#  USTC-ICTS/PCFT-20-18 Modular Invariant Quark and Lepton Models in Double
Covering of $S_{4}$ Modular Group

Xiang-Gan Liu1,2111E-mail<EMAIL_ADDRESS>Chang-Yuan
Yao3222E-mail<EMAIL_ADDRESS>Gui-Jun Ding1,2333E-mail:
<EMAIL_ADDRESS>
1Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China
2Interdisciplinary Center for Theoretical Study and Department of Modern
Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
3School of Physics, Nankai University, Tianjin 300071, China

###### Abstract

We perform a comprehensive analysis of the homogeneous finite modular group
$\Gamma^{\prime}_{4}\equiv S^{\prime}_{4}$ which is the double covering of
$S_{4}$ group. The weight 1 modular forms of level 4 are constructed in terms
of Dedekind eta function, and they transform as a triplet
$\mathbf{\hat{3}^{\prime}}$ of $S^{\prime}_{4}$. The integral weight modular
forms until weight 6 are built from the tensor products of weight 1 modular
forms. We perform a systematical classification of $S^{\prime}_{4}$ modular
models for lepton masses and mixing with/without generalized CP, where the
left-handed leptons are assigned to triplet of $S^{\prime}_{4}$ and right-
handed charged leptons transform as singlets under $S^{\prime}_{4}$, and we
consider both scenarios where the neutrino masses arise from Weinberg operator
or type I seesaw mechanism. The phenomenological implications of the minimal
models for lepton masses, mixing angles, CP violation phases and neutrinoless
double decay are discussed. The $S^{\prime}_{4}$ modular symmetry is extended
to quark sector, we present several predictive models which use nine or ten
free parameters including real and imaginary parts of $\tau$ to describe quark
masses and CKM mixing matrix. We give a quark-lepton unified model which can
explain the flavor structure of quarks and leptons simultaneously for a common
value of $\tau$.

##  1 Introduction

The origin of fermion masses and the mixing matrices is one of the greatest
challenges for modern particle physics. Neutrino oscillation provides new
clues for the understanding of the flavor problem. It is known that neutrino
mixing angles show a pattern which is completely different than that of quark
mixing: all quarks mixing angles are small, while for the lepton sector two
mixing angles $\theta_{12}$, $\theta_{23}$ are large, the third one
$\theta_{13}$ is small and it is comparable to the size of the quark Cabibbo
mixing angle [1]. The evidence of CP violation in neutrino oscillation is
reported recently [2]. Given the successful use of symmetries in various
fields of physics, it was conceived that the flavor structure of quarks and
leptons is dictated by certain flavor symmetry, and different kinds of flavor
symmetry groups (abelian, non-abelian, continuous, discrete, global, local,
linearly or non-linearly realized) have been considered so far. In particular,
it turns that the discrete non-abelian flavor symmetry is quite suitable to
reproduce the large lepton mixing angles, a huge number of models have been
constructed, see [3] for recent review. If discrete flavor symmetry is
combined with generalized CP symmetry [4, 5], one can predict leptonic CP
violation phase. It is notable that a unified description of the observed
structure of the quark and lepton mixing can be achieved if the flavor and CP
symmetries are broken down to $Z_{2}\times CP$ in neutrino, charged lepton, up
quark and down quark sectors, and the minimal flavor group is the dihedral
group $D_{14}$ [6, 7, 8, 9].

In any realistic model based on discrete flavor symmetry, the flavor symmetry
is spontaneously broken by the vacuum expectation values (VEVs) of a set of
scalar fields called flavons which are standard model singlet albeit
transforming non-trivially under the flavor symmetry group. The VEVs of flavon
are typically aligned along certain directions in flavor space, and the vacuum
alignment determines the flavor structure of quarks and leptons. One has to
intelligently design the flavon energy density to achieve the required vacuum
alignment as the global minimum of the scalar potential. In most models,
discrete flavor symmetry is accompanied by additional symmetries, either
discrete like $Z_{N}$ or continuous like $U(1)$, to ensure the needed vacuum
alignment and to reproduce the observed mass hierarchies. Hence the flavor
symmetry breaking sector introduces many independent parameters, makes the
flavor model rather complicated. Moreover, high dimensional operators
compatible with symmetry the model can lead to corrections to leading order
results such the predictability of the moded is spoiled in some sense.

Recently modular invariance as flavor symmetry has suggested to understand the
neutrino masses and lepton flavor mixing [10]. Modular symmetry naturally
appears in torus and orbifold compactifications of string theory. In this
approach, flavon fields are not absolute requirement, the flavor symmetry
symmetry can be uniquely broken by the VEV of the modulus $\tau$. Hence the
vacuum alignment problem is simplified considerably although a moduli
stabilization mechanism is needed. In modular invariant models, the Yukawa
couplings transform nontrivially under the modular symmetry and they are just
modular forms which are holomorphic functions of $\tau$. In the limit of exact
supersymmetry, the superpotenial is completely fixed by modular symmetry.
Furthermore, modular invariant models can be quite predictive, typical minimal
modular models describe the neutrino masses, mixing angles and CP violating
phases in terms of five free real parameters including the real and imaginary
parts of $\tau$.

The finite modular group $\Gamma_{N}=SL(2,Z)/\Gamma(N)$ arising from the
quotient of the $SL(2,Z)$ modular group by congruence subgroups $\Gamma(N)$
have been utilized for the flavor symmetry of quarks and leptons. Some models
for lepton masses and flavor mixing have been constructed at level $N=2$[11,
12, 13, 14], level $N=3$ [10, 15, 11, 16, 17, 18, 12, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], level $N=4$ [36, 37, 38, 39, 40,
41, 42, 29], level $N=5$ [43, 44, 41] and level $N=7$ [45]. The quark masses
and mixing parameters can also be addressed by using modular symmetry [33, 18,
21, 35], and the fermion mass hierarchies can naturally arise as a result of a
weighton which is a standard model singlet field with non-zero modular weight
[34]. Modular symmetry has been discussed in the context of $SU(5)$ grand
unification theory [17, 13]. It is notable that the dynamics of modular
symmetry could be tested at present and future neutrino oscillation
experiments [46]. The modular symmetry has been extended to consistently
include generalized CP symmetry under which the complex modulus $\tau$
transforms as $\tau\rightarrow-\tau^{*}$ [47, 48, 49, 50, 51]. The interplay
between flavor symmetry, CP symmetry and modular invariance was recently
analyzed in string theory [48, 52]. Extension to the direct product of
multiple modular symmetry has been proposed [38, 40]. We have generalized the
modular invariance approach to include the odd weight modular forms which can
be organized into irreducible representations of the homogeneous finite
modular group $\Gamma^{\prime}_{N}$ [23]. $\Gamma^{\prime}_{N}$ is generally
the double covering of the inhomogeneous finite modular group $\Gamma_{N}$.
Texture zeros of fermion mass matrices can be naturally obtained from
$\Gamma^{\prime}_{N}$, the masses and mixing of quarks and leptons can be
addressed in $\Gamma^{\prime}_{3}\cong T^{\prime}$ [44]. The integral weight
modular forms provide new opportunity for modular symmetry model building, and
$\Gamma^{\prime}_{3}\cong T^{\prime}$ has been studied in [23, 44]. In the
present work, we shall consider the next homogeneous finite modular group
$\Gamma^{\prime}_{4}\equiv S^{\prime}_{4}$.

We intend to perform a systematical analysis of lepton models based on
$S^{\prime}_{4}$ modular symmetry with/without generalized CP. For normal
ordering neutrino masses, we find that fifteen viable models which can
describe the neutrino masses, mixing angles and CP violation phases in terms
of five real parameters $|g_{2}/g_{1}|$, $\arg{(g_{2}/g_{1})}$,
$\texttt{Re}(\tau)$, $\texttt{Im}(\tau)$, and $g^{2}_{1}v_{u}^{2}/\Lambda$.
After imposing the generalized CP symmetry, the phase $\arg{(g_{2}/g_{1})}$ is
constrained to be $0$ or $\pi$, six out of the fifteen models can produce a
good fit to the data. The neutrino mass spectrum tends to be quasi-degenerate
in previous models based on inhomogeneous finite modular group $\Gamma_{N}$,
nevertheless the neutrino masses are much lighter in these $S^{\prime}_{4}$
models. Moreover, we extend the $S^{\prime}_{4}$ modular symmetry to the quark
sector. The rich structure of the modular forms at level 4 allows many
possibilities to accommodate the experimental on quark masses and CKM matrix.
After scanning the assignments for quark and lepton fields, we find a model
which can describe the flavor structure of quarks and leptons simultaneously
for a common value of $\tau$.

The remainder of the paper is organized as follows. In section 2, we briefly
review the basic aspects of modular symmetry, we show that the inhomogeneous
finite modular group $\Gamma_{N}$ is isomorphic to the quotient of the
homogeneous finite modular group $\Gamma^{\prime}_{N}$ over the center
$\\{I,-I\\}$, i.e. $\Gamma_{N}\cong\Gamma^{\prime}_{N}/\\{I,-I\\}$. The
integral weight modular forms at level 4 are constructed up to weight 6 in
section 3, and they are arranged into different irreducible representations of
$S^{\prime}_{4}$. The generalized CP symmetry compatible with $S^{\prime}_{4}$
modular symmetry is discussed in section 4. We find that the generalized CP
symmetry requires all the coupling constants real in our working basis. In
section 5, we perform a systematical classification of $S^{\prime}_{4}$
modular models for lepton masses and mixing, where the left-handed leptons are
assigned to triplet of $S^{\prime}_{4}$ and right-handed charged leptons
transform as singlets under $S^{\prime}_{4}$, and the neutrino masses are
described are described by the Weinberg operator or through the type I seesaw
mechanism. The $S^{\prime}_{4}$ modular symmetry is utilized to address the
flavor problem of quark masses hierarchies and CKM mixing matrix, and several
models with small number of free parameters are presented in section 6. We
give a quark-lepton unification model in section 7, which can explain the
masses and mixing patterns of quark and lepton for a common value of $\tau$.
Section 8 concludes the paper. Appendix A gives the necessary group theory of
$S^{\prime}_{4}$ as well as the Clebsch-Gordan (CG) coefficients.

##  2 Modular symmetry and modular forms

The modular group $\overline{\Gamma}$ is isomorphic to the projective special
linear group $PSL(2,\mathbb{Z})$ of $2\times 2$ matrices with integer
coefficients and unit determinant,

$\overline{\Gamma}\cong
PSL(2,\mathbb{Z})=\left\\{\left.\pm\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\right|a,b,c,d\in\mathbb{Z},ad-bc=1\right\\}\,,$ (1)

where the pairs of matrices $A$ and $-A$ are identified. Hence
$PSL(2,\mathbb{Z})$ is the quotient of the two-dimensional special linear
group $\Gamma\equiv SL(2,\mathbb{Z})$ over the integers by its center
$\\{I,-I\\}$, i.e., $\overline{\Gamma}=PSL(2,\mathbb{Z})\cong
SL(2,\mathbb{Z})/\\{I,-I\\}$, where $I$ is two dimensional unit matrix. The
modular group acts on the upper-half complex plane
$\mathcal{H}=\\{\tau\in\mathbb{C}|\texttt{Im}(\tau)>0\\}$ by fractional linear
transformations,

$\tau\mapsto\gamma\tau=\gamma(\tau)=\frac{a\tau+b}{c\tau+d},~{}~{}~{}\texttt{Im}(\tau)>0\,,$
(2)

which implies

$\displaystyle I(\tau)=\tau\,,$
$\displaystyle\texttt{Im}(\gamma(\tau))=\frac{\texttt{Im}\tau}{|c\tau+d|^{2}}>0\,,$
$\displaystyle(\gamma\gamma^{\prime})(\tau)=\gamma(\gamma^{\prime}(\tau)),~{}~{}\text{for}~{}~{}\forall\gamma,\gamma^{\prime}\in\overline{\Gamma}\,.$
(3)

Hence every fractional linear transformation corresponds to a modular group
element $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$, and $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}$ and $-\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$ represent the
same fractional linear transformation. The modular group $\overline{\Gamma}$
has infinity group elements which can be obtained as a combination of the two
fundamental transformations

$S(\tau)=-\frac{1}{\tau},~{}~{}~{}~{}~{}T(\tau)=\tau+1\,,$ (4)

with the corresponding matrices

$S=\begin{pmatrix}0~{}&1\\\
-1~{}&0\end{pmatrix},~{}~{}~{}~{}T=\begin{pmatrix}1~{}&1\\\
0~{}&1\end{pmatrix}\,.$ (5)

We check immediately that in $\overline{\Gamma}$ we have the relations

$S^{2}=(ST)^{3}=I\,,$ (6)

and also $(TS)^{3}=I$ which is equivalent to $(ST)^{3}=I$ if $S^{2}=I$. The
corresponding relations in $\Gamma$ are $S^{2}=-I,(ST^{3})=I$ so that
$S^{4}=(ST)^{3}=I$. The $\overline{\Gamma}$ orbit of every
$\tau\in\mathcal{H}$ has a unique representative in the standard fundamental
domain $\mathcal{D}$,

$\mathcal{D}=\\{\tau|\texttt{Im}(\tau)>0,|\texttt{Re}(\tau)|<\frac{1}{2},|\tau|>1\\}\,,$
(7)

which is bounded by the vertical lines $\texttt{Re}(\tau)=-\frac{1}{2}$,
$\texttt{Re}(\tau)=\frac{1}{2}$ and the circle $|\tau|=1$ in the upper half
plane $\mathcal{H}$. The transformations $S$ and $T$ can map any point in
$\mathcal{H}$ into the fundamental domain $\mathcal{D}$, and no two points
inside $\mathcal{D}$ differ by a linear fraction transformation. The
transformation $T$ pairs the two vertical lines
$\texttt{Re}(\tau)=\pm\frac{1}{2}$, and the transformation $S$ maps the arc of
$|\tau|=1$ from $i$ to $e^{\pi i/3}$ into the arc from $i$ to $e^{2\pi i/3}$ .

The principal congruence subgroup of level $N$ is defined as

$\displaystyle\Gamma(N)$ $\displaystyle=$
$\displaystyle\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma,\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=\begin{pmatrix}1&0\\\
0&1\end{pmatrix}~{}~{}(\texttt{mod}~{}N)\right\\}\,,$
$\displaystyle\overline{\Gamma}(N)$ $\displaystyle=$
$\displaystyle\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\overline{\Gamma},\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=\pm\begin{pmatrix}1&0\\\
0&1\end{pmatrix}~{}~{}(\texttt{mod}~{}N)\right\\}\,,$ (8)

which are normal subgroups of $\Gamma$ and $\overline{\Gamma}$ respectively.
Obviously we have $T^{N}\in\Gamma(N)$, $\Gamma=\Gamma(1)$,
$\overline{\Gamma}=\overline{\Gamma}(1)$, $\overline{\Gamma}=\Gamma/\\{\pm
I\\}$, $\overline{\Gamma}(2)=\Gamma(2)/\\{\pm I\\}$. For $N>2$, we have
$-1\neq 1~{}(\texttt{mod}~{}N)$ and thus $I\notin\Gamma(N)$, consequently
$\Gamma(N)=\overline{\Gamma}(N)$. The finite modular group is the quotient of
modular group over its principal congruence subgroup [53, 23],

$\displaystyle\textit{inhomogeneous finite modular group}~{}:{}$
$\displaystyle\Gamma_{N}\equiv\overline{\Gamma}/\overline{\Gamma}(N)\,,$ (9)
$\displaystyle\textit{homogeneous finite modular group}~{}:{}$
$\displaystyle\Gamma^{\prime}_{N}\equiv\Gamma/\Gamma(N)\,.$

We see $\Gamma_{2}\cong\Gamma^{\prime}_{2}$, $\Gamma^{\prime}_{N}$ is the
double covering of $\Gamma_{N}$ for $N>2$ and $\Gamma_{N}$ is isomorphic to
the quotient of $\Gamma^{\prime}_{N}$ over its center $\\{I,-I\\}$, i.e.,
$\Gamma_{N}\cong\Gamma^{\prime}_{N}/\\{I,-I\\}$ [23]. Hence
$\Gamma^{\prime}_{N}$ has double the number of group elements as $\Gamma_{N}$
with $|\Gamma^{\prime}_{N}|=2|\Gamma_{N}|$. The homogeneous finite modular
group $\Gamma^{\prime}_{N}$ can be obtained from $\Gamma_{N}$ by including
another generator $R$ which is related to $-I\in SL(2,\mathbb{Z})$ and
commutes with all elements of the $SL(2,\mathbb{Z})$ group. For $N\leq 5$, the
multiplication rules of the finite modular groups are444The multiplication
rules of $\Gamma^{\prime}_{N}$ can also be written as
$S^{4}=(ST)^{3}=T^{N}=1,S^{2}T=TS^{2}$. [23],

$\displaystyle\Gamma_{N}~{}:{}$ $\displaystyle S^{2}=(ST)^{3}=T^{N}=1\,,$ (10)
$\displaystyle\Gamma^{\prime}_{N}~{}:{}$ $\displaystyle
S^{2}=R,~{}~{}(ST)^{3}=T^{N}=R^{2}=1,~{}~{}RT=TR\,.$

It is remarkable that $\Gamma_{N}$ and $\Gamma^{\prime}_{N}$ for $N\leq 5$ is
isomorphic to permutation groups and their double covering, i.e.,
$\Gamma_{2}=\Gamma^{\prime}_{2}\cong S_{3}$, $\Gamma_{3}\cong A_{4}$,
$\Gamma^{\prime}_{3}\cong T^{\prime}$, $\Gamma_{4}\cong S_{4}$,
$\Gamma_{4}\cong S^{\prime}_{4}$, $\Gamma_{5}\cong A_{5}$,
$\Gamma^{\prime}_{5}\cong A^{\prime}_{5}$. For $N>5$, additional relations
besides those in Eq. (10) are needed to render the groups $\Gamma_{N}$ and
$\Gamma^{\prime}_{N}$ finite [53, 45].

