#  HUPD-2306 $S_{4}$ Lepton Flavor Model with 3HDM

Yukimura Izawa<EMAIL_ADDRESS>

1Physics Program, Graduate School of Advanced Science
and Engineering, Hiroshima University,
Higashi-Hiroshima 739-8526, Japan
2International Institute for Sustainability with Knotted Chiral Meta
Matter/WPI-SKCM2, Hiroshima University,
2-313 Kagamiyama, Higashi-Hiroshima City, Hiroshima, Japan 739-0046

( Abstract
We prsent a lepton flavor model with $S_{4}$ and $U(1)_{\mathrm{FN}}$
symmetry. The left-handed leptons are assigned as a triplet under the $S_{4}$
symmetry, the right-handed electron and muon as $S_{4}$ doublets, and the
right-handed tauon as an $S_{4}$ singlet. We introduce three right-handed
Majorana neutrinos that are charged under the $S_{4}$ symmetry and two scalar
fields. Additionally, the three Higgs doublets are assigned as an $S_{4}$
triplet, and we analyze the scalar potential to get the conditions for the
Higgs VEVs. We numerically calculate the PMNS matrix, and give strong
predictions for the mixing angle $0.486<\sin\theta_{23}<0.603$ and the Dirac
CP phase $60.12^{\circ}<|\delta_{CP}|<76.47^{\circ}$. Our predictions for the
lightest neutrino mass $m_{light}\simeq 5.53~{}[\mathrm{meV}]$ and the
effective Majorana neutrino mass $m_{ee}\simeq 6.33~{}[\mathrm{meV}]$ from the
neutrinoless double beta decay experiments are relatively close to upper
limits. Lastly, we obtain the sum of neutrino masses $m_{1}+m_{2}+m_{3}\simeq
66.1~{}[\mathrm{meV}]$, and two Majorana phases $\eta_{1}$ and $\eta_{2}$. )

##  1 Introduction

The Standard Model (SM) of elementary particles successfully describes the
known particles and their interactions. It has explained the mechanism of
electroweak symmetry breaking and the mechanism of mass acquisition for gauge
bosons and fermions. With the discovery of the Higgs particle in 2012 [1, 2],
the SM is believed to be a correct theory. However, there are some mysteries
in the SM, for example the generational structure of fermions. That structure
comes from the magnitude of the Yukawa interactions, which describe the
interaction between charged fermions and the Higgs boson in the SM. The Yukawa
interactions are simply parameterized to match experimental data. Therefore,
it is not possible to theoretically explain the difference in masses and
mixing of flavors between generations in the SM. In particular, the flavor
mixing of leptons is predicted to be larger than that of quarks and the
magnitude of the CP phase of the lepton sector is not yet precisely known.

Leptons have another mystery related to neutrinos. In the SM the neutrinos
have zero mass, because there are no right-handed neutrinos. However, the
discovery of neutrino oscillations has shown that neutrinos have tiny masses.
Although neutrino masses can not be explained by the SM, the neutrino mass
matrix is very important when considering lepton flavor mixing. In addition,
it is not yet known whether neutrinos are Dirac or Majorana particles. If the
neutrinos are Majorana particles, then two additional CP phases called
Majorana phases are added, the magnitude of which is not yet known.

Therefore, many studies have been conducted to understand the generation
structure. For example, the Froggatt-Nielsen (FN) mechanism introduced global
a $U(1)_{\mathrm{FN}}$ symmetry [3] to provide a natural explanation for the
fermion mass hierarchies. Also, several studies of non-Abelian discrete
symmetries (see for review [4, 5, 6]) have been used to explain flavor mixing
[7, 8, 9, 10, 11, 12, 13, 14]. The Yukawa couplings are controlled by these
symmetries, and it leads to a natural explanation for the mixing angles of
lepton.

Much work has also been done to explain neutrino masses [15, 16, 17, 18, 19,
20]. The masses of the lighter left-handed neutrinos are often explained using
heavy right-handed Majorana neutrinos and the seesaw mechanism. Majorana
neutrinos are being searched for using neutrinoless double beta
($0\nu\beta\beta$) decay at experiments like KamLAND-Zen [21, 22]. These
experiments have provided an upper limit to the effective Majorana neutrino
mass. According to this paper [22], the inverted ordering region is being
explored.

In building our flavor model, in addition to the non-Abelian discrete and the
$U(1)_{\mathrm{FN}}$ symmetries, we focus on the Higgs sector. Many multi-
Higgs models have appeared in recent times. It is natural to assume that not
only fermions but also Higgs sector has a similar generation structure, and
there are many previous studies of that type [23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. We consider a three Higgs doublets
model(3HDM) [41], which means the number of Higgs doublets is three and has a
flavor symmetry. This means we assume the Higgs sector follows the same
generation structure as the fermions. In a previous study, we considered an
$A_{4}$ symmetry, which is one possible non-Abelian discrete symmetry, and the
3HDM to construct a flavor model [40]. In that study, we could reproduce the
lepton mixing angles and give predictions for the CP violating phase and the
lightest neutrino mass. However, we only reproduced the neutrino masses in the
inverted hierarchy. As mentioned above, the inverted hierarchy is currently
being explored and unfavor [22].

In this paper, we consider the $S_{4}$ symmetry that is also a possible non-
Abelian discrete symmetry. The advantage of the $S_{4}$ symmetry is that the
$S_{4}$ symmetry contains one doublet that is not included in the $A_{4}$
symmetry. Because of that we can build an $S_{4}$ lepton flavor model with
$U(1)_{\mathrm{FN}}$ and 3HDM, which reproduce neutrino masses in the normal
hierarchy. Additionally, our model predicts the mixing angles and the
magnitude of Dirac CP phase of leptons. We also predict the lightest neutrino
mass, the sum of neutrino masses, the effective Majorana neutrino mass at the
$0\nu\beta\beta$ decay experiment and the two Majorana phases in normal-
ordering.

The rest of this paper is organized as follows. In Section 2, we present the
lepton flavor model and obtain mass matrices. In Section 3, we show the
numerical analysis of our flavor model. In Section 4, we show the scalar
potential analysis. In Section 5, we summarize this paper. We give the brief
introduction of the $S_{4}$ symmetry and show the multiplication rule of the
$S_{4}$ group in Appendix A. We shortly introduce the 3HDM with $S_{4}$
symmetry in Appendix B.

##  2 Lepton flavor model with $S_{4}$ symmetry

In this section, we present a lepton flavor model with $S_{4}$ symmetry, and
we show the mass matrices of charged leptons and neutrinos.

The $S_{4}$ symmetry is the symmetry of the $S_{4}$ group, which is the
symmetric group of order 4. The $S_{4}$ group has two types of singlets
$\mathbf{1}$, $\mathbf{1^{\prime}}$, one type of doublet $\mathbf{2}$, and two
types of triplets $\mathbf{3}$, $\mathbf{3^{\prime}}$, as noted in Appendix A.
We assign the left-handed leptons as an $S_{4}$ triplet, the right-handed
electron and muon as $S_{4}$ doublets, and the right-handed tauon as an
$S_{4}$ singlet. Then, we suppose three right-handed Majorana neutrinos to
obtain neutrino masses with the seesaw mechanism. Uniquely, the right-handed
electron neutrino is assigned as an $S_{4}$ singlet, and the right-handed muon
neutrino and tauon neutrinos are assigned as $S_{4}$ doublets.

