$ charges in
Eq. (40) are defined by

$\displaystyle X_{0}(\mathbf{8})$ $\displaystyle\equiv$ $\displaystyle{\rm
diag}(-\frac{1}{4}\,,-\frac{1}{4}\,,-\frac{1}{4}\,,-\frac{1}{4}\,,+\frac{1}{4}\,,+\frac{1}{4}\,,+\frac{1}{4}\,,+\frac{1}{4})\,,$
(41a) $\displaystyle X_{1}(\mathbf{4}_{c})$ $\displaystyle\equiv$
$\displaystyle{\rm
diag}(-\frac{1}{12}+X_{0}\,,-\frac{1}{12}+X_{0}\,,-\frac{1}{12}+X_{0}\,,\frac{1}{4}+X_{0})\,,$
(41b) $\displaystyle X_{2}(\mathbf{4}_{W})$ $\displaystyle\equiv$
$\displaystyle{\rm
diag}(\frac{1}{12}+X_{1}\,,\frac{1}{12}+X_{1}\,,\frac{1}{12}+X_{1}\,,-\frac{1}{4}+X_{1})\,,$
(41c) $\displaystyle Y(\mathbf{4}_{W})$ $\displaystyle\equiv$
$\displaystyle{\rm
diag}(\frac{1}{4}+X_{1}\,,\frac{1}{4}+X_{1}\,,-\frac{1}{4}+X_{1}\,,-\frac{1}{4}+X_{1})\,.$
(41d)

Following the above symmetry breaking pattern and charge quantizations in Eqs.
(41), one can decompose the minimal set of Higgs fields as

$\displaystyle\mathbf{\overline{8_{H}}}^{\rho}$ $\displaystyle\supset$
$\displaystyle(\mathbf{\overline{4}}\,,\mathbf{1}\,,+\frac{1}{4})_{\mathbf{H}}^{\rho}\oplus(\mathbf{1}\,,\mathbf{\overline{4}}\,,-\frac{1}{4})_{\mathbf{H}}^{\rho}\,,$
(42a)
$\displaystyle(\mathbf{\overline{4}}\,,\mathbf{1}\,,+\frac{1}{4})_{\mathbf{H}}^{\rho}$
$\displaystyle=$
$\displaystyle(\mathbf{\overline{3}}\,,\mathbf{1}\,,+\frac{1}{3})_{\mathbf{H}}^{\rho}\oplus(\mathbf{1}\,,\mathbf{1}\,,0)_{\mathbf{H}}^{\rho}\,,$
(42b)
$\displaystyle(\mathbf{1}\,,\mathbf{\overline{4}}\,,-\frac{1}{4})_{\mathbf{H}}^{\rho}$
$\displaystyle=$
$\displaystyle(\mathbf{1}\,,\mathbf{\overline{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{\rho}\oplus(\mathbf{1}\,,\mathbf{1}\,,0)_{\mathbf{H}}^{\rho^{\prime}}=(\mathbf{1}\,,\mathbf{\overline{2}}\,,-\frac{1}{2})_{\mathbf{H}}^{\rho}\oplus(\mathbf{1}\,,\mathbf{1}\,,0)_{\mathbf{H}}^{\rho}\oplus(\mathbf{1}\,,\mathbf{1}\,,0)_{\mathbf{H}}^{\rho^{\prime}}\,,$
(42c) $\displaystyle\mathbf{70_{H}}$ $\displaystyle\supset$
$\displaystyle(\mathbf{4}\,,\mathbf{\overline{4}}\,,+\frac{1}{2})_{\mathbf{H}}\supset(\mathbf{1}\,,\mathbf{\overline{4}}\,,+\frac{3}{4})_{\mathbf{H}}$
(42d) $\displaystyle\supset$
$\displaystyle(\mathbf{1}\,,\mathbf{\overline{3}}\,,+\frac{2}{3})_{\mathbf{H}}^{\prime\prime\prime}=(\mathbf{1}\,,\mathbf{\overline{2}}\,,+\frac{1}{2})_{\mathbf{H}}^{\prime\prime\prime}\oplus(\mathbf{1}\,,\mathbf{1}\,,+1)_{\mathbf{H}}\,.$

Both the $\mathbf{\overline{8_{H}}}^{\rho}$ and the $\mathbf{70_{H}}$ contain
the EWSB components of
$(\mathbf{1}\,,\mathbf{\overline{2}}\,,-\frac{1}{2})_{\mathbf{H}}^{\rho}$ and
$(\mathbf{1}\,,\mathbf{\overline{2}}\,,+\frac{1}{2})_{\mathbf{H}}$,
respectively. Besides, the four $\mathbf{\overline{8_{H}}}^{\rho}$ Higgs
fields contain three singlet components for the intermediate symmetry breaking
in Eq. (40). A more careful counting by the anomaly-free condition at each
stage of symmetry breaking shows that the Higgs spectrum is left with one EW
Higgs doublets from the $\mathbf{\overline{8_{H}}}^{\rho}$ and one from the
$\mathbf{70_{H}}$ if one adopted the “fermion-Higgs matching pattern” of
symmetry breaking, as can be expected for the $n_{g}=3$ case. In this regard,
to have a realistic Higgs spectrum at the EW scale, a “fermion-Higgs
mismatching pattern” of the intermediate symmetry breaking can be generally
expected.

##  3 The Higgs sector of the ${\rm SU}(6)$

In this section, we describe the Higgs sector according to the symmetry
breaking pattern in Eq. (2.1), which consists of ${\rm SU}(3)_{W}$ anti-
fundamentals of
$\Phi_{\mathbf{\overline{3}}\,,\rho}\equiv(\mathbf{1}\,,\mathbf{\overline{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{\rho}\subset\mathbf{\overline{6}}_{\mathbf{H}}^{\rho}$
(with $\rho=1\,,2$) and
$\Phi_{\mathbf{\overline{3}}}^{\prime}\equiv(\mathbf{1}\,,\mathbf{\overline{3}}\,,+\frac{2}{3})_{\mathbf{H}}\subset\mathbf{15_{H}}$
after the ${\rm SU}(6)$ GUT symmetry breaking.

###  3.1 The Higgs potential

The most general ${\rm SU}(6)$ Higgs potential contains Higgs fields of
$(\mathbf{\overline{6_{H}}}^{\rho}\,,\mathbf{15_{H}}\,,\mathbf{35_{H}})$. The
adjoint Higgs field of $\mathbf{35_{H}}$ is responsible for the GUT symmetry
breaking of ${\rm SU}(6)\to\mathcal{G}_{331}$. For our purpose, only the Higgs
fields of $(\mathbf{\overline{6_{H}}}^{\rho}\,,\mathbf{15_{H}})$ will be
relevant for the sequential symmetry breakings. At the GUT scale, the
following terms can be expected in the Higgs potential

$\displaystyle V$ $\displaystyle\supset$ $\displaystyle
m_{11}^{2}|\mathbf{\overline{6_{H}}}^{1}|^{2}+m_{22}^{2}|\mathbf{\overline{6_{H}}}^{2}|^{2}-\Big{(}m_{12}^{2}\mathbf{\overline{6_{H}}}^{1}(\mathbf{\overline{6_{H}}}^{2})^{\dagger}+H.c.\Big{)}+\frac{\lambda_{1}}{2}|\mathbf{\overline{6_{H}}}^{1}|^{4}+\frac{\lambda_{2}}{2}|\mathbf{\overline{6_{H}}}^{2}|^{4}$
(43) $\displaystyle+$
$\displaystyle\lambda_{3}|\mathbf{\overline{6_{H}}}^{1}|^{2}|\mathbf{\overline{6_{H}}}^{2}|^{2}+\lambda_{4}(\mathbf{\overline{6_{H}}}^{1\,{\dagger}}\mathbf{\overline{6_{H}}}^{2})(\mathbf{\overline{6_{H}}}^{2\,{\dagger}}\mathbf{\overline{6_{H}}}^{1})+\frac{\lambda_{5}}{2}\Big{[}(\mathbf{\overline{6_{H}}}^{1\,{\dagger}}\mathbf{\overline{6_{H}}}^{2})^{2}+H.c.\Big{]}$
$\displaystyle+$ $\displaystyle
m^{2}|\mathbf{15_{H}}|^{2}+\lambda|\mathbf{15_{H}}|^{4}$ $\displaystyle+$
$\displaystyle\Big{(}\kappa_{1}|\mathbf{\overline{6_{H}}}^{1}|^{2}+\kappa_{2}|\mathbf{\overline{6_{H}}}^{2}|^{2}\Big{)}|\mathbf{15_{H}}|^{2}+\kappa_{3}(\mathbf{\overline{6_{H}}}^{1\,{\dagger}}\mathbf{15_{H}})(\mathbf{15_{H}}^{\dagger}\mathbf{\overline{6_{H}}}^{1})+\kappa_{4}(\mathbf{\overline{6_{H}}}^{2\,{\dagger}}\mathbf{15_{H}})(\mathbf{15_{H}}^{\dagger}\mathbf{\overline{6_{H}}}^{2})$
$\displaystyle+$
$\displaystyle\Big{(}\nu\,\mathbf{\overline{6_{H}}}^{[1}\mathbf{\overline{6_{H}}}^{2]}\mathbf{15_{H}}+H.c.\Big{)}\,.$

The Higgs potential contains the mass squared parameters of
$(m_{11}^{2}\,,m_{22}^{2}\,,m_{12}^{2}\,,m^{2})$, dimension-one parameter of
$\nu$, and dimensionless self couplings of
$(\lambda_{1\,,...\,,5}\,,\lambda\,,\kappa_{1\,,...\,,4})$. After the GUT
symmetry breaking, we assume all ${\rm SU}(3)_{c}$ colored components of
$(\mathbf{\overline{6_{H}}}^{\rho}\,,\mathbf{15_{H}})$ obtain heavy masses of
$\Lambda_{\rm GUT}$. The residual massless Higgs fields transforming under the
${\rm SU}(3)_{W}\otimes{\rm U}(1)_{X}$ symmetry form the following Higgs
potential

$\displaystyle V_{\rm tot}$ $\displaystyle=$ $\displaystyle
V(\Phi_{\mathbf{\overline{3}}\,,\rho})+V(\Phi_{\mathbf{\overline{3}}}^{\prime})+V(\Phi_{\mathbf{\overline{3}}\,,\rho}\,,\Phi_{\mathbf{\overline{3}}}^{\prime})\,,$
(44a) $\displaystyle V(\Phi_{\mathbf{\overline{3}}\,,\rho})$ $\displaystyle=$
$\displaystyle
m_{11}^{2}|\Phi_{\mathbf{\overline{3}}\,,1}|^{2}+m_{22}^{2}|\Phi_{\mathbf{\overline{3}}\,,2}|^{2}-\Big{(}m_{12}^{2}\Phi_{\mathbf{\overline{3}}\,,1}^{\dagger}\Phi_{\mathbf{\overline{3}}\,,2}+H.c.\Big{)}$
(44b) $\displaystyle+$
$\displaystyle\frac{\lambda_{1}}{2}|\Phi_{\mathbf{\overline{3}}\,,1}|^{4}+\frac{\lambda_{2}}{2}|\Phi_{\mathbf{\overline{3}}\,,2}|^{4}+\lambda_{3}|\Phi_{\mathbf{\overline{3}}\,,1}|^{2}|\Phi_{\mathbf{\overline{3}}\,,2}|^{2}$
$\displaystyle+$
$\displaystyle\lambda_{4}(\Phi_{\mathbf{\overline{3}}\,,1}^{\dagger}\Phi_{\mathbf{\overline{3}}\,,2})(\Phi_{\mathbf{\overline{3}}\,,2}^{\dagger}\Phi_{\mathbf{\overline{3}}\,,1})+\frac{\lambda_{5}}{2}\Big{[}(\Phi_{\mathbf{\overline{3}}\,,1}^{\dagger}\Phi_{\mathbf{\overline{3}}\,,2})^{2}+H.c.\Big{]}\,,$
$\displaystyle V(\Phi_{\mathbf{\overline{3}}}^{\prime})$ $\displaystyle=$
$\displaystyle
m^{2}|\Phi_{\mathbf{\overline{3}}}^{\prime}|^{2}+\lambda|\Phi_{\mathbf{\overline{3}}}^{\prime}|^{4}\,,$
(44c) $\displaystyle
V(\Phi_{\mathbf{\overline{3}}\,,\rho}\,,\Phi_{\mathbf{\overline{3}}}^{\prime})$
$\displaystyle=$
$\displaystyle\Big{(}\kappa_{1}|\Phi_{\mathbf{\overline{3}}\,,1}|^{2}+\kappa_{2}|\Phi_{\mathbf{\overline{3}}\,,2}|^{2}\Big{)}|\Phi_{\mathbf{\overline{3}}}^{\prime}|^{2}$
(44d) $\displaystyle+$
$\displaystyle\kappa_{3}(\Phi_{\mathbf{\overline{3}}\,,1}^{\dagger}\Phi_{\mathbf{\overline{3}}}^{\prime})(\Phi_{\mathbf{\overline{3}}}^{\prime\,{\dagger}}\Phi_{\mathbf{\overline{3}}\,,1})+\kappa_{4}(\Phi_{\mathbf{\overline{3}}\,,2}^{\dagger}\Phi_{\mathbf{\overline{3}}}^{\prime})(\Phi_{\mathbf{\overline{3}}}^{\prime\,{\dagger}}\Phi_{\mathbf{\overline{3}}\,,2})$
$\displaystyle+$
$\displaystyle\Big{(}\nu\epsilon_{IJK}(\Phi_{\mathbf{\overline{3}}\,,1})_{I}(\Phi_{\mathbf{\overline{3}}\,,2})_{J}(\Phi_{\mathbf{\overline{3}}}^{\prime})_{K}+H.c.\Big{)}\,.$

The last $\nu$-term in Eq. (44d) is inevitable by both the gauge symmetry and
the emergent global ${\rm SU}(2)_{F}$ symmetry in Eq. (2), with
$I\,,J\,,K=1\,,2\,,3$ being the ${\rm SU}(3)_{W}$ anti-fundamental indices.