The modular form $f(\tau)$ of level $N$ and weight $k$ is a holomorphic
function on $\mathcal{H}$ and at all cusps, and it is required to satisfy the
following modular transformation property

$f\left(\gamma\tau\right)=(c\tau+d)^{k}f(\tau)~{}~{}\text{for
all}~{}~{}\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\Gamma(N)\,.$ (11)

The modular forms of level $N$ and weight $k$ span a linear space denoted by
$M_{k}(\Gamma(N))$, and the dimension formula is [54, 55]

$\text{dim}M_{k}(\Gamma(N))=\frac{(k-1)N+6}{24}N^{2}\prod_{p|N}(1-\frac{1}{p^{2}})\,,$
(12)

for $N>2$, where the product is over the prime divisors $p$ of $N$. For level
$N=4$, we have $\text{dim}M_{k}(\Gamma(N))=2k+1$. As has been proved in [23],
one can always find a basis of $M_{k}(\Gamma(N))$ such that the weight $k$
modular forms of level $N$ can be decomposed into different irreducible
representations of $\Gamma^{\prime}_{N}$ up to the automorphy factor
$(c\tau+d)^{k}$. To be more specific, the transformation of the weight $k$
modular form multiplet
$Y^{(k)}_{\mathbf{r}}(\tau)=(f_{1}(\tau),\,f_{2}(\tau),\,\dots)^{T}$ at level
$N$ can be described by an irreducible representation $\rho_{\mathbf{r}}$ of
$\Gamma^{\prime}_{N}$,

$Y^{(k)}_{\mathbf{r}}(\gamma\tau)=(c\tau+d)^{k}\rho_{\mathbf{r}}(\gamma)Y^{(k)}_{\mathbf{r}}(\tau)\,,$
(13)

where $\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$ is a representative
element of $\Gamma^{\prime}_{N}$. In a given representation basis of
$\Gamma^{\prime}_{N}$, the modular multiplet $Y^{(k)}_{\mathbf{r}}$ can be
fixed up to an overall irrelevant constant by applying Eq. (13) for the
generators $S$ and $T$.

##  3 Modular forms of level $N=4$

Applying the general dimension formula in Eq. (12) for $N=4$, we find the
modular space $M_{k}(\Gamma(4))$ has dimension $2k+1$. The modular space
$M_{k}(\Gamma(4))$ has been constructed explicitly by making use of Dedekind
eta function [55],

$M_{k}(\Gamma(4))=\bigoplus_{a+b=2k,\,a,b\geq
0}\mathbb{C}\frac{\eta^{2b-2a}(4\tau)\eta^{5a-b}(2\tau)}{\eta^{2a}(\tau)}\,,$
(14)

where $\eta(\tau)$ is the famous Dedekind eta function defined by

$\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}\left(1-q^{n}\right),~{}~{}~{}q=e^{i2\pi\tau}\,.$
(15)

The Dedekind eta function is a crucial example of a half-integral weight
modular form, having weight $1/2$ and level $1$. The eta function satisfies
the well-known transformation formulas [56, 54, 57],

$\displaystyle\eta(\tau)\stackrel{{\scriptstyle
S}}{{\longmapsto}}\eta(-1/\tau)=\sqrt{-i\tau}~{}\eta(\tau),$ (16)
$\displaystyle\eta(\tau)\stackrel{{\scriptstyle
T}}{{\longmapsto}}\eta(\tau+1)=e^{i\pi/12}\eta(\tau)\,.$

As shown in Eq. (14), we can choose the three linearly independent basis
vectors of the weight $1$ modular space of level $4$ as

$\displaystyle
e_{1}(\tau)=\frac{\eta^{4}(4\tau)}{\eta^{2}(2\tau)},~{}~{}~{}e_{2}(\tau)=\frac{\eta^{10}(2\tau)}{\eta^{4}(4\tau)\eta^{4}(\tau)},~{}~{}~{}e_{3}(\tau)=\frac{\eta^{4}(2\tau)}{\eta^{2}(2\tau)}\,.$
(17)

The $q-$expansion of $e_{1}(\tau),e_{2}(\tau),e_{3}(\tau)$ reads:

$\displaystyle
e_{1}(\tau)=\sqrt{q}\left(1+2q^{2}+q^{4}+2q^{6}+2q^{8}+3q^{12}+2q^{14}+2q^{18}+2q^{20}+\dots\right),$
$\displaystyle
e_{2}(\tau)=1+4q+4q^{2}+4q^{4}+8q^{5}+4q^{8}+4q^{9}+8q^{10}+8q^{13}+4q^{16}+\dots,$
$\displaystyle
e_{3}(\tau)=q^{1/4}\left(1+2q+q^{2}+2q^{3}+2q^{4}+3q^{6}+2q^{7}+2q^{9}+2q^{10}+\dots\right)\,.$
(18)

From the identities of the eta function in Eq. (16), we know that
$e_{1,2,3}(\tau)$ transform under the actions of $S$ and $T$ as follow,

$\displaystyle e_{1}(\tau)\stackrel{{\scriptstyle
T}}{{\longmapsto}}-e_{1}(\tau),~{}~{}e_{2}(\tau)\stackrel{{\scriptstyle
T}}{{\longmapsto}}e_{2},~{}~{}e_{3}(\tau)\stackrel{{\scriptstyle
T}}{{\longmapsto}}ie_{3}\,.$ $\displaystyle e_{1}(\tau)\stackrel{{\scriptstyle
S}}{{\longmapsto}}\frac{1}{8}(-i\tau)(4e_{1}+e_{2}-4e_{3}),$ $\displaystyle
e_{2}(\tau)\stackrel{{\scriptstyle
S}}{{\longmapsto}}\frac{1}{8}(-i\tau)(16e_{1}+4e_{2}+16e_{3}),$ $\displaystyle
e_{3}(\tau)\stackrel{{\scriptstyle
S}}{{\longmapsto}}\frac{1}{8}(-i\tau)(-8e_{1}+2e_{2})\,.$ (19)

As shown in Eq. (13), it is always possible to choose a set of basis in
$M_{k}(\Gamma(4))$ such that the basis vectors can be arranged into several
modular multiplets which transform in irreducible representations of
$\Gamma^{\prime}_{4}\equiv S^{\prime}_{4}$. Thus for the weight 1 modular
forms of level 4, solving the condition of Eq. (13), we find the original
basis $e_{1,2,3}(\tau)$ can be arranged into triplet modular form
$Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}$ transforming as a triplet
$\mathbf{\hat{3}^{\prime}}$ of $S^{\prime}_{4}$,

$Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}(\tau)\equiv\begin{pmatrix}Y_{1}(\tau)\\\
Y_{2}(\tau)\\\ Y_{3}(\tau)\end{pmatrix}\,,$ (20)

where $Y_{1,2,3}(\tau)$ are linear combinations of $e_{1,2,3}(\tau)$ as
follows

$\displaystyle Y_{1}(\tau)$
$\displaystyle=4\sqrt{2}\,e_{1}(\tau)+\sqrt{2}\,ie_{2}(\tau)+2\sqrt{2}(1-i)e_{3}(\tau),$
$\displaystyle Y_{2}(\tau)$
$\displaystyle=-2\sqrt{2}(1+\sqrt{3})\omega^{2}e_{1}(\tau)-\frac{1-\sqrt{3}}{\sqrt{2}}i\omega^{2}e_{2}(\tau)+2\sqrt{2}(1-i)\omega^{2}e_{3}(\tau),$
$\displaystyle Y_{3}(\tau)$ $\displaystyle=2\sqrt{2}(\sqrt{3}-1)\omega
e_{1}(\tau)-\frac{1+\sqrt{3}}{\sqrt{2}}i\omega
e_{2}(\tau)+2\sqrt{2}(1-i)\omega e_{3}(\tau).$ (21)

It is straightforward to check that
$Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}(\tau)$ transforms under $S$ and $T$ as

$Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}(-1/\tau)=-\tau\rho_{\mathbf{\hat{3}^{\prime}}}(S)Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}(\tau),\qquad
Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}(\tau+1)=\rho_{\mathbf{\hat{3}^{\prime}}}(T)Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}(\tau)\,,$
(22)

where the representation matrices $\rho_{\mathbf{\hat{3}^{\prime}}}(S)$ and
$\rho_{\mathbf{\hat{3}^{\prime}}}(T)$ in our working basis are summarized in
table 7. From the expansion formulas of $e_{1,2,3}$ in Eq. (18), we know the
$q-$expansion of $Y_{1,2,3}$:

$\displaystyle Y_{1}(\tau)$ $\displaystyle=$
$\displaystyle\sqrt{2}\,i+2\sqrt{2}(1-i)q^{1/4}+4\sqrt{2}q^{1/2}+4\sqrt{2}\,iq+4\sqrt{2}(1-i)q^{5/4}+4\sqrt{2}\,iq^{2}$
$\displaystyle+2\sqrt{2}(1-i)q^{9/4}+8\sqrt{2}q^{5/2}+4\sqrt{2}(1-i)q^{13/4}+4\sqrt{2}\,iq^{4}+\dots\,,$
$\displaystyle Y_{2}(\tau)+Y_{3}(\tau)$ $\displaystyle=$
$\displaystyle\frac{3+i}{\sqrt{2}}-2\sqrt{2}(1-i)q^{1/4}+2\sqrt{2}(1+3i)q^{1/2}+2\sqrt{2}\,(3+i)q-4\sqrt{2}(1-i)q^{5/4}$
$\displaystyle+2\sqrt{2}(3+i)q^{2}-2\sqrt{2}(1-i)q^{9/4}+4\sqrt{2}(1+3i)q^{5/2}-4\sqrt{2}(1-i)q^{13/4}$
$\displaystyle+2\sqrt{2}(3+i)q^{4}+\dots\,,$ $\displaystyle
Y_{2}(\tau)-Y_{3}(\tau)$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{3}{2}}\,(1+i)\Big{[}1+4q^{1/4}-4q^{1/2}+4q+8q^{5/4}+4q^{2}+4q^{9/4}-8q^{5/2}$
(23) $\displaystyle+8q^{13/4}+4q^{4}+\dots\,\Big{]}\,.$

From the expressions of $q-$ expansion, we see that the modular forms
$Y_{1,2,3}$ satisfy the following constraint,

$\left(Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{1^{\prime}}}=Y_{1}^{2}+2Y_{2}Y_{3}=0\,.$
(24)

The higher weight modular forms can be constructed from the tensor products of
lower weight modular forms with the help of the CG coefficients of
$S^{\prime}_{4}$ in Appendix A, and they are homogeneous polynomials of
$Y_{1,2,3}$. Using the contraction rules for
$\mathbf{\hat{3}^{\prime}}\otimes\mathbf{\hat{3}^{\prime}}\rightarrow\mathbf{1^{\prime}}\oplus\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3^{\prime}}$,
we find that the weight 2 modular forms of level 4 decompose
$\mathbf{2}\oplus\mathbf{3}$ under $S^{\prime}_{4}$

$\displaystyle
Y^{(2)}_{\mathbf{2}}=\left(Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{2}}=\begin{pmatrix}-Y_{2}^{2}-2Y_{1}Y_{3}\\\
Y_{3}^{2}+2Y_{1}Y_{2}\end{pmatrix},$ $\displaystyle
Y^{(2)}_{\mathbf{3}}=\left(Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{3}}=\begin{pmatrix}2Y_{1}^{2}-2Y_{2}Y_{3}\\\
2Y_{3}^{2}-2Y_{1}Y_{2}\\\ 2Y_{2}^{2}-2Y_{1}Y_{3}\end{pmatrix}\,.$ (25)

Note
$\left(Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{3^{\prime}}}=(0,\,0,\,0)^{T}$
which arises from the antisymmetric CG coefficients. Likewise, the weight 3
modular forms can be obtained from the tensor products of
$Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}$ with $Y^{(2)}_{\mathbf{2}}$ and
$Y^{(2)}_{\mathbf{3}}$, and they are arranged into a singlet
$\mathbf{\hat{1}^{\prime}}$ and two triplets $\mathbf{\hat{3}}$ and
$\mathbf{\hat{3}^{\prime}}$ under $S^{\prime}_{4}$,

$\displaystyle
Y^{(3)}_{\mathbf{\hat{1}^{\prime}}}=\left(Y^{(2)}_{\mathbf{3}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{\hat{1}^{\prime}}}=2(Y_{1}^{3}+Y_{2}^{3}+Y_{3}^{3}-3Y_{1}Y_{2}Y_{3}),$
$\displaystyle
Y^{(3)}_{\mathbf{\hat{3}}}=\left(Y^{(2)}_{\mathbf{3}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{\hat{3}}}=\begin{pmatrix}2(2Y_{1}^{3}-Y_{2}^{3}-Y_{3}^{3})\\\
6Y_{3}(Y_{2}^{2}-Y_{1}Y_{3})\\\ 6Y_{2}(Y_{3}^{2}-Y_{1}Y_{2})\end{pmatrix},$
$\displaystyle
Y^{(3)}_{\mathbf{\hat{3}^{\prime}}}=\left(Y^{(2)}_{\mathbf{3}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{\hat{3}^{\prime}}}=\begin{pmatrix}2(Y_{2}^{3}-Y_{3}^{3})\\\
2(-2Y_{1}^{2}Y_{2}+Y_{2}^{2}Y_{3}+Y_{1}Y_{3}^{2})\\\
2(2Y_{1}^{2}Y_{3}-Y_{1}Y_{2}^{2}-Y_{2}Y_{3}^{2})\end{pmatrix}\,.$ (26)

We have three additional contractions between weight 1 and 2 modular forms,
nevertheless they are not independent from
$Y^{(3)}_{\mathbf{\hat{1}^{\prime}}}$, $Y^{(3)}_{\mathbf{\hat{3}}}$ and
$Y^{(3)}_{\mathbf{\hat{3}^{\prime}}}$,

$\displaystyle\left(Y^{(2)}_{\mathbf{3}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{\hat{2}}}=\begin{pmatrix}0\\\
0\end{pmatrix}\,,$ (27a)
$\displaystyle\left(Y^{(2)}_{\mathbf{2}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{\hat{3}^{\prime}}}=\begin{pmatrix}-Y_{2}^{3}+Y_{3}^{3}\\\
2Y_{1}^{2}Y_{2}-Y_{2}^{2}Y_{3}-Y_{1}Y_{3}^{2})\\\
-2Y_{1}^{2}Y_{3}+Y_{1}Y_{2}^{2}+Y_{2}Y_{3}^{2}\end{pmatrix}=-\frac{1}{2}Y^{(3)}_{\mathbf{\hat{3}^{\prime}}}\,,$
(27b)
$\displaystyle\left(Y^{(2)}_{\mathbf{2}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{\hat{3}}}=\begin{pmatrix}Y_{2}^{3}+Y_{3}^{3}+4Y_{1}Y_{2}Y_{3}\\\
2Y_{1}^{2}Y_{2}+Y_{2}^{2}Y_{3}+3Y_{1}Y_{3}^{2}\\\
3Y_{1}Y_{2}^{2}+2Y_{1}^{2}Y_{3}+Y_{2}Y_{3}^{2}\end{pmatrix}=-\frac{1}{2}Y^{(3)}_{\mathbf{\hat{3}}}\,.$
(27c)

The last relation follows from the constraint in Eq. (24). In a similar
manner, we can find out the linearly independent modular forms of higher
weights and corresponding constraints. In the following, we present the
explicit forms of the modular forms. The weight 4 modular space has dimension
$2\times 4+1=9$, they arrange into the irreducible presentations $\mathbf{1}$,
$\mathbf{2}$, $\mathbf{3}$ and $\mathbf{3^{\prime}}$ of $S^{\prime}_{4}$,

$\displaystyle
Y^{(4)}_{\mathbf{1}}=\left(Y^{(3)}_{\mathbf{\hat{3}}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{1}}=4(Y_{1}^{4}-2Y_{1}(Y_{2}^{3}+Y_{3}^{3})+3Y_{2}^{2}Y_{3}^{2}),$
$\displaystyle
Y^{(4)}_{\mathbf{2}}=\left(Y^{(3)}_{\mathbf{\hat{3}}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{2}}=\begin{pmatrix}-2Y_{3}^{4}+4Y_{1}^{3}Y_{3}+4Y_{2}^{3}Y_{3}-6Y_{1}^{2}Y_{2}^{2}\\\
-2Y_{2}^{4}+4Y_{1}^{3}Y_{2}+4Y_{2}Y_{3}^{3}-6Y_{1}^{2}Y_{3}^{2}\end{pmatrix},$
$\displaystyle
Y^{(4)}_{\mathbf{3}}=\left(Y^{(3)}_{\mathbf{\hat{3}}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{3}}=\begin{pmatrix}6Y_{1}(-Y_{2}^{3}+Y_{3}^{3})\\\
6Y_{1}Y_{3}(Y_{2}^{2}-Y_{1}Y_{3})+2Y_{2}(-2Y_{1}^{3}+Y_{2}^{3}+Y_{3}^{3})\\\
6Y_{1}Y_{2}(Y_{1}Y_{2}-Y_{3}^{2})-2Y_{3}(-2Y_{1}^{3}+Y_{2}^{3}+Y_{3}^{3})\end{pmatrix},$
$\displaystyle
Y^{(4)}_{\mathbf{3^{\prime}}}=\left(Y^{(3)}_{\mathbf{\hat{3}}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{3^{\prime}}}=\begin{pmatrix}2(4Y_{1}^{4}-6Y_{2}^{2}Y_{3}^{2}+Y_{1}(Y_{2}^{3}+Y_{3}^{3}))\\\
2(Y_{2}^{4}-2Y_{1}^{3}Y_{2}+7Y_{2}Y_{3}^{3}+3Y_{1}^{2}Y_{3}^{2}-9Y_{1}Y_{2}^{2}Y_{3})\\\
2(Y_{3}^{4}-2Y_{1}^{3}Y_{3}+7Y_{2}^{3}Y_{3}+3Y_{1}^{2}Y_{2}^{2}-9Y_{1}Y_{2}Y_{3}^{2})\end{pmatrix}.$
(28)