Here, we introduce the three Higgs doublets that are assigned as an $S_{4}$
triplet. This means the Higgs sector has a similar generation structure to
fermions. Additionally, we introduce two scalar fields $\Theta$ and $X$, where
the field $\Theta$ is an $S_{4}$ singlet and the field $X$ is an $S_{4}$
doublet. This allows us to preform a precise reproduction of experimental
results. We impose the $U(1)_{\mathrm{FN}}$ symmetry on the added scalar
fields and right-handed lepton doublet, and we limit the couplings of scalar
fields. In Table 1, we summarize the particle assignments of the $SU(2)_{L}$,
$S_{4}$ and $U(1)_{\mathrm{FN}}$ symmetries.

| $\bar{\ell}=(\bar{\ell}_{e},\bar{\ell}_{\mu},\bar{\ell}_{\tau})$ | $\ell_{R}=(e_{R},\mu_{R})$ | $\tau_{R}$ | $\nu_{eR}$ | $\nu_{R}=(\nu_{\mu R},\nu_{\tau R})$ | $\phi=(\phi_{1},\phi_{2},\phi_{3})$ | $\Theta$ | $X$
---|---|---|---|---|---|---|---|---
$SU(2)_{L}$ | $2$ | $1$ | $1$ | $1$ | $1$ | $2$ | $1$ | $1$
$S_{4}$ | ${\bf 3}$ | ${\bf 2}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 2}$ | ${\bf 3}$ | ${\bf 1}$ | ${\bf 2}$
$U(1)_{\mathrm{FN}}$ | $0$ | $+1$ | $0$ | $0$ | $0$ | $0$ | $-1$ | $-1$
Table 1: The charge assignments of $SU(2)_{L}\times S_{4}\times
U(1)_{\mathrm{FN}}$ symmetry in our model.

We can write down the Lagrangian for Yukawa interactions and Majorana mass
term in our model. The $SU(2)_{L}\times S_{4}\times U(1)_{\mathrm{FN}}$
invariant Lagrangian is,

$\mathcal{L}_{Y}=\mathcal{L}_{\ell}+\mathcal{L}_{D}+\mathcal{L}_{M}+h.c.,$ (1)

where,

$\displaystyle\mathcal{L}_{\ell}$
$\displaystyle=\frac{y_{e\mu}}{\Lambda}\bar{\ell}\phi\ell_{R}\Theta+y_{\tau}\bar{\ell}\phi\tau_{R}+\frac{y_{\ell}}{\Lambda}\bar{\ell}\phi\ell_{R}X,$
(2) $\displaystyle\mathcal{L}_{D}$
$\displaystyle=y_{De}\bar{\ell}\tilde{\phi}\nu_{eR}+y_{D\mu\tau}\bar{\ell}\tilde{\phi}\nu_{R},$
(3) $\displaystyle\mathcal{L}_{M}$
$\displaystyle=\frac{1}{2}M_{eR}\bar{\nu}_{eR}^{C}\nu_{eR}+\frac{1}{2}M_{\mu\tau
R}\bar{\nu}_{R}^{C}\nu_{R}.$ (4)

Note that $y_{e\mu},y_{\tau},y_{\ell}$, $y_{De}$ and $y_{D\mu\tau}$ are Yukawa
couplings and, $M_{eR}$ and $M_{\mu\tau R}$ are the right-handed Majorana
neutrino masses. We consider the Yukawa coupling $y_{e\mu},y_{\tau},y_{\ell}$
as real numbers, and $y_{De}$ and $y_{D\mu\tau}$ to be complex numbers. At the
high-energy scale, we take the real VEVs of $\Theta$ and $X$ as
$\langle\Theta\rangle=\Theta_{0},\langle X\rangle=(\mathrm{X}_{1},0)$. Then,
after spontaneous symmetry breaking (SSB), the three Higgs doublets have the
real VEVs $\langle\phi_{i}\rangle=\begin{pmatrix}0\\\
\frac{1}{\sqrt{2}}v_{i}\end{pmatrix}$ $(i=1,2,3)$. For the charged lepton
sector Eq. (2), we derive the Yukawa interactions following the $S_{4}$
multiplication rules Eq. (A)-(A) in Appendix A;

$\displaystyle\frac{y_{e\mu}}{\Lambda}\bar{\ell}\phi\ell_{R}\Theta$
$\displaystyle=\frac{y_{e\mu}\Theta}{\Lambda}\begin{pmatrix}\bar{\ell}_{e}\\\
\bar{\ell}_{\mu}\\\
\bar{\ell}_{\tau}\end{pmatrix}\times\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\
\phi_{3}\end{pmatrix}\times\begin{pmatrix}e_{R}\\\ \mu_{R}\end{pmatrix}$
$\displaystyle=\frac{y_{e\mu}\Theta}{\Lambda}\Big{[}\frac{1}{\sqrt{2}}(\bar{\ell}_{\mu}\phi_{2}-\bar{\ell}_{\tau}\phi_{3})e_{R}+\frac{1}{\sqrt{6}}(-2\bar{\ell}_{e}\phi_{1}+\bar{\ell}_{\mu}\phi_{2}+\bar{\ell}_{\tau}\phi_{3})\mu_{R}\Big{]}$
$\displaystyle\rightarrow\frac{y_{e\mu}\Theta_{0}}{\Lambda}\Big{[}\frac{1}{2}(\bar{\mu}_{L}v_{2}-\bar{\tau}_{L}v_{3})e_{R}+\frac{1}{2\sqrt{3}}(-2\bar{e}_{L}v_{1}+\bar{\mu}_{L}v_{2}+\bar{\tau}_{L}v_{3})\mu_{R}\Big{]},$
(5) $\displaystyle y_{\tau}\bar{\ell}\phi\tau_{R}$
$\displaystyle=y_{\tau}\begin{pmatrix}\bar{\ell}_{e}\\\ \bar{\ell}_{\mu}\\\
\bar{\ell}_{\tau}\end{pmatrix}\times\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\
\phi_{3}\end{pmatrix}\tau_{R}$
$\displaystyle=y_{\tau}(\bar{\ell}_{e}\phi_{1}+\bar{\ell}_{\mu}\phi_{2}+\bar{\ell}_{\tau}\phi_{3})\tau_{R}$
$\displaystyle\rightarrow\frac{y_{\tau}}{\sqrt{2}}(\bar{e}_{L}v_{1}+\bar{\mu}_{L}v_{2}+\bar{\tau}_{L}v_{3})\tau_{R},$
(6) $\displaystyle\frac{y_{\ell}}{\Lambda}\bar{\ell}\phi\ell_{R}X$
$\displaystyle=\frac{y_{\ell}}{\Lambda}\begin{pmatrix}\bar{\ell}_{e}\\\
\bar{\ell}_{\mu}\\\
\bar{\ell}_{\tau}\end{pmatrix}\times\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\
\phi_{3}\end{pmatrix}\times\begin{pmatrix}e_{R}\\\
\mu_{R}\end{pmatrix}\times\begin{pmatrix}X_{1}\\\ X_{2}\end{pmatrix}$
$\displaystyle=\frac{y_{\ell
1}}{\Lambda}(\bar{\ell}_{e}\phi_{1}+\bar{\ell}_{\mu}\phi_{2}+\bar{\ell}_{\tau}\phi_{3})(e_{R}X_{1}+\mu_{R}X_{2})$
$\displaystyle\quad\,+\frac{y_{\ell
2}}{\Lambda}\Big{[}\frac{1}{\sqrt{2}}(\bar{\ell}_{\mu}\phi_{2}-\bar{\ell}_{\tau}\phi_{3})(e_{R}X_{2}+\mu_{R}X_{1})+\frac{1}{\sqrt{6}}(-2\bar{\ell}_{e}\phi_{1}+\bar{\ell}_{\mu}\phi_{2}+\bar{\ell}_{\tau}\phi_{3})(e_{R}X_{1}-\mu_{R}X_{2})\Big{]}$
$\displaystyle\rightarrow\frac{y_{\ell
1}}{\sqrt{2}\Lambda}(\bar{e}_{L}v_{1}+\bar{\mu}_{L}v_{2}+\bar{\tau}_{L}v_{3})e_{R}\mathrm{X}_{1}$
$\displaystyle\quad\,+\frac{y_{\ell
2}}{\Lambda}\Big{[}\frac{1}{2}(\bar{\mu}_{L}v_{2}-\bar{\tau}_{L}v_{3})\mu_{R}\mathrm{X}_{1}+\frac{1}{2\sqrt{3}}(-2\bar{e}_{L}v_{1}+\bar{\mu}_{L}v_{2}+\bar{\tau}_{L}v_{3})e_{R}\mathrm{X}_{1}\Big{]}.$
(7)