We denote the Higgs fields under the ${\rm SU}(3)_{W}\otimes{\rm U}(1)_{X}$
representations as follows

$\displaystyle\Phi_{\mathbf{\overline{3}}\,,\rho}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\sqrt{2}\phi_{\rho}^{-}\\\
\phi_{\rho}-i\eta_{\rho}\\\
h_{\rho}-i\pi_{\rho}\end{array}\right)\,,\quad\Phi_{\mathbf{\overline{3}}}^{\prime}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}h_{u}+i\pi_{u}\\\
\sqrt{2}\chi^{+}\\\ \sqrt{2}\chi^{\prime\,+}\end{array}\right)\,,$ (51)

where the electric charges are given according to Eq. (6). According to the
VEV assignment in Eq. (28), we expect the non-vanishing Higgs VEVs of

$\displaystyle\langle
h_{1}\rangle=V_{1}=V_{331}c_{\tilde{\beta}}\,,\quad\langle
h_{2}\rangle=V_{2}=V_{331}s_{\tilde{\beta}}\,,\quad\langle
h_{u}\rangle=v_{u}\,,$ (52)

with

$\displaystyle t_{\tilde{\beta}}$ $\displaystyle\equiv$
$\displaystyle\frac{V_{2}}{V_{1}}$ (53)

parametrizing the ratio between two $331$ symmetry breaking VEVs. Accordingly,
the minimization of the Higgs potential in Eqs. (44) leads to the following
conditions

$\displaystyle\frac{\partial V}{\partial V_{1}}=0$ $\displaystyle\Rightarrow$
$\displaystyle
m_{11}^{2}=m_{12}^{2}t_{\tilde{\beta}}-\frac{\lambda_{1}}{2}V_{1}^{2}-\frac{\lambda_{3}+\lambda_{4}+\lambda_{5}}{2}V_{2}^{2}-\frac{\kappa_{1}}{2}v_{u}^{2}\,,$
(54a) $\displaystyle\frac{\partial V}{\partial V_{2}}=0$
$\displaystyle\Rightarrow$ $\displaystyle
m_{22}^{2}=\frac{m_{12}^{2}}{t_{\tilde{\beta}}}-\frac{\lambda_{2}}{2}V_{2}^{2}-\frac{\lambda_{3}+\lambda_{4}+\lambda_{5}}{2}V_{1}^{2}-\frac{\kappa_{2}}{2}v_{u}^{2}\,,$
(54b) $\displaystyle\frac{\partial V}{\partial v_{u}}=0$
$\displaystyle\Rightarrow$ $\displaystyle m^{2}=-\lambda
v_{u}^{2}-\frac{1}{2}(\kappa_{1}V_{1}^{2}+\kappa_{2}V_{2}^{2})\,.$ (54c)

Note that the $\nu$-term that mixes the $\Phi_{\mathbf{\overline{3}}\,,\rho}$
and the $\Phi_{\mathbf{\overline{3}}}^{\prime}$ does not enter the
minimization condition with the VEV assignment in Eq. (28), while this term
will be important in generating the fermion masses. Correspondingly, this
leads to an unwanted tadpole term of
$-\frac{1}{2\sqrt{2}}\nu(\phi_{1}V_{2}-\phi_{2}V_{1})v_{u}$ in the Higgs
potential.

To resolve the tadpole problem, the only way is to develop EWSB VEVs of

$\displaystyle\langle\phi_{1}\rangle=u_{1}=v_{\phi}c_{\beta^{\prime}}\,,\quad\langle\phi_{2}\rangle=u_{2}=v_{\phi}s_{\beta^{\prime}}\,,$
(55)

presumably with $v_{\phi}\sim\mathcal{O}(v_{\rm EW})$ as was considered in
Refs. [40, 41, 42, 43]. The Nambu-Goldstone boson (NGB) of $\xi^{0}$ can be
obtained from the following derivative terms

$\displaystyle\sim$
$\displaystyle\frac{i}{2\sqrt{2}}g_{3L}(N_{\mu}-\bar{N}_{\mu})\partial^{\mu}\Big{[}(u_{1}h_{1}+u_{2}h_{2})-(V_{1}\phi_{1}+V_{2}\phi_{2})\Big{]}$
(56) $\displaystyle\Rightarrow$
$\displaystyle\xi^{0}=c_{\theta}(c_{\tilde{\beta}}\phi_{1}+s_{\tilde{\beta}}\phi_{2})-s_{\theta}(c_{\beta^{\prime}}h_{1}+s_{\beta^{\prime}}h_{2})\,,$

with

$\displaystyle t_{\theta}$ $\displaystyle\equiv$
$\displaystyle\frac{v_{\phi}}{V_{331}}$ (57)

parametrizing the ratio between two symmetry-breaking scales in each ${\rm
SU}(3)_{W}$ anti-fundamental Higgs of $\Phi_{\mathbf{\overline{3}}\,,\rho}$.
With a natural assumption of the following VEV orthogonal relation of

$\displaystyle\sum_{\rho}u_{\rho}V_{\rho}=0\Rightarrow\beta^{\prime}=\tilde{\beta}-\frac{\pi}{2}\,,$
(58)

the potential mixing between the $(W_{\mu}^{\pm}\,,C_{\mu}^{\pm})$, as well as
the $(N_{\mu}\,,\bar{N}_{\mu}\,,Z_{\mu}^{\prime})$, can be avoided. This can
be confirmed with the explicit gauge fields in terms of a $3\times 3$ matrix
given in Eq. (221). Thus, the VEVs in Eq. (55) become

$\displaystyle\langle\phi_{1}\rangle=u_{1}=v_{\phi}s_{\tilde{\beta}}\,,\quad\langle\phi_{2}\rangle=u_{2}=-v_{\phi}c_{\tilde{\beta}}\,.$
(59)

The minimization conditions of the Higgs potential in Eqs. (54) are modified
to be

$\displaystyle\frac{\partial V}{\partial V_{1}}=0$ $\displaystyle\Rightarrow$
$\displaystyle
m_{11}^{2}=m_{12}^{2}t_{\tilde{\beta}}-\frac{\lambda_{1}}{2}(V_{1}^{2}+u_{1}^{2})-\frac{\lambda_{3}}{2}(V_{2}^{2}+u_{2}^{2})$
(60a)
$\displaystyle-\frac{\lambda_{4}+\lambda_{5}}{2}(u_{1}u_{2}+V_{1}V_{2})t_{\tilde{\beta}}-\frac{\kappa_{1}}{2}v_{u}^{2}+\frac{\nu
u_{2}v_{u}}{\sqrt{2}V_{1}}\,,$ $\displaystyle\frac{\partial V}{\partial
V_{2}}=0$ $\displaystyle\Rightarrow$ $\displaystyle
m_{22}^{2}=\frac{m_{12}^{2}}{t_{\tilde{\beta}}}-\frac{\lambda_{2}}{2}(V_{2}^{2}+u_{2}^{2})-\frac{\lambda_{3}}{2}(V_{1}^{2}+u_{1}^{2})$
(60b)
$\displaystyle-\frac{\lambda_{4}+\lambda_{5}}{2}(u_{1}u_{2}+V_{1}V_{2})\frac{1}{t_{\tilde{\beta}}}-\frac{\kappa_{2}}{2}v_{u}^{2}-\frac{\nu
u_{1}v_{u}}{\sqrt{2}V_{2}}\,,$ $\displaystyle\frac{\partial V}{\partial
v_{u}}=0$ $\displaystyle\Rightarrow$ $\displaystyle-m^{2}=\lambda
v_{u}^{2}+\frac{\kappa_{1}}{2}(V_{1}^{2}+u_{1}^{2})+\frac{\kappa_{2}}{2}(V_{2}^{2}+u_{2}^{2})$
(60c) $\displaystyle+\frac{\nu}{\sqrt{2}v_{u}}(u_{1}V_{2}-u_{2}V_{1})\,,$
$\displaystyle\frac{\partial V}{\partial u_{1}}=0$ $\displaystyle\Rightarrow$
$\displaystyle
m_{11}^{2}=-\frac{m_{12}^{2}}{t_{\tilde{\beta}}}-\frac{\lambda_{1}}{2}(V_{1}^{2}+u_{1}^{2})-\frac{\lambda_{3}}{2}(V_{2}^{2}+u_{2}^{2})$
(60d)
$\displaystyle+\frac{\lambda_{4}+\lambda_{5}}{2}(u_{1}u_{2}+V_{1}V_{2})\frac{1}{t_{\tilde{\beta}}}-\frac{\kappa_{1}}{2}v_{u}^{2}-\frac{\nu
V_{2}v_{u}}{\sqrt{2}u_{1}}\,,$ $\displaystyle\frac{\partial V}{\partial
u_{2}}=0$ $\displaystyle\Rightarrow$ $\displaystyle
m_{22}^{2}=-m_{12}^{2}t_{\tilde{\beta}}-\frac{\lambda_{2}}{2}(V_{2}^{2}+u_{2}^{2})-\frac{\lambda_{3}}{2}(V_{1}^{2}+u_{1}^{2})$
(60e)
$\displaystyle+\frac{\lambda_{4}+\lambda_{5}}{2}(u_{1}u_{2}+V_{1}V_{2})t_{\tilde{\beta}}-\frac{\kappa_{2}}{2}v_{u}^{2}+\frac{\nu
V_{1}v_{u}}{\sqrt{2}u_{2}}\,.$

By equating Eqs. (60a) with (60d), and Eqs. (60b) with (60e), we have a
constraint of

$\displaystyle t_{\theta}^{3}-\frac{\sqrt{2}\nu
v_{u}}{(\lambda_{4}+\lambda_{5})V_{331}^{2}}t_{\theta}^{2}+\big{(}\frac{2m_{12}^{2}}{s_{\tilde{\beta}}c_{\tilde{\beta}}(\lambda_{4}+\lambda_{5})V_{331}^{2}}-1\big{)}t_{\theta}+\frac{\sqrt{2}\nu
v_{u}}{(\lambda_{4}+\lambda_{5})V_{331}^{2}}=0\,,$ (61)

with the relation in Eq. (58). Consider the scale hierarchy of
$m_{12}^{2}\sim\mathcal{O}(V_{331}^{2})$, $\nu\sim v_{u}\sim\mathcal{O}(v_{\rm
EW})$. It is straightforward to expect $t_{\theta}\sim\mathcal{O}((v_{\rm
EW}/V_{331})^{2})$ from Eq. (61). Thus, a natural scale of $v_{\phi}$ can be
further suppressed from the EW scale of $v_{\rm EW}$, such as
$\sim\mathcal{O}(1)\,\rm GeV$. This was not noted in the previous Refs. [40,
41, 42, 43] with the similar VEV assignment. Here and after, we will consider
the following parameter inputs 999A negative $\nu$ is chosen so that the
physical Higgs boson mass squares are positive in the current context. This
may not be necessary with additional terms included in the Higgs potential.

$\displaystyle\nu\sim-\mathcal{O}(100)\,{\rm GeV}\,,\quad v_{u}\approx
246\,{\rm GeV}\,,\quad v_{\phi}\sim\mathcal{O}(1)\,{\rm GeV}\,,\quad
V_{331}\sim\mathcal{O}(10)\,{\rm TeV}$ $\displaystyle
t_{\tilde{\beta}}\sim\mathcal{O}(1)\,,\quad
m_{12}^{2}\sim\mathcal{O}(100)\,{\rm TeV}^{2}\,,$ (62)

instead of performing the detailed parameter scans. The choice of
$V_{331}\sim\mathcal{O}(10)\,{\rm TeV}$ will become clear from the
$(b\,,\tau)$ Yukawa couplings with the $125\,\rm GeV$ SM-like Higgs boson.
Notice that in the decoupling limit of $m_{12}\sim V_{331}\to\infty$, one
naturally has $t_{\theta}\to 0$ and $v_{\phi}\to 0$ from Eq. (61).

###  3.2 The charged and CP-odd Higgs bosons

The charged Higgs bosons of
$\Phi_{\pm}=(\phi_{1}^{\pm}\,,\phi_{2}^{\pm}\,,\chi^{\pm}\,,\chi^{\prime\,\pm})$
form the mass squared matrix of

$\displaystyle(\mathcal{M}_{\pm}^{2})_{4\times 4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}M_{\phi_{1}^{+}\phi_{1}^{-}}^{2}&M_{\phi_{1}^{+}\phi_{2}^{-}}^{2}&M_{\phi_{1}^{+}\chi^{-}}^{2}&M_{\phi_{1}^{+}\chi^{\prime\,-}}^{2}\\\\[5.69054pt]
M_{\phi_{1}^{-}\phi_{2}^{+}}^{2}&M_{\phi_{2}^{+}\phi_{2}^{-}}^{2}&M_{\phi_{2}^{+}\chi^{-}}^{2}&M_{\phi_{2}^{+}\chi^{\prime\,-}}^{2}\\\\[5.69054pt]
M_{\phi_{1}^{-}\chi^{+}}^{2}&M_{\phi_{2}^{-}\chi^{+}}^{2}&M_{\chi^{+}\chi^{-}}^{2}&M_{\chi^{+}\chi^{\prime\,-}}^{2}\\\\[5.69054pt]
M_{\phi_{1}^{-}\chi^{\prime\,+}}^{2}&M_{\phi_{2}^{-}\chi^{\prime\,+}}^{2}&M_{\chi^{-}\chi^{\prime\,+}}^{2}&M_{\chi^{\prime\,+}\chi^{\prime\,-}}^{2}\\\
\end{array}\right)\sim\left(\begin{array}[]{cccc}1&1&\epsilon&\epsilon\\\
1&1&\epsilon&\epsilon\\\ \epsilon&\epsilon&\epsilon^{2}&\epsilon^{2}\\\
\epsilon&\epsilon&\epsilon^{2}&1\\\ \end{array}\right)V_{331}^{2}\,,$ (71)

with a small parameter given by
$\epsilon\equiv\frac{v_{u}}{V_{331}}\sim\frac{v_{\phi}}{\nu}\sim\mathcal{O}(0.01)$.
The orthogonal transformations to gauge eigenstates of $\Phi_{\pm}$ are
expressed as follows

$\displaystyle\left(\begin{array}[]{c}G^{\pm}\\\ G^{\prime\,\pm}\\\
H_{1}^{\pm}\\\ H_{2}^{\pm}\end{array}\right)$ $\displaystyle=$
$\displaystyle\widetilde{V}_{\pm}\cdot\left(\begin{array}[]{c}\phi_{1}^{\pm}\\\
\phi_{2}^{\pm}\\\ \chi^{\pm}\\\ \chi^{\prime\,\pm}\end{array}\right)\,,$ (80)
$\displaystyle\widetilde{V}_{\pm}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}c_{\tau_{1}}&0&0&-s_{\tau_{1}}\\\
0&1&0&0\\\ 0&0&1&0\\\ s_{\tau_{1}}&0&0&c_{\tau_{1}}\\\
\end{array}\right)\cdot\left(\begin{array}[]{cccc}1&0&0&0\\\
0&c_{\tau_{2}}&-s_{\tau_{2}}&0\\\ 0&s_{\tau_{2}}&c_{\tau_{2}}&0\\\ 0&0&0&1\\\
\end{array}\right)\cdot\left(\begin{array}[]{cccc}c_{\tilde{\beta}}&s_{\tilde{\beta}}&0&0\\\
-s_{\tilde{\beta}}&c_{\tilde{\beta}}&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\
\end{array}\right)$ (93) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}c_{\tau_{1}}c_{\tilde{\beta}}&c_{\tau_{1}}s_{\tilde{\beta}}&0&-s_{\tau_{1}}\\\
-c_{\tau_{2}}s_{\tilde{\beta}}&c_{\tau_{2}}c_{\tilde{\beta}}&-s_{\tau_{2}}&0\\\
-s_{\tau_{2}}s_{\tilde{\beta}}&s_{\tau_{2}}c_{\tilde{\beta}}&c_{\tau_{2}}&0\\\
s_{\tau_{1}}c_{\tilde{\beta}}&s_{\tau_{1}}s_{\tilde{\beta}}&0&c_{\tau_{1}}\\\
\end{array}\right)\,,$ (98)

with $t_{\tau_{1}}\equiv\frac{v_{u}}{V_{331}}$ and
$t_{\tau_{2}}\equiv\frac{v_{\phi}}{v_{u}}$. Two corresponding non-zero
eigenvalues for two charged Higgs bosons of $H_{1\,,2}^{\pm}$ are given by