The weight 5 modular forms of level 4 decompose as
$\mathbf{\hat{2}}\oplus\mathbf{\hat{3}}\oplus\mathbf{\hat{3}^{\prime}}\oplus\mathbf{\hat{3}^{\prime}}$
under $S^{\prime}_{4}$, and they are given by

$\displaystyle
Y^{(5)}_{\mathbf{\hat{2}}}=\left(Y^{(4)}_{\mathbf{3^{\prime}}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{\hat{2}}}=\begin{pmatrix}2(Y_{2}^{5}+2Y_{1}^{4}Y_{3}+2Y_{1}Y_{3}^{4}+Y_{2}^{2}Y_{3}^{3}+Y_{1}^{3}Y_{2}^{2}-Y_{1}Y_{2}^{3}Y_{3}-6Y_{1}^{2}Y_{2}Y_{3}^{2})\\\
2(Y_{3}^{5}+2Y_{1}^{4}Y_{2}+2Y_{1}Y_{2}^{4}+Y_{1}^{3}Y_{3}^{2}+Y_{2}^{3}Y_{3}^{2}-Y_{1}Y_{2}Y_{3}^{3}-6Y_{1}^{2}Y_{2}^{2}Y_{3})\end{pmatrix},$
$\displaystyle
Y^{(5)}_{\mathbf{\hat{3}}}=\left(Y^{(4)}_{\mathbf{3}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{\hat{3}}}=\begin{pmatrix}18Y_{1}^{2}(-Y_{2}^{3}+Y_{3}^{3})\\\
4Y_{1}^{4}Y_{2}+4Y_{1}(Y_{2}^{4}-5Y_{2}Y_{3}^{3})+14Y_{1}^{3}Y_{3}^{2}-4Y_{3}^{2}(Y_{2}^{3}+Y_{3}^{3})+6Y_{1}^{2}Y_{2}^{2}Y_{3}\\\
-4Y_{1}^{4}Y_{3}-4Y_{1}(Y_{3}^{4}-5Y_{2}^{3}Y_{3})-14Y_{1}^{3}Y_{2}^{2}+4Y_{2}^{2}(Y_{2}^{3}+Y_{3}^{3})-6Y_{1}^{2}Y_{2}Y_{3}^{2}\end{pmatrix},$
$\displaystyle
Y^{(5)}_{\mathbf{\hat{3}^{\prime}},I}=\left(Y^{(4)}_{\mathbf{2}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{\hat{3}^{\prime}}}=\begin{pmatrix}8Y_{1}^{3}Y_{2}Y_{3}-6Y_{1}^{2}(Y_{2}^{3}+Y_{3}^{3})+2Y_{2}Y_{3}(Y_{2}^{3}+Y_{3}^{3})\\\
4Y_{1}^{4}Y_{2}-2Y_{1}Y_{2}^{4}-6Y_{1}^{2}Y_{2}^{2}Y_{3}-2Y_{1}^{3}Y_{3}^{2}+4Y_{2}^{3}Y_{3}^{2}+4Y_{1}Y_{2}Y_{3}^{3}-2Y_{3}^{5}\\\
-2(Y_{1}^{3}Y_{2}^{2}+Y_{2}^{5}-2Y_{1}^{4}Y_{3}+3Y_{1}^{2}Y_{2}Y_{3}^{2}-2Y_{2}^{2}Y_{3}^{3}+Y_{1}(-2Y_{2}^{3}Y_{3}+Y_{3}^{4}))\end{pmatrix},$
$\displaystyle
Y^{(5)}_{\mathbf{\hat{3}^{\prime}},II}=\left(Y^{(4)}_{\mathbf{1}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{\hat{3}^{\prime}}}=\begin{pmatrix}4Y_{1}(Y_{1}^{4}+3Y_{2}^{2}Y_{3}^{2}-2Y_{1}(Y_{2}^{3}+Y_{3}^{3}))\\\
4Y_{2}(Y_{1}^{4}+3Y_{2}^{2}Y_{3}^{2}-2Y_{1}(Y_{2}^{3}+Y_{3}^{3}))\\\
4Y_{3}(Y_{1}^{4}+3Y_{2}^{2}Y_{3}^{2}-2Y_{1}(Y_{2}^{3}+Y_{3}^{3}))\end{pmatrix}\,,$
(29)

where $Y^{(5)}_{\mathbf{\hat{3}^{\prime}},I}$ and
$Y^{(5)}_{\mathbf{\hat{3}^{\prime}},II}$ denote two weight 5 modular forms
transforming as triplet $\mathbf{\hat{3}^{\prime}}$ of $S^{\prime}_{4}$, and
they can also taken to be any two linearly independent combinations of
$Y^{(5)}_{\mathbf{\hat{3}^{\prime}},I}$ and
$Y^{(5)}_{\mathbf{\hat{3}^{\prime}},II}$. Finally there are 13 independent
weight 6 modular forms of level 5, and they can be arranged into the following
irreducible representations of $S^{\prime}_{4}$,

$\displaystyle
Y^{(6)}_{\mathbf{1^{\prime}}}=\left(Y^{(5)}_{\mathbf{\hat{3}^{\prime}},I}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{1^{\prime}}}$
$\displaystyle~{}~{}~{}~{}~{}~{}=-2(Y_{2}^{6}+Y_{3}^{6}-8Y_{1}^{4}Y_{2}Y_{3}+6Y_{1}^{2}Y_{2}^{2}Y_{3}^{2}-4Y_{2}^{3}Y_{3}^{3}+4Y_{1}^{3}(Y_{2}^{3}+Y_{3}^{3})-2Y_{1}Y_{2}Y_{3}(Y_{2}^{3}+Y_{3}^{3}))\,,$
$\displaystyle
Y^{(6)}_{\mathbf{1}}=\left(Y^{(5)}_{\mathbf{\hat{3}}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{1}}=4(Y_{2}^{3}-Y_{3}^{3})(-8Y_{1}^{3}+Y_{2}^{3}+Y_{3}^{3}+6Y_{1}Y_{2}Y_{3})\,,$
$\displaystyle
Y^{(6)}_{\mathbf{2}}=\left(Y^{(5)}_{\mathbf{\hat{3}^{\prime}},II}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{2}}=\begin{pmatrix}-4(Y_{2}^{2}+2Y_{1}Y_{3})(Y_{1}^{4}+3Y_{2}^{2}Y_{3}^{2}-2Y_{1}(Y_{2}^{3}+Y_{3}^{3}))\\\
4(Y_{3}^{2}+2Y_{1}Y_{2})(Y_{1}^{4}+3Y_{2}^{2}Y_{3}^{2}-2Y_{1}(Y_{2}^{3}+Y_{3}^{3}))\end{pmatrix},$
$\displaystyle
Y^{(6)}_{\mathbf{3},I}=\left(Y^{(5)}_{\mathbf{\hat{3}^{\prime}},I}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{3}}$
$\displaystyle=\begin{pmatrix}2(Y_{2}^{6}+Y_{3}^{6}+4Y_{1}^{4}Y_{2}Y_{3}+6Y_{1}^{2}Y_{2}^{2}Y_{3}^{2}-4Y_{2}^{3}Y_{3}^{3}-5Y_{1}^{3}(Y_{2}^{3}+Y_{3}^{3})+Y_{1}Y_{2}Y_{3}(Y_{2}^{3}+Y_{3}^{3}))\\\
-2(2Y_{1}^{5}Y_{2}-5Y_{1}^{4}Y_{3}^{2}+3Y_{1}^{3}Y_{2}^{2}Y_{3}+3Y_{2}^{2}Y_{3}(Y_{2}^{3}-Y_{3}^{3})+Y_{1}^{2}(5Y_{2}Y_{3}^{3}-4Y_{2}^{4})+Y_{1}(Y_{3}^{5}-2Y_{2}^{3}Y_{3}^{2}))\\\
-2(2Y_{1}^{5}Y_{3}-5Y_{1}^{4}Y_{2}^{2}+3Y_{1}^{3}Y_{2}Y_{3}^{2}+3Y_{2}Y_{3}^{2}(Y_{3}^{3}-Y_{2}^{3})+Y_{1}(Y_{2}^{5}-2Y_{2}^{2}Y_{3}^{3})+Y_{1}^{2}(5Y_{2}^{3}Y_{3}-4Y_{3}^{4}))\end{pmatrix}\,,$
$\displaystyle
Y^{(6)}_{\mathbf{3},II}=\left(Y^{(5)}_{\mathbf{\hat{3}^{\prime}},II}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{3}}=\begin{pmatrix}8(Y_{1}^{2}-Y_{2}Y_{3})(Y_{1}^{4}-2Y_{1}(Y_{2}^{3}+Y_{3}^{3})+3Y_{2}^{2}Y_{3}^{2})\\\
8(Y_{3}^{2}-Y_{1}Y_{2})(Y_{1}^{4}-2Y_{1}(Y_{2}^{3}+Y_{3}^{3})+3Y_{2}^{2}Y_{3}^{2})\\\
8(Y_{2}^{2}-Y_{1}Y_{3})(Y_{1}^{4}-2Y_{1}(Y_{2}^{3}+Y_{3}^{3})+3Y_{2}^{2}Y_{3}^{2})\end{pmatrix},$
$\displaystyle
Y^{(6)}_{\mathbf{3^{\prime}}}=\left(Y^{(5)}_{\mathbf{\hat{3}^{\prime}},I}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\right)_{\mathbf{3^{\prime}}}$
$\displaystyle=\begin{pmatrix}-2(Y_{2}^{3}-Y_{3}^{3})(Y_{1}^{3}+Y_{2}^{3}+Y_{3}^{3}-3Y_{1}Y_{2}Y_{3})\\\
2(2Y_{1}^{5}Y_{2}-7Y_{1}^{3}Y_{2}^{2}Y_{3}-Y_{1}^{4}Y_{3}^{2}-Y_{2}^{2}Y_{3}(Y_{2}^{3}+Y_{3}^{3})+Y_{1}^{2}(2Y_{2}^{4}+5Y_{2}Y_{3}^{3})+Y_{1}(2Y_{2}^{3}Y_{3}^{2}-Y_{3}^{5}))\\\
2(Y_{1}^{4}Y_{2}^{2}+7Y_{1}^{3}Y_{2}Y_{3}^{2}-2Y_{1}^{5}Y_{3}+Y_{2}Y_{3}^{2}(Y_{2}^{3}+Y_{3}^{3})-Y_{1}^{2}(5Y_{2}^{3}Y_{3}+2Y_{3}^{4})+Y_{1}(Y_{2}^{5}-2Y_{2}^{2}Y_{3}^{3}))\end{pmatrix}.$
(30)

We note that the results of even weight modular forms obtained here coincide
with those of our previous work [29] up to some overall constants.
Specifically the following relations are fulfilled:

$\displaystyle
Y^{(2)}_{\mathbf{2}}=\frac{3}{2}(3i+\sqrt{3})\widetilde{Y}^{(2)}_{\mathbf{2}},~{}~{}Y^{(2)}_{\mathbf{3}}=(3-3i\sqrt{3})\widetilde{Y}^{(2)}_{\mathbf{3}}\,,~{}~{}Y^{(4)}_{\mathbf{1}}=-27e^{3\pi
i/4}(i+\sqrt{3})\widetilde{Y}^{(4)}_{\mathbf{1}},$ $\displaystyle
Y^{(4)}_{\mathbf{2}}=27e^{3\pi
i/4}(i+\sqrt{3})\widetilde{Y}^{(4)}_{\mathbf{2}},~{}~{}Y^{(4)}_{\mathbf{3}}=18\sqrt{3}e^{\pi
i/4}\widetilde{Y}^{(4)}_{\mathbf{3}},~{}~{}Y^{(4)}_{\mathbf{3^{\prime}}}=18\sqrt{3}e^{\pi
i/4}\widetilde{Y}^{(4)}_{\mathbf{3^{\prime}}}\,,$ $\displaystyle
Y^{(6)}_{\mathbf{1}}=-162\sqrt{6}(1+i)\widetilde{Y}^{(6)}_{\mathbf{1}},~{}~{}Y^{(6)}_{\mathbf{1^{\prime}}}=-81\sqrt{6}(1+i)\widetilde{Y}^{(6)}_{\mathbf{1^{\prime}}},~{}~{}Y^{(6)}_{\mathbf{2}}=81\sqrt{6}(1+i)\widetilde{Y}^{(6)}_{\mathbf{2}},$
$\displaystyle Y^{(6)}_{\mathbf{3},I}=-162e^{7\pi
i/12}\widetilde{Y}^{(6)}_{\mathbf{3},II},~{}~{}Y^{(6)}_{\mathbf{3},II}=-324e^{7\pi
i/12}\widetilde{Y}^{(6)}_{\mathbf{3},I},~{}~{}Y^{(6)}_{\mathbf{3}^{\prime}}=-162e^{7\pi
i/12}\widetilde{Y}^{(6)}_{\mathbf{3}^{\prime}}\,,$ (31)

where the modular forms in [29] are denoted with a symbol
“$~{}\widetilde{}~{}$ ”. The modular forms of level 4 up to weight 6 are
summarized in table 1. We notice that all the odd weight modular forms are in
hatted irreducible representations of $S^{\prime}_{4}$ while the even weight
modular forms are in unhatted irreducible representations of $S^{\prime}_{4}$.
Note that generator $R$ is represented by unit matrix and the $S^{\prime}_{4}$
group can not be distinguished from $S_{4}$ in unhatted irreducible
representations. For notation simplicity of model construction in the
following, we denote the components of modular multiplets as follows,

$\displaystyle Y^{(2)}_{\mathbf{2}}\equiv\begin{pmatrix}Y^{(2)}_{1}\\\
Y^{(2)}_{2}\end{pmatrix},~{}Y^{(2)}_{\mathbf{3}}\equiv\begin{pmatrix}Y^{(2)}_{3}\\\
Y^{(2)}_{4}\\\
Y^{(2)}_{5}\end{pmatrix},~{}Y^{(3)}_{\mathbf{\hat{1}^{\prime}}}\equiv
Y^{(3)}_{1},~{}Y^{(3)}_{\mathbf{\hat{3}}}\equiv\begin{pmatrix}Y^{(3)}_{2}\\\
Y^{(3)}_{3}\\\
Y^{(3)}_{4}\end{pmatrix},~{}Y^{(3)}_{\mathbf{\hat{3}^{\prime}}}\equiv\begin{pmatrix}Y^{(3)}_{5}\\\
Y^{(3)}_{6}\\\ Y^{(3)}_{7}\end{pmatrix}\,,$ $\displaystyle
Y^{(4)}_{\mathbf{1}}\equiv
Y^{(4)}_{1},~{}~{}Y^{(4)}_{\mathbf{2}}\equiv\begin{pmatrix}Y^{(4)}_{2}\\\
Y^{(4)}_{3}\end{pmatrix},~{}~{}Y^{(4)}_{\mathbf{3}}\equiv\begin{pmatrix}Y^{(4)}_{4}\\\
Y^{(4)}_{5}\\\
Y^{(4)}_{6}\end{pmatrix},~{}~{}Y^{(4)}_{\mathbf{3^{\prime}}}\equiv\begin{pmatrix}Y^{(4)}_{7}\\\
Y^{(4)}_{8}\\\ Y^{(4)}_{9}\end{pmatrix}\,,$ $\displaystyle
Y^{(5)}_{\mathbf{\hat{2}}}\equiv\begin{pmatrix}Y^{(5)}_{1}\\\
Y^{(4)}_{2}\end{pmatrix},~{}~{}Y^{(5)}_{\mathbf{\hat{3}}}\equiv\begin{pmatrix}Y^{(5)}_{3}\\\
Y^{(5)}_{4}\\\
Y^{(5)}_{5}\end{pmatrix},~{}~{}Y^{(5)}_{\mathbf{\hat{3}^{\prime}},I}\equiv\begin{pmatrix}Y^{(5)}_{6}\\\
Y^{(5)}_{7}\\\
Y^{(5)}_{8}\end{pmatrix},~{}~{}Y^{(5)}_{\mathbf{\hat{3}^{\prime}},II}\equiv\begin{pmatrix}Y^{(5)}_{9}\\\
Y^{(5)}_{10}\\\ Y^{(5)}_{11}\end{pmatrix}\,,$ $\displaystyle
Y^{(6)}_{\mathbf{1^{\prime}}}\equiv Y^{(6)}_{1},~{}Y^{(6)}_{\mathbf{1}}\equiv
Y^{(6)}_{2},~{}Y^{(6)}_{\mathbf{2}}\equiv\begin{pmatrix}Y^{(6)}_{3}\\\
Y^{(6)}_{4}\end{pmatrix},~{}Y^{(6)}_{\mathbf{3},I}\equiv\begin{pmatrix}Y^{(6)}_{5}\\\
Y^{(6)}_{6}\\\
Y^{(6)}_{7}\end{pmatrix},~{}Y^{(6)}_{\mathbf{3},II}\equiv\begin{pmatrix}Y^{(6)}_{8}\\\
Y^{(6)}_{9}\\\ Y^{(6)}_{10}\end{pmatrix},$ $\displaystyle
Y^{(6)}_{\mathbf{3^{\prime}}}\equiv\begin{pmatrix}Y^{(6)}_{11}\\\
Y^{(6)}_{12}\\\ Y^{(6)}_{13}\end{pmatrix}\,,$ (32)

Modular weight $k$ | Modular forms $Y^{(k)}_{\mathbf{r}}$
---|---
$k=1$ | $Y^{(1)}_{\mathbf{\hat{3^{\prime}}}}$
$k=2$ | $Y^{(2)}_{\mathbf{2}},~{}Y^{(2)}_{\mathbf{3}}$
$k=3$ | $Y^{(3)}_{\mathbf{\hat{1^{\prime}}}},~{}Y^{(3)}_{\mathbf{\hat{3}}},~{}Y^{(3)}_{\mathbf{\hat{3}^{\prime}}}$
$k=4$ | $Y^{(4)}_{\mathbf{1}},~{}Y^{(4)}_{\mathbf{2}},~{}Y^{(4)}_{\mathbf{3}},~{}Y^{(4)}_{\mathbf{3}^{\prime}}$
$k=5$ | $Y^{(5)}_{\mathbf{\hat{2}}},~{}Y^{(5)}_{\mathbf{\hat{3}}},~{}Y^{(5)}_{\mathbf{\hat{3}^{\prime}},I},Y^{(5)}_{\mathbf{\hat{3}^{\prime}},II}$
$k=6$ | $Y^{(6)}_{\mathbf{1^{\prime}}},~{}Y^{(6)}_{\mathbf{1}},~{}Y^{(6)}_{\mathbf{2}},~{}Y^{(6)}_{\mathbf{3},I},~{}Y^{(6)}_{\mathbf{3},II},~{}Y^{(6)}_{\mathbf{3}^{\prime}}$
Table 1: Summary of modular forms of level $N=4$ up to weight 6, the subscript
$\mathbf{r}$ denote the transformation property under homogeneous finite
modular group $S^{\prime}_{4}$.