The mass matrix from Eq. (5) and Eq. (6) is $M_{\ell 1}$, where

$\displaystyle M_{\ell
1}=\begin{pmatrix}0&-\frac{1}{\sqrt{3}}\frac{y_{e\mu}\Theta_{0}}{\Lambda}v_{1}&\frac{1}{\sqrt{2}}y_{\tau}v_{1}\\\
\frac{1}{2}\frac{y_{e\mu}\Theta_{0}}{\Lambda}v_{2}&\frac{1}{2\sqrt{3}}\frac{y_{e\mu}\Theta_{0}}{\Lambda}v_{2}&\frac{1}{\sqrt{2}}y_{\tau}v_{2}\\\
-\frac{1}{2}\frac{y_{e\mu}\Theta_{0}}{\Lambda}v_{3}&\frac{1}{2\sqrt{3}}\frac{y_{e\mu}\Theta_{0}}{\Lambda}v_{3}&\frac{1}{\sqrt{2}}y_{\tau}v_{3}\end{pmatrix}_{LR\,.}$
(8)

The mass matrix from Eq. (7) is $M_{\ell 2}$, where

$\displaystyle M_{\ell 2}=\begin{pmatrix}(\frac{1}{\sqrt{2}}y_{\ell
1}-\frac{1}{\sqrt{3}}y_{\ell 2})v_{1}\frac{\mathrm{X}_{1}}{\Lambda}&0&0\\\
(\frac{1}{\sqrt{2}}y_{\ell 1}+\frac{1}{2\sqrt{3}}y_{\ell
2})v_{2}\frac{\mathrm{X}_{1}}{\Lambda}&\frac{1}{2}y_{\ell
2}v_{2}\frac{\mathrm{X}_{1}}{\Lambda}&0\\\ (\frac{1}{\sqrt{2}}y_{\ell
1}+\frac{1}{2\sqrt{3}}y_{\ell
2})v_{3}\frac{\mathrm{X}_{1}}{\Lambda}&-\frac{1}{2}y_{\ell
2}v_{3}\frac{\mathrm{X}_{1}}{\Lambda}&0\\\ \end{pmatrix}_{LR\,.}$ (9)

Then, we obtain the charged lepton mass matrix $M_{\ell}$ as

$\displaystyle M_{\ell}$ $\displaystyle=M_{\ell 1}+M_{\ell 2}$
$\displaystyle=\begin{pmatrix}(\frac{1}{\sqrt{2}}y_{\ell
1}-\frac{1}{\sqrt{3}}y_{\ell
2})v_{1}\frac{\mathrm{X}_{1}}{\Lambda}&-\frac{1}{\sqrt{3}}\frac{y_{e\mu}\Theta_{0}}{\Lambda}v_{1}&\frac{1}{\sqrt{2}}y_{\tau}v_{1}\\\
\frac{1}{2}\frac{y_{e\mu}\Theta_{0}}{\Lambda}v_{2}+(\frac{1}{\sqrt{2}}y_{\ell
1}+\frac{1}{2\sqrt{3}}y_{\ell
2})v_{2}\frac{\mathrm{X}_{1}}{\Lambda}&\frac{1}{2\sqrt{3}}\frac{y_{e\mu}\Theta_{0}}{\Lambda}v_{2}+\frac{1}{2}y_{\ell
2}v_{2}\frac{\mathrm{X}_{1}}{\Lambda}&\frac{1}{\sqrt{2}}y_{\tau}v_{2}\\\
-\frac{1}{2}\frac{y_{e\mu}\Theta_{0}}{\Lambda}v_{3}+(\frac{1}{\sqrt{2}}y_{\ell
1}+\frac{1}{2\sqrt{3}}y_{\ell
2})v_{3}\frac{\mathrm{X}_{1}}{\Lambda}&\frac{1}{2\sqrt{3}}\frac{y_{e\mu}\Theta_{0}}{\Lambda}v_{3}-\frac{1}{2}y_{\ell
2}v_{3}\frac{\mathrm{X}_{1}}{\Lambda}&\frac{1}{\sqrt{2}}y_{\tau}v_{3}\end{pmatrix}_{LR\,.}$
(10)

Next, we calculate the Dirac neutrino mass matrix from the Lagrangian for
Dirac neutrino Yukawa interactions Eq. (3). As with the charged lepton mass
matrix, the Dirac neutrino mass matrix $M_{D}$ is obtained via the following,

$\displaystyle
M_{D}=\begin{pmatrix}\frac{1}{\sqrt{2}}y_{De}v_{1}&0&-\frac{1}{\sqrt{3}}y_{D\mu\tau}v_{1}\\\
\frac{1}{\sqrt{2}}y_{De}v_{2}&\frac{1}{2}y_{D\mu\tau}v_{2}&\frac{1}{2\sqrt{3}}y_{D\mu\tau}v_{2}\\\
\frac{1}{\sqrt{2}}y_{De}v_{3}&-\frac{1}{2}y_{D\mu\tau}v_{3}&\frac{1}{2\sqrt{3}}y_{D\mu\tau}v_{3}\end{pmatrix}_{.}$
(11)

We also derive the right-handed Majorana neutrino mass matrix $M_{R}$ from Eq.
(4),

$\displaystyle M_{R}=\begin{pmatrix}M_{eR}&0&0\\\ 0&M_{\mu\tau R}&0\\\
0&0&M_{\mu\tau R}\end{pmatrix}_{.}$ (12)

After these calculations of neutrino mass matrices, we use the type-I seesaw
mechanism and obtain the left-handed Majorana neutrino mass matrix $m_{\nu}$:

$\displaystyle m_{\nu}=-M_{D}M_{R}^{-}1M_{D}^{T},$ (13)
$\displaystyle(m_{\nu})_{ij}=-\frac{e^{2i\phi_{De}}|y_{De}|^{2}v_{i}^{2}}{2M_{eR}}-\frac{e^{2i\phi_{D\mu\tau}}|y_{D\mu\tau}|^{2}v_{i}^{2}}{3M_{\mu\tau
R}},\qquad\,\,i=j\,(i,j=1,2,3),$
$\displaystyle(m_{\nu})_{ij}=-\frac{e^{2i\phi_{De}}|y_{De}|^{2}v_{i}v_{j}}{2M_{eR}}+\frac{e^{2i\phi_{D\mu\tau}}|y_{D\mu\tau}|^{2}v_{i}v_{j}}{6M_{\mu\tau
R}},\quad i\neq j\,(i,j=1,2,3),$

where, $\phi_{De}$ and $\phi_{D\mu\tau}$ are the complex phases from the
Yukawa couplings $y_{De}$ and $y_{D\mu\tau}$.