$\displaystyle
M_{H_{1}^{\pm}}^{2}=\frac{\mathcal{B}}{16V_{331}v_{u}v_{\phi}}\Big{(}1-\sqrt{1+\frac{\mathcal{A}}{\mathcal{B}^{2}}}\Big{)}\,,\quad
M_{H_{2}^{\pm}}^{2}=\frac{\mathcal{B}}{16V_{331}v_{u}v_{\phi}}\Big{(}1+\sqrt{1+\frac{\mathcal{A}}{\mathcal{B}^{2}}}\Big{)}\,,$
(99) $\displaystyle\mathcal{A}=32\sqrt{2}\nu
v_{u}v_{\phi}(v_{u}^{2}+v_{\phi}^{2})(\kappa_{3}+\kappa_{4}+(\kappa_{3}-\kappa_{4})c_{2\tilde{\beta}})V_{331}^{5}-64v_{\phi}^{2}(v_{u}^{2}+v_{\phi}^{2})(2\nu^{2}+\kappa_{3}\kappa_{4}v_{u}^{2})V_{331}^{4}$
$\displaystyle+$ $\displaystyle 32\sqrt{2}\nu
v_{u}v_{\phi}(v_{u}^{2}+v_{\phi}^{2})((v_{u}^{2}+v_{\phi}^{2})(\kappa_{3}+\kappa_{4})+(v_{u}^{2}-v_{\phi}^{2})(\kappa_{3}-\kappa_{4})c_{2\tilde{\beta}})V_{331}^{3}$
$\displaystyle-$ $\displaystyle
64v_{u}^{2}v_{\phi}^{2}(v_{u}^{2}+v_{\phi}^{2})(2\nu^{2}+\kappa_{3}\kappa_{4}v_{u}^{2})V_{331}^{2}-32\sqrt{2}\nu
v_{u}^{3}v_{\phi}^{3}(v_{u}^{2}+v_{\phi}^{2})((\kappa_{3}-\kappa_{4})c_{2\tilde{\beta}}-\kappa_{3}-\kappa_{4})V_{331}$
$\displaystyle\sim$ $\displaystyle\nu v_{\phi}v_{u}^{3}V_{331}^{5}\,,$
$\displaystyle\mathcal{B}=2V_{331}v_{u}v_{\phi}\Big{[}c_{2\tilde{\beta}}(\kappa_{3}-\kappa_{4})(V_{331}^{2}-v_{\phi}^{2})+(\kappa_{3}+\kappa_{4})(V_{331}^{2}+v_{\phi}^{2})\Big{]}$
$\displaystyle-$ $\displaystyle 8\sqrt{2}\nu
V_{331}^{2}v_{\phi}^{2}-4\sqrt{2}\nu
v_{u}^{2}(V_{331}^{2}+v_{\phi}^{2})+4V_{331}v_{u}^{3}v_{\phi}(\kappa_{3}+\kappa_{4})$
$\displaystyle\sim$ $\displaystyle v_{u}v_{\phi}V_{331}^{3}\,.$

A simple expansion of Eq. (99) in terms of the mass hierarchy assumed in Eq.
(3.1) leads to the approximated mass scales of

$\displaystyle M_{H_{1}^{\pm}}^{2}\sim\mathcal{O}(V_{331}^{2})\,,\quad
M_{H_{2}^{\pm}}^{2}\sim\mathcal{O}(V_{331}^{2})\,.$ (100)

The CP-odd Higgs bosons form the mass squared matrix of

$\displaystyle(\mathcal{M}_{0^{-}}^{2})_{5\times 5}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccc}M_{\pi_{u}\pi_{u}}^{2}&M_{\pi_{u}\eta_{1}}^{2}&M_{\pi_{u}\eta_{2}}^{2}&M_{\pi_{u}\pi_{1}}^{2}&M_{\pi_{u}\pi_{2}}^{2}\\\
M_{\pi_{u}\eta_{1}}^{2}&M_{\eta_{1}\eta_{1}}^{2}&M_{\eta_{1}\eta_{2}}^{2}&M_{\pi_{1}\eta_{1}}^{2}&M_{\pi_{2}\eta_{1}}^{2}\\\
M_{\pi_{u}\eta_{2}}^{2}&M_{\eta_{1}\eta_{2}}^{2}&M_{\eta_{2}\eta_{2}}^{2}&M_{\pi_{1}\eta_{2}}^{2}&M_{\pi_{2}\eta_{2}}^{2}\\\
M_{\pi_{u}\pi_{1}}^{2}&M_{\pi_{1}\eta_{1}}^{2}&M_{\pi_{1}\eta_{2}}^{2}&M_{\pi_{1}\pi_{1}}^{2}&M_{\pi_{1}\pi_{2}}^{2}\\\
M_{\pi_{u}\pi_{2}}^{2}&M_{\pi_{2}\eta_{1}}^{2}&M_{\pi_{2}\eta_{2}}^{2}&M_{\pi_{1}\pi_{2}}^{2}&M_{\pi_{2}\pi_{2}}^{2}\\\
\end{array}\right)$ (106) $\displaystyle\sim$
$\displaystyle\left(\begin{array}[]{ccccc}\epsilon^{2}&\epsilon&\epsilon&\epsilon^{3}&\epsilon^{3}\\\
\epsilon&1&1&\epsilon^{2}&\epsilon^{2}\\\
\epsilon&1&1&\epsilon^{2}&\epsilon^{2}\\\
\epsilon^{3}&\epsilon^{2}&\epsilon^{2}&1&1\\\
\epsilon^{3}&\epsilon^{2}&\epsilon^{2}&1&1\\\
\end{array}\right)V_{331}^{2}\,,$ (112)

in the basis of $\Phi_{0^{-}}=(\pi_{u}\,,\eta_{1\,,2}\,,\pi_{1\,,2})$. We find
three zero eigenvalues corresponding to three massless NGBs with the
constraint in Eq. (61). The orthogonal transformations to $\Phi_{0^{-}}$ are
expressed as follows

$\displaystyle\left(\begin{array}[]{c}G^{0}\\\ G^{0\,\prime}\\\
G^{0\,\prime\prime}\\\ A^{0}\\\ A^{0\,\prime}\end{array}\right)$
$\displaystyle=$
$\displaystyle\widetilde{V}_{0^{-}}\cdot\left(\begin{array}[]{c}\pi_{u}\\\
\eta_{1}\\\ \eta_{2}\\\ \pi_{1}\\\ \pi_{2}\\\
\end{array}\right)\,,\quad\widetilde{V}_{0^{-}}=\left(\begin{array}[]{ccccc}0&c_{\theta}c_{\tilde{\beta}}&c_{\theta}s_{\tilde{\beta}}&s_{\theta}s_{\tilde{\beta}}&-s_{\theta}c_{\tilde{\beta}}\\\
0&-s_{\theta}s_{\tilde{\beta}}&s_{\theta}c_{\tilde{\beta}}&c_{\theta}c_{\tilde{\beta}}&c_{\theta}s_{\tilde{\beta}}\\\
c_{\tau}&s_{\tau}c_{\theta}s_{\tilde{\beta}}&-s_{\tau}c_{\theta}c_{\tilde{\beta}}&s_{\tau}s_{\theta}c_{\tilde{\beta}}&s_{\tau}s_{\theta}s_{\tilde{\beta}}\\\
-s_{\tau}&c_{\tau}c_{\theta}s_{\tilde{\beta}}&-c_{\tau}c_{\theta}c_{\tilde{\beta}}&c_{\tau}s_{\theta}c_{\tilde{\beta}}&c_{\tau}s_{\theta}s_{\tilde{\beta}}\\\
0&s_{\theta}c_{\tilde{\beta}}&s_{\theta}s_{\tilde{\beta}}&-c_{\theta}s_{\tilde{\beta}}&c_{\theta}c_{\tilde{\beta}}\\\
\end{array}\right)\,,$ (128)

with
$t_{\tau}\equiv\frac{v_{\phi}V_{331}}{\sqrt{V_{331}^{2}+v_{\phi}^{2}}v_{u}}\approx\frac{v_{\phi}}{v_{u}}$.
Two corresponding non-zero eigenvalues for two CP-odd Higgs bosons are given
by

$\displaystyle
M_{A^{0}}^{2}=\frac{-\sqrt{2}\nu(V_{331}^{2}+v_{\phi}^{2})v_{u}^{2}+(\lambda_{4}-\lambda_{5})V_{331}v_{\phi}(V_{331}^{2}+v_{\phi}^{2})v_{u}}{2V_{331}v_{\phi}v_{u}}\sim\mathcal{O}(V_{331}^{2})\,,$
(129) $\displaystyle
M_{A^{0\,\prime}}^{2}=-\nu\frac{V_{331}^{2}(v_{u}^{2}+v_{\phi}^{2})+v_{\phi}^{2}v_{u}^{2}}{\sqrt{2}V_{331}v_{\phi}v_{u}}\sim\mathcal{O}(V_{331}^{2})\,.$

Hence, we do not expect the discovery of these two CP-odd Higgs bosons at the
current LHC direct searches.

###  3.3 The CP-even Higgs bosons

There are five CP-even Higgs fields of $(h_{u}\,,\phi_{1\,,2}\,,h_{1\,,2})$ in
the gauge eigenbasis, and one of their linear combination will be identified
as the NGB. Their masses and mixings play the key role in generating the
bottom quark and tau lepton masses, as well as determining their Yukawa
couplings with the SM-like Higgs boson. It takes two steps to obtain their
mass eigenstates. To see that, we first perform the orthonormal
transformations to $(\phi_{1\,,2}\,,h_{1\,,2})$ as follows

$\displaystyle\left(\begin{array}[]{c}\xi^{0}\\\ \phi^{0}\\\ h_{1}^{\prime}\\\
h_{2}^{\prime}\\\ \end{array}\right)$ $\displaystyle=$
$\displaystyle\widetilde{V}_{0^{+}}\cdot\left(\begin{array}[]{c}\phi_{1}\\\
\phi_{2}\\\ h_{1}\\\ h_{2}\\\
\end{array}\right)\,,\quad\widetilde{V}_{0^{+}}=\small{\left(\begin{array}[]{cccc}c_{\theta}c_{\tilde{\beta}}&c_{\theta}s_{\tilde{\beta}}&-s_{\theta}s_{\tilde{\beta}}&s_{\theta}c_{\tilde{\beta}}\\\
-c_{\theta}s_{\tilde{\beta}}&c_{\theta}c_{\tilde{\beta}}&-s_{\theta}c_{\tilde{\beta}}&-s_{\theta}s_{\tilde{\beta}}\\\
s_{\theta}c_{\tilde{\beta}}&s_{\theta}s_{\tilde{\beta}}&c_{\theta}s_{\tilde{\beta}}&-c_{\theta}c_{\tilde{\beta}}\\\
-s_{\theta}s_{\tilde{\beta}}&s_{\theta}c_{\tilde{\beta}}&c_{\theta}c_{\tilde{\beta}}&c_{\theta}s_{\tilde{\beta}}\\\
\end{array}\right)}\,,$ (142)

with $\xi^{0}$ being the massless NGB. Under the basis of
$(h_{u}\,,\phi^{0}\,,h_{1\,,2}^{\prime})$, the remaining four CP-even Higgs
fields form the mass squared matrix and can be expanded as follows

$\displaystyle(\mathcal{M}_{0^{+}}^{2})_{4\times 4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}M_{h_{u}h_{u}}^{2}&M_{h_{u}\phi^{0}}^{2}&M_{h_{u}h_{1}^{\prime}}^{2}&M_{h_{u}h_{2}^{\prime}}^{2}\\\\[5.69054pt]
M_{h_{u}\phi^{0}}^{2}&M_{\phi^{0}\phi^{0}}^{2}&M_{\phi^{0}h_{1}^{\prime}}^{2}&M_{\phi^{0}h_{2}^{\prime}}^{2}\\\\[5.69054pt]
M_{h_{u}h_{1}^{\prime}}^{2}&M_{\phi^{0}h_{1}^{\prime}}^{2}&M_{h_{1}^{\prime}h_{1}^{\prime}}^{2}&M_{h_{1}^{\prime}h_{2}^{\prime}}^{2}\\\\[5.69054pt]
M_{h_{u}h_{2}^{\prime}}^{2}&M_{\phi^{0}h_{2}^{\prime}}^{2}&M_{h_{1}^{\prime}h_{2}^{\prime}}^{2}&M_{h_{2}^{\prime}h_{2}^{\prime}}^{2}\\\
\end{array}\right)\sim\left(\begin{array}[]{cccc}\epsilon^{2}&\epsilon&\epsilon&\epsilon\\\
\epsilon&1&\epsilon^{2}&\epsilon^{2}\\\ \epsilon&\epsilon^{2}&1&1\\\
\epsilon&\epsilon^{2}&1&1\\\ \end{array}\right)V_{331}^{2}$ (151)
$\displaystyle=$
$\displaystyle(\mathcal{M}_{0^{+}}^{2})^{(0)}+(\mathcal{M}_{0^{+}}^{2})^{(1)}+(\mathcal{M}_{0^{+}}^{2})^{(2)}\,,$
(152)

with a small parameter given by
$\epsilon\equiv\frac{v_{u}}{V_{331}}\sim\mathcal{O}(0.01)$. The further
diagonalization of Eq. (151) transforms into mass eigenstates of
$(H_{u}\,,H_{\phi}\,,H_{1}\,,H_{2})$ such that

$\displaystyle\left(\begin{array}[]{c}H_{u}\\\ H_{\phi}\\\ H_{1}\\\
H_{2}\end{array}\right)=V_{0^{+}}\cdot\left(\begin{array}[]{c}h_{u}\\\
\phi^{0}\\\ h_{1}^{\prime}\\\ h_{2}^{\prime}\\\ \end{array}\right)\,,\quad
V_{0^{+}}(\mathcal{M}_{0^{+}}^{2})V_{0^{+}}^{T}={\rm
diag}(M_{H_{u}}^{2}\,,M_{H_{\phi}}^{2}\,,M_{H_{1}}^{2}\,,M_{H_{2}}^{2})\,.$
(161)

Among them, $H_{u}$ is the lightest CP-even Higgs boson with mass of $125$
GeV, while others have masses of $\sim\mathcal{O}(V_{331})$.