##  4 Generalized CP consistent with $S^{\prime}_{4}$ modular symmetry

In order to consistently implement CP symmetry in the context of modular
symmetry, the complex modulus $\tau$ should transform under the action CP as
[47, 48, 49, 50, 51]

$\tau\stackrel{{\scriptstyle\mathcal{CP}}}{{\longmapsto}}-\tau^{*}\,,$ (33)

up to modular transformations. A generic chiral superfield $\Phi(x)$ assigned
to an irreducible representation $\mathbf{r}$ of the finite modular group
$\Gamma^{\prime}_{N}$ transforms under the action of $\Gamma^{\prime}_{N}$ as

$\Phi(x)\stackrel{{\scriptstyle\gamma}}{{\longmapsto}}(c\tau+d)^{-k}\rho_{\mathbf{r}}(\gamma)\Phi(x),~{}~{}~{}\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma\,,$ (34)

where $-k$ is the modular weight of $\Phi$. We impose CP symmetry on the
modular invariant theory. A generalized CP transformation acts on the chiral
superfield $\Phi(x)$ as

$\Phi(x)\stackrel{{\scriptstyle\mathcal{CP}}}{{\longmapsto}}X_{\mathbf{r}}\overline{\Phi}(\mathcal{P}x)\,,$
(35)

where $\mathcal{P}x=(t,-\vec{x})$, and a bar denotes the hermitian conjugate
superfield, $X_{\mathbf{r}}$ is not necessarily diagonal and it in general
acts in a non-trivial way on the flavor space. As has been shown in [47],
constraints on the choice of $X_{\mathbf{r}}$ arise from the requirement that
the subsequent application of the CP transformation, the modular symmetry and
the inverse CP transformation should be represented by another element of the
modular symmetry group, i.e.

$X_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(\gamma)X^{-1}_{\mathbf{r}}=\rho_{\mathbf{r}}(u(\gamma))\,,$
(36)

where $u(\gamma)$ is an outer automorphism of the modular group,

$\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\longmapsto
u(\gamma)=\begin{pmatrix}a&-b\\\ -c&d\end{pmatrix}\,.$ (37)

Eq. (36) is the so-called consistency condition which CP and modular
symmetries have to obey in order to give a consistent definition of
generalized CP transformations in setting with modular symmetry. It is notable
that the consistency condition Eq. (36) should be satisfied for all
irreducible representations of the finite modular group $\Gamma^{\prime}_{N}$.
We see that the CP transformation $X_{\mathbf{r}}$ maps the modular group
element $\gamma$ onto another element $u(\gamma)$ and the group structure of
the modular symmetry is preserved, i.e.
$u(\gamma_{1}\gamma_{2})=u(\gamma_{1})u(\gamma_{2})$. Hence it is sufficient
to impose Eq. (36) on the generators $S$ and $T$,

$X_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(S)X^{-1}_{\mathbf{r}}=\rho^{\dagger}_{\mathbf{r}}(S),\qquad
X_{\mathbf{r}}\rho^{*}_{\mathbf{r}}(T)X^{-1}_{\mathbf{r}}=\rho^{\dagger}_{\mathbf{r}}(T)\,,$
(38)

where the identities $u(S)=S^{-1}$ and $u(T)=T^{-1}$ are used. The consistency
condition in Eq. (36) determines the CP transformation $X_{\mathbf{r}}$ up to
an overall phase for a given irreducible representation $\mathbf{r}$. As
regards the double covering group $S^{\prime}_{4}$ with the basis given in
table 7, solving the consistency conditions of Eq. (38), we find that the
generalized CP transformation $X_{\mathbf{r}}$ coincides with the
representation matrix of $S$,

$X_{\mathbf{r}}=\rho_{\mathbf{r}}(S)\,,$ (39)

which is a combination of the modular symmetry transformation $S$ and the
canonical CP transformation. Furthermore, we have checked that the modular
forms $Y^{(k)}_{\mathbf{r}}(\tau)$ of section LABEL:sec:ModF transforms in the
same way as $\Phi(x)$ under CP:

$Y^{(k)}_{\mathbf{r}}(\tau)\stackrel{{\scriptstyle\mathcal{CP}}}{{\longmapsto}}Y^{(k)}_{\mathbf{r}}(-\tau^{*})=X_{\mathbf{r}}[Y^{(k)}_{\mathbf{r}}(\tau)]^{*},~{}~{}\text{with}~{}~{}X_{\mathbf{r}}=\rho_{\mathbf{r}}(S)\,.$
(40)

Since the theory is required to be modular invariant, the CP transformation in
Eq. (39) is essentially equivalent to the canonical CP transformation. As
shown in Appendix A, all the CG coefficients in our working basis are real,
therefore the generalized CP symmetry would constrain all the couplings in the
Lagrangian to be real.

##  5 Lepton models based on $S^{\prime}_{4}$ modular symmetry

We work in the framework of the modular invariant supersymmetric theory [58,
59, 10]. In the setting of $N=1$ global supersymmetry, the action can be
generally written as

$\mathcal{S}=\int d^{4}xd^{2}\theta
d^{2}\bar{\theta}\,\mathcal{K}(\Phi_{I},\bar{\Phi}_{I},\tau,\bar{\tau})+\left[\int
d^{4}xd^{2}\theta\mathcal{W}(\Phi_{I},\tau)+h.c.\right]\,,$ (41)

where $K(\Phi_{I},\bar{\Phi}_{I},\tau,\bar{\tau})$ is Kähler potential, it is
real gauge invariant function of the chiral superfields $\Phi$ and their
hermitian conjugates $\bar{\Phi}$. $W(\Phi_{I},\tau)$ refers to the
superpotential, and it is a holomorphic gauge invariant function of the chiral
superfields $\Phi$. The whole action $\mathcal{S}$ should be modular
invariant. The transformation properties of $\Phi_{I}$ are specified by its
modular weight $-k_{I}$ and the representation $\mathbf{r}_{I}$ under
$\Gamma^{\prime}_{N}$,

$\tau\rightarrow\gamma\tau=\frac{a\tau+b}{c\tau+d},~{}~{}~{}~{}~{}\Phi_{I}\rightarrow(c\tau+d)^{-k_{I}}\rho_{\mathbf{r}_{I}}(\gamma)\Phi_{I}\,.$
(42)

Following Ref. [10], we take the Kähler potential to be the minimal form,

$\mathcal{K}(\Phi_{I},\bar{\Phi}_{I},\tau,\bar{\tau})=-h\Lambda^{2}\log(-i\tau+i\bar{\tau})+\sum_{I}(-i\tau+i\bar{\tau})^{-k_{I}}|\Phi_{I}|^{2}\,,$
(43)

where $h$ is a positive constant. After the modulus $\tau$ gets a vacuum
expectation, this Kähler potential gives the kinetic terms for the scalar
components of the supermultiplet $\Phi_{I}$ and the modulus field $\tau$. The
Kähler potential is strongly constrained in some models based on string theory
[60, 61, 62], and the above minimal Kähler potential as the leading order
contribution could possibly be achieved. The superpotential $\mathcal{W}$ can
be expanded into power series of supermultiplets $\Phi_{I}$

$\mathcal{W}(\Phi_{I},\tau)=\sum_{n}Y_{I_{1}...I_{n}}(\tau)\Phi_{I_{1}}...\Phi_{I_{n}}\,.$
(44)

Modular invariance requires the function $Y_{I_{1}...I_{n}}(\tau)$ should be a
modular form of weight $k_{Y}$ of level $N$ and in the representation
$\mathbf{r}_{Y}$ of $\Gamma^{\prime}_{N}$:

$Y(\tau)\to
Y(\gamma\tau)=(c\tau+d)^{k_{Y}}\rho_{\mathbf{r}_{Y}}(\gamma)Y(\tau)\,,$ (45)

where $k_{Y}$ and $\mathbf{r}_{Y}$ should satisfy the conditions

$k_{Y}=k_{1}+...+k_{n},~{}~{}~{}\rho_{\mathbf{r}_{Y}}\otimes\rho_{\mathbf{r}_{I_{1}}}\otimes\ldots\otimes\rho_{\mathbf{r}_{I_{n}}}\ni\mathbf{1}\,.$
(46)

In the present work, we shall study the modular symmetry group of level $N=4$,
and a comprehensive analysis of lepton models with $S^{\prime}_{4}$ modular
symmetry is performed in the following. Without mentioning explicitly, we
shall consider the cases involving weight 1, weight 2 and weight 3 modular
forms. The cases with higher weight modular forms can be discussed in the same
fashion.

###  5.1 Charged lepton sector

The left-handed lepton doublet fields are assigned to transforms as triplet
$\mathbf{3}$, $\mathbf{3^{\prime}}$,
$\mathbf{\hat{3}},\,\mathbf{\hat{3}^{\prime}}$ of $S^{\prime}_{4}$, and we
assume the right-handed charged leptons transforms as singlet $\mathbf{1}$,
$\mathbf{1^{\prime}}$, $\mathbf{\hat{1}}$ or $\mathbf{\hat{1}^{\prime}}$ of
$\Gamma^{\prime}_{4}$. Thus the most general superpotential for the charged
lepton masses can be written as :

$\mathcal{W}_{e}\,=\alpha(E^{c}_{1}Lf_{E_{1}}(Y))_{\mathbf{1}}H_{d}+\beta(E^{c}_{2}Lf_{E_{2}}(Y))_{\mathbf{1}}H_{d}+\gamma(E^{c}_{3}Lf_{E_{3}}(Y))_{\mathbf{1}}H_{d}\,.$
(47)

The modular forms $f_{E_{1}}(Y)$, $f_{E_{2}}(Y)$ and $f_{E_{3}}(Y)$ should
transform as three dimensional irreducible representations under
$S^{\prime}_{4}$, and their explicit forms depend on the weight and
representation assignments for $L$ and $E^{c}_{1,2,3}$. In order to charged
lepton mass matrix with rank less than three otherwise at least one charged
lepton would be massless, $f_{E_{1}}(Y)$, $f_{E_{2}}(Y)$ and $f_{E_{3}}(Y)$
must be different modular multiplets. For illustration, we consider modular
forms of weight less than four, consequently $f_{E_{1}}(Y)$, $f_{E_{2}}(Y)$,
$f_{E_{3}}(Y)$ can only be $Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}$,
$Y^{(2)}_{\mathbf{3}}$, $Y^{(3)}_{\mathbf{\hat{3}}}$ and
$Y^{(3)}_{\mathbf{\hat{3}^{\prime}}}$. It is remarkable that the CG
coefficients for the contraction triplet$\otimes$ triplet$\rightarrow$singlet
are all the same in our basis. As a consequence, there are only four different
structures of charged lepton mass matrix if the weights of the relevant
modular forms are less than four.

  * (i)

$f_{E_{1}}(Y)=Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}$,
$f_{E_{2}}(Y)=Y^{(2)}_{\mathbf{3}}$, $f_{E_{3}}(Y)=Y^{(3)}_{\mathbf{\hat{3}}}$
In this case, there are four different representation assignments which give
rise to the same charged lepton mass matrix:

$\begin{cases}\rho_{L}=\mathbf{3},~{}\rho_{E^{c}_{1}}=\mathbf{\hat{1}},~{}\rho_{E^{c}_{2}}=\mathbf{1},~{}\rho_{E^{c}_{3}}=\mathbf{\hat{1}^{\prime}}\,,\\\
\rho_{L}=\mathbf{3^{\prime}},~{}\rho_{E^{c}_{1}}=\mathbf{\hat{1}^{\prime}},~{}\rho_{E^{c}_{2}}=\mathbf{1^{\prime}},~{}\rho_{E^{c}_{3}}=\mathbf{\hat{1}}\,,\\\
\rho_{L}=\mathbf{\hat{3}},~{}\rho_{E^{c}_{1}}=\mathbf{1},~{}\rho_{E^{c}_{2}}=\mathbf{\hat{1}^{\prime}},~{}\rho_{E^{c}_{3}}=\mathbf{1^{\prime}}\,,\\\
\rho_{L}=\mathbf{\hat{3}^{\prime}},~{}\rho_{E^{c}_{1}}=\mathbf{1^{\prime}},~{}\rho_{E^{c}_{2}}=\mathbf{\hat{1}},~{}\rho_{E^{c}_{3}}=\mathbf{1}\,.\end{cases}$
(48)

The superpotential for the charged lepton masses are given by,

$\displaystyle\mathcal{W}_{e}=\alpha(E^{c}_{1}LY^{(1)}_{\mathbf{\hat{3}^{\prime}}})_{\mathbf{1}}H_{d}+\beta(E^{c}_{2}LY^{(2)}_{\mathbf{3}})_{\mathbf{1}}H_{d}+\gamma(E^{c}_{3}LY^{(3)}_{\mathbf{\hat{3}}})_{\mathbf{1}}H_{d}$
$\displaystyle~{}~{}~{}~{}=\alpha
E^{c}_{1}(L_{1}Y_{1}+L_{3}Y_{2}+L_{2}Y_{3})H_{d}+\beta
E^{c}_{2}(L_{1}Y^{(2)}_{3}+L_{3}Y^{(2)}_{4}+L_{2}Y^{(2)}_{5})H_{d}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\gamma
E^{c}_{3}(L_{1}Y^{(3)}_{2}+L_{3}Y^{(3)}_{3}+L_{2}Y^{(3)}_{4})H_{d}\,.$ (49)

The condition of modular weight cancellation requires

$k_{E_{1}}=k_{E_{2}}-1=k_{E_{3}}-2=1-k_{L}\,.$ (50)

  * (ii)

$f_{E_{1}}(Y)=Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}$,
$f_{E_{2}}(Y)=Y^{(2)}_{\mathbf{3}}$,
$f_{E_{3}}(Y)=Y^{(3)}_{\mathbf{\hat{3}^{\prime}}}$
There are also four representation assignments for the lepton fields,

$\begin{cases}\rho_{L}=\mathbf{3},~{}\rho_{E^{c}_{1}}=\mathbf{\hat{1}},~{}\rho_{E^{c}_{2}}=\mathbf{1},~{}\rho_{E^{c}_{3}}=\mathbf{\hat{1}}\,,\\\
\rho_{L}=\mathbf{3^{\prime}},~{}\rho_{E^{c}_{1}}=\mathbf{\hat{1}^{\prime}},~{}\rho_{E^{c}_{2}}=\mathbf{1^{\prime}},~{}\rho_{E^{c}_{3}}=\mathbf{\hat{1}^{\prime}}\,,\\\
\rho_{L}=\mathbf{\hat{3}},~{}\rho_{E^{c}_{1}}=\mathbf{1},~{}\rho_{E^{c}_{2}}=\mathbf{\hat{1}^{\prime}},~{}\rho_{E^{c}_{3}}=\mathbf{1}\,,\\\
\rho_{L}=\mathbf{\hat{3}^{\prime}},~{}\rho_{E^{c}_{1}}=\mathbf{1^{\prime}},~{}\rho_{E^{c}_{2}}=\mathbf{\hat{1}},~{}\rho_{E^{c}_{3}}=\mathbf{1^{\prime}}\,.\end{cases}$
(51)