##  3 Numerical analysis

In this section, we numerically obtain the PMNS matrix from the leptons mass
matrices. We show the numerical results such as the mixing angles of leptons,
Dirac CP phase, the lightest neutrino mass, the sum of neutrino masses, the
effective Majorana neutrino mass at the $0\nu\beta\beta$ decay experiment, and
two Majorana phases in normal-ordering.

First, we get the PMNS matrix representing the lepton flavor mixing from the
mass matrices of charged leptons and neutrinos obtained in the section 2. We
find the unitary matrix $V_{\ell}$ that diagonalizes the charged lepton mass
matrix. The charged lepton mass matrix $M_{\ell}$ in Eq. (10) becomes

$\displaystyle M_{\ell}=\begin{pmatrix}(\frac{1}{\sqrt{2}}y_{\ell
1}-\frac{1}{\sqrt{3}}y_{\ell
2})v_{1}\mathrm{X^{\prime}}_{1}&-\frac{1}{\sqrt{3}}y^{\prime}_{e\mu}v_{1}&\frac{1}{\sqrt{2}}y_{\tau}v_{1}\\\
\frac{1}{2}y^{\prime}_{e\mu}v_{2}+(\frac{1}{\sqrt{2}}y_{\ell
1}+\frac{1}{2\sqrt{3}}y_{\ell
2})v_{2}\mathrm{X^{\prime}}_{1}&\frac{1}{2\sqrt{3}}y^{\prime}_{e\mu}v_{2}+\frac{1}{2}y_{\ell
2}v_{2}\mathrm{X^{\prime}}_{1}&\frac{1}{\sqrt{2}}y_{\tau}v_{2}\\\
-\frac{1}{2}y^{\prime}_{e\mu}v_{3}+(\frac{1}{\sqrt{2}}y_{\ell
1}+\frac{1}{2\sqrt{3}}y_{\ell
2})v_{3}\mathrm{X^{\prime}}_{1}&\frac{1}{2\sqrt{3}}y^{\prime}_{e\mu}v_{3}-\frac{1}{2}y_{\ell
2}v_{3}\mathrm{X^{\prime}}_{1}&\frac{1}{\sqrt{2}}y_{\tau}v_{3}\end{pmatrix}_{LR,}$
(14)

where, $\mathrm{X^{\prime}}_{1}\equiv\frac{\mathrm{X}_{1}}{\Lambda}$, and
$y^{\prime}_{e\mu}\equiv\frac{y_{e\mu}\Theta_{0}}{\Lambda}$. We use
$M_{\ell}M_{\ell}^{\dagger}$ to find the unitary matrix $V_{\ell}$ that
diagonalizes $M_{\ell}M_{\ell}^{\dagger}$. Similarly, we evaluate the neutrino
sector in the same way. The neutrino mass matrix $m_{\nu}$ in Eq. (13) is
rewritten to be

$\displaystyle(m_{\nu})_{ij}=-\frac{1}{2}e^{2i\phi_{De}}|y^{\prime}_{De}|^{2}v_{i}^{2}-\frac{1}{3}e^{2i\phi_{D\mu\tau}}|y^{\prime}_{D\mu\tau}|^{2}v_{i}^{2},\qquad\,\,i=j\,(i,j=1,2,3),$
$\displaystyle(m_{\nu})_{ij}=-\frac{1}{2}e^{2i\phi_{De}}|y^{\prime}_{De}|^{2}v_{i}v_{j}+\frac{1}{6}e^{2i\phi_{D\mu\tau}}|y^{\prime}_{D\mu\tau}|^{2}v_{i}v_{j},\quad
i\neq j\,(i,j=1,2,3),$

where, $|y^{\prime}_{De}|^{2}\equiv\frac{|y_{De}|^{2}}{M_{eR}}$, and
$|y^{\prime}_{D\mu\tau}|^{2}\equiv\frac{|y_{D\mu\tau}|^{2}}{M_{\mu\tau R}}$.
We use $m_{\nu}m_{\nu}^{\dagger}$ to find the unitary matrix $V_{\nu}$ that
diagonalizes $m_{\nu}m_{\nu}^{\dagger}$.

Next, we perform the numerical analysis with the physical input parameters the
masses of the charged leptons $m_{e},m_{\mu},m_{\tau}$, and the neutrino mass-
squared differences $\Delta m^{2}_{21},\Delta m^{2}_{31}$. We assume normal
hierarchy for the neutrinos and take, from NuFIT 5.2 [42, 43], the neutrino
mass-squared differences $\Delta m^{2}_{21}=7.41\times
10^{-5}~{}\mathrm{eV^{2}}$, $\Delta m^{2}_{31}=2.507\times
10^{-3}~{}\mathrm{eV^{2}}$ in Table 2. Additionally, we use the PDG data [44]
for the masses of the charged leptons.

Normal Ordering | bfp $\pm 1\sigma$ |  $3\sigma$ range
---|---|---
$\sin^{2}\theta_{12}$ | $0.303^{+0.012}_{-0.012}$ | $0.270\to 0.341$
$\theta_{12}/^{\circ}$ | $33.41^{+0.75}_{-0.72}$ | $31.31\to 35.74$
$\sin^{2}\theta_{23}$ | $0.451^{+0.019}_{-0.016}$ | $0.408\to 0.603$
$\theta_{23}/^{\circ}$ | $42.2^{+1.1}_{-0.9}$ | $39.7\to 51.0$
$\sin^{2}\theta_{13}$ | $0.02225^{+0.00056}_{-0.00059}$ | $0.02052\to 0.02398$
$\theta_{13}/^{\circ}$ | $8.58^{+0.11}_{-0.11}$ | $8.23\to 8.91$
$\delta_{\mathrm{CP}}/^{\circ}$ | $232^{+36}_{-26}$ | $144\to 350$
$\frac{\Delta m_{21}^{2}}{10^{-5}[\mathrm{eV}^{2}]}$ | $7.41^{+0.21}_{-0.20}$ | $6.82\to 8.03$
$\frac{\Delta m_{31}^{2}}{10^{-3}[\mathrm{eV}^{2}]}$ | $+2.507^{+0.026}_{-0.027}$ |  $+2.427\to-2.590$
Table 2: NuFIT 5.2 data [42, 43] in the normal ordering, where $\Delta
m^{2}_{ij}$ is the mass-squared differences between $m_{i}$ and $m_{j}$.