To have positive definite eigenvalues for CP-even Higgs boson mass squares in
Eq. (151), one cannot have the $\nu$ parameter as large as $V_{331}$. That is
why we chose $\nu\sim\mathcal{O}(100)\,{\rm GeV}$ in Eq. (3.1). However, a
$\nu$-problem emerges, namely, why is a mass parameter from a $331$-invariant
Higgs potential takes a value comparable to the EW scale. This problem is
analogous to the well-known $\mu$-problem in the minimal supersymmetric
Standard Model (MSSM) [68]. One can thus expect this $\nu$-term to originate
from some non-renormalizable terms in the realistic non-minimal GUTs with
$n_{g}=3$ 101010In the MSSM case, this was known as the Kim-Nilles mechanism
for the $\mu$-problem [69, 70].. This type of terms are inevitable due to the
gravitational effects that break the global ${\rm U}(1)$ symmetry explicitly
111111In the axion models, this effect leads to what is known as the PQ
quality problem [18, 19, 20].. Taking the ${\rm SU}(8)$ GUT as an example
again, one such possible $d=5$ non-renormalizable term is expected to be

$\displaystyle{\rm SU}(8)\leavevmode\nobreak\ $ $\displaystyle:$
$\displaystyle\leavevmode\nobreak\ \frac{1}{M_{\rm
pl}}\epsilon_{\rho_{1}...\rho_{4}}\mathbf{\overline{8_{H}}}^{\rho_{1}}...\mathbf{\overline{8_{H}}}^{\rho_{4}}\cdot\mathbf{70_{H}}$
(162) $\displaystyle\supset$ $\displaystyle\frac{1}{M_{\rm
pl}}\epsilon_{\rho_{1}...\rho_{4}}\,(\mathbf{\overline{4}}\,,\mathbf{1}\,,+\frac{1}{4})_{\mathbf{H}}^{\rho_{1}}\otimes(\mathbf{1}\,,\mathbf{\overline{4}}\,,-\frac{1}{4})_{\mathbf{H}}^{\rho_{2}}\otimes(\mathbf{1}\,,\mathbf{\overline{4}}\,,-\frac{1}{4})_{\mathbf{H}}^{\rho_{3}}\otimes(\mathbf{1}\,,\mathbf{\overline{4}}\,,-\frac{1}{4})_{\mathbf{H}}^{\rho_{4}}\otimes(\mathbf{4}\,,\mathbf{\overline{4}}\,,+\frac{1}{2})_{\mathbf{H}}$
$\displaystyle\supset$ $\displaystyle\frac{V_{441}}{M_{\rm
pl}}\epsilon_{\rho_{2}...\rho_{4}}\,(\mathbf{1}\,,\mathbf{\overline{4}}\,,-\frac{1}{4})_{\mathbf{H}}^{\rho_{2}}\otimes(\mathbf{1}\,,\mathbf{\overline{4}}\,,-\frac{1}{4})_{\mathbf{H}}^{\rho_{3}}\otimes(\mathbf{1}\,,\mathbf{\overline{4}}\,,-\frac{1}{4})_{\mathbf{H}}^{\rho_{4}}\otimes(\mathbf{1}\,,\mathbf{\overline{4}}\,,+\frac{3}{4})_{\mathbf{H}}$
$\displaystyle\sim$ $\displaystyle\frac{V_{441}V_{341}}{M_{\rm
pl}}\epsilon_{\rho_{3}\rho_{4}}\,(\mathbf{1}\,,\mathbf{\overline{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{\rho_{3}}\otimes(\mathbf{1}\,,\mathbf{\overline{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{\rho_{4}}\otimes(\mathbf{1}\,,\mathbf{\overline{3}}\,,+\frac{2}{3})_{\mathbf{H}}^{\prime\prime\prime}\,.$

Here, the decompositions are achieved according to Eqs. (42). Obviously, this
non-renormalizable term induced by the gravitational effect reproduces what we
considered as the $\nu$-term in Eq. (44d). Thus, the value of
$\nu\sim\mathcal{O}(100)\,{\rm GeV}$ in Eq. (3.1) can be naturally realized
with $V_{441}\sim\mathcal{O}(10^{12})\,{\rm GeV}$ and
$V_{341}\sim\mathcal{O}(10^{9})\,{\rm GeV}$.

With the hierarchies of mass parameter in Eq. (3.1), the diagonalization of
mass matrix in Eq. (151) can be achieved in terms of perturbation. Hence, we
express the mixing matrix in Eq. (161) as

$\displaystyle V_{0^{+}}$ $\displaystyle=$
$\displaystyle\widetilde{U}\widetilde{U}^{(0)}\,.$ (163)

At the leading order, it is straightforward to diagonalize the
$(\mathcal{M}_{0^{+}}^{2})^{(0)}$ by an orthogonal matrix of

$\displaystyle\widetilde{U}^{(0)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\mathbb{I}_{2\times 2}&&\\\
&c_{\alpha}&-s_{\alpha}\\\ &s_{\alpha}&c_{\alpha}\\\ \end{array}\right)\,,$
(167)

into

$\displaystyle\widetilde{U}^{(0)}\cdot(\mathcal{M}_{0^{+}}^{2})^{(0)}\cdot\widetilde{U}^{(0)\,T}={\rm
diag}(0\,,M_{\phi^{0}\phi^{0}}^{2}\,,\widetilde{M}_{h_{1}^{\prime}h_{1}^{\prime}}^{2}\,,\widetilde{M}_{h_{2}^{\prime}h_{2}^{\prime}}^{2})\,,$
(168)

where
$t_{\alpha}=[M_{h_{1}^{\prime}h_{1}^{\prime}}^{2}-M_{h_{2}^{\prime}h_{2}^{\prime}}^{2}-\sqrt{(M_{h_{1}^{\prime}h_{1}^{\prime}}^{2}-M_{h_{2}^{\prime}h_{2}^{\prime}}^{2})^{2}+4M_{h_{1}^{\prime}h_{2}^{\prime}}^{4}}]/(2M_{h_{1}^{\prime}h_{2}^{\prime}}^{2})$.
The mixing matrix of $\widetilde{U}$ for the higher-order terms can be
expanded up to $\mathcal{O}(\epsilon^{2})$ as

$\displaystyle\widetilde{U}$ $\displaystyle=$
$\displaystyle\mathbb{I}_{4\times
4}+\widetilde{U}^{(1)}+\widetilde{U}^{(2)}+\mathcal{O}(\epsilon^{3})\,,$ (169)

with $\widetilde{U}^{(1)}\sim\mathcal{O}(\epsilon)$ and
$\widetilde{U}^{(2)}\sim\mathcal{O}(\epsilon^{2})$. Similarly, $V_{0^{+}}$ can
also be expanded as:

$\displaystyle V_{0^{+}}$ $\displaystyle=$
$\displaystyle\widetilde{U}^{(0)}+V_{0^{+}}^{(1)}+V_{0^{+}}^{(2)}+\mathcal{O}(\epsilon^{3})\,.$
(170)

For our later usage, we find that the $V_{0^{+}}^{(1)}$ is expressed as
follows

$\displaystyle V_{0^{+}}^{(1)}$ $\displaystyle=$
$\displaystyle\small{\left(\begin{array}[]{cccc}&-\frac{M_{h_{u}\phi^{0}}^{2}}{M_{\phi^{0}\phi^{0}}^{2}}&-\frac{M_{h_{u}h_{1}^{\prime}}^{2}}{M_{h_{1}^{\prime}h_{1}^{\prime}}^{2}}&-\frac{M_{h_{u}h_{2}^{\prime}}^{2}}{M_{h_{2}^{\prime}h_{2}^{\prime}}^{2}}\\\
\frac{M_{h_{u}\phi^{0}}^{2}}{M_{\phi^{0}\phi^{0}}^{2}}&&&\\\\[5.69054pt]
\frac{M_{h_{u}h_{1}^{\prime}}^{2}}{M_{h_{1}^{\prime}h_{1}^{\prime}}^{2}}&&&\\\\[5.69054pt]
\frac{M_{h_{u}h_{2}^{\prime}}^{2}}{M_{h_{2}^{\prime}h_{2}^{\prime}}^{2}}&&&\\\
\end{array}\right)\sim\left(\begin{array}[]{cccc}&-\epsilon&-\epsilon&-\epsilon\\\
\epsilon&&&\\\ \epsilon&&&\\\ \epsilon&&&\\\ \end{array}\right)}\,,$ (179)

with
$(0\,,0\,,\widetilde{M}_{h_{u}h_{1}^{\prime}}^{2}\,,\widetilde{M}_{h_{u}h_{2}^{\prime}}^{2})=\widetilde{U}^{(0)}\cdot(0\,,0\,,M_{h_{u}h_{1}^{\prime}}^{2}\,,M_{h_{u}h_{2}^{\prime}}^{2})$.
By using the perturbation expansion in Eq. (170), we find the SM-like CP-even
Higgs boson mass of

$\displaystyle M_{H_{u}}^{2}$ $\displaystyle=$
$\displaystyle(2\lambda-\frac{\nu
v_{\phi}V_{331}}{\sqrt{2}v_{u}^{3}})v_{u}^{2}-[(V_{0^{+}}^{(1)})_{12}^{2}M_{\phi^{0}\phi^{0}}^{2}]-[(V_{0^{+}}^{(1)})_{13}^{2}M_{h_{1}^{\prime}h_{1}^{\prime}}^{2}]-[(V_{0^{+}}^{(1)})_{14}^{2}M_{h_{2}^{\prime}h_{2}^{\prime}}^{2}]\,.$
(180)

Since the mixing elements are
$(V_{0^{+}}^{(1)})_{ij}\sim\mathcal{O}(\epsilon)$, all terms here are of the
EW scales.

###  3.4 Summary of the Higgs spectrum

| $M_{H_{u}}$ | $M_{H_{\phi}}$ | $M_{H_{1}}$ | $M_{H_{2}}$ | $M_{A^{0}}$ | $M_{A^{0\,\prime}}$ | $M_{H_{1}^{\pm}}$ | $M_{H_{2}^{\pm}}$
---|---|---|---|---|---|---|---|---
A | $125\,{\rm GeV}$ | $13.2\,{\rm TeV}$ | $14.1\,{\rm TeV}$ | $10.7\,{\rm TeV}$ | $13.4\,{\rm TeV}$ | $13.2\,{\rm TeV}$ | $13.2\,{\rm TeV}$ | $7.1\,{\rm TeV}$
B | $125\,{\rm GeV}$ | $29.5\,{\rm TeV}$ | $50.1\,{\rm TeV}$ | $33.3\,{\rm TeV}$ | $2.9\,{\rm TeV}$ | $2.9\,{\rm TeV}$ | $2.9\,{\rm TeV}$ | $15.8\,{\rm TeV}$
| $\lambda$ | $\lambda_{1}$ | $\lambda_{2}$ | $\lambda_{3}$ | $\lambda_{4}$ | $\lambda_{5}$ | $\tan\tilde{\beta}$ |
A | $0.51$ | $2.0$ | $1.0$ | $1.0$ | $0.2$ | $0.1$ | $1.7$ |
B | $0.13$ | $1.0$ | $1.0$ | $1.0$ | $0.1$ | $0.1$ | $10.0$ |
| $\kappa_{1}$ | $\kappa_{2}$ | $\kappa_{3}$ | $\kappa_{4}$ | $v_{u}$ | $v_{\phi}$ | $\nu$ | $V_{331}$
A | $0.8$ | $1.0$ | $1.0$ | $1.0$ | $246\,{\rm GeV}$ | $1.0\,{\rm GeV}$ | $-100\,{\rm GeV}$ | $10\,{\rm TeV}$
B | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $246\,{\rm GeV}$ | $2.0\,{\rm GeV}$ | $-200\,{\rm GeV}$ | $50\,{\rm TeV}$
Table 2:  The Higgs spectrum and the parameters in the Higgs potential.

In the end of this section, we briefly summarize the Higgs spectrum in the
current context. The symmetry breaking of
$\mathcal{G}_{331}\to\mathcal{G}_{\rm SM}$ and the sequential EWSB require
eight NGBs, while the Higgs sector contains three ${\rm SU}(3)_{W}$ anti-
fundamentals of $\Phi_{\mathbf{\overline{3}}\,,\rho}$ and
$\Phi_{\mathbf{\overline{3}}}^{\prime}$. Therefore, we have ten real scalars
in all. Through the above analysis, we find the $331$ Higgs spectrum is
consist of: two charged Higgs bosons of $H_{1\,,2}^{\pm}$ from Eq. (100), two
CP-odd Higgs bosons of $(A^{0}\,,A^{0\,\prime})$ from Eq. (129), and four CP-
even Higgs bosons of $(H_{u}\,,H_{\phi}\,,H_{1}\,,H_{2})$ from Eqs. (151) and
(161). The explicit expressions for Higgs mass matrix in Eqs. (71), (106), and
(151) will be given in App. A.3. At the EW scale, our Higgs spectrum only
contains one CP-even Higgs boson of $H_{u}$, while all other Higgs boson
masses are decoupled. Therefore, our effective theory at the EW scale is
distinct from the 2HDM, where a total of four Higgs bosons with masses of the
EW scale are generally expected. Here, we list two benchmark models for the
Higgs spectrum in Tab. 2 to demonstrate our results explicitly.

##  4 Bottom quark and tau lepton masses in the ${\rm SU}(6)$

###  4.1 The Yukawa couplings

By taking the Higgs VEVs in Eqs. (52) and (55), we have the following mass
matrices for the down-type $(b\,,B)$ quarks and charged $(\tau\,,E)$ leptons

$\displaystyle(Y_{\mathcal{D}})_{\rho\sigma}\Big{[}(\mathbf{3}\,,\mathbf{3}\,,0)_{\mathbf{F}}\otimes(\mathbf{1}\,,\mathbf{\bar{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{\rho}\otimes(\mathbf{\bar{3}}\,,\mathbf{1}\,,+\frac{1}{3})_{\mathbf{F}}^{\sigma}$
(181e) $\displaystyle\oplus$
$\displaystyle(\mathbf{1}\,,\mathbf{\bar{3}}\,,+\frac{2}{3})_{\mathbf{F}}\otimes(\mathbf{1}\,,\mathbf{\bar{3}}\,,-\frac{1}{3})_{\mathbf{H}}^{\rho}\otimes(\mathbf{1}\,,\mathbf{\bar{3}}\,,-\frac{1}{3})_{\mathbf{F}}^{\sigma}\Big{]}+H.c.$
$\displaystyle\Rightarrow$
$\displaystyle(\bar{b}_{L}\,,\bar{B}_{L})\cdot\mathcal{M}_{\mathcal{B}}\cdot\left(\begin{array}[]{c}b_{R}\\\
B_{R}\\\
\end{array}\right)+(\bar{\tau}_{L}\,,\bar{E}_{L})\cdot\mathcal{M}_{\mathcal{E}}\cdot\left(\begin{array}[]{c}\tau_{R}\\\
E_{R}\\\ \end{array}\right)+H.c.\,,$ $\displaystyle\mathcal{M}_{\mathcal{B}}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}(Y_{\mathcal{D}})_{11}u_{1}+(Y_{\mathcal{D}})_{12}u_{2}&(Y_{\mathcal{D}})_{21}u_{1}+(Y_{\mathcal{D}})_{22}u_{2}\\\
(Y_{\mathcal{D}})_{11}V_{1}+(Y_{\mathcal{D}})_{12}V_{2}&(Y_{\mathcal{D}})_{21}V_{1}+(Y_{\mathcal{D}})_{22}V_{2}\\\
\end{array}\right)\sim\left(\begin{array}[]{cc}\epsilon^{2}&\epsilon^{2}\\\
1&1\\\ \end{array}\right)V_{331}\,,$ (181j)
$\displaystyle\mathcal{M}_{\mathcal{E}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}(Y_{\mathcal{D}})_{11}u_{1}+(Y_{\mathcal{D}})_{12}u_{2}&-(Y_{\mathcal{D}})_{11}V_{1}-(Y_{\mathcal{D}})_{12}V_{2}\\\
(Y_{\mathcal{D}})_{21}u_{1}+(Y_{\mathcal{D}})_{22}u_{2}&-(Y_{\mathcal{D}})_{21}V_{1}-(Y_{\mathcal{D}})_{22}V_{2}\\\
\end{array}\right)\sim\left(\begin{array}[]{cc}\epsilon^{2}&-1\\\
\epsilon^{2}&-1\\\ \end{array}\right)V_{331}\,.$ (181o)

Given the seesaw-like mass matrices according to the mass hierarchy given in
Eq. (3.1), a suppressed bottom quark and tau lepton masses of
$\sim\mathcal{O}(1)\,{\rm GeV}$ can be realized with the natural Yukawa
couplings of $(Y_{\mathcal{D}})_{ij}\sim\mathcal{O}(1)$.