The superpotential for the charged lepton masses takes the following form,

$\displaystyle\mathcal{W}_{e}=\alpha(E^{c}_{1}LY^{(1)}_{\mathbf{\hat{3}^{\prime}}})_{\mathbf{1}}H_{d}+\beta(E^{c}_{2}LY^{(2)}_{\mathbf{3}})_{\mathbf{1}}H_{d}+\gamma(E^{c}_{3}LY^{(3)}_{\mathbf{\hat{3}^{\prime}}})_{\mathbf{1}}H_{d}$
$\displaystyle~{}~{}~{}~{}=\alpha
E^{c}_{1}(L_{1}Y_{1}+L_{3}Y_{2}+L_{2}Y_{3})H_{d}+\beta
E^{c}_{2}(L_{1}Y^{(2)}_{3}+L_{3}Y^{(2)}_{4}+L_{2}Y^{(2)}_{5})H_{d}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\gamma
E^{c}_{3}(L_{1}Y^{(3)}_{5}+L_{3}Y^{(3)}_{6}+L_{2}Y^{(3)}_{7})H_{d}\,.$ (52)

Modular invariance imposes the following constraints on modular weights,

$k_{E_{1}}=k_{E_{2}}-1=k_{E_{3}}-2=1-k_{L}\,.$ (53)

  * (iii)

$f_{E_{1}}(Y)=Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}$,
$f_{E_{2}}(Y)=Y^{(3)}_{\mathbf{\hat{3}^{\prime}}}$,
$f_{E_{3}}(Y)=Y^{(3)}_{\mathbf{\hat{3}}}$
Similar to previous cases, the lepton fields can be assigned to

$\begin{cases}\rho_{L}=\mathbf{3},~{}\rho_{E^{c}_{1}}=\mathbf{\hat{1}},~{}\rho_{E^{c}_{2}}=\mathbf{\hat{1}},~{}\rho_{E^{c}_{3}}=\mathbf{\hat{1}^{\prime}}\,,\\\
\rho_{L}=\mathbf{3^{\prime}},~{}\rho_{E^{c}_{1}}=\mathbf{\hat{1}^{\prime}},~{}\rho_{E^{c}_{2}}=\mathbf{\hat{1}^{\prime}},~{}\rho_{E^{c}_{3}}=\mathbf{\hat{1}}\,,\\\
\rho_{L}=\mathbf{\hat{3}},~{}\rho_{E^{c}_{1}}=\mathbf{1},~{}\rho_{E^{c}_{2}}=\mathbf{1},~{}\rho_{E^{c}_{3}}=\mathbf{1^{\prime}}\,,\\\
\rho_{L}=\mathbf{\hat{3}^{\prime}},~{}\rho_{E^{c}_{1}}=\mathbf{1^{\prime}},~{}\rho_{E^{c}_{2}}=\mathbf{1^{\prime}},~{}\rho_{E^{c}_{3}}=\mathbf{1}\,.\end{cases}$
(54)

The superpotential for the charged lepton masses is of the form,

$\displaystyle\mathcal{W}_{e}=\alpha(E^{c}_{1}LY^{(1)}_{\mathbf{\hat{3}^{\prime}}})_{\mathbf{1}}H_{d}+\beta(E^{c}_{2}LY^{(3)}_{\mathbf{\hat{3}^{\prime}}})_{\mathbf{1}}H_{d}+\gamma(E^{c}_{3}LY^{(3)}_{\mathbf{\hat{3}}})_{\mathbf{1}}H_{d}$
$\displaystyle~{}~{}~{}~{}=\alpha
E^{c}_{1}(L_{1}Y_{1}+L_{3}Y_{2}+L_{2}Y_{3})H_{d}+\beta
E^{c}_{2}(L_{1}Y^{(3)}_{5}+L_{3}Y^{(3)}_{6}+L_{2}Y^{(3)}_{7})H_{d}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\gamma
E^{c}_{3}(L_{1}Y^{(3)}_{2}+L_{3}Y^{(3)}_{3}+L_{2}Y^{(3)}_{4})H_{d}\,,$ (55)

with the modular weights

$k_{E_{1}}=k_{E_{2}}-2=k_{E_{3}}-2=1-k_{L}\,.$ (56)

  * (iv)

$f_{E_{1}}(Y)=Y^{(3)}_{\mathbf{\hat{3}^{\prime}}}$,
$f_{E_{2}}(Y)=Y^{(2)}_{\mathbf{3}}$, $f_{E_{3}}(Y)=Y^{(3)}_{\mathbf{\hat{3}}}$
Likewise, we have four different representation assignments which gives the
same superpotential $\mathcal{W}_{e}$ as well as the the same charged lepton
mass matrix,

$\begin{cases}\rho_{L}=\mathbf{3},~{}\rho_{E^{c}_{1}}=\mathbf{\hat{1}},~{}\rho_{E^{c}_{2}}=\mathbf{1},~{}\rho_{E^{c}_{3}}=\mathbf{\hat{1}^{\prime}}\,,\\\
\rho_{L}=\mathbf{3^{\prime}},~{}\rho_{E^{c}_{1}}=\mathbf{\hat{1}^{\prime}},~{}\rho_{E^{c}_{2}}=\mathbf{1^{\prime}},~{}\rho_{E^{c}_{3}}=\mathbf{\hat{1}}\,,\\\
\rho_{L}=\mathbf{\hat{3}},~{}\rho_{E^{c}_{1}}=\mathbf{1},~{}\rho_{E^{c}_{2}}=\mathbf{\hat{1}^{\prime}},~{}\rho_{E^{c}_{3}}=\mathbf{1^{\prime}}\,,\\\
\rho_{L}=\mathbf{\hat{3}^{\prime}},~{}\rho_{E^{c}_{1}}=\mathbf{1^{\prime}},~{}\rho_{E^{c}_{2}}=\mathbf{\hat{1}},~{}\rho_{E^{c}_{3}}=\mathbf{1}\,.\end{cases}$
(57)

The superpotential for the charged lepton masses reads as,

$\displaystyle\mathcal{W}_{e}=\alpha(E^{c}_{1}LY^{(3)}_{\mathbf{\hat{3}^{\prime}}})_{\mathbf{1}}H_{d}+\beta(E^{c}_{2}LY^{(2)}_{\mathbf{3}})_{\mathbf{1}}H_{d}+\gamma(E^{c}_{3}LY^{(3)}_{\mathbf{\hat{3}}})_{\mathbf{1}}H_{d}$
$\displaystyle~{}~{}~{}~{}=\alpha
E^{c}_{1}(L_{1}Y^{(3)}_{5}+L_{3}Y^{(3)}_{6}+L_{2}Y^{(3)}_{7})H_{d}+\beta
E^{c}_{2}(L_{1}Y^{(2)}_{3}+L_{3}Y^{(2)}_{4}+L_{2}Y^{(2)}_{5})H_{d}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\gamma
E^{c}_{3}(L_{1}Y^{(3)}_{2}+L_{3}Y^{(3)}_{3}+L_{2}Y^{(3)}_{4})H_{d}\,.$ (58)

The modular weights $k_{L}$ and $k_{E_{1},E_{2},E_{3}}$ satisfy the
constraints

$k_{E_{1}}-1=k_{E_{2}}=k_{E_{3}}-1=2-k_{L}\,.$ (59)

It is straightforward to read out the predicted charged lepton mass matrix for
each case discussed above, and results are summarized in table 2. We can
exchange the assignments for the right-handed charged lepton fields
$E^{c}_{1,2,3}$, accordingly the rows of the charged lepton mass matrix would
be permutated. However, the hermitian combination $M^{\dagger}_{e}M_{e}$ are
left invariant such that the predictions for charged lepton mass and the
unitary rotation $U_{e}$ are unchanged, where $U_{e}$ diagonalize the charged
lepton mass matrix via
$U^{\dagger}_{e}M^{\dagger}_{e}M_{e}U_{e}=\text{diag}(m^{2}_{e},m^{2}_{\mu},m^{2}_{\tau})$.

Cases |  rep assignments |  weights | Charged lepton mass matrix
---|---|---|---
$(\rho_{L},\rho_{E^{c}_{1}},\rho_{E^{c}_{2}},\rho_{E^{c}_{3}})$ |  $k_{L}+k_{E^{c}_{1,2,3}}$
$C_{1}$ | $\begin{cases}(\mathbf{3},\mathbf{\hat{1}},\mathbf{1},\mathbf{\hat{1}^{\prime}})\\\ (\mathbf{3^{\prime}},\mathbf{\hat{1}^{\prime}},\mathbf{1^{\prime}},\mathbf{\hat{1}})\\\ (\mathbf{\hat{3}},\mathbf{1},\mathbf{\hat{1}^{\prime}},\mathbf{1^{\prime}})\\\ (\mathbf{\hat{3}^{\prime}},\mathbf{1^{\prime}},\mathbf{\hat{1}},\mathbf{1})\end{cases}$ | $(1,2,3)$ |  $M_{e}=\begin{pmatrix}\alpha\,Y_{1}~{}&~{}\alpha\,Y_{3}~{}&~{}\alpha\,Y_{2}\\\ \beta Y^{(2)}_{3}~{}&~{}\beta Y^{(2)}_{5}~{}&~{}\beta Y^{(2)}_{4}\\\ \gamma Y^{(3)}_{2}~{}&~{}\gamma Y^{(3)}_{4}~{}&~{}\gamma Y^{(3)}_{3}\\\ \end{pmatrix}v_{d}$
$C_{2}$ | $\begin{cases}(\mathbf{3},\mathbf{\hat{1}},\mathbf{1},\mathbf{\hat{1}})\\\ (\mathbf{3^{\prime}},\mathbf{\hat{1}^{\prime}},\mathbf{1^{\prime}},\mathbf{\hat{1}^{\prime}})\\\ (\mathbf{\hat{3}},\mathbf{1},\mathbf{\hat{1}^{\prime}},\mathbf{1})\\\ (\mathbf{\hat{3}^{\prime}},\mathbf{1^{\prime}},\mathbf{\hat{1}},\mathbf{1^{\prime}})\end{cases}$ | $(1,2,3)$ |  $M_{e}=\begin{pmatrix}\alpha\,Y_{1}~{}&~{}\alpha\,Y_{3}~{}&~{}\alpha\,Y_{2}\\\ \beta Y^{(2)}_{3}~{}&~{}\beta Y^{(2)}_{5}~{}&~{}\beta Y^{(2)}_{4}\\\ \gamma Y^{(3)}_{5}~{}&~{}\gamma Y^{(3)}_{7}~{}&~{}\gamma Y^{(3)}_{6}\\\ \end{pmatrix}v_{d}$
$C_{3}$ | $\begin{cases}(\mathbf{3},\mathbf{\hat{1}},\mathbf{\hat{1}},\mathbf{\hat{1}^{\prime}})\\\ (\mathbf{3^{\prime}},\mathbf{\hat{1}^{\prime}},\mathbf{\hat{1}^{\prime}},\mathbf{\hat{1}})\\\ (\mathbf{\hat{3}},\mathbf{1},\mathbf{1},\mathbf{1^{\prime}})\\\ (\mathbf{\hat{3}^{\prime}},\mathbf{1^{\prime}},\mathbf{1^{\prime}},\mathbf{1})\end{cases}$ | $(1,3,3)$ |  $M_{e}=\begin{pmatrix}\alpha\,Y_{1}~{}&~{}\alpha\,Y_{3}~{}&~{}\alpha\,Y_{2}\\\ \beta Y^{(3)}_{5}~{}&~{}\beta Y^{(3)}_{7}~{}&~{}\beta Y^{(3)}_{6}\\\ \gamma Y^{(3)}_{2}~{}&~{}\gamma Y^{(3)}_{4}~{}&~{}\gamma Y^{(3)}_{3}\\\ \end{pmatrix}v_{d}$
$C_{4}$ | $\begin{cases}(\mathbf{3},\mathbf{\hat{1}},\mathbf{1},\mathbf{\hat{1}^{\prime}})\\\ (\mathbf{3^{\prime}},\mathbf{\hat{1}^{\prime}},\mathbf{1^{\prime}},\mathbf{\hat{1}})\\\ (\mathbf{\hat{3}},\mathbf{1},\mathbf{\hat{1}^{\prime}},\mathbf{1^{\prime}})\\\ (\mathbf{\hat{3}^{\prime}},\mathbf{1^{\prime}},\mathbf{\hat{1}},\mathbf{1})\end{cases}$ | $(3,2,3)$ |  $M_{e}=\begin{pmatrix}\alpha\,Y^{(3)}_{5}~{}&~{}\alpha\,Y^{(3)}_{7}~{}&~{}\alpha\,Y^{(3)}_{6}\\\ \beta Y^{(2)}_{3}~{}&~{}\beta Y^{(2)}_{5}~{}&~{}\beta Y^{(2)}_{4}\\\ \gamma Y^{(3)}_{2}~{}&~{}\gamma Y^{(3)}_{4}~{}&~{}\gamma Y^{(3)}_{3}\\\ \end{pmatrix}v_{d}$

Table 2:  The modular $S^{\prime}_{4}$ models in charged lepton sector for
different weight and representation assignments, where the charged lepton mass
matrix $M_{e}$ is given in the convention $E^{c}\,M_{e}\,L$ with
$v_{d}=\langle H^{0}_{d}\rangle$.

###  5.2 Neutrino sector

In neutrino sector, we assume that neutrinos are Majorana particles, and we
consider two scenarios that the neutrino masses are described by the effective
Weinberg operator or arise from the type-I seesaw mechanism. The left-handed
lepton doublets would be assigned to transform as triplet under
$S^{\prime}_{4}$. Guided by the principle of minimality and simplicity, we
shall consider modular multiplets with weight less than four similar to the
charged lepton sector. For the cases involving higher weight modular forms,
more modular invariant operators accompanied by free coupling constants would
be allowed, and the predictive power of the models would be reduced.

####  5.2.1 Weinberg operator

From the $S^{\prime}_{4}$ Kronecker products
$\mathbf{3}\otimes\mathbf{3}=\mathbf{3^{\prime}}\otimes\mathbf{3^{\prime}}=\mathbf{1}\oplus\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3^{\prime}}$,
$\mathbf{\hat{3}}\otimes\mathbf{\hat{3}}=\mathbf{\hat{3}^{\prime}}\otimes\mathbf{\hat{3}^{\prime}}=\mathbf{1^{\prime}}\oplus\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3^{\prime}}$,
we know that the operator $LLH_{u}H_{u}$ can not couple with odd weight
modular forms such as $Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}$,
$Y^{(3)}_{\mathbf{\hat{1^{\prime}}}}$, $Y^{(3)}_{\mathbf{\hat{3}}}$,
$Y^{(3)}_{\mathbf{\hat{3}^{\prime}}}$ to form a $S^{\prime}_{4}$ singlet. At
the lowest order, the weight $2$ modular multiplets $Y^{(2)}_{\mathbf{2}}$,
$Y^{(2)}_{\mathbf{3}}$ enter into the Weinberg operator, and the
superpotential for neutrino masses are as follows.

  * •

$\rho_{L}=\mathbf{3}$ or $\mathbf{3^{\prime}}$

$\displaystyle\mathcal{W}_{\nu}=\frac{g_{1}}{\Lambda}((LL)_{\mathbf{3}}Y^{(2)}_{\mathbf{3}})_{\mathbf{1}}H_{u}H_{u}+\frac{g_{2}}{\Lambda}((LL)_{\mathbf{2}}Y^{(2)}_{2})_{\mathbf{1}}H_{u}H_{u}$
$\displaystyle~{}~{}~{}~{}=\big{[}g_{1}(2L_{1}L_{2}+L^{2}_{3})Y^{(2)}_{1}+g_{1}(2L_{1}L_{3}+L^{2}_{2})Y^{(2)}_{2}\big{]}\frac{H^{2}_{u}}{\Lambda}\,.$
(60)

The modular weight $k_{L}$ should be equal to 1, i.e. $k_{L}=1$. From the CG
coefficients of $\mathbf{3}\otimes\mathbf{3}\rightarrow\mathbf{3}$ and
$\mathbf{3^{\prime}}\otimes\mathbf{3^{\prime}}\rightarrow\mathbf{3}$, we know
that the contraction $(LL)_{\mathbf{3}}$ is a antisymmetric combination of
lepton fields $L$, while Lorentz invariance requires that the Majorana mass
term $((LL)_{\mathbf{3}}Y^{(2)}_{\mathbf{3}})_{\mathbf{1}}H_{u}H_{u}$ should
be symmetric with respect to $L$. As a result, the term proportional to
$g_{2}$ is vanishing, and the corresponding neutrino mass matrix $M_{\nu}$
read as

$M_{\nu}=g\begin{pmatrix}0~{}&~{}Y^{(2)}_{1}~{}&~{}Y^{(2)}_{2}\\\
Y^{(2)}_{1}~{}&~{}Y^{(2)}_{2}~{}&~{}0\\\
Y^{(2)}_{2}~{}&~{}0~{}&~{}Y^{(2)}_{1}\\\
\end{pmatrix}\dfrac{v^{2}_{u}}{\Lambda}\,.$ (61)

where $v_{u}=\langle H^{0}_{u}\rangle$

  * •

$\rho_{L}=\mathbf{\hat{3}}$ or $\mathbf{\hat{3}^{\prime}}$

$\displaystyle\mathcal{W}_{\nu}=\frac{g_{1}}{\Lambda}((LL)_{\mathbf{2}}Y^{(2)}_{\mathbf{2}})_{\mathbf{1}}H_{u}H_{u}+\frac{g_{2}}{\Lambda}((LL)_{\mathbf{3}}Y^{(2)}_{\mathbf{3}})_{\mathbf{1}}H_{u}H_{u}\,$
$\displaystyle~{}~{}~{}~{}=\big{[}g_{1}(2L_{1}L_{2}+L^{2}_{3})Y^{(2)}_{1}-g_{1}(2L_{1}L_{3}+L^{2}_{2})Y^{(2)}_{2}+g_{2}(2L^{2}_{1}-2L_{2}L_{3})Y^{(2)}_{3}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+g_{2}(2L^{2}_{2}-2L_{1}L_{3})Y^{(2)}_{4}+g_{2}(2L^{2}_{3}-2L_{1}L_{2})Y^{(2)}_{5}\big{]}\frac{H^{2}_{u}}{\Lambda}\,,$
(62)

with the weight $k_{L}=1$. The light neutrino mass matrix $M_{\nu}$ is of the
form

$M_{\nu}=\begin{pmatrix}2g_{2}Y^{(2)}_{3}~{}&g_{1}Y^{(2)}_{1}-g_{2}Y^{(2)}_{5}~{}&-g_{1}Y^{(2)}_{2}-g_{2}Y^{(2)}_{4}\\\
g_{1}Y^{(2)}_{1}-g_{2}Y^{(2)}_{5}~{}&-g_{1}Y^{(2)}_{2}+2g_{2}Y^{(2)}_{4}~{}&-g_{2}Y^{(2)}_{3}\\\
-g_{1}Y^{(2)}_{2}-g_{2}Y^{(2)}_{4}~{}&-g_{2}Y^{(2)}_{3}~{}&g_{1}Y^{(2)}_{1}+2g_{2}Y^{(2)}_{5}\\\
\end{pmatrix}\dfrac{v^{2}_{u}}{\Lambda}\,.$ (63)

####  5.2.2 Type-I seesaw mechanism

Three generations of right-handed neutrinos are introduced in the present work
and they are assumed to transforms as triplet under the $S^{\prime}_{4}$. Then
the most general superpotential in neutrino sector can be written as

$\mathcal{W}_{\nu}=g\left(N^{c}LH_{u}f_{D}(Y)\right)_{\mathbf{1}}+\Lambda\left(N^{c}N^{c}f_{M}(Y)\right)_{\mathbf{1}}\,,$
(64)

where $f_{N}(Y)$ and $f_{M}(Y)$ are modular multiplets. Similar to the case of
Weinberg operator, from the Kronecker products of two triplets, we know that
$f_{M}(Y)$ can be $\tau-$independent constant555There are no non-trivial
modular forms of weight zero. or weight 2 modular form.