Our model parameters are

$\displaystyle v_{1},v_{2},v_{3},y^{\prime}_{e\mu},y_{\tau},y_{\ell 1},y_{\ell
2},\mathrm{X^{\prime}}_{1},m_{1},|y^{\prime}_{De}|,|y^{\prime}_{D\mu\tau}|\,\mathrm{and}\,\phi_{De},$

where, $v_{1}$, $v_{2}$, and $v_{3}$ are the VEVs of Higgs that satisfy
$v_{1}^{2}+v_{2}^{2}+v_{3}^{2}=v^{2}$ with $v\approx 246~{}\mathrm{GeV}$ [44].
For the charged lepton mass matrix $M_{\ell}M_{\ell}^{\dagger}$, the Yukawa
couplings $y_{\ell 1}$ and $y_{\ell 2}$ are restricted to be between $-\pi$
and $\pi$. The remaining variables $y^{\prime}_{e\mu}$, $y_{\tau}$, and
$\mathrm{X^{\prime}}_{1}$ are determined such that the charged lepton masses
$m_{e}$, $m_{\mu}$, and $m_{\tau}$ are reproduced when
$M_{\ell}M_{\ell}^{\dagger}$ is diagonalized. For the neutrino sector, only
the two type of mass-squared differences are known, and the absolute masses
are not known. Therefore, of the three neutrino masses $m_{1}$, $m_{2}$ and
$m_{3}$, we vary $m_{1}$ within the range satisfied by the experiment data
[45]. From that the remaining two masses $m_{2}$ and $m_{3}$ are fixed.

Next, the variables $|y^{\prime}_{De}|$, $|y^{\prime}_{D\mu\tau}|$,
$\phi_{De}$ of the neutrino mass matrix $m_{\nu}m_{\nu}^{\dagger}$ are
determined by reproducing the neutrino masses $m_{1},m_{2}$ and $m_{3}$ when
$m_{\nu}m_{\nu}^{\dagger}$ is diagonalized. Here, $\phi_{D\mu\tau}$ can be
absorbed as a whole phase when calculating $m_{\nu}m_{\nu}^{\dagger}$, so it
has no physical meaning. From the above, we can numerically obtain the unitary
matrices $V_{\ell}$ and $V_{\nu}$ that diagonalize
$M_{\ell}M_{\ell}^{\dagger}$ and $m_{\nu}m_{\nu}^{\dagger}$. The PMNS matrix
$U$, which represents the lepton flavor mixing is obtained by

$\displaystyle U=V^{\dagger}_{\ell}V_{\nu}.$ (15)

We denote the $ij$ components of $U$ as $U_{ij}$. Then we can derive the
mixing angles $\theta_{12}$, $\theta_{23}$, and $\theta_{13}$ as follows;

$\displaystyle\tan\theta_{12}$ $\displaystyle=\frac{|U_{12}|}{|U_{11}|},$ (16)
$\displaystyle\tan\theta_{23}$ $\displaystyle=\frac{|U_{23}|}{|U_{33}|},$ (17)
$\displaystyle\sin\theta_{13}$ $\displaystyle=|U_{13}|.$ (18)

Therefore, we can get the PMNS matrix $U$ by Eq. (15) and also numerically
obtain the mixing angles from Eqs. (16) - (18). From NuFIT 5.2 [42, 43], the
magnitude of the mixing angles is fixed in the range shown in Table 2.

Our results that lay within the range of the mixing angles are shown in Table
2. First, the relation of the VEVs of Higgs, $v_{1}$, $v_{2}$, and $v_{3}$ are
shown in the Figure 1.

(a)

(b)

Figure 1: (a) The relation between $v_{1}$ and $v_{2}$. (b) The relation
between $v_{1}$ and $v_{3}$.

The range of $v_{1}$ is almost $38.18~{}[\mathrm{GeV}]\leq|v_{1}|\leq
55.67~{}[\mathrm{GeV}]$ and takes a relatively small value compared to $v_{2}$
and $v_{3}$. The $v_{2}$ and $v_{3}$ are almost $89.51~{}[\mathrm{GeV}]\sim
106.40~{}[\mathrm{GeV}]$ or $218.27~{}[\mathrm{GeV}]\sim
223.30~{}[\mathrm{GeV}]$. Our results for the range of $y^{\prime}_{e\mu}$,
$y_{\tau}$, and $\mathrm{X^{\prime}}_{1}$ are $6.93\times 10^{-4}\leq
y^{\prime}_{e\mu}\leq 9.91\times 10^{-4}$, $1.99\times 10^{-3}\leq
y_{\tau}\leq 1.01\times 10^{-2}$, and $2.25\times
10^{-4}\leq\mathrm{X^{\prime}}_{1}\leq 4.21\times 10^{-2}$ respectively.
Notice the $y^{\prime}_{e\mu}$ is restricted to a narrow range. However,
$y_{\tau}$ and $\mathrm{X^{\prime}}_{1}$ are approximately $10^{1}$ or
$10^{2}$ wide. Then, the results for the ranges $|y^{\prime}_{De}|$ and
$|y^{\prime}_{D\mu\tau}|$ are $7.90\times
10^{-13}~{}[\mathrm{eV}^{-1/2}]\leq|y^{\prime}_{De}|\leq 8.80\times
10^{-13}~{}[\mathrm{eV}^{-1/2}]$ and $1.81\times
10^{-12}~{}[\mathrm{eV}^{-1/2}]\leq|y^{\prime}_{D\mu\tau}|\leq 1.84\times
10^{-12}~{}[\mathrm{eV}^{-1/2}]$ respectively. If we consider the Yukawa
couplings of Dirac neutrinos are on the order of $10^{-1}$, the masses of the
right-handed Majorana neutrinos, $M_{eR}$ and $M_{\mu\tau R}$, are on the
order of $10^{15}~{}[\mathrm{GeV}]$ and $10^{13}~{}[\mathrm{GeV}]$. Recall we
are taking the full range of $-\pi$ to $\pi$ for $y_{\ell 1}$ and $y_{\ell
2}$. The results of $m_{1}$ and $\phi_{De}$ will be shown later.

Next, within the allowed parameter region from above, we predict the lepton
mixing angles, the Dirac CP phase $\delta_{CP}$, the effective Majorana
neutrino mass $m_{ee}$ for $0\nu\beta\beta$ decay experiments, the sum of
neutrino masses, and the Majorana phases $\eta_{1},\eta_{2}$.

Figure 2: The relation between $\sin^{2}\theta_{23}$ and $\delta_{CP}$. In our
model, the positive and negative CP phases are obtained equally. The white
area is the 3$\sigma$ range from NuFIT 5.2 [42, 43].

Our prediction for the mixing angle $\sin^{2}\theta_{23}$ and Dirac CP phase
$\delta_{CP}$ is shown in the Figure 2. The Dirac CP phase $\delta_{CP}$ can
be obtained from the two equations. One equation is the Jarlskog invariant
[46] $J_{CP}$, which is written as

$\displaystyle J_{CP}=\mathrm{Im}[U_{11}U^{*}_{21}U^{*}_{12}U_{22}].$ (19)

Then, we can get $\delta_{CP}$ by using $J_{CP}$,

$\displaystyle\sin\delta_{CP}=\frac{J_{CP}}{s_{12}c_{12}s_{23}c_{23}s_{13}c^{2}_{13}},$
(20)

where, $s_{ij}=\sin\theta_{ij}$ and $c_{ij}=\cos\theta_{ij}$. The other
equation to obtain $\delta_{CP}$ is

$\displaystyle\cos\delta_{CP}=\frac{s^{2}_{12}s^{2}_{23}+c^{2}_{12}s^{2}_{13}c^{2}_{23}-|U_{31}|^{2}}{2s_{12}c_{12}s_{23}c_{23}s_{13}}.$
(21)

We find the $|\delta_{CP}|$ in our model falls in the range $60.12^{\circ}\sim
76.47^{\circ}$. In addition, we can predict $\sin\theta_{23}$ to be in the
range $0.486\sim 0.603$. The results of $\sin\theta_{12}$ and
$\sin\theta_{13}$ take the full range of possible values. Then, the prediction
of the lightest neutrino mass $m_{light}$ and the effective Majorana neutrino
mass $m_{ee}$ at the $0\nu\beta\beta$ decay experiment is shown in the Figure
3.