###  4.2 The bottom quark mass

Specifically, we first illustrate the bottom quark mass, and the tau lepton
mass can be obtained straightforwardly. In general, the mass matrix in Eq.
(181j) can be diagonalized through the bi-unitary transformation as

$\displaystyle
U_{L}^{\mathcal{B}}\mathcal{M}_{\mathcal{B}}(U_{R}^{\mathcal{B}})^{\dagger}=\left(\begin{array}[]{cc}m_{b}&0\\\
0&m_{B}\\\ \end{array}\right)\,,$ (184) $\displaystyle
U_{L/R}^{\mathcal{B}}=\left(\begin{array}[]{cc}c_{L/R}&-s_{L/R}\\\
s_{L/R}&c_{L/R}\\\
\end{array}\right)\,,\quad\left(\begin{array}[]{c}b_{L/R}^{\prime}\\\
B_{L/R}^{\prime}\end{array}\right)=U_{L/R}^{\mathcal{B}}\cdot\left(\begin{array}[]{c}b_{L/R}\\\
B_{L/R}\end{array}\right)\,,$ (191)

with $(b^{\prime}\,,B^{\prime})$ being the mass eigenstates. We find that the
corresponding Yukawa couplings are expressed in terms of masses and mixing
angles as follows

$\displaystyle(Y_{\mathcal{D}})_{11}$ $\displaystyle=$
$\displaystyle\sqrt{2}\Big{[}(c_{L}c_{R}m_{b}+s_{L}s_{R}m_{B})\frac{s_{\tilde{\beta}}}{v_{\phi}}+(-s_{L}c_{R}m_{b}+c_{L}s_{R}m_{B})\frac{c_{\tilde{\beta}}}{V_{331}}\Big{]}\,,$
(192a) $\displaystyle(Y_{\mathcal{D}})_{12}$ $\displaystyle=$
$\displaystyle\sqrt{2}\Big{[}(c_{L}c_{R}m_{b}+s_{L}s_{R}m_{B})\frac{-c_{\tilde{\beta}}}{v_{\phi}}+(-s_{L}c_{R}m_{b}+c_{L}s_{R}m_{B})\frac{s_{\tilde{\beta}}}{V_{331}}\Big{]}\,,$
(192b) $\displaystyle(Y_{\mathcal{D}})_{21}$ $\displaystyle=$
$\displaystyle\sqrt{2}\Big{[}(-c_{L}s_{R}m_{b}+s_{L}c_{R}m_{B})\frac{s_{\tilde{\beta}}}{v_{\phi}}+(s_{L}s_{R}m_{b}+c_{L}c_{R}m_{B})\frac{c_{\tilde{\beta}}}{V_{331}}\Big{]}\,,$
(192c) $\displaystyle(Y_{\mathcal{D}})_{22}$ $\displaystyle=$
$\displaystyle\sqrt{2}\Big{[}(-c_{L}s_{R}m_{b}+s_{L}c_{R}m_{B})\frac{-c_{\tilde{\beta}}}{v_{\phi}}+(s_{L}s_{R}m_{b}+c_{L}c_{R}m_{B})\frac{s_{\tilde{\beta}}}{V_{331}}\Big{]}\,.$
(192d)

Under the reasonable limit of $\varphi_{L/R}\to 0$ and $t_{\tilde{\beta}}\sim
1$, we find the natural Yukawa couplings of
$(Y_{\mathcal{D}})_{11}\sim(Y_{\mathcal{D}})_{12}\sim
m_{b}/v_{\phi}\sim\mathcal{O}(1)$ and
$(Y_{\mathcal{D}})_{21}\sim(Y_{\mathcal{D}})_{22}\sim
m_{B}/V_{331}\sim\mathcal{O}(1)$.

By performing the orthogonal transformation in Eq. (142), we find the
following bottom quark Yukawa couplings with the CP-even Higgs bosons

$\displaystyle-\mathcal{L}_{Y}^{\mathcal{Q}\,,0^{+}}$ $\displaystyle\supset$
$\displaystyle-
m_{b}\overline{b_{L}^{\prime}}b_{R}^{\prime}\Big{[}(c_{L}^{2}\frac{c_{\theta}}{v_{\phi}}+s_{L}^{2}\frac{s_{\theta}}{V_{331}})\phi^{0}+s_{L}c_{L}(\frac{c_{\theta}}{v_{\phi}}+\frac{s_{\theta}}{V_{331}})h_{1}^{\prime}+(c_{L}^{2}\frac{s_{\theta}}{v_{\phi}}-s_{L}^{2}\frac{c_{\theta}}{V_{331}})h_{2}^{\prime}\Big{]}+H.c.$
(193) $\displaystyle\approx$ $\displaystyle-
m_{b}\overline{b_{L}^{\prime}}b_{R}^{\prime}\Big{(}\frac{c_{L}^{2}}{v_{\phi}}\phi^{0}+\frac{s_{L}c_{L}}{v_{\phi}}h_{1}^{\prime}+\frac{c_{L}^{2}-s_{L}^{2}}{V_{331}}h_{2}^{\prime}\Big{)}+H.c.\,,$

from Eq. (252b), and with the approximation given under the decoupling limit
of $v_{\phi}/V_{331}\to 0$. By using the orthogonal transformation to CP-even
Higgs fields in Eq. (161), the bottom quark Yukawa coupling with the SM-like
Higgs boson of $H_{u}$ reads

$\displaystyle-\mathcal{L}_{Y}[H_{u}]$ $\displaystyle\supset$ $\displaystyle-
m_{b}\overline{b_{L}^{\prime}}b_{R}^{\prime}\Big{[}(c_{L}^{2}\frac{c_{\theta}}{v_{\phi}}+s_{L}^{2}\frac{s_{\theta}}{V_{331}})(V_{0^{+}}^{(1)})_{12}+s_{L}c_{L}(\frac{c_{\theta}}{v_{\phi}}+\frac{s_{\theta}}{V_{331}})(V_{0^{+}}^{(1)})_{13}$
(194) $\displaystyle+$
$\displaystyle(c_{L}^{2}\frac{s_{\theta}}{v_{\phi}}-s_{L}^{2}\frac{c_{\theta}}{V_{331}})(V_{0^{+}}^{(1)})_{14}\Big{]}H_{u}+H.c.$
$\displaystyle\approx$ $\displaystyle-
m_{b}\overline{b_{L}^{\prime}}b_{R}^{\prime}\Big{[}\frac{c_{L}^{2}}{v_{\phi}}(V_{0^{+}}^{(1)})_{12}+\frac{s_{L}c_{L}}{v_{\phi}}(V_{0^{+}}^{(1)})_{13}+\frac{c_{L}^{2}-s_{L}^{2}}{V_{331}}(V_{0^{+}}^{(1)})_{14}\Big{]}H_{u}+H.c.$
$\displaystyle\approx$ $\displaystyle-
m_{b}\overline{b_{L}^{\prime}}b_{R}^{\prime}\Big{[}\frac{c_{L}^{2}}{v_{\phi}}(V_{0^{+}}^{(1)})_{12}+\frac{s_{L}c_{L}}{v_{\phi}}(V_{0^{+}}^{(1)})_{13}\Big{]}H_{u}+H.c.\,,$

with the mixing matrices in Eqs. (161), (163), (179) for the CP-even Higgs
bosons. Likewise, we find the Yukawa couplings of the SM-like Higgs boson of
$H_{u}$ with the heavy $B^{\prime}$ quark as

$\displaystyle-\mathcal{L}_{Y}[H_{u}]$ $\displaystyle\approx$ $\displaystyle-
m_{B}\overline{B_{L}^{\prime}}B_{R}^{\prime}\Big{[}s_{L}^{2}c_{\theta}\frac{s_{\beta}^{2}+s_{\beta}c_{\beta}}{v_{\phi}}(V_{0^{+}}^{(1)})_{12}-s_{L}c_{L}\frac{c_{\theta}}{v_{\phi}}(V_{0^{+}}^{(1)})_{13}\Big{]}H_{u}+H.c.\,.$
(195)

One can expect two constraints from the SM sector, namely,

  * $\mathfrak{X}$

the EW charged currents mediated by $W^{\pm}$,

  * $\mathfrak{Y}$

all SM-like Higgs boson couplings, including $H_{u}\bar{f}f$, $H_{u}VV$
($V=W^{\pm}\,,Z$), $H_{u}gg$, and $H_{u}\gamma\gamma$.

From the EW charged currents given in terms of the gauge eigenstates in Eq.
(248), it is straightforward to find that $V_{tb}=c_{L}$. It is thus natural
to take the limit of $c_{L}\to 1$, according to the measurement of the CKM
mixing angle of $|V_{tb}|=1.013\pm 0.030$ [71]. Under this limit, the SM-like
Higgs boson couplings to the heavy $B^{\prime}$ quark vanishes in Eq. (195) as
$\varphi_{L}\to 0$. Thus, the potential heavy $B^{\prime}$ quark contributions
to the effective $H_{u}gg$ and $H_{u}\gamma\gamma$ couplings are vanishing in
this limit. Let us return to the bottom quark Yukawa coupling in Eq. (194)
when $\varphi_{L}\to 0$, it is further simplified to

$\displaystyle-\mathcal{L}_{Y}[H_{u}]$ $\displaystyle\approx$
$\displaystyle-\frac{m_{b}}{v_{\phi}}(V_{0^{+}}^{(1)})_{12}\,\overline{b_{L}^{\prime}}b_{R}^{\prime}H_{u}+H.c.\,.$
(196)

By requiring that the tree-level $H_{u}\bar{b}^{\prime}b^{\prime}$ Yukawa
coupling in Eq. (194) is the same as the SM prediction [72, 73], we find the
relation of

$\displaystyle\frac{v_{\rm EW}}{v_{\phi}}(V_{0^{+}}^{(1)})_{12}\approx
1\Rightarrow\frac{v_{\rm EW}^{2}}{v_{\phi}V_{331}}\approx 1\,,$ (197)

with the mixing angle of $(V_{0^{+}}^{(1)})_{12}\sim\epsilon=\frac{v_{\rm
EW}}{V_{331}}$ in Eq. (179). For simplicity, the sub-leading correction term
suppressed by $1/V_{331}$ in Eq. (194) is neglected. Apparently, this relation
leads to the natural new physics scale for the $331$ symmetry of

$\displaystyle\boxed{V_{331}\sim\mathcal{O}(10)\,\rm TeV}\,,$ (198)

with the reasonable choice of $v_{\phi}\sim\mathcal{O}(1)\,{\rm GeV}$ for the
bottom quark Yukawa coupling. This confirms our previous assumption of the
benchmark parameter input in Eq. (3.1). We have also checked that a new
physics scale of $V_{331}$ in Eq. (198) is even consistent with the most
stringent limit to the rare flavor-changing lepton decay process of ${\rm
Br}(\mu\to e\gamma)$ [74] when generalizing to the three-generational case
[45].

###  4.3 The tau lepton mass

The tau lepton mass and Yukawa couplings follow closely from the bottom quark
case, and we present the discussion here for completeness. The general
$\mathcal{E}=(\tau\,,E)$ mass matrix in Eq. (181o) is related to the
$\mathcal{B}=(b\,,B)$ mass matrix in Eq. (181j) by

$\displaystyle\mathcal{M}_{\mathcal{E}}$ $\displaystyle=$
$\displaystyle\mathcal{M}_{\mathcal{B}}^{T}\cdot\sigma_{3}\,.$ (199)

It is straightforward to find that the bi-unitary transformation for the
$\mathcal{E}=(\tau\,,E)$ is simply related to those for the
$\mathcal{B}=(b\,,B)$ as below

$\displaystyle
U_{L}^{\mathcal{E}}\mathcal{M}_{\mathcal{E}}(U_{R}^{\mathcal{E}})^{\dagger}=\left(\begin{array}[]{cc}m_{\tau}&0\\\
0&m_{E}\\\
\end{array}\right)\,,\quad\left(\begin{array}[]{c}\tau_{L/R}^{\prime}\\\
E_{L/R}^{\prime}\end{array}\right)=U_{L/R}^{\mathcal{E}}\cdot\left(\begin{array}[]{c}\tau_{L/R}\\\
E_{L/R}\end{array}\right)$ (206) $\displaystyle
U_{L}^{\mathcal{E}}=\left(\begin{array}[]{cc}c_{R}&-s_{R}\\\ s_{R}&c_{R}\\\
\end{array}\right)\,,\quad
U_{R}^{\mathcal{E}}=\left(\begin{array}[]{cc}c_{L}&s_{L}\\\ -s_{L}&c_{L}\\\
\end{array}\right)\,.$ (211)

Immediately, this leads a result of $s_{R}=0$ from leptonic sector of the EW
charged currents in Eq. (248). Analogous to Eqs. (192), the Yukawa couplings
can also be expressed as

$\displaystyle(Y_{\mathcal{D}})_{11}$ $\displaystyle=$
$\displaystyle\sqrt{2}\Big{[}(c_{L}c_{R}m_{\tau}-s_{L}s_{R}m_{E})\frac{s_{\tilde{\beta}}}{v_{\phi}}+(s_{L}c_{R}m_{\tau}+c_{L}s_{R}m_{E})\frac{-c_{\tilde{\beta}}}{V_{331}}\Big{]}\,,$
(212a) $\displaystyle(Y_{\mathcal{D}})_{12}$ $\displaystyle=$
$\displaystyle\sqrt{2}\Big{[}(c_{L}c_{R}m_{\tau}-s_{L}s_{R}m_{E})\frac{-c_{\tilde{\beta}}}{v_{\phi}}+(s_{L}c_{R}m_{\tau}+c_{L}s_{R}m_{E})\frac{-s_{\tilde{\beta}}}{V_{331}}\Big{]}\,,$
(212b) $\displaystyle(Y_{\mathcal{D}})_{21}$ $\displaystyle=$
$\displaystyle\sqrt{2}\Big{[}(-c_{L}s_{R}m_{\tau}-s_{L}c_{R}m_{E})\frac{s_{\tilde{\beta}}}{v_{\phi}}+(s_{L}s_{R}m_{\tau}-c_{L}c_{R}m_{E})\frac{c_{\tilde{\beta}}}{V_{331}}\Big{]}\,,$
(212c) $\displaystyle(Y_{\mathcal{D}})_{22}$ $\displaystyle=$
$\displaystyle\sqrt{2}\Big{[}(c_{L}s_{R}m_{\tau}+s_{L}c_{R}m_{E})\frac{c_{\tilde{\beta}}}{v_{\phi}}+(s_{L}s_{R}m_{\tau}-c_{L}c_{R}m_{E})\frac{s_{\tilde{\beta}}}{V_{331}}\Big{]}\,.$
(212d)

Obviously, Eqs. (192) and Eqs. (212) lead to the degenerate fermion mass
predictions of $m_{b}=m_{\tau}$ and $m_{B}=-m_{E}$ 121212The relative negative
sign in $m_{B}=-m_{E}$ can always be rotated away by redefining the right-
handed component of $E$.. Thus, the $b$-$\tau$ mass unification issue cannot
be addressed at the tree level. Their mass splitting can be attributed to the
renormalization group running. This was first discussed in the context of the
Georgi-Glashow ${\rm SU}(5)$ model [75]. However, results therein cannot be
naively applied to the $(b\,,\tau)$ mass ratio in the non-minimal GUTs. To
fully evaluate their mass splitting, we expect two prerequisites of: (i)
evaluation of the intermediate symmetry breaking scales from an appropriate
GUT group, and (ii) the identification of the SM fermion representations with
the extended color and weak symmetries. Both are distinctive features of the
non-minimal GUTs, and we defer to analyze the details in the future work. By
performing the orthogonal transformation in Eq. (142), we find the tau lepton
Yukawa coupling with the SM-like Higgs boson the same as that for the bottom
quark case in Eq. (194), with $m_{b}\to m_{\tau}$. Therefore, the scale of
$V_{331}$ in Eq. (198) can be similarly determined from the tau lepton, given
the current LHC measurements of the $H_{u}\tau\tau$ coupling [76, 77].