  * •

$f_{M}(Y)=1$
In this case, the right-handed neutrinos can transform as $\mathbf{3}$ or
$\mathbf{3^{\prime}}$ under $S^{\prime}_{4}$ (i.e., $\rho_{N^{c}}=\mathbf{3}$
or $\mathbf{3^{\prime}}$) and their modular weight should be vanishing with
$k_{N^{c}}=0$. The heavy neutrino mass term is

$\displaystyle\mathcal{W}_{N}=\Lambda(N^{c}N^{c})_{\mathbf{1}}=\Lambda(N^{c}_{1}N^{c}_{1}+N^{c}_{2}N^{c}_{3}+N^{c}_{3}N^{c}_{2})\,,$
(65)

which leads to the following heavy neutrino mass matrix,

$M_{N}=\begin{pmatrix}1~{}&~{}0~{}&~{}0\\\ 0~{}&~{}0~{}&~{}1\\\
0~{}&~{}1~{}&~{}0\end{pmatrix}\Lambda\,.$ (66)

  * •

$f_{M}(Y)=Y^{(2)}_{\mathbf{2}},\,Y^{(2)}_{\mathbf{3}}$
If the right-handed neutrinos are assigned to transform as unhatted triplet
$\rho_{N^{c}}=\mathbf{3}$ or $\mathbf{3^{\prime}}$ with $k_{N^{c}}=1$, we have

$\displaystyle\mathcal{W}_{N}=\Lambda((N^{c}N^{c})_{\mathbf{2}}Y^{(2)}_{\mathbf{2}})_{\mathbf{1}}+\Lambda^{\prime}((N^{c}N^{c})_{\mathbf{3}}Y^{(2)}_{\mathbf{3}})_{\mathbf{1}}$
(67)
$\displaystyle~{}~{}~{}~{}=\Lambda\big{[}(2N^{c}_{1}N^{c}_{2}+N^{c}_{3}N^{c}_{3})Y^{(2)}_{1}+(2N^{c}_{1}N^{c}_{3}+N^{c}_{2}N^{c}_{2})Y^{(2)}_{2}\big{]}\,.$

Notice that the term
$[(N^{c}N^{c})_{\mathbf{3}}Y^{(2)}_{\mathbf{3}}]_{\mathbf{1}}$ is vanishing
because the contractions for both
$\mathbf{3}\otimes\mathbf{3}\rightarrow\mathbf{3}$ and
$\mathbf{3^{\prime}}\otimes\mathbf{3^{\prime}}\rightarrow\mathbf{3}$ are
antisymmetric combinations. The corresponding heavy Majorana mass matrix
$M_{N}$ can be easily read out as

$M_{N}=\begin{pmatrix}0~{}&~{}Y^{(2)}_{1}~{}&~{}Y^{(2)}_{2}\\\
Y^{(2)}_{1}~{}&~{}Y^{(2)}_{2}~{}&~{}0\\\
Y^{(2)}_{2}~{}&~{}0~{}&~{}Y^{(2)}_{1}\end{pmatrix}\Lambda\,.$ (68)

On the other hand, we can also assign the right-handed neutrinos to hatted
triplets $\rho_{N^{c}}=\mathbf{\hat{3}}$ or $\mathbf{\hat{3}^{\prime}}$ with
$k_{N^{c}}=1$. Then the superpotential $\mathcal{W}_{N}$ is

$\displaystyle\mathcal{W}_{N}$ $\displaystyle=$
$\displaystyle\Lambda[(N^{c}N^{c})_{\mathbf{2}}Y^{(2)}_{\mathbf{2}}]_{\mathbf{1}}+\Lambda^{\prime}[(N^{c}N^{c})_{\mathbf{3}}Y^{(2)}_{\mathbf{3}}]_{\mathbf{1}}$
(69) $\displaystyle=$
$\displaystyle\Lambda\big{[}(2N^{c}_{1}N^{c}_{2}+N^{c}_{3}N^{c}_{3})Y^{(2)}_{1}-(2N^{c}_{1}N^{c}_{3}+N^{c}_{2}N^{c}_{2})Y^{(2)}_{2}]+\Lambda^{\prime}[(2N^{c}_{1}N^{c}_{1}-2N^{c}_{2}N^{c}_{3})Y^{(2)}_{3}$
$\displaystyle~{}~{}+(2N^{c}_{2}N^{c}_{2}-2N^{c}_{1}N^{c}_{3})Y^{(2)}_{4}+(2N^{c}_{3}N^{c}_{3}-2N^{c}_{1}N^{c}_{2})Y^{(2)}_{5}\big{]}\,,$

which gives rise to

$M_{N}=\begin{pmatrix}2\Lambda^{\prime}Y^{(2)}_{3}~{}&\Lambda
Y^{(2)}_{1}-\Lambda^{\prime}Y^{(2)}_{5}~{}&-\Lambda
Y^{(2)}_{2}-\Lambda^{\prime}Y^{(2)}_{4}\\\ \Lambda
Y^{(2)}_{1}-\Lambda^{\prime}Y^{(2)}_{5}~{}&-\Lambda
Y^{(2)}_{2}+2\Lambda^{\prime}Y^{(2)}_{4}~{}&-\Lambda^{\prime}Y^{(2)}_{3}\\\
-\Lambda
Y^{(2)}_{2}-\Lambda^{\prime}Y^{(2)}_{4}~{}&-\Lambda^{\prime}Y^{(2)}_{3}~{}&\Lambda
Y^{(2)}_{1}+2\Lambda^{\prime}Y^{(2)}_{5}\\\ \end{pmatrix}\,.$ (70)

Now we proceed to discuss the neutrino Yukawa interaction term
$g\left(N^{c}LH_{u}f_{D}\left(Y\right)\right)_{\mathbf{1}}$. The modular form
$f_{D}(Y)$ is fixed by the assignments for $L$ and $N^{c}$, it can $1$,
$Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}$, $Y^{(2)}_{\mathbf{2}}$,
$Y^{(2)}_{\mathbf{3}}$, $Y^{(3)}_{\mathbf{\hat{1}^{\prime}}}$,
$Y^{(3)}_{\mathbf{\hat{3}}}$ and $Y^{(3)}_{\mathbf{\hat{3}^{\prime}}}$ up to
weight 3. We shall report the predictions for Dirac neutrino mass matrix for
each possible cases.

  * (i)

$f_{N}(Y)=1$

In this case, left-handed lepton doublet $L$ and right-handed neutrinos $N$
contract to a singlet, hence their assignments can be
$(\rho_{N^{c}},\,\rho_{L})=(\mathbf{3},\,\mathbf{3})$ or
$(\mathbf{3^{\prime}},\,\mathbf{3^{\prime}})$ or
$(\mathbf{\hat{3}},\,\mathbf{\hat{3}^{\prime}})$ or
$(\mathbf{\hat{3}^{\prime}},\,\mathbf{\hat{3}})$. The Dirac neutrino mass term
is

$\displaystyle\mathcal{W}_{D}=g(N^{c}L)_{\mathbf{1}}H_{u}=g[(L_{1}N^{c}_{1}+L_{2}N^{c}_{3}+L_{3}N^{c}_{2})H_{u}\,,$
(71)

with $k_{N}+k_{L}=0$. Consequently the Dirac neutrino mass matrix read as

$M_{D}=g\begin{pmatrix}1~{}&0~{}&0\\\ 0~{}&0~{}&1\\\
0~{}&1~{}&0\end{pmatrix}v_{u}\,.$ (72)

  * (ii)

$f_{N}(Y)=Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}$
There are eight possible assignments for $\rho_{L}$ and $\rho_{N^{c}}$, and
they can divided into two categories. In the case of
$(\rho_{N^{c}},\,\rho_{L})=(\mathbf{3},\,\mathbf{\hat{3}})$,
$(\mathbf{3^{\prime}},\,\mathbf{\hat{3}^{\prime}})$ or
$(\mathbf{\hat{3}},\,\mathbf{3})$ ,
$(\mathbf{\hat{3}^{\prime}},\,\mathbf{3^{\prime}})$, we have

$\displaystyle\mathcal{W}_{D}=g\Big{(}(N^{c}L)_{\mathbf{\hat{3}}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\Big{)}_{\mathbf{1}}H_{u}$
$\displaystyle~{}~{}~{}~{}=\pm
g[(L_{2}N^{c}_{3}-L_{3}N^{c}_{2})Y_{1}+(L_{3}N^{c}_{1}-L_{1}N^{c}_{3})Y_{2}+(L_{1}N^{c}_{2}-L_{2}N^{c}_{1})Y_{3}]H_{u}\,,$
(73)

with the modular weights $k_{N^{c}}+k_{L}=1$. We can read out the Dirac
neutrino mass matrix is

$M_{D}=g\begin{pmatrix}0~{}&-Y_{3}~{}&Y_{2}\\\ Y_{3}~{}&0~{}&-Y_{1}\\\
-Y_{2}~{}&Y_{1}~{}&0\\\ \end{pmatrix}v_{u}\,.$ (74)

For the second type of assignments
$(\rho_{N^{c}},\,\rho_{L})=(\mathbf{3},\,\mathbf{\hat{3}^{\prime}})$,
$(\mathbf{3^{\prime}},\,\mathbf{\hat{3}})$,
$(\mathbf{\hat{3}^{\prime}},\,\mathbf{3})$ or
$(\mathbf{\hat{3}},\,\mathbf{3^{\prime}})$ with $k_{N^{c}}+k_{L}=1$, we find

$\displaystyle\mathcal{W}_{D}$ $\displaystyle=$ $\displaystyle
g\Big{(}(N^{c}L)_{\mathbf{\hat{3}}}Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}\Big{)}_{\mathbf{1}}H_{u}$
(75) $\displaystyle=$ $\displaystyle
g[(2L_{1}N^{c}_{1}-L_{2}N^{c}_{3}-L_{3}N^{c}_{2})Y_{1}+(2L_{2}N^{c}_{2}-L_{1}N^{c}_{3}-L_{3}N^{c}_{1})Y_{2}$
$\displaystyle+(2L_{3}N^{c}_{3}-L_{1}N^{c}_{2}-L_{2}N^{c}_{1})Y_{3}]H_{u}\,,$

which leads to

$M_{D}=g\begin{pmatrix}2Y_{1}&-Y_{3}&-Y_{2}\\\ -Y_{3}&2Y_{2}&-Y_{1}\\\
-Y_{2}&-Y_{1}&2Y_{3}\end{pmatrix}v_{u}\,.$ (76)

  * (iii)

$f_{D}(Y)=Y^{(2)}_{\mathbf{2}},\,Y^{(2)}_{\mathbf{3}}$
The modular weights of $L$ and $N^{c}$ should compensate that of $f_{D}(Y)$,
they satisfy the condition $k_{N^{c}}+k_{L}=2$. For the assignments
$(\rho_{N^{c}},\,\rho_{L})=(\mathbf{3},\,\mathbf{3})$ ,
$(\mathbf{3^{\prime}},\,\mathbf{3^{\prime}})$ ,
$(\mathbf{\hat{3}},\,\mathbf{\hat{3}^{\prime}})$ or
$(\mathbf{\hat{3}^{\prime}},\,\mathbf{\hat{3}})$, we have

$\displaystyle\mathcal{W}_{D}$ $\displaystyle=$
$\displaystyle\Big{(}g_{1}(N^{c}L)_{\mathbf{2}}Y^{(2)}_{\mathbf{2}}\Big{)}_{\mathbf{1}}H_{u}+g_{2}\Big{(}(N^{c}L)_{\mathbf{3}}Y^{(2)}_{\mathbf{3}}\Big{)}_{\mathbf{1}}H_{u}$
(77) $\displaystyle=$ $\displaystyle
g_{1}[(L_{2}N^{c}_{1}+L_{1}N^{c}_{2}+L_{3}N^{c}_{3})Y^{(2)}_{1}+(L_{3}N^{c}_{1}+L_{1}N^{c}_{3}+L_{2}N^{c}_{2})Y^{(2)}_{2}]H_{u}$
$\displaystyle+g_{2}[(L_{2}N^{c}_{3}-L_{3}N^{c}_{2})Y^{(2)}_{3}+(L_{3}N^{c}_{1}-L_{1}N^{c}_{3})Y^{(2)}_{4}+(L_{1}N^{c}_{2}-L_{2}N^{c}_{1})Y^{(2)}_{5}]H_{u}\,.$

Accordingly the Dirac neutrino mass matrix is of the following form

$M_{D}=\begin{pmatrix}0~{}&g_{1}Y^{(2)}_{1}-g_{2}Y^{(2)}_{5}~{}&g_{1}Y^{(2)}_{2}+g_{2}Y^{(2)}_{4}\\\
g_{1}Y^{(2)}_{1}+g_{2}Y^{(2)}_{5}~{}&g_{1}Y^{(2)}_{2}~{}&-g_{2}Y^{(2)}_{3}\\\
g_{1}Y^{(2)}_{2}-g_{2}Y^{(2)}_{4}~{}&g_{2}Y^{(2)}_{3}~{}&g_{1}Y^{(2)}_{1}\\\
\end{pmatrix}v_{u}\,.$ (78)

We can also assign $N^{c}$ and $L$ to the $S^{\prime}_{4}$ triplets
$(\rho_{N^{c}},\,\rho_{L})=(\mathbf{\hat{3}},\,\mathbf{\hat{3}})$,
$(\mathbf{\hat{3}^{\prime}},\,\mathbf{\hat{3}^{\prime}})$ ,
$(\mathbf{3},\,\mathbf{3^{\prime}})$ or $(\mathbf{3^{\prime}},\,\mathbf{3})$,
and thus

$\displaystyle\mathcal{W}_{D}$ $\displaystyle=$
$\displaystyle\Big{(}g_{1}(N^{c}L)_{\mathbf{2}}Y^{(2)}_{\mathbf{2}}\Big{)}_{\mathbf{1}}H_{u}+g_{2}\Big{(}(N^{c}L)_{\mathbf{3}}Y^{(2)}_{\mathbf{3}}\Big{)}_{\mathbf{1}}H_{u}$
(79) $\displaystyle=$ $\displaystyle
g_{1}[(L_{2}N^{c}_{1}+L_{1}N^{c}_{2}+L_{3}N^{c}_{3})Y^{(2)}_{1}-(L_{3}N^{c}_{1}+L_{1}N^{c}_{3}+L_{2}N^{c}_{2})Y^{(2)}_{2}]H_{u}$
$\displaystyle+g_{2}[(2L_{1}N^{c}_{1}-L_{2}N^{c}_{3}-L_{3}N^{c}_{2})Y^{(2)}_{3}+(2L_{2}N^{c}_{2}-L_{3}N^{c}_{1}-L_{1}N^{c}_{3})Y^{(2)}_{4}$
$\displaystyle+(2L_{3}N^{c}_{3}-L_{1}N^{c}_{2}-L_{2}N^{c}_{1})Y^{(2)}_{5}]H_{u}\,.$

The Dirac neutrino mass matrix read as

$M_{D}=\begin{pmatrix}2g_{2}Y^{(2)}_{3}~{}&g_{1}Y^{(2)}_{1}-g_{2}Y^{(2)}_{5}~{}&-g_{1}Y^{(2)}_{2}-g_{2}Y^{(2)}_{4}\\\
g_{1}Y^{(2)}_{1}-g_{2}Y^{(2)}_{5}~{}&-g_{1}Y^{(2)}_{2}+2g_{2}Y^{(2)}_{4}~{}&-g_{2}Y^{(2)}_{3}\\\
-g_{1}Y^{(2)}_{2}-g_{2}Y^{(2)}_{4}~{}&-g_{2}Y^{(2)}_{3}~{}&g_{1}Y^{(2)}_{1}+2g_{2}Y^{(2)}_{5}\\\
\end{pmatrix}v_{u}\,.$ (80)