Figure 3: The relation between the lightest neutrino mass $m_{light}$ and the
effective Majorana neutrino mass $m_{ee}$ at the $0\nu\beta\beta$ decay
experiment. The blue area is model independent analyzes for the in normal
ordering of the neutrino mass hierarchy. The gray area is upper limit on the
effective Majorana neutrino mass of $36~{}[\mathrm{meV}]$ at $90\%$ C.L. [22].
The yellow area is upper limit on the lightest neutrino mass of
$30.1~{}[\mathrm{meV}]$ which is estimated in [45].

We derive the effective Majorana neutrino mass $m_{ee}$ using

$\displaystyle m_{ee}=|\Sigma_{i}U^{2}_{1i}m_{i}|.$ (22)

We find $m_{ee}\simeq 6.33[\mathrm{meV}]$ and $m_{light}\simeq
5.53[\mathrm{meV}]$. Both of those results lay relatively close to upper
limits. Therefore, we consider this model is a relatively easy to confirm by
the near future experiments. Lastly, the relation between the sum of neutrino
masses and the complex phase of the Yukawa coupling of Dirac neutrino
$\phi_{De}$ is shown in the Figure 4.

(a)

(b)

Figure 4: (a) The relation between the sum of neutrino masses and the complex
phase of the Yukawa coupling of Dirac neutrino $\phi_{De}$. (b) The relation
between Majorana phases $\eta_{1}$ and $\eta_{2}$.

For the comparison of the sum of neutrino masses, we find the sum of neutrino
masses $m_{1}+m_{2}+m_{3}\simeq 66.1~{}[\mathrm{meV}]$, which is much smaller
than the experimental limit from cosmology [45] is $0.12~{}[\mathrm{eV}]$. We
derived $|\phi_{De}|$ to be approximately $146^{\circ}$. Finally, we show the
prediction of Majorana phases $\eta_{1}$ and $\eta_{2}$ in the Figure 4. The
Majorana phases $\eta_{1}$ and $\eta_{2}$ can be derived as

$\displaystyle
e^{i\eta_{1}}=\frac{U_{11}U^{*}_{13}}{c_{12}c_{13}s_{13}e^{i\delta_{CP}}},\quad
e^{i\eta_{2}}=\frac{U_{12}U^{*}_{13}}{s_{12}c_{13}s_{13}e^{i\delta_{CP}}}.$
(23)

Because it has not yet been determined whether the neutrino is a Dirac or
Majorana particle, this is only a possible prediction for magnitude of the
Majorana phases $\eta_{1}$ and $\eta_{2}$.

##  4 Potential analysis

In this section, we analyze the scalar potential. The scalar potential in our
model (Table 1) can be written as,

$\displaystyle
V=\mu^{2}\phi^{\dagger}\phi+\lambda(\phi^{\dagger}\phi)^{2}+c\phi^{\dagger}\phi
X^{\dagger}X+k\phi^{\dagger}\phi\Theta^{\dagger}\Theta+(g\phi^{\dagger}\phi\Theta^{\dagger}X+h.c.),$
(24)

where $\lambda>0$. We calculate this potential using the multiplication rules
of the $S_{4}$ symmetry in Eq. (A)-(A) from Appendix A. Specifically, from Eq.
(39) in Appendix B, the first and second term in Eq. (24) are derived as
below,

$\displaystyle\mu^{2}\phi^{\dagger}\phi+\lambda(\phi^{\dagger}\phi)^{2}=$
$\displaystyle\mu^{2}(|\phi_{1}|^{2}+|\phi_{2}|^{2}+|\phi_{3}|^{2})$
$\displaystyle+(\lambda_{1}+\frac{2}{3}\lambda_{2})(|\phi_{1}|^{4}+|\phi_{2}|^{4}+|\phi_{3}|^{4})$
$\displaystyle+(2\lambda_{1}-\frac{2}{3}\lambda_{2}+2\lambda_{3}-2\lambda_{4})(|\phi_{1}\phi_{2}|^{2}+|\phi_{2}\phi_{3}|^{2}+|\phi_{3}\phi_{1}|^{2})$
$\displaystyle+(\lambda_{3}+\lambda_{4})\left[(\phi_{1}\phi_{2}^{\dagger})^{2}+(\phi_{2}\phi_{3}^{\dagger})^{2}+(\phi_{3}\phi_{1}^{\dagger})^{2}+h.c.\right].$
(25)

Then, we obtain the third term in Eq. (24),

$\displaystyle c\phi^{\dagger}\phi X^{\dagger}X=$ $\displaystyle
c\begin{pmatrix}\phi^{\dagger}_{1}\\\ \phi^{\dagger}_{2}\\\
\phi^{\dagger}_{3}\end{pmatrix}\times\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\
\phi_{3}\end{pmatrix}\times\begin{pmatrix}X^{\dagger}_{1}\\\
X^{\dagger}_{2}\end{pmatrix}\times\begin{pmatrix}X_{1}\\\ X_{2}\end{pmatrix}$
$\displaystyle=$ $\displaystyle
c_{1}(|\phi_{1}|^{2}+|\phi_{2}|^{2}+|\phi_{3}|^{2})(|X_{1}|^{2}+|X_{2}|^{2})$
$\displaystyle+c_{2}\left[\frac{1}{\sqrt{2}}(|\phi_{2}|^{2}-|\phi_{3}|^{2})(X_{1}^{\dagger}X_{2}+X^{\dagger}_{2}X_{1})+\frac{1}{\sqrt{6}}(2|\phi_{1}|^{2}+|\phi_{2}|^{2}+|\phi_{3}|^{2})(|X_{1}|^{2}-|X_{2}|^{2})\right].$
(26)

Similarly, we obtain the forth term and fifth term in Eq. (24),

$\displaystyle k\phi^{\dagger}\phi\Theta^{\dagger}\Theta=$ $\displaystyle
k\begin{pmatrix}\phi^{\dagger}_{1}\\\ \phi^{\dagger}_{2}\\\
\phi^{\dagger}_{3}\end{pmatrix}\times\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\
\phi_{3}\end{pmatrix}|\Theta|^{2}$ $\displaystyle=$ $\displaystyle
k|\Theta|^{2}(|\phi_{1}|^{2}+|\phi_{2}|^{2}+|\phi_{3}|^{2}),$ (27)
$\displaystyle g(\phi^{\dagger}\phi\Theta^{\dagger}X+h.c.)=$
$\displaystyle\left(g\begin{pmatrix}\phi^{\dagger}_{1}\\\
\phi^{\dagger}_{2}\\\
\phi^{\dagger}_{3}\end{pmatrix}\times\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\
\phi_{3}\end{pmatrix}\Theta^{\dagger}\begin{pmatrix}X_{1}\\\
X_{2}\end{pmatrix}+h.c.\right)$ $\displaystyle=$
$\displaystyle\left[g\left(\frac{1}{\sqrt{2}}(|\phi_{2}|^{2}-|\phi_{3}|^{2})\Theta^{\dagger}X_{1}+\frac{1}{\sqrt{6}}(2|\phi_{1}|^{2}+|\phi_{2}|^{2}+|\phi_{3}|^{2})\Theta^{\dagger}X_{2}\right)+h.c.\right].$
(28)