###  4.4 The possible radiative mechanism

We comment on the possible radiative fermion mass generation in the current
scenario, which was proposed and considered to produce fermion mass
hierarchies in various context [78, 79, 80, 81, 82, 83, 84, 85, 86]. In such a
paradigm, the general assumption is that some light fermion masses can be
radiatively generated with vanishing tree-level masses. Specifically, we
should check that whether the bottom quark and tau lepton masses can be
generated with $m_{b}=0$ in Eq. (184) and $m_{\tau}=0$ in Eq. (206). Let us
consider the $\mathcal{B}=(b\,,B)$ case without loss of generality, the Yukawa
couplings are reduced to the following expressions

$\displaystyle(Y_{\mathcal{D}})_{11}$ $\displaystyle=$
$\displaystyle\frac{m_{B}}{v_{\phi}}s_{R}(s_{L}s_{\tilde{\beta}}+t_{\theta}c_{L}c_{\tilde{\beta}})\,,$
(213a) $\displaystyle(Y_{\mathcal{D}})_{12}$ $\displaystyle=$
$\displaystyle\frac{m_{B}}{v_{\phi}}s_{R}(t_{\theta}c_{L}s_{\tilde{\beta}}-s_{L}c_{\tilde{\beta}})=(Y_{\mathcal{D}})_{22}t_{R}\,,$
(213b) $\displaystyle(Y_{\mathcal{D}})_{21}$ $\displaystyle=$
$\displaystyle\frac{m_{B}}{v_{\phi}}c_{R}(s_{L}s_{\tilde{\beta}}+t_{\theta}c_{L}c_{\tilde{\beta}})=(Y_{\mathcal{D}})_{11}/t_{R}\,,$
(213c) $\displaystyle(Y_{\mathcal{D}})_{22}$ $\displaystyle=$
$\displaystyle\frac{m_{B}}{v_{\phi}}c_{R}(t_{\theta}c_{L}s_{\tilde{\beta}}-s_{L}c_{\tilde{\beta}})\,,$
(213d)

under the vanishing tree-level mass of $m_{b}=0$. The bottom quark and its
heavy partner $B$ can be mediated through the flavor-changing neutral vector
bosons of $(N_{\mu}\,,\bar{N}_{\mu})$ as in Eq. (247a), while this only
happens for the left-handed components. Thus, the neutral vector bosons of
$(N_{\mu}\,,\bar{N}_{\mu})$ cannot lead to a radiative mass terms as was
suggested in Refs. [80, 81]. The remaining possibility may be due to the
mediation from the Higgs sector, as in Ref. [84]. By taking the $m_{b}=0$ in
Eqs. (252a) and (252b), the neutral Higgs bosons can only mediate the left-
handed $b^{\prime}$ and right-handed $B^{\prime}$. Thus, it is impossible to
generate a radiative mass term of $m_{b}^{\rm
rad}\overline{b_{L}^{\prime}}b_{R}^{\prime}+H.c.$ with a vanishing tree-level
$m_{b}=0$. The same argument also applies to the $\mathcal{E}=(\tau\,,E)$ case
with the $m_{\tau}=0$ limit.

##  5 Conclusions

In this work, we study the bottom quark and tau lepton mass generations in the
framework of one-generational ${\rm SU}(6)$ GUT. The symmetry breaking stage
of $\mathcal{G}_{331}\to\mathcal{G}_{\rm SM}$ is found to be general for more
realistic non-minimal GUTs with $n_{g}=3$. A different assignment to the Higgs
VEVs from the previous studies is considered so that the bottom quark and tau
lepton can obtain tree-level masses with $\sim\mathcal{O}(1)$ Yukawa
couplings. We consider this fermion-Higgs mismatching pattern to be general,
such as in more realistic unified model with the ${\rm SU}(8)$ symmetry. Thus,
we prevent the pattern leading to multiple EW Higgs doublets, which is very
problematic with the ongoing LHC searches. An automatically generated small
Higgs VEVs of $\sim\mathcal{O}(1)\,\rm GeV$ is found to be possible as long as
a gauge-invariant $\nu$-term in the Higgs potential can be of
$\sim\mathcal{O}(100)\,\rm GeV$. Notice that this $\nu$-term is also invariant
under the emergent global symmetry of Eq. (2), which emerges automatically
from the anomaly-free condition. By requiring the Yukawa coupling of the SM-
like Higgs boson to SM fermions of $y_{f}^{\rm SM}=\frac{\sqrt{2}m_{f}}{v_{\rm
EW}}$, we find the $331$ symmetry-breaking scale of $V_{331}\sim 10\,\rm TeV$
in the current context. This was not mentioned in the previous context. With
the distinct VEV assignments in Eqs. (52) and (55), we find a Higgs sector
consisting of one single CP-even Higgs boson at the EW scale. All other Higgs
bosons have masses of $\sim{\cal O}(V_{331})$, as we have described in Sec. 3.
Therefore, the effective theory at the EW scale contains only one SM-like CP-
even Higgs boson, and is not described by a 2HDM.

Historically, it was proposed that three-generational SM fermions may be
embedded non-trivially in a non-minimal GUT [3]. Through our recent analysis
[21, 87], we find that the ${\rm SU}(8)$ GUT can be the minimal model that
have three-generational SM fermions transform differently under the extended
gauge symmetries beyond the EW scale. Through the current discussion, we wish
to mention the relations between the ${\rm SU}(6)$ toy model and the realistic
${\rm SU}(8)$ model. First, the ${\rm SU}(6)$ subgroup of the ${\cal G}_{331}$
can be generic in the context of the ${\rm SU}(8)$ GUT, as was shown in Eq.
(40). Therefore, the results such as the ${\cal G}_{331}$ gauge sector and
part of the Higgs sector in the current discussion can become useful in the
context of the ${\rm SU}(8)$ model. Second, the symmetry breaking pattern can
be generalized, where the seemingly unnatural $\nu$-term in Eq. (44d) that
generates the EWSB VEVs for the $(b\,,\tau)$ masses are natural due to the
gravitational effect in the ${\rm SU}(8)$ model. This means a potential
relation between the gravitational effect and the flavor sector, which was
never mentioned in any previous GUT literature according to our knowledge.
Since the one-generational ${\rm SU}(6)$ GUT is a toy model, there are several
issues beyond the scope of the current discussions. They include: (i) the
$b$-$\tau$ mass unification, (ii) the three-generational SM fermion mixings.
Furthermore, the SM fermions in the non-minimal GUTs are usually accompanied
with heavy partners from the ${\rm SU}(N)$ anti-fundamentals. They can be
mediated through the heavy charged and/or neutral vector bosons as well as
heavy Higgs bosons during the intermediate symmetry breaking stages of the
non-minimal GUT symmetry. It is therefore necessary to carry out detailed
analysis of their experimental implications in some rare flavor-changing
processes. All these issues will be studied elsewhere when extending to more
realistic non-minimal GUTs such as the ${\rm SU}(8)$, where three-generational
SM fermions are embedded non-trivially.

## Acknowledgements

We would like to thank Wenbin Yan, Chang-Yuan Yao, and Ye-Ling Zhou for
enlightening discussions. N.C. would like to thank Tibet University, Shanghai
Institute of Optics and Fine Mechanics (CAS), and Peking University for
hospitality when preparing this work. N.C. is partially supported by the
National Natural Science Foundation of China (under Grants No. 12035008 and
No. 12275140), and Nankai University. Y.N.M. is partially supported by the
National Natural Science Foundation of China (under Grant No. 12205227), the
Fundamental Research Funds for the Central Universities (WUT: 2022IVA052), and
Wuhan University of Technology.

##  Appendix A The gauge symmetry breaking in the $331$ model

In this section, we summarize the necessary results of the gauge symmetry
breaking of ${\cal G}_{331}\to{\cal G}_{\rm SM}$ for the current discussions
as well as for the future studies.

###  A.1 The $331$ gauge bosons

The kinematic terms for the ${\rm SU}(3)_{W}$ Higgs fields are

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\sum_{\rho}|D_{\mu}\Phi_{\mathbf{\overline{3}}\,,\rho}|^{2}+|D_{\mu}\Phi_{\mathbf{\overline{3}}}^{\prime}|^{2}\,.$
(214)

Generically, the covariant derivative for the ${\rm SU}(3)_{W}$ fundamental
representation is defined according to the convention in Refs. [35, 44]
131313In some $331$ literatures, e.g. Ref. [37], the ${\rm U}(1)$ charge is
defined with a $3\times 3$ unity matrix of $\frac{1}{\sqrt{6}}\mathbb{I}_{3}$.

$\displaystyle
D_{\mu}\Phi_{\mathbf{3}}\equiv(\partial_{\mu}-ig_{3L}A_{\mu}^{a}\frac{\lambda^{a}}{2}-ig_{X}X\mathbb{I}_{3}X_{\mu})\Phi_{\mathbf{3}}\,,$
(215)

with $\lambda^{a}\,(a=1\,,...\,,8)$ being the ${\rm SU}(3)$ Gell-Mann
matrices. For the ${\rm SU}(3)_{W}$ anti-fundamental representation, the
covariant is defined as

$\displaystyle
D_{\mu}\Phi_{\mathbf{\overline{3}}}\equiv(\partial_{\mu}+ig_{3L}A_{\mu}^{a}\frac{(\lambda^{a})^{*}}{2}-ig_{X}X\mathbb{I}_{3}X_{\mu})\Phi_{\mathbf{\overline{3}}}$
(216) $\displaystyle=$
$\displaystyle(\partial_{\mu}+ig_{3L}A_{\mu}^{a}\frac{(\lambda^{a})^{T}}{2}-ig_{X}X\mathbb{I}_{3}X_{\mu})\Phi_{\mathbf{\overline{3}}}\,,$

with the hermiticity of $(\lambda^{a})^{\dagger}=\lambda^{a}$. Note that the
definitions in Eqs. (215) and (216) are applicable to the ${\rm SU}(3)_{W}$
fermions.

Explicitly, we express the gauge fields in terms of a $3\times 3$ matrix as
follows

$\displaystyle
g_{3L}A_{\mu}^{a}\frac{\lambda^{a}}{2}+g_{X}X\mathbb{I}_{3}X_{\mu}$ (221)
$\displaystyle=$
$\displaystyle\small{\frac{1}{2}\left(\begin{array}[]{ccc}g_{3L}(A_{\mu}^{3}+\frac{1}{\sqrt{3}}A_{\mu}^{8})+2g_{X}XX_{\mu}&g_{3L}(A_{\mu}^{1}-iA_{\mu}^{2})&g_{3L}(A_{\mu}^{4}-iA_{\mu}^{5})\\\
g_{3L}(A_{\mu}^{1}+iA_{\mu}^{2})&g_{3L}(-A_{\mu}^{3}+\frac{1}{\sqrt{3}}A_{\mu}^{8})+2g_{X}XX_{\mu}&g_{3L}(A_{\mu}^{6}-iA_{\mu}^{7})\\\
g_{3L}(A_{\mu}^{4}+iA_{\mu}^{5})&g_{3L}(A_{\mu}^{6}+iA_{\mu}^{7})&-\frac{2g_{3L}}{\sqrt{3}}A_{\mu}^{8}+2g_{X}XX_{\mu}\\\
\end{array}\right)}$ $\displaystyle\,.$

One can identify the charged gauge bosons of
$W_{\mu}^{\pm}\equiv\frac{1}{\sqrt{2}}(A_{\mu}^{1}\mp iA_{\mu}^{2})$,
$C_{\mu}^{\pm}\equiv\frac{1}{\sqrt{2}}(A_{\mu}^{4}\mp iA_{\mu}^{5})$, and the
neutral gauge bosons of
$N_{\mu}\equiv\frac{1}{\sqrt{2}}(A_{\mu}^{6}-iA_{\mu}^{7})$, and
$\bar{N}_{\mu}\equiv\frac{1}{\sqrt{2}}(A_{\mu}^{6}+iA_{\mu}^{7})$. The
electric charges of gauge bosons can be obtained by the relation of
$[\hat{Q}\,,A_{\mu}^{a}\lambda^{a}]=Q_{A}^{IJ}(A_{\mu}^{a}\lambda^{a})_{IJ}$,
with $X=0$ (since the ${\rm SU}(3)_{W}$ gauge bosons do not take the ${\rm
U}(1)_{X}$ charges) in the electric charge operator given in Eq. (6).