  * (iv)

$f_{D}(Y)=Y^{(3)}_{\mathbf{\hat{1}^{\prime}}},\,Y^{(3)}_{\mathbf{\hat{3}}},\,Y^{(3)}_{\mathbf{\hat{3}^{\prime}}}$
The weight cancellation requires $k_{L}$ and $k_{N^{c}}$ fulfill the condition
$k_{N^{c}}+k_{L}=3$. Invariance of the neutrino Yukawa coupling under
$S^{\prime}_{4}$ entails $N^{c}$ and $L$ should contract to
$\mathbf{\hat{1}}$, $\mathbf{\hat{3}}$ and $\mathbf{\hat{3}^{\prime}}$.
Therefore $N^{c}$ and $L$ can be assigned to
$(\rho_{N^{c}},\,\rho_{L})=(\mathbf{3},\,\mathbf{\hat{3}})$,
$(\mathbf{3^{\prime}},\,\mathbf{\hat{3}^{\prime}})$,
$(\mathbf{\hat{3}},\,\mathbf{3})$ or
$(\mathbf{\hat{3}^{\prime}},\,\mathbf{3^{\prime}})$, then the superpotential
$\mathcal{W}_{D}$ is of the form

$\displaystyle\mathcal{W}_{D}$ $\displaystyle=$ $\displaystyle
g_{1}\Big{(}(N^{c}L)_{\mathbf{\hat{3}}}Y^{(3)}_{\mathbf{\hat{3}^{\prime}}}\Big{)}_{\mathbf{1}}H_{u}+g_{2}\Big{(}(N^{c}L)_{\mathbf{\hat{3}^{\prime}}}Y^{(3)}_{\mathbf{\hat{3}}}\Big{)}_{\mathbf{1}}H_{u}+g_{3}\Big{(}(N^{c}L)_{\mathbf{\hat{1}}}Y^{(3)}_{\mathbf{\hat{1}^{\prime}}}\Big{)}_{\mathbf{1}}H_{u}$
(81) $\displaystyle=$ $\displaystyle\pm
g_{1}[(L_{2}N^{c}_{3}-L_{3}N^{c}_{2})Y^{(3)}_{5}+(L_{3}N^{c}_{1}-L_{1}N^{c}_{3})Y^{(3)}_{6}+(L_{1}N^{c}_{2}-L_{2}N^{c}_{1})Y^{(3)}_{7}]H_{u}$
$\displaystyle+g_{2}[(2L_{1}N^{c}_{1}-L_{2}N^{c}_{3}-L_{3}N^{c}_{2})Y^{(3)}_{2}+(2L_{2}N^{c}_{2}-L_{1}N^{c}_{3}-L_{3}N^{c}_{1})Y^{(3)}_{3}$
$\displaystyle+(2L_{3}N^{c}_{3}-L_{1}N^{c}_{2}-L_{2}N^{c}_{1})Y^{(3)}_{4}]H_{u}+g_{3}[L_{1}N^{c}_{1}+L_{2}N^{c}_{3}+L_{3}N^{c}_{2}]H_{u}\,,$

which gives rise to

$\displaystyle
M_{D}=\begin{pmatrix}2g_{2}Y^{(3)}_{2}+g_{3}Y^{(3)}_{1}~{}&-g_{1}Y^{(3)}_{7}-g_{2}Y^{(3)}_{4}~{}&g_{1}Y^{(3)}_{6}-g_{2}Y^{(3)}_{3}\\\
g_{1}Y^{(3)}_{7}-g_{2}Y^{(3)}_{4}~{}&2g_{2}Y^{(3)}_{3}~{}&-g_{1}Y^{(3)}_{5}-g_{2}Y^{(3)}_{2}+g_{3}Y^{(3)}_{1}\\\
-g_{1}Y^{(3)}_{6}-g_{2}Y^{(3)}_{3}~{}&g_{1}Y^{(3)}_{5}-g_{2}Y^{(3)}_{2}+g_{3}Y^{(3)}_{1}~{}&2g_{2}Y^{(3)}_{4}\\\
\end{pmatrix}v_{u}\,.$ (82)

We can also assign $N^{c}$ and $L$ to transform as
$(\rho_{N^{c}},\,\rho_{L})=(\mathbf{3},\,\mathbf{\hat{3}^{\prime}})$,
$(\mathbf{3^{\prime}},\,\mathbf{\hat{3}})$,
$(\mathbf{\hat{3}^{\prime}},\,\mathbf{3})$ or
$(\mathbf{\hat{3}},\,\mathbf{3^{\prime}})$, then we have

$\displaystyle\mathcal{W}_{D}$ $\displaystyle=$ $\displaystyle
g_{1}\Big{(}(N^{c}L)_{\mathbf{\hat{3}}}Y^{(3)}_{\mathbf{\hat{3}^{\prime}}}\Big{)}_{\mathbf{1}}H_{u}+g_{2}\Big{(}(N^{c}L)_{\mathbf{\hat{3}^{\prime}}}Y^{(3)}_{\mathbf{\hat{3}}}\Big{)}_{\mathbf{1}}H_{u}$
(83) $\displaystyle=$ $\displaystyle
g_{1}[(2L_{1}N^{c}_{1}-L_{2}N^{c}_{3}-L_{3}N^{c}_{2})Y^{(3)}_{5}+(2L_{2}N^{c}_{2}-L_{1}N^{c}_{3}-L_{3}N^{c}_{1})Y^{(3)}_{6}$
$\displaystyle+(2L_{3}N^{c}_{3}-L_{1}N^{c}_{2}-L_{2}N^{c}_{1})Y^{(3)}_{7}]H_{u}+g_{2}[(L_{2}N^{c}_{3}-L_{3}N^{c}_{2})Y^{(3)}_{2}$
$\displaystyle+(L_{3}N^{c}_{1}-L_{1}N^{c}_{3})Y^{(3)}_{3}+(L_{1}N^{c}_{2}-L_{2}N^{c}_{1})Y^{(3)}_{4}]H_{u}\,.$

The Dirac neutrino mass matrix is determined to be

$M_{D}=\begin{pmatrix}2g_{1}Y^{(3)}_{5}&-g_{1}Y^{(3)}_{7}-g_{2}Y^{(3)}_{4}&-g_{1}Y^{(3)}_{6}+g_{2}Y^{(3)}_{3}\\\
-g_{1}Y^{(3)}_{7}+g_{2}Y^{(3)}_{4}&2g_{1}Y^{(3)}_{6}&-g_{1}Y^{(3)}_{5}-g_{2}Y^{(3)}_{2}\\\
-g_{1}Y^{(3)}_{6}-g_{2}Y^{(3)}_{3}&-g_{1}Y^{(3)}_{5}+g_{2}Y^{(3)}_{2}&2g_{1}Y^{(3)}_{7}\\\
\end{pmatrix}v_{u}\,.$ (84)

For all the above type-I seesaw models, the effective light neutrino mass
matrix is given by the seesaw formula,

$M_{\nu}=-M_{D}^{T}M_{N}^{-1}M_{D}\,.$ (85)

We are interested in the models with less free parameters, and we list the
possible neutrino models in table 3 for which the resulting light neutrino
mass matrices contain less than four free parameters excluding the modulus
$\tau$.

Cases | $W_{1}$ | $W_{2}$ | $S_{1}$ | $S_{2}$ | $S_{3}$ | $S_{4}$
---|---|---|---|---|---|---
Irrep$(\rho_{N^{c}},\,\rho_{L})$ | $\begin{cases}(-,\mathbf{3})\\\ (-,\mathbf{3^{\prime}})\end{cases}$ | $\begin{cases}(-,\mathbf{\hat{3}})\\\ (-,\mathbf{\hat{3}^{\prime}})\end{cases}$ | $\begin{cases}(\mathbf{3},\,\mathbf{3})\\\ (\mathbf{3^{\prime}},\,\mathbf{3^{\prime}})\end{cases}$ | $\begin{cases}(\mathbf{\hat{3}},\,\mathbf{\hat{3}^{\prime}})\\\ (\mathbf{\hat{3}^{\prime}},\,\mathbf{\hat{3}})\end{cases}$
weight$(k_{N^{c}},k_{L})$ | $(-,1)$ | $(-,1)$ | $(-1,1)$ | $(2,0)$ | $(1,1)$ | $(-1,1)$
Neutrino | Eq. (61) | Eq. (63) | Eq. (68) | Eq. (66) | Eq. (68) | Eq. (70)
mass matrices | Eq. (72) | Eq. (78) | Eq. (78) | Eq. (72)
Cases | $S_{5}$ | $S_{6}$ | $S_{7}$ | $S_{8}$ | $S_{9}$ | $S_{10}$
Irrep$(\rho_{N^{c}},\,\rho_{L})$ | $\begin{cases}(\mathbf{3},\,\mathbf{3^{\prime}})\\\ (\mathbf{3^{\prime}},\,\mathbf{3})\end{cases}$ | $\begin{cases}(\mathbf{3},\,\mathbf{\hat{3}})\\\ (\mathbf{3^{\prime}},\,\mathbf{\hat{3}^{\prime}})\end{cases}$
weight$(k_{N^{c}},k_{L})$ | $(2,0)$ | $(1,1)$ | $(1,0)$ | $(0,1)$ | $(3,0)$ | $(2,1)$
Neutrino | Eq. (66) | Eq. (68) | Eq. (66) | Eq. (68) | Eq. (66) | Eq. (68)
mass matrices | Eq. (80) | Eq. (80) | Eq. (74) | Eq. (74) | Eq. (82) | Eq. (82)
Cases | $S_{11}$ | $S_{12}$ | $S_{13}$ | $S_{14}$ | $S_{15}$ | $S_{16}$
Irrep$(\rho_{N^{c}},\,\rho_{L})$ | $\begin{cases}(\mathbf{\hat{3}},\,\mathbf{3})\\\ (\mathbf{\hat{3}^{\prime}},\,\mathbf{3^{\prime}})\end{cases}$ | $\begin{cases}(\mathbf{3},\,\mathbf{\hat{3}^{\prime}})\\\ (\mathbf{3^{\prime}},\,\mathbf{\hat{3}})\end{cases}$ | $\begin{cases}(\mathbf{\hat{3}^{\prime}},\,\mathbf{3})\\\ (\mathbf{\hat{3}},\,\mathbf{3^{\prime}})\end{cases}$
weight$(k_{N^{c}},k_{L})$ | $(0,1)$ | $(1,0)$ | $(0,1)$ | $(3,0)$ | $(2,1)$ | $(0,1)$
Neutrino | Eq. (70) | Eq. (66) | Eq. (68) | Eq. (66) | Eq. (68) | Eq. (70)
mass matrices | Eq. (74) | Eq. (76) | Eq. (76) | Eq. (84) | Eq. (84) | Eq. (76)

Table 3:  Summary of neutrino models with less than four free parameters
excluding $\tau$, $W_{1,2}$ and $S_{i}$ ($i=1,\ldots,16$) denote the models in
which neutrino masses arise from Weinberg operator and type-I seesaw mechanism
respectively.

###  5.3 Numerical results

In short, the charged lepton can take four possible forms shown in table 2 if
only modular forms of weight less than 4 are considered, and there are
eighteen neutrino models with parameters less than 4, as summarized in table
3. Combining charged lepton sector with neutrino sector, we obtain totally
$4\times 18=72$ lepton models which are denoted as $C_{i}$-$W_{1}$,
$C_{i}$-$W_{2}$ and $C_{i}$-$S_{j}$ with the indices $i=1,\ldots,4$,
$j=1,\ldots,16$. We see that for the four cases $C_{1,2,3,4}$ the charged
lepton mass matrix $M_{e}$ depends on three parameters $\alpha$, $\beta$ and
$\gamma$ which can be made real by redefining the phases of the right-handed
charged leptons $E^{c}_{1,2,3}$. The three parameters $\alpha$, $\beta$ and
$\gamma$ are in one-to-one correspondence with the charged lepton masses. The
electron, muon, tau masses can be reproduced by adjusting the parameters
$\alpha$, $\beta$ and $\gamma$. We confront each model with the neutrino
oscillation data and charged lepton masses, we perform a conventional
$\chi^{2}$ analysis to optimize the model parameters and determine how well
each model can be compatible with the observations. The overall mass scale
$\alpha v_{d}$ in the charged lepton mass matrix and $g^{2}v^{2}_{u}/\Lambda$
in the neutrino mass matrix can be fixed by requiring the electron mass and
the mass splitting $\Delta m^{2}_{21}$ are reproduced. Since the overall
factor of the mass matrix doesn’t affect the predictions for mass ratios,
mixing angles and CP violating phases, we construct the $\chi^{2}$ function
using the lepton mixing angles $\theta_{12}$, $\theta_{13}$, $\theta_{23}$ and
the mass ratios $m_{e}/m_{\mu}$, $m_{\mu}/m_{\tau}$, $\Delta m^{2}_{21}/\Delta
m^{2}_{31}$. The neutrino oscillation parameters are taken from the latest
global fit results of NuFIT v4.1 including the atmospheric neutrino data from
Super-Kamiokande [63]. Since the current data somewhat prefer normal ordering
(NO) over inverted neutrino (IO) mass ordering, we shall focus on NO neutrino
masses in the numerical analysis. The best fit values and $1\sigma$ ranges of
the three lepton mixing angles CP violating phase $\delta_{CP}$ and the
neutrino mass squared differences are as follows

$\displaystyle\sin^{2}\theta_{12}=0.310^{+0.013}_{-0.012},~{}~{}~{}\sin^{2}\theta_{13}=0.02237^{+0.00066}_{-0.00065},~{}~{}~{}\sin^{2}\theta_{23}=0.563^{+0.018}_{-0.024},$
(86)
$\displaystyle\delta^{l}_{CP}/\pi=1.2278^{+0.2167}_{-0.1556},~{}~{}~{}\frac{\Delta
m^{2}_{21}}{10^{-5}\text{eV}^{2}}=7.39^{+0.21}_{-0.20},~{}~{}~{}\frac{\Delta
m^{2}_{31}}{10^{-3}\text{eV}^{2}}=2.528^{+0.029}_{-0.031}\,.$

The ratios of charged lepton masses taken from [64],

$m_{e}/m_{\mu}=0.0048\pm 0.0002,\quad m_{\mu}/m_{\tau}=0.0565\pm 0.0045$ (87)

The leptonic Dirac CP phase $\delta^{l}_{CP}$ is not measured precisely at
present and the indication of a preferred value of $\delta^{l}_{CP}$ from
global data analyses is rather weak, we don’t include the information of
$\delta^{l}_{CP}$ in the $\chi^{2}$ function. The absolute values of all
coupling constants are scanned in the region $[0,10^{4}]$ and the phases are
freely varied in the range $[0,2\pi]$, and the modulus $\tau$ is restricted in
the right-hand part of the fundamental domain $\mathcal{D}$ with
$0\leq\texttt{Re}(\tau)\leq 0.5$, the reason of why not scanning the complete
fundamental domain is explained below. We numerically minimize the $\chi^{2}$
function by using the minimization algorithms incorporated in the package
MINUIT developed by CERN to determine the optimum values of the input
parameters. We find that 15 models can give very good fit to the data for
certain values of input parameters. We display the best fit values of the
input parameter for which the $\chi^{2}$ function reach a global minimum
$\chi^{2}_{\text{min}}$ in table 4, and we also give the predictions for
lepton mixing parameters and neutrino masses at the best fitting point in
table 4. For all the 15 phenomenologically viable models, the light neutrino
mass matrix $M_{\nu}$ depends on a single complex parameter $g_{2}/g_{1}$ and
the complex modulus $\tau$ besides the overall scale
$g^{2}_{1}v^{2}_{u}/\Lambda$. Hence the three lepton mixing angles, Dirac and
Majorana CP phases and three light neutrino masses are completely determined
by five real parameters $|g_{2}/g_{1}|$, $\arg{(g_{2}/g_{1})}$,
$\texttt{Re}(\tau)$, $\texttt{Im}(\tau)$, and $g^{2}_{1}v_{u}^{2}/\Lambda$.
The number of free parameters is four less than the that of observables,
therefore these models are quite predictive. Note that the models
$C_{i}$-$S_{9}$ and $C_{i}$-$S_{10}$ contain more free parameters,
consequently we don’t not show the numerical results of these model here. As
can be seen from table 4, the three lepton mixing angles $\theta_{12}$,
$\theta_{13}$, $\theta_{23}$ and the neutrino mass squared difference $\Delta
m^{2}_{21}$, $\Delta m^{2}_{31}$ fall in the $1\sigma$ experimental range for
these 15 viable models except the models $C_{1}$-$S_{3}$ and $C_{4}$-$S_{3}$
where $\sin^{2}\theta_{23}=0.4949$ is outside the $1\sigma$ region but still
in the $3\sigma$ region.

Furthermore, we notice that the modular forms have the property
$Y^{(k)}_{\mathbf{r}}(-\tau^{*})=\rho_{\mathbf{r}}(S)[Y^{(k)}_{\mathbf{r}}(\tau)]^{*}$
as shown in Eq. (40). Therefore if we make the replacment
$\tau\rightarrow-\tau^{*}$, $g_{1,2}\rightarrow g^{*}_{1,2}$ and perform the
$S$ transformation on both lepton and right-handed neutrino fields, the
charged lepton and neutrino mass matrices would become their complex
conjugate. Therefore under such transformation, lepton masses and mixing
angles are unchanged while the signs of all CP violating phases are flipped.
As a consequence, the complex modulus $\tau$ is limited in the right-hand part
of the fundamental domain $\mathcal{D}$ with $0\leq\texttt{Re}(\tau)\leq 0.5$
when we scan over the parameter space in numerical minimisation. The
predictions of the mixing parameters in the left-hand part of $\mathcal{D}$
with $-0.5\leq\texttt{Re}(\tau)\leq 0$ can be easily obtained by reversing the
overall signs of the Dirac and Majorana CP phases. Hence all the numerical
results given in table 4 should understand to come in pair with opposite CP
violating phases.