Here, we assume the VEVs of $X$ and $\Theta$ to $(\mathrm{X}_{1},0)$ and
$\Theta_{0}$, and the VEV of $\phi_{i}$ is $(0,v_{i}/\sqrt{2})\,(i=1,2,3)$.
Then, we consider the minimum conditions of the potential, and we obtain the
following equations,

$\displaystyle v_{1}$
$\displaystyle=\pm\sqrt{-\frac{\mu^{2}+c_{1}\mathrm{X}_{1}^{2}+k\Theta_{0}^{2}}{3\lambda_{1}+4\lambda_{3}}-\frac{2}{3\sqrt{6}}\frac{3\lambda_{1}+2\lambda_{2}}{(3\lambda_{1}+4\lambda_{4})(\lambda_{2}-2\lambda_{3})}c_{2}\mathrm{X}_{1}^{2}},$
(29) $\displaystyle v_{2}$
$\displaystyle=\pm\sqrt{-\frac{\mu^{2}+c_{1}\mathrm{X}_{1}^{2}+k\Theta_{0}^{2}}{3\lambda_{1}+4\lambda_{3}}+\frac{1}{3\sqrt{6}}\frac{3\lambda_{1}-4\lambda_{2}+12\lambda_{3}}{(3\lambda_{1}+4\lambda_{4})(\lambda_{2}-2\lambda_{3})}c_{2}\mathrm{X}_{1}^{2}-\frac{1}{\sqrt{2}}\frac{g\Theta_{0}\mathrm{X}_{1}}{\lambda_{2}-2\lambda_{3}}},$
(30) $\displaystyle v_{3}$
$\displaystyle=\pm\sqrt{-\frac{\mu^{2}+c_{1}\mathrm{X}_{1}^{2}+k\Theta_{0}^{2}}{3\lambda_{1}+4\lambda_{3}}+\frac{1}{3\sqrt{6}}\frac{3\lambda_{1}-4\lambda_{2}+12\lambda_{3}}{(3\lambda_{1}+4\lambda_{4})(\lambda_{2}-2\lambda_{3})}c_{2}\mathrm{X}_{1}^{2}+\frac{1}{\sqrt{2}}\frac{g\Theta_{0}\mathrm{X}_{1}}{\lambda_{2}-2\lambda_{3}}}.$
(31)

Since there are many parameters for the scalar potential, the result in the
Figure 1 can be realized. One worry could be how the masses of the neutral and
charged Higgs contribute to the flavor-changing neutral currents(FCNC). We
expect the masses can be heavy enough to suppress the FCNC and use this model
by adjusting the parameters of the potential. However, that is beyond the
scope of this work.

##  5 Summary

We have built a lepton flavor model with $S_{4}$ and $U(1)_{\mathrm{FN}}$
symmetries. The left-handed leptons are assigned as an $S_{4}$ triplet and the
right-handed tauon as an $S_{4}$ singlet. Then, we assign the right-handed
electron and muon as an $S_{4}$ doublet and have $U(1)_{\mathrm{FN}}$ charge.
We have assumed there are three right-handed Majorana neutrinos that have an
$S_{4}$ symmetric charge and two types of scalar fields to reproduce the
lepton flavor mixing. We have introduced three Higgs doublets that were
assigned as an $S_{4}$ triplet, and calculated the charged lepton mass matrix
and the left-handed Majorana neutrino mass matrix.

We have performed a numerical analysis to obtain the PMNS matrix, and gave
predictions for the mixing angle $\theta_{23}$ and the Dirac CP phase
$\delta_{CP}$. The results were $0.486<\sin\theta_{23}<0.603$ and
$60.12^{\circ}<|\delta_{CP}|<76.47^{\circ}$, which is a strong prediction for
the $|\delta_{CP}|$. We also gave predictions for the lightest neutrino mass
$m_{light}$, the effective Majorana neutrino mass $m_{ee}$ at the
$0\nu\beta\beta$ decay experiment, the sum of neutrino masses and two Majorana
phases $\eta_{1}$ and $\eta_{2}$. The results were $m_{light}\simeq
5.53~{}[\mathrm{meV}]$, $m_{ee}\simeq 6.33~{}[\mathrm{meV}]$,
$m_{1}+m_{2}+m_{3}\simeq 66.1~{}[\mathrm{meV}]$. For $m_{ee}$ and $m_{light}$,
we were able to obtain values relatively close to current experimentally upper
limits. Finally, we analyzed the scalar potential to get the conditions for
the Higgs VEVs.

Our model could be extended to the quark sector. Where we could impose the
$S_{4}$ symmetry on the quarks as well, and consider how to assign the $S_{4}$
charge and built a flavor model. The model would be tested by the precise
observations of the CP phase in the quark sector. It is interesting to study
more phenomenological aspects on the flavor physics based on flavor symmetry
with multi-Higgs in the near future.

Acknowledgement

We thank M. Tanimoto, Y. Shimizu, N. Benoit, Y. Watanabe and S. Takeshita for
useful discussions. This work was supported by JST, the establishment of
university fellowships towards the creation of science technology innovation,
Grant Number JPMJFS2129.

## Appendix

##  Appendix A $S_{4}$ symmetry

We give a brief introduction of the $S_{4}$ symmetry. The $S_{4}$ symmetry is
the symmetry of the $S_{4}$ group, which is the symmetric group of order 4.
The $S_{4}$ group consists of all permutations among four objects
$(x_{1},x_{2},x_{3},x_{4})$. Therefore, the number of elements of the $S_{4}$
group are $4!=24$

$\displaystyle(x_{1},x_{2},x_{3},x_{4})\rightarrow(x_{i},x_{j},x_{k},x_{l}).$
(32)

The $S_{4}$ group is the smallest non-Abelian discrete symmetry group that has
five irreducible representations. There are two types of singlets
$\mathbf{1}$, $\mathbf{1^{\prime}}$, one type of doublet $\mathbf{2}$, and two
types of triplets $\mathbf{3}$, $\mathbf{3^{\prime}}$. The $S_{4}$ symmetry
represents the symmetry of cubic geometry.

Next, we show the multiplication rule of the $S_{4}$ symmetry:

$\displaystyle\mathbf{3}\otimes\mathbf{3}$
$\displaystyle=\mathbf{1}\oplus\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3^{\prime}},$
$\displaystyle\mathbf{3^{\prime}}\otimes\mathbf{3^{\prime}}$
$\displaystyle=\mathbf{1}\oplus\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3^{\prime}},$
$\displaystyle\mathbf{3}\otimes\mathbf{3^{\prime}}$
$\displaystyle=\mathbf{1^{\prime}}\oplus\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3^{\prime}},$
$\displaystyle\mathbf{2}\otimes\mathbf{2}$
$\displaystyle=\mathbf{1}\oplus\mathbf{1^{\prime}}\oplus\mathbf{2},$
$\displaystyle\mathbf{2}\otimes\mathbf{3}$
$\displaystyle=\mathbf{3}\oplus\mathbf{3^{\prime}},$ (33)
$\displaystyle\mathbf{2}\otimes\mathbf{3^{\prime}}$
$\displaystyle=\mathbf{3}\oplus\mathbf{3^{\prime}},$
$\displaystyle\mathbf{3}\otimes\mathbf{1^{\prime}}$
$\displaystyle=\mathbf{3^{\prime}},$
$\displaystyle\mathbf{3^{\prime}}\otimes\mathbf{1^{\prime}}$
$\displaystyle=\mathbf{3},$
$\displaystyle\mathbf{2}\otimes\mathbf{1^{\prime}}$
$\displaystyle=\mathbf{2},$ $\displaystyle\begin{pmatrix}a_{1}\\\
a_{2}\end{pmatrix}_{\mathbf{2}}\otimes\begin{pmatrix}b_{1}\\\
b_{2}\end{pmatrix}_{\mathbf{2}}$
$\displaystyle=\left(a_{1}b_{1}+a_{2}b_{2}\right)_{\mathbf{1}}\oplus\left(-a_{1}b_{2}+a_{2}b_{1}\right)_{\mathbf{1^{\prime}}}$
$\displaystyle\oplus\begin{pmatrix}a_{1}b_{2}+a_{2}b_{1}\\\
a_{1}b_{1}-a_{2}b_{2}\end{pmatrix}_{\mathbf{2}},$ (34)
$\displaystyle\begin{pmatrix}a_{1}\\\ a_{2}\\\
a_{3}\end{pmatrix}_{\mathbf{3}}\otimes\begin{pmatrix}b_{1}\\\ b_{2}\\\
b_{3}\end{pmatrix}_{\mathbf{3}}$
$\displaystyle=\left(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}\right)_{\mathbf{1}}$
$\displaystyle\oplus\begin{pmatrix}\frac{1}{\sqrt{2}}(a_{2}b_{2}-a_{3}b_{3})\\\
\frac{1}{\sqrt{6}}(-2a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3})\end{pmatrix}_{\mathbf{2}}$
$\displaystyle\oplus\begin{pmatrix}a_{3}b_{2}+a_{2}b_{3}\\\
a_{1}b_{3}+a_{3}b_{1}\\\
a_{2}b_{1}+a_{1}b_{2}\end{pmatrix}_{\mathbf{3}}\oplus\begin{pmatrix}a_{3}b_{2}-a_{2}b_{3}\\\
a_{1}b_{3}-a_{3}b_{1}\\\
a_{2}b_{1}-a_{1}b_{2}\end{pmatrix}_{\mathbf{3^{\prime}}}.$ (35)

More details are shown in the review [4, 5].

##  Appendix B 3HDM with $S_{4}$ symmetry

We briefly introduce the 3HDM with $S_{4}$ symmetry. In general 3HDM means
that the SM Higgs sector is extended to include three Higgs doublets, denoted
as

$\displaystyle\phi_{1}=\begin{pmatrix}\phi^{+}_{1}\\\
\phi^{0}_{1}\end{pmatrix},\phi_{2}=\begin{pmatrix}\phi^{+}_{2}\\\
\phi^{0}_{2}\end{pmatrix},\phi_{3}=\begin{pmatrix}\phi^{+}_{3}\\\
\phi^{0}_{3},\end{pmatrix}$ (36)

where the Higgs doublets $\phi_{i}$, with $i=1$, $2$, $3$, have the same
$U(1)_{Y}$ hypercharge. One Higgs doublet consists of four real scalar fields.
Thus, there are twelve real scalar fields in the 3HDM.

In this study, we consider the Higgs fields to also have a flavor symmetry.
Therefore, we regard three Higgs fields as a $S_{4}$ triplet,

$\displaystyle\phi=\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\
\phi_{3}\end{pmatrix}.$ (37)

In general the scalar potential is given by

$\displaystyle V=\mu^{2}\phi^{\dagger}\phi+\lambda(\phi^{\dagger}\phi)^{2},$
(38)

where, $\mu^{2}<0,\lambda>0$. We get this potential by using the
multiplication rule of the $S4$ symmetry

$\displaystyle V=$ $\displaystyle\mu^{2}\begin{pmatrix}\phi^{\dagger}_{1}\\\
\phi^{\dagger}_{2}\\\
\phi^{\dagger}_{3}\end{pmatrix}\times\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\
\phi_{3}\end{pmatrix}+\lambda\begin{pmatrix}\phi^{\dagger}_{1}\\\
\phi^{\dagger}_{2}\\\
\phi^{\dagger}_{3}\end{pmatrix}\times\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\
\phi_{3}\end{pmatrix}\times\begin{pmatrix}\phi^{\dagger}_{1}\\\
\phi^{\dagger}_{2}\\\
\phi^{\dagger}_{3}\end{pmatrix}\times\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\
\phi_{3}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\mu^{2}(|\phi_{1}|^{2}+|\phi_{2}|^{2}+|\phi_{3}|^{2})+\lambda_{1}(|\phi_{1}|^{2}+|\phi_{2}|^{2}+|\phi_{3}|^{2})^{2}$
$\displaystyle+\frac{2}{3}\lambda_{2}(|\phi_{1}|^{4}+|\phi_{2}|^{4}+|\phi_{3}|^{4}-|\phi_{1}|^{2}|\phi_{2}|^{2}-|\phi_{2}|^{2}|\phi_{3}|^{2}-|\phi_{3}|^{2}|\phi_{1}|^{2})$
$\displaystyle+\lambda_{3}\left[(\phi^{\dagger}_{1}\phi_{2})^{2}+(\phi^{\dagger}_{2}\phi_{3})^{2}+(\phi^{\dagger}_{3}\phi_{1})^{2}+|\phi_{1}|^{2}|\phi_{2}|^{2}+|\phi_{2}|^{2}|\phi_{3}|^{2}+|\phi_{3}|^{2}|\phi_{1}|^{2}+h.c.\right]$
$\displaystyle+\lambda_{4}\left[(\phi^{\dagger}_{1}\phi_{2})^{2}+(\phi^{\dagger}_{2}\phi_{3})^{2}+(\phi^{\dagger}_{3}\phi_{1})^{2}-|\phi_{1}|^{2}|\phi_{2}|^{2}-|\phi_{2}|^{2}|\phi_{3}|^{2}-|\phi_{3}|^{2}|\phi_{1}|^{2}+h.c.\right]$
$\displaystyle=$
$\displaystyle\mu^{2}(|\phi_{1}|^{2}+|\phi_{2}|^{2}+|\phi_{3}|^{2})$
$\displaystyle+(\lambda_{1}+\frac{2}{3}\lambda_{2})(|\phi_{1}|^{4}+|\phi_{2}|^{4}+|\phi_{3}|^{4})$
$\displaystyle+(2\lambda_{1}-\frac{2}{3}\lambda_{2}+2\lambda_{3}-2\lambda_{4})(|\phi_{1}\phi_{2}|^{2}+|\phi_{2}\phi_{3}|^{2}+|\phi_{3}\phi_{1}|^{2})$
$\displaystyle+(\lambda_{3}+\lambda_{4})\left[(\phi_{1}\phi_{2}^{\dagger})^{2}+(\phi_{2}\phi_{3}^{\dagger})^{2}+(\phi_{3}\phi_{1}^{\dagger})^{2}+h.c.\right].$
(39)

Then, we may write the VEVs of Higgs doublets $\phi_{i}$ where $i=1$, $2$, $3$
as

$\displaystyle\langle\phi_{i}\rangle=\begin{pmatrix}0\\\
\frac{1}{\sqrt{2}}v_{i}\end{pmatrix}.$ (40)

By taking into account the minimization conditions on the potential in Eq.
(39), the following relation about the VEVs of Higgs is obtained,

$\displaystyle
v_{1}^{2}=v_{2}^{2}=v_{3}^{2}=\frac{-\mu^{2}}{3\lambda_{1}+4\lambda_{3}}.$
(41)

We have considered the VEVs to be real numbers.

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