The charged and neutral $331$-gauge boson mass squares at the tree level read

$\displaystyle
m_{C_{\mu}^{\pm}}^{2}=m_{N_{\mu}\,,\bar{N}_{\mu}}^{2}=\frac{1}{4}g_{3L}^{2}V_{331}^{2}\,,$
(222)

with the VEV assignment in Eq. (52) for simplity. The other neutral gauge
boson is due to the linear combination of $(A^{8}_{\mu}\,,X_{\mu})$, whose
mass matrix is

$\displaystyle\frac{1}{2}\cdot\frac{(V_{331})^{2}}{9}(A^{8\,\mu}\,,X^{\mu})\left(\begin{array}[]{cc}3g_{3L}^{2}&-\sqrt{3}g_{3L}g_{X}\\\
-\sqrt{3}g_{3L}g_{X}&g_{X}^{2}\\\
\end{array}\right)\cdot\left(\begin{array}[]{c}A^{8}_{\mu}\\\ X_{\mu}\\\
\end{array}\right)\,,$ (227) $\displaystyle\Rightarrow$ $\displaystyle
m_{Z_{\mu}^{\prime}}^{2}=\frac{1}{9}(g_{X}^{2}+3g_{3L}^{2})(V_{331})^{2}\,.$
(228)

It is straightforward to define a mixing angle $\theta_{X}$ for the $331$
symmetry breaking as

$\displaystyle\left(\begin{array}[]{c}Z^{\prime}_{\mu}\\\ B_{\mu}\\\
\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}c_{X}&-s_{X}\\\ s_{X}&c_{X}\\\
\end{array}\right)\cdot\left(\begin{array}[]{c}A^{8}_{\mu}\\\ X_{\mu}\\\
\end{array}\right)\,,$ (235)

with

$\displaystyle t_{X}$ $\displaystyle\equiv$
$\displaystyle\tan\theta_{X}=\frac{g_{X}}{\sqrt{3}g_{3L}}\,.$ (236)

Thus, $Z^{\prime}_{\mu}$ and $B_{\mu}$ can be expressed in terms of
$A_{\mu}^{8}$ and $X_{\mu}$ as

$\displaystyle
Z^{\prime}_{\mu}=\frac{\sqrt{3}g_{3L}A_{\mu}^{8}-g_{X}X_{\mu}}{\sqrt{g_{X}^{2}+3g_{3L}^{2}}}\,,\quad
B_{\mu}=\frac{g_{X}A_{\mu}^{8}+\sqrt{3}g_{3L}X_{\mu}}{\sqrt{g_{X}^{2}+3g_{3L}^{2}}}\,.$
(237)

The ${\rm U}(1)_{Y}$ coupling of $\alpha_{Y}$ is related to the ${\rm
SU}(3)_{W}\otimes{\rm U}(1)_{X}$ couplings of $(\alpha_{3L}\,,\alpha_{X})$ as

$\displaystyle\alpha_{Y}^{-1}=\frac{1}{3}\alpha_{3L}^{-1}+\alpha_{X}^{-1}\,,\quad\frac{1}{3}\alpha_{3L}^{-1}=\alpha_{Y}^{-1}s_{X}^{2}\,,\quad\alpha_{X}^{-1}=\alpha_{Y}^{-1}c_{X}^{2}\,.$
(238)

Correspondingly, the diagonal components of the ${\rm SU}(3)_{W}\otimes{\rm
U}(1)_{X}$ covariant derivative in Eq. (221) become

$\displaystyle\frac{1}{2}{\rm
diag}\Big{(}g_{3L}A_{\mu}^{3}+g_{Y}(\frac{1}{3}+2X)B_{\mu}\,,-g_{3L}A_{\mu}^{3}+g_{Y}(\frac{1}{3}+2X)B_{\mu}\,,g_{Y}(-\frac{2}{3}+2X)B_{\mu}\Big{)}$
(239) $\displaystyle+$ $\displaystyle\frac{g_{Y}}{6}{\rm
diag}\Big{(}-6Xt_{X}+\frac{1}{t_{X}}\,,-6Xt_{X}+\frac{1}{t_{X}}\,,-6Xt_{X}-\frac{2}{t_{X}}\Big{)}Z^{\prime}_{\mu}\,.$

Clearly, the $A_{\mu}^{3}$ and $B_{\mu}$ terms from first two components
recover the covariant derivatives in the EW theory with $X=\frac{1}{3}$. The
off-diagonal components in Eq. (221) become

$\displaystyle\frac{g_{3L}}{\sqrt{2}}\left(\begin{array}[]{ccc}0&W_{\mu}^{+}&C_{\mu}^{+}\\\
W_{\mu}^{-}&0&N_{\mu}\\\ C_{\mu}^{-}&\bar{N}_{\mu}&0\\\ \end{array}\right)\,.$
(243)

Below, we call five massive gauge bosons of

$\displaystyle\Big{\\{}C_{\mu}^{\pm}\,,N_{\mu}\,,\bar{N}_{\mu}\,,Z_{\mu}^{\prime}\Big{\\}}$
(244)

at this stage of symmetry breaking as the $331$ gauge bosons, while the
remaining ones of $\Big{\\{}W_{\mu}^{\pm}\,,A_{\mu}^{3}\,,B_{\mu}\Big{\\}}$
are the usual EW gauge bosons.

###  A.2 The gauge couplings of fermions

The ${\rm SU}(3)_{W}\otimes{\rm U}(1)_{X}$ covariant derivatives for chiral
fermions in Tab. 1 are expressed as follows in terms of gauge eigenstates

$\displaystyle iD_{\mu}\mathcal{B}_{R}^{\rho}\supset
g_{X}(-\frac{1}{3})X_{\mu}\mathcal{B}_{R}^{\rho}\,,$ (245a) $\displaystyle
iD_{\mu}\mathcal{L}_{L}^{\rho}\supset(-g_{3L}A_{\mu}^{a}\frac{(\lambda^{a})^{T}}{2}+g_{X}X_{\mu}\mathbb{I}_{3}(-\frac{1}{3}))\mathcal{L}_{L}^{\rho}\,,$
(245b) $\displaystyle iD_{\mu}t_{R}\supset g_{X}(+\frac{2}{3})X_{\mu}t_{R}\,,$
(245c) $\displaystyle
iD_{\mu}\mathcal{E}_{R}\supset(+g_{3L}A_{\mu}^{a}\frac{\lambda^{a}}{2}-g_{X}X_{\mu}\mathbb{I}_{3}\frac{2}{3})\mathcal{E}_{R}\,,$
(245d) $\displaystyle iD_{\mu}\mathcal{Q}_{L}\supset
g_{3L}A_{\mu}^{a}\frac{\lambda^{a}}{2}\mathcal{Q}_{L}\,.$ (245e)

Analogous to the SM, we should find the charged currents and neutral currents
for the ${\rm SU}(3)_{W}\otimes{\rm U}(1)_{X}$ gauge bosons.

The flavor-changing ${\rm SU}(3)_{W}\otimes{\rm U}(1)_{X}$ charged currents
are mediated by $C_{\mu}^{\pm}$ as follows

$\displaystyle\mathcal{L}_{{\rm SU}(3)_{W}}^{\rm CC}$ $\displaystyle=$
$\displaystyle\frac{g_{3L}}{\sqrt{2}}\Big{[}\overline{t}_{L}\gamma^{\mu}B_{L}+\overline{N_{R}}\gamma^{\mu}\tau_{R}-\overline{\widetilde{N}_{L}^{1}}\gamma^{\mu}\tau_{L}-\overline{\widetilde{N}_{L}^{2}}\gamma^{\mu}E_{L}\Big{]}C_{\mu}^{+}+H.c.\,.$
(246)

The ${\rm SU}(3)_{W}\otimes{\rm U}(1)_{X}$ neutral currents contain both the
flavor-changing components mediated by $(N_{\mu}\,,\bar{N}_{\mu})$, and the
flavor-conserving components mediated by $Z^{\prime}_{\mu}$. In the chiral
basis, they read

$\displaystyle\mathcal{L}_{{\rm SU}(3)_{W}}^{{\rm NC\,,\bcancel{\rm F}}}$
$\displaystyle=$
$\displaystyle\frac{g_{3L}}{\sqrt{2}}\Big{[}\overline{\widetilde{N}_{L}^{1}}\gamma^{\mu}\nu_{L}+\overline{\widetilde{N}_{L}^{2}}\gamma^{\mu}N_{L}+\overline{E}_{R}\gamma^{\mu}\tau_{R}+\overline{b}_{L}\gamma^{\mu}B_{L}\Big{]}N_{\mu}$
(247a) $\displaystyle+$
$\displaystyle\frac{g_{3L}}{\sqrt{2}}\Big{[}\overline{\nu}_{L}\gamma^{\mu}\widetilde{N}_{L}^{1}+\overline{N}_{L}\gamma^{\mu}\widetilde{N}^{2}_{L}+\overline{\tau}_{R}\gamma^{\mu}E_{R}+\overline{B_{L}}\gamma^{\mu}b_{L}\Big{]}\bar{N}_{\mu}\,$
$\displaystyle\mathcal{L}_{{\rm SU}(3)_{W}}^{\rm NC\,,F}$ $\displaystyle=$
$\displaystyle
g_{Y}\Big{[}\overline{t}_{L}\gamma^{\mu}(\frac{1}{6t_{X}})t_{L}+\overline{b}_{L}\gamma^{\mu}(\frac{1}{6t_{X}})b_{L}+\overline{B}_{L}\gamma^{\mu}(-\frac{1}{3t_{X}})B_{L}$
(247b) $\displaystyle+$
$\displaystyle\overline{t}_{R}\gamma^{\mu}(-\frac{2}{3}t_{X})t_{R}+\overline{b}_{R}\gamma^{\mu}(\frac{1}{3}t_{X})b_{R}+\overline{B}_{R}\gamma^{\mu}(\frac{1}{3}t_{X})B_{R}$
$\displaystyle+$
$\displaystyle\overline{\tau}_{L}\gamma^{\mu}(\frac{1}{3}t_{X}-\frac{1}{6t_{X}})\tau_{L}+\overline{\nu}_{L}\gamma^{\mu}(\frac{1}{3}t_{X}-\frac{1}{6t_{X}})\nu_{L}$
$\displaystyle+$
$\displaystyle\overline{E}_{L}\gamma^{\mu}(\frac{1}{3}t_{X}-\frac{1}{6t_{X}})E_{L}+\overline{N}_{L}\gamma^{\mu}(\frac{1}{3}t_{X}-\frac{1}{6t_{X}})N_{L}+\overline{N}_{R}\gamma^{\mu}(\frac{2}{3}t_{X}+\frac{1}{6t_{X}})N_{R}$
$\displaystyle+$
$\displaystyle\overline{\widetilde{N}_{L}^{1}}\gamma^{\mu}(\frac{1}{3}t_{X}+\frac{1}{3t_{X}})\widetilde{N}_{L}^{1}+\overline{\widetilde{N}^{2}_{L}}\gamma^{\mu}(\frac{1}{3}t_{X}+\frac{1}{3t_{X}})\widetilde{N}^{2}_{L}$
$\displaystyle+$
$\displaystyle\overline{E}_{R}\gamma^{\mu}(\frac{2}{3}t_{X}+\frac{1}{6t_{X}})E_{R}+\overline{\tau}_{R}\gamma^{\mu}(\frac{2}{3}t_{X}-\frac{1}{3t_{X}})\tau_{R}\Big{]}Z_{\mu}^{\prime}\,.$

Apparently, the EW charged currents should reproduce the SM case, which are

$\displaystyle\mathcal{L}_{{\rm SU}(2)_{W}}^{{\rm CC}}$ $\displaystyle=$
$\displaystyle\frac{g_{3L}}{\sqrt{2}}\Big{[}\overline{t}_{L}\gamma^{\mu}b_{L}+\overline{\tau}_{L}\gamma^{\mu}\nu_{L}\Big{]}W_{\mu}^{+}+H.c.$
(248)

###  A.3 The mass matrices of the Higgs bosons

The matrix elements for the charged Higgs bosons in Eq. (71) read

$\displaystyle M_{\phi_{1}^{+}\phi_{1}^{-}}^{2}$ $\displaystyle=$
$\displaystyle\frac{v_{u}\Big{[}\sqrt{2}\nu\left(s_{\tilde{\beta}}^{2}(v_{\phi}^{2}-V_{331}^{2})-v_{\phi}^{2}\right)+\kappa_{3}V_{331}v_{\phi}v_{u}\Big{]}}{2V_{331}v_{\phi}}\,,$
(249a) $\displaystyle M_{\phi_{1}^{\pm}\phi_{2}^{\mp}}^{2}$ $\displaystyle=$
$\displaystyle\frac{\nu
v_{u}s_{\tilde{\beta}}c_{\tilde{\beta}}(V_{331}^{2}-v_{\phi}^{2})}{\sqrt{2}V_{331}v_{\phi}}\,,$
(249b) $\displaystyle M_{\phi_{2}^{+}\phi_{2}^{-}}^{2}$ $\displaystyle=$
$\displaystyle\frac{v_{u}\Big{[}\sqrt{2}\nu\left(c_{\tilde{\beta}}^{2}(v_{\phi}^{2}-V_{331}^{2})-v_{\phi}^{2}\right)+\kappa_{4}V_{331}v_{\phi}v_{u}\Big{]}}{2V_{331}v_{\phi}}\,,$
(249c) $\displaystyle M_{\phi_{1}^{\pm}\chi_{1}^{\mp}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}s_{\tilde{\beta}}(\kappa_{3}v_{\phi}v_{u}-\sqrt{2}\nu
V_{331})\,,$ (249d) $\displaystyle M_{\phi_{1}^{\pm}\chi_{2}^{\mp}}^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2}c_{\tilde{\beta}}(\kappa_{3}V_{331}v_{u}-\sqrt{2}\nu
v_{\phi})\,,$ (249e) $\displaystyle M_{\phi_{2}^{\pm}\chi_{1}^{\mp}}^{2}$
$\displaystyle=$ $\displaystyle\frac{1}{2}c_{\tilde{\beta}}(\sqrt{2}\nu
V_{331}-\kappa_{4}v_{\phi}v_{u})\,,$ (249f) $\displaystyle
M_{\phi_{2}^{\pm}\chi_{2}^{\mp}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}s_{\tilde{\beta}}(\kappa_{4}V_{331}v_{u}-\sqrt{2}\nu
v_{\phi})\,,$ (249g) $\displaystyle M_{\chi_{1}^{+}\chi_{1}^{-}}^{2}$
$\displaystyle=$ $\displaystyle\frac{v_{\phi}(-\sqrt{2}\nu
V_{331}+\kappa_{3}v_{\phi}v_{u}s_{\tilde{\beta}}^{2}+\kappa_{4}v_{\phi}v_{u}c_{\tilde{\beta}}^{2})}{2v_{u}}\,,$
(249h) $\displaystyle M_{\chi_{1}^{\pm}\chi_{2}^{\mp}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}V_{331}v_{\phi}s_{2\tilde{\beta}}(\kappa_{3}-\kappa_{4})\,,$
(249i) $\displaystyle M_{\chi_{2}^{+}\chi_{2}^{-}}^{2}$ $\displaystyle=$
$\displaystyle\frac{V_{331}(\kappa_{3}V_{331}v_{u}c_{\tilde{\beta}}^{2}+\kappa_{4}V_{331}v_{u}s_{\tilde{\beta}}^{2}-\sqrt{2}\nu
v_{\phi})}{2v_{u}}\,.$ (249j)

The matrix elements for the CP-odd Higgs bosons in Eq. (106) read

$\displaystyle M_{\pi_{u}\pi_{u}}^{2}$ $\displaystyle=$
$\displaystyle-\frac{\nu V_{331}v_{\phi}}{\sqrt{2}v_{u}}\,,$ (250a)
$\displaystyle M_{\pi_{u}\eta_{1}}^{2}$ $\displaystyle=$
$\displaystyle\frac{\nu V_{331}s_{\tilde{\beta}}}{\sqrt{2}}\,,$ (250b)
$\displaystyle M_{\pi_{u}\eta_{2}}^{2}$ $\displaystyle=$
$\displaystyle-\frac{\nu V_{331}c_{\tilde{\beta}}}{\sqrt{2}}\,,$ (250c)
$\displaystyle M_{\pi_{u}\pi_{1}}^{2}$ $\displaystyle=$
$\displaystyle\frac{\nu v_{\phi}c_{\tilde{\beta}}}{\sqrt{2}}\,,$ (250d)
$\displaystyle M_{\pi_{u}\pi_{2}}^{2}$ $\displaystyle=$
$\displaystyle\frac{\nu v_{\phi}s_{\tilde{\beta}}}{\sqrt{2}}\,,$ (250e)
$\displaystyle M_{\eta_{1}\eta_{1}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}v_{\phi}^{2}c_{\tilde{\beta}}^{2}(\lambda_{4}-\lambda_{5})-\frac{\nu
v_{u}\Big{[}s_{\tilde{\beta}}^{2}(V_{331}^{2}-v_{\phi}^{2})+v_{\phi}^{2}\Big{]}}{\sqrt{2}V_{331}v_{\phi}}\,,$
(250f) $\displaystyle M_{\eta_{1}\eta_{2}}^{2}$ $\displaystyle=$
$\displaystyle\frac{s_{\tilde{\beta}}c_{\tilde{\beta}}\Big{[}\sqrt{2}\nu(V_{331}^{2}-v_{\phi}^{2})v_{u}+V_{331}v_{\phi}^{3}(\lambda_{4}-\lambda_{5})\Big{]}}{2V_{331}v_{\phi}}\,,$
(250g) $\displaystyle M_{\pi_{1}\eta_{1}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}V_{331}v_{\phi}s_{2\tilde{\beta}}(\lambda_{5}-\lambda_{4})\,,$
(250h) $\displaystyle M_{\pi_{2}\eta_{1}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}V_{331}v_{\phi}c_{\tilde{\beta}}^{2}(\lambda_{4}-\lambda_{5})-\sqrt{2}\nu
v_{u}\Big{]}\,,$ (250i) $\displaystyle M_{\eta_{2}\eta_{2}}^{2}$
$\displaystyle=$ $\displaystyle\frac{\nu
v_{u}\Big{[}c_{\tilde{\beta}}^{2}(v_{\phi}^{2}-V_{331}^{2})-v_{\phi}^{2}\Big{]}}{\sqrt{2}V_{331}v_{\phi}}+\frac{1}{2}v_{\phi}^{2}s_{\tilde{\beta}}^{2}(\lambda_{4}-\lambda_{5})\,,$
(250j) $\displaystyle M_{\pi_{1}\eta_{2}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}V_{331}v_{\phi}s_{\tilde{\beta}}^{2}(\lambda_{5}-\lambda_{4})+\frac{\nu
v_{u}}{\sqrt{2}}\,,$ (250k) $\displaystyle M_{\pi_{2}\eta_{2}}^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{4}V_{331}v_{\phi}s_{2\tilde{\beta}}(\lambda_{4}-\lambda_{5})\,,$
(250l) $\displaystyle M_{\pi_{1}\pi_{1}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}V_{331}^{2}s_{\tilde{\beta}}^{2}(\lambda_{4}-\lambda_{5})+\frac{\nu
v_{u}\Big{[}s_{\tilde{\beta}}^{2}(v_{\phi}^{2}-V_{331}^{2})-v_{\phi}^{2}\Big{]}}{\sqrt{2}V_{331}v_{\phi}}\,,$
(250m) $\displaystyle M_{\pi_{1}\pi_{2}}^{2}$ $\displaystyle=$
$\displaystyle\frac{s_{\tilde{\beta}}c_{\tilde{\beta}}\Big{[}V_{331}^{3}v_{\phi}(\lambda_{5}-\lambda_{4})+\sqrt{2}\nu(V_{331}^{2}-v_{\phi}^{2})v_{u}\Big{]}}{2V_{331}v_{\phi}}\,,$
(250n) $\displaystyle M_{\pi_{2}\pi_{2}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}V_{331}^{2}c_{\tilde{\beta}}^{2}(\lambda_{4}-\lambda_{5})+\frac{\nu
v_{u}\Big{[}c_{\tilde{\beta}}^{2}(v_{\phi}^{2}-V_{331}^{2})-v_{\phi}^{2}\Big{]}}{\sqrt{2}V_{331}v_{\phi}}\,.$
(250o)

The matrix elements for the CP-even Higgs bosons in Eq. (151) read

$\displaystyle M_{h_{u}h_{u}}^{2}$ $\displaystyle=$ $\displaystyle 2\lambda
v_{u}^{2}-\frac{\nu V_{331}v_{\phi}}{\sqrt{2}v_{u}}\,,$ (251a) $\displaystyle
M_{h_{u}\phi^{0}}^{2}$ $\displaystyle=$ $\displaystyle-\frac{\sqrt{2}\nu
V_{331}^{2}+2V_{331}v_{\phi}v_{u}(\kappa_{1}+\kappa_{2})+\sqrt{2}\nu
v_{\phi}^{2}}{2\sqrt{V_{331}^{2}+v_{\phi}^{2}}}\,,$ (251b) $\displaystyle
M_{\phi^{0}\phi^{0}}^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{V_{331}v_{\phi}\left(V_{331}^{2}+v_{\phi}^{2}\right)}\Big{[}-V_{331}^{3}v_{\phi}^{3}(\lambda_{1}+\lambda_{2}+2\lambda_{3})+\frac{\nu}{\sqrt{2}}(V_{331}^{2}-v_{\phi}^{2})^{2}v_{u}\Big{]}\,,$
(251c) $\displaystyle M_{h_{u}h_{1}^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle
v_{u}s_{\tilde{\beta}}c_{\tilde{\beta}}(\kappa_{1}-\kappa_{2})\sqrt{V_{331}^{2}+v_{\phi}^{2}}\,,$
(251d) $\displaystyle M_{h_{u}h_{2}^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{v_{u}\Big{[}s_{\tilde{\beta}}^{2}(\kappa_{2}V_{331}^{2}-\kappa_{1}v_{\phi}^{2})+c_{\tilde{\beta}}^{2}(\kappa_{1}V_{331}^{2}-\kappa_{2}v_{\phi}^{2})\Big{]}}{\sqrt{V_{331}^{2}+v_{\phi}^{2}}}\,,$
(251e) $\displaystyle M_{\phi^{0}h_{1}^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle
V_{331}v_{\phi}s_{\tilde{\beta}}c_{\tilde{\beta}}(\lambda_{2}-\lambda_{1})\,,$
(251f) $\displaystyle M_{\phi^{0}h_{2}^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}V_{331}v_{\phi}c_{2\tilde{\beta}}(\lambda_{2}-\lambda_{1})-\frac{(V_{331}^{2}-v_{\phi}^{2})\left(V_{331}v_{\phi}(\lambda_{1}+\lambda_{2}+2\lambda_{3})+2\sqrt{2}\nu
v_{u}\right)}{V_{331}^{2}+v_{\phi}^{2}}\Big{]}\,,$ (251g) $\displaystyle
M_{h_{1}^{\prime}h_{1}^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{8V_{331}v_{\phi}}\left(V_{331}^{2}+v_{\phi}^{2}\right)(V_{331}v_{\phi}c_{4\tilde{\beta}}(\lambda_{1}+\lambda_{2}-2(\lambda_{3}+\lambda_{4}+\lambda_{5}))$
(251h) $\displaystyle-$ $\displaystyle
V_{331}v_{\phi}(\lambda_{1}+\lambda_{2}+2(-\lambda_{3}+\lambda_{4}+\lambda_{5}))+4\sqrt{2}\nu
v_{u})\,,$ $\displaystyle M_{h_{1}^{\prime}h_{2}^{\prime}}^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{4}s_{2\tilde{\beta}}\Big{[}c_{2\tilde{\beta}}\left(V_{331}^{2}+v_{\phi}^{2}\right)(\lambda_{1}+\lambda_{2}-2(\lambda_{3}+\lambda_{4}+\lambda_{5}))+(\lambda_{1}-\lambda_{2})(V_{331}^{2}-v_{\phi}^{2})\Big{]}\,,$
(251i) $\displaystyle M_{h_{2}^{\prime}h_{2}^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{8\left(V_{331}^{2}+v_{\phi}^{2}\right)}\Big{[}(V_{331}^{4}+v_{\phi}^{4})(3\lambda_{1}+3\lambda_{2}+2(\lambda_{3}+\lambda_{4}+\lambda_{5}))+c_{4\tilde{\beta}}\left(V_{331}^{2}+v_{\phi}^{2}\right)^{2}(\lambda_{1}+\lambda_{2}$
(251j) $\displaystyle-$ $\displaystyle
2(\lambda_{3}+\lambda_{4}+\lambda_{5}))+4c_{2\tilde{\beta}}(\lambda_{1}-\lambda_{2})\left(V_{331}^{4}-v_{\phi}^{4}\right)$
$\displaystyle-$ $\displaystyle
2V_{331}^{2}v_{\phi}^{2}(\lambda_{1}+\lambda_{2}+6\lambda_{3}-2(\lambda_{4}+\lambda_{5}))-16\sqrt{2}\nu
V_{331}v_{\phi}v_{u})\Big{]}\,.$

###  A.4 The Yukawa couplings of fermions

In terms of the fermion mass eigenstates of $(b^{\prime}\,,B^{\prime})$, the
CP-odd and CP-even Higgs Yukawa couplings are expressed as follows

$\displaystyle-\mathcal{L}_{Y}^{\mathcal{Q}\,,0^{-}}$ $\displaystyle\supset$
$\displaystyle
im_{b}\overline{b_{L}^{\prime}}b_{R}^{\prime}\Big{(}\zeta_{1}\eta_{1}+\zeta_{2}\eta_{2}+\zeta_{3}\pi_{1}+\zeta_{4}\pi_{2}\Big{)}$
(252a) $\displaystyle+$ $\displaystyle
im_{B}\overline{b_{L}^{\prime}}B_{R}^{\prime}\Big{(}\zeta_{1}^{\prime}\eta_{1}+\zeta_{2}^{\prime}\eta_{2}+\zeta_{3}^{\prime}\pi_{1}+\zeta_{4}^{\prime}\pi_{2}\Big{)}$
$\displaystyle+$ $\displaystyle
im_{b}\overline{B_{L}^{\prime}}b_{R}^{\prime}\Big{(}-\zeta_{3}\eta_{1}-\zeta_{4}\eta_{2}+\zeta_{1}\pi_{1}+\zeta_{2}\pi_{2}\Big{)}$
$\displaystyle+$ $\displaystyle
im_{B}\overline{B_{L}^{\prime}}B_{R}^{\prime}\Big{(}-\zeta_{3}^{\prime}\eta_{1}-\zeta_{4}^{\prime}\eta_{2}+\zeta_{1}^{\prime}\pi_{1}+\zeta_{2}^{\prime}\pi_{2}\Big{)}+H.c.\,,$
$\displaystyle-\mathcal{L}_{Y}^{\mathcal{Q}\,,0^{+}}$ $\displaystyle\supset$
$\displaystyle
m_{b}\overline{b_{L}^{\prime}}b_{R}^{\prime}\Big{(}\zeta_{1}\phi_{1}+\zeta_{2}\phi_{2}+\zeta_{3}h_{1}+\zeta_{4}h_{2}\Big{)}$
(252b) $\displaystyle+$ $\displaystyle
m_{B}\overline{b_{L}^{\prime}}B_{R}^{\prime}\Big{(}\zeta_{1}^{\prime}\phi_{1}+\zeta_{2}^{\prime}\phi_{2}+\zeta_{3}^{\prime}h_{1}+\zeta_{4}^{\prime}h_{2}\Big{)}$
$\displaystyle+$ $\displaystyle
m_{b}\overline{B_{L}^{\prime}}b_{R}^{\prime}\Big{(}-\zeta_{3}\phi_{1}-\zeta_{4}\phi_{2}+\zeta_{1}h_{1}+\zeta_{2}h_{2}\Big{)}$
$\displaystyle+$ $\displaystyle
m_{B}\overline{B_{L}^{\prime}}B_{R}^{\prime}\Big{(}-\zeta_{3}^{\prime}\phi_{1}-\zeta_{4}^{\prime}\phi_{2}+\zeta_{1}^{\prime}h_{1}+\zeta_{2}^{\prime}h_{2}\Big{)}+H.c.\,,$

where we parametrize the couplings as follows

$\displaystyle\zeta_{1}=c_{L}^{2}\frac{s_{\tilde{\beta}}}{v_{\phi}}-s_{L}c_{L}\frac{c_{\tilde{\beta}}}{V_{331}}\,,\quad\zeta_{1}^{\prime}=s_{L}c_{L}\frac{s_{\tilde{\beta}}}{v_{\phi}}+c_{L}^{2}\frac{c_{\tilde{\beta}}}{V_{331}}\,,$
(253a)
$\displaystyle\zeta_{2}=-c_{L}^{2}\frac{c_{\tilde{\beta}}}{v_{\phi}}-s_{L}c_{L}\frac{s_{\tilde{\beta}}}{V_{331}}\,,\quad\zeta_{2}^{\prime}=-s_{L}c_{L}\frac{c_{\tilde{\beta}}}{v_{\phi}}+c_{L}^{2}\frac{s_{\tilde{\beta}}}{V_{331}}\,,$
(253b)
$\displaystyle\zeta_{3}=-s_{L}c_{L}\frac{s_{\tilde{\beta}}}{v_{\phi}}+s_{L}^{2}\frac{c_{\tilde{\beta}}}{V_{331}}\,,\quad\zeta_{3}^{\prime}=-s_{L}^{2}\frac{s_{\tilde{\beta}}}{v_{\phi}}-s_{L}c_{L}\frac{c_{\tilde{\beta}}}{V_{331}}\,,$
(253c)
$\displaystyle\zeta_{4}=s_{L}c_{L}\frac{c_{\tilde{\beta}}}{v_{\phi}}+s_{L}^{2}\frac{s_{\tilde{\beta}}}{V_{331}}\,,\quad\zeta_{4}^{\prime}=s_{L}^{2}\frac{c_{\tilde{\beta}}}{v_{\phi}}-s_{L}c_{L}\frac{s_{\tilde{\beta}}}{V_{331}}\,.$
(253d)

For the fermion mass eigenstates of $(\tau^{\prime}\,,E^{\prime})$, their
Yukawa couplings can be obtained by replacing
$(m_{b}\,,m_{B})\to(m_{\tau}\,,-m_{E})$ in the above Eqs. (252a) and (252b).

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