It is known that the neutrino mass spectrum tends to be nearly degenerate in
modular invariant models based on inhomogeneous finite modular group. As can
be seen from table 4, a remarkable feature of these modular $S^{\prime}_{4}$
models is that the neutrino masses are hierarchical except the models
$C_{2}$-$W_{2}$ and $C_{3}$-$W_{2}$. From the predictions for neutrino masses,
mixing angles and CP violating phases in table 4, we can pin down the
effective neutrino mass $|m_{ee}|$ relevant to neutrinoless double beta decay.
We displayed the the lightest neutrino mass and $|m_{ee}|$ of each viable
model in figure 1, where the experiemental bound of KamLAND-Zen [65] and the
expected sensitivities of future experiments [66, 67, 68, 69, 70, 71] are
indicated by the horizontal lines. For the models $C_{1}$-$S_{6}$ and
$C_{3}$-$S_{6}$, the effective Majorana mass are $|m_{ee}|\simeq 2.864\times
10^{-6}$ meV and $|m_{ee}|=8.620\times 10^{-7}$ meV respectively with the
lightest neutrino mass $m_{1}\simeq 6.953$ meV. Hence the corresponding points
are not visible in the figure. As indicated in the figure, we expect that some
of these predictions will be tested in next generation neutrinoless double
beta decay experiments.

If we require the theory to be invariant under both $S^{\prime}_{4}$ modular
symmetry and the generalized CP symmetry, all the couplings would be
restricted to be real in our working basis, as shown in section 4. Thus the
number of free parameters in a model would be reduced further. For the 15
viable models listed in table 4, the generalized CP symmetry enforces both
coupling constants $g_{1}$ and $g_{2}$ to be real. As a consequence, the light
neutrino mass matrix would be described by four real parameters $g_{2}/g_{1}$,
$\texttt{Re}(\tau)$, $\texttt{Im}(\tau)$ and the overall scale
$g^{2}_{1}v_{u}^{2}/\Lambda$. In particular, the complex modulus $\tau$ and
modular forms would be sources all CP violation phases. We find only six out
of the 15 models are compatible with data, the numerical results are shown in
table 5.

| Models | Best fit values of the input parameters for NO | $\chi^{2}_{\mathrm{min}}$
---|---|---
$\texttt{Re}\langle\tau\rangle$ | $\texttt{Im}\langle\tau\rangle$ | $\beta/\alpha$ | $\gamma/\alpha$ | $|g_{2}/g_{1}|$ | $\arg{(g_{2}/g_{1})}/\pi$ |  $\alpha v_{d}$/MeV |  $\dfrac{g^{2}_{1}v_{u}^{2}}{\Lambda}/$meV
$C_{1}$-$W_{2}$ | 0.3656 | 1.1638 | 670.6170 | 13.7484 | 0.7008 | 0.026 | 0.1995 | 2.746 | 2.277
$C_{2}$-$W_{2}$ | 0.4600 | 0.8911 | 34.6012 | 203.4790 | 0.8046 | 1.827 | 0.170 | 5.274 | 0.005
$C_{3}$-$W_{2}$ | 0.4519 | 0.8957 | 196.249 | 57.8262 | 0.7826 | 0.181 | 0.171 | 4.970 | $7.57\times 10^{-5}$
$C_{4}$-$W_{2}$ | 0.3658 | 1.1639 | 4536.51 | 92.9859 | 0.7007 | 0.026 | 0.0295 | 2.746 | 2.261
$C_{1}$-$S_{3}$ | 0.3860 | 1.3025 | 717.989 | 13.6791 | 0.5157 | 1.806 | 0.2030 | 4.573 | 8.211
$C_{4}$-$S_{3}$ | 0.3860 | 1.3025 | 6074.95 | 115.738 | 0.5157 | 0.807 | 0.0240 | 4.573 | 8.211
$C_{1}$-$S_{5}$ | 0.1470 | 0.9994 | 0.0001 | 0.0031 | 0.3405 | 0.350 | 653.091 | 0.233 | $7.78\times 10^{-6}$
$C_{4}$-$S_{5}$ | 0.0582 | 1.0131 | 9584.42 | 253.791 | 0.3889 | 0.307 | 0.0123 | 0.214 | $2.27\times 10^{-5}$
$C_{1}$-$S_{6}$ | 0.1764 | 0.9915 | 0.0016 | 0.8839 | 1.5584 | 1.865 | 38.325 | 0.086 | 1.581
$C_{3}$-$S_{6}$ | 0.1763 | 0.9914 | 0.0003 | 0.8838 | 1.5576 | 1.865 | 38.320 | 0.085 | 1.577
$C_{1}$-$S_{15}$ | 0.4792 | 1.171 | 44.6105 | 215.725 | 1.828 | 1.357 | 0.1934 | 0.036 | $1.66\times 10^{-5}$
$C_{4}$-$S_{15}$ | 0.4881 | 1.1629 | 244.836 | 1192.49 | 1.7919 | 0.636 | 0.0349 | 0.036 | $1.01\times 10^{-5}$
$C_{1}$-$S_{16}$ | 0.2609 | 1.1527 | 608.789 | 13.2779 | 0.2412 | 0.094 | 0.2178 | 7.324 | 4.323
$C_{3}$-$S_{16}$ | 0.3065 | 1.0220 | 229.712 | 15.8054 | 0.2119 | 0.067 | 0.186 | 8.083 | 4.323
$C_{4}$-$S_{16}$ | 0.2673 | 1.150 | 5679.84 | 123.527 | 0.2342 | 1.906 | 0.0233 | 7.226 | 4.309

| Models | Predictions for mixing parameters and neutrino masses at best fitting point
---|---
$\sin^{2}\theta_{12}$ | $\sin^{2}\theta_{13}$ | $\sin^{2}\theta_{23}$ | $\delta^{l}_{CP}/\pi$ | $\alpha_{21}/\pi$ | $\alpha_{31}/\pi$ |  $m_{1}$/meV |  $m_{2}$/meV |  $m_{3}$/meV |  $|m_{ee}|$/meV
$C_{1}$-$W_{2}$ | 0.3100 | 0.02184 | 0.5326 | 1.329 | 1.509 | 0.525 | 32.491 | 33.609 | 59.818 | 25.735
$C_{2}$-$W_{2}$ | 0.3100 | 0.02237 | 0.5643 | 0.416 | 1.987 | 0.992 | 115.066 | 115.387 | 125.574 | 115.022
$C_{3}$-$W_{2}$ | 0.3100 | 0.02237 | 0.5628 | 0.432 | 0.005 | 1.006 | 104.766 | 105.118 | 116.207 | 104.886
$C_{4}$-$W_{2}$ | 0.3100 | 0.02184 | 0.5327 | 1.328 | 1.509 | 0.525 | 32.478 | 33.597 | 59.810 | 25.723
$C_{1}$-$S_{3}$ | 0.3136 | 0.02254 | 0.4949 | 0.997 | 1.342 | 1.055 | 26.348 | 27.715 | 56.764 | 14.591
$C_{4}$-$S_{3}$ | 0.3136 | 0.02254 | 0.4949 | 0.997 | 1.342 | 1.055 | 26.347 | 27.714 | 56.764 | 14.590
$C_{1}$-$S_{5}$ | 0.3100 | 0.02237 | 0.5630 | 0.439 | 0.078 | 0.729 | 19.363 | 21.185 | 53.879 | 20.394
$C_{4}$-$S_{5}$ | 0.3100 | 0.02237 | 0.5630 | 0.852 | 0.092 | 0.307 | 19.539 | 21.346 | 53.946 | 19.231
$C_{1}$-$S_{6}$ | 0.3220 | 0.02227 | 0.5435 | 1.001 | 1.000 | 1.002 | 6.953 | 11.056 | 50.611 | $2.864\times 10^{-6}$
$C_{3}$-$S_{6}$ | 0.3221 | 0.02226 | 0.5435 | 1.001 | 1.000 | 1.001 | 6.954 | 11.057 | 50.642 | $8.620\times 10^{-7}$
$C_{1}$-$S_{15}$ | 0.3100 | 0.02237 | 0.5629 | 1.173 | 1.646 | 1.556 | 18.802 | 20.674 | 53.680 | 15.841
$C_{4}$-$S_{15}$ | 0.3100 | 0.02237 | 0.5630 | 0.583 | 0.352 | 0.383 | 19.164 | 21.004 | 53.807 | 15.639
$C_{1}$-$S_{16}$ | 0.3008 | 0.02147 | 0.5630 | 1.676 | 1.578 | 1.734 | 5.065 | 9.977 | 49.550 | 4.927
$C_{3}$-$S_{16}$ | 0.3008 | 0.02147 | 0.5630 | 0.664 | 0.762 | 1.537 | 7.498 | 11.407 | 49.857 | 4.579
$C_{4}$-$S_{16}$ | 0.3008 | 0.02147 | 0.5630 | 0.298 | 1.410 | 0.242 | 4.982 | 9.936 | 49.543 | 4.848

Table 4: The best fit values of the input parameters at the minimum of the $\chi^{2}$ under the assumption of NO neutrino masses. We give the predictions for neutrino mixing angles $\theta_{12}$, $\theta_{13}$, and Dirac CP violating phase $\delta^{l}_{CP}$ as well as Majorana CP violating phases $\alpha_{21}$, $\alpha_{31}$, and the light neutrino masses $m_{1,2,3}$ and the effective mass $|m_{ee}|$ in neutrinoless double decay. Notice in the CP dual point $\tau\rightarrow-\tau^{*}$, $g_{1,2}\rightarrow g^{*}_{1,2}$, the signs of Dirac and Majorana CP phases are reversed while the predictions for lepton mixing angles and neutrino masses are unchanged.  Figure 1: The predictions for lightest neutrino mass $m_{1}$ and the effective Majorana mass $|m_{ee}|$ for the 15 phenomenologically viable models at the best fit points shown in table 4. The blue (red) lines denote the most general allowed regions for NO (IO) where the neutrino oscillation parameters are freely varied in their $3\sigma$ regions [63]. The vertical grey exclusion band denotes the bound on the lightest neutrino mass coming from the cosmological data $\Sigma_{i}m_{i}<0.120\,\text{eV}$ at $95\%$ confidence level obtained by the Planck collaboration [72]. The values of $|m_{ee}|$ in the models $C_{1}$-$S_{6}$ and $C_{3}$-$S_{6}$ are too tiny to be visible.  Models with gCP |  $C_{1}$-$S_{5}$ |  $C_{4}$-$S_{5}$ |  $C_{3}$-$S_{6}$ |  $C_{1}$-$S_{16}$ |  $C_{3}$-$S_{16}$ |  $C_{4}$-$S_{16}$
---|---|---|---|---|---|---
$\texttt{Re}\langle\tau\rangle$ | 0.1997 | 0.2118 | 0.1745 | 0.3028 | 0.3166 | 0.3028
$\texttt{Im}\langle\tau\rangle$ | 0.9969 | 0.9709 | 0.9847 | 1.1351 | 1.0086 | 1.1351
$\beta/\alpha$ | 0.000087 | 435.027 | 0.00033 | 629.354 | 227.194 | 4958.27
$\gamma/\alpha$ | 0.0031 | 1696.69 | 0.8776 | 13.470 | 16.5263 | 106.115
$g_{2}/g_{1}$ | $-0.0066$ | 0.6520 | 1.6939 | 0.1779 | 0.1899 | 0.1780
$\alpha v_{d}$/MeV | 654.213 | 0.0199 | 38.2 | 0.2079 | 0.1843 | 0.0264
$\dfrac{g^{2}_{1}v_{u}^{2}}{\Lambda}$/meV | 0.371 | 0.160 | 0.072 | 6.564 | 7.851 | 6.565
$\sin^{2}\theta_{12}$ | 0.3105 | 0.3145 | 0.3234 | 0.3008 | 0.3008 | 0.3008
$\sin^{2}\theta_{13}$ | 0.02239 | 0.02289 | 0.02230 | 0.02147 | 0.02147 | 0.02147
$\sin^{2}\theta_{23}$ | 0.5057 | 0.4491 | 0.5461 | 0.5630 | 05630 | 0.5630
$\delta_{CP}/\pi$ | 0.540 | 1.789 | 1.000 | 0 | 0.893 | 0
$\alpha_{21}/\pi$ | 0.086 | 0.837 | 1.000 | 1.422 | 0.765 | 1.422
$\alpha_{31}/\pi$ | 1.057 | 1.715 | 1.000 | 0 | 1.787 | 0
$m_{e}/m_{\mu}$ | 0.0048 | 0.0048 | 0.0048 | 0.0048 | 0.0048 | 0.0048
$m_{\mu}/m_{\tau}$ | 0.0564 | 0.0561 | 0.0565 | 0.0565 | 0.0565 | 0.0565
$m_{1}$/meV | 14.658 | 37.383 | 7.006 | 4.456 | 7.456 | 4.456
$m_{2}$/meV | 16.993 | 38.359 | 11.090 | 9.683 | 11.379 | 9.683
$m_{3}$/meV | 52.380 | 62.724 | 50.629 | 49.491 | 49.850 | 49.492
$\sum_{i}m_{i}$/meV | 84.031 | 138.466 | 68.725 | 63.630 | 68.685 | 63.631
$|m_{ee}|$/meV | $16.074$ | $17.290$ | $8.61\times 10^{-7}$ | $4.400$ | $4.332$ | $4.400$
$\chi^{2}_{\mathrm{min}}$ | 5.694 | 23.279 | 1.601 | 4.328 | 4.331 | 4.328
Table 5:  The best fit values of the input parameters after imposing
generalized CP. We give the predictions for lepton mixing parameters and
neutrino masses at the best fit points. Notice in the CP dual point
$\tau\rightarrow-\tau^{*}$, the signs of Dirac and Majorana CP phases are
reversed while the predictions for lepton mixing angles and neutrino masses
are unchanged.

In order to show the predictive ability of the model after adding generalized
CP symmetry, for comparison, we take the model $C_{1}$-$S_{5}$ as an example
and give detailed numerical scanning results , the correlation graph with or
without generalized CP symmetry are shown below. From these figure 3, figure
4, figure 5, figure 6 and figure 7, it can be seen that for the model
$C_{1}$-$S_{5}$, the points compatible with experiment data are all located
near the boundary of the fundamental domain $\mathcal{D}$, and there are five
independent and disconnected regions in the parameter space. The correlation
between some observables in each region is very strong, and the patterns of
different regions show very different behaviors. After gCP symmetry is added,
only two independent and disconnected regions are left in the allowed
parameter space of model $C_{1}$-$S_{5}$. And they all shrink in the
$\langle\tau\rangle$ plane, one of them is very small, almost a point. These
shows the strong restriction of gCP on the parameter space.

Figure 2: All allowed region of the modulus $\langle\tau\rangle$ in the
fundamental domain $\mathcal{D}$ for the model $C_{1}$-$S_{5}$ without gCP.
There are five disconnected parameter regions, moreover, two of them (region
II and III) coincide on the $\langle\tau\rangle$ plane as shown in the second
figure. The color of points from red to blue represents the $\chi^{2}$ from
small to large, as shown in the color bar. Note that we only show the points
where $\texttt{Re}\langle\tau\rangle$ is between $0$ and $0.5$ in
$\mathcal{D}$, because other points can be connected with it through modular
transformation or CP transformation. Figure 3: The predictions for the
correlations among the input free parameters, neutrino mixing angles, CP
violation phases and neutrino masses in the region I of model $C_{1}$-$S_{5}$
without gCP symmetry. Figure 4: The predictions for the correlations among the
input free parameters, neutrino mixing angles, CP violation phases and
neutrino masses in the region II of model $C_{1}$-$S_{5}$ without gCP
symmetry. Figure 5: The predictions for the correlations among the input free
parameters, neutrino mixing angles, CP violation phases and neutrino masses in
the region III of model $C_{1}$-$S_{5}$ without gCP symmetry. Figure 6: The
predictions for the correlations among the input free parameters, neutrino
mixing angles, CP violation phases and neutrino masses in the region IV of
model $C_{1}$-$S_{5}$ without gCP symmetry. Figure 7: The predictions for the
correlations among the input free parameters, neutrino mixing angles, CP
violation phases and neutrino masses in the region V of model $C_{1}$-$S_{5}$
without gCP symmetry. Figure 8: The predictions for the correlations among the
input free parameters, neutrino mixing angles, CP violation phases and
neutrino masses in the model $C_{1}$-$S_{5}$ with gCP symmetry.

##  6 Quark models based on $S^{\prime}_{4}$ modular symmetry

In this section, we will exploit the $S^{\prime}_{4}$ modular symmetry to
understand the quark mass hierarchies and the observed pattern of hierarchial
quark mixing angles and CP violating phase encoded in the CKM matrix. We aim
to construct viable quark mass models with a minimal amount of free
parameters. The quark fields can be assigned to triplet of $S^{\prime}_{4}$,
the direct product of a doublet and a singlet, or the direct sum of three
singlets. Similar to what we have done in the charged lepton sector, we can
classify the structures of the quark mass matrix for each assignment. We have
constructed the tens of thousands possible quark models by using Wolfram
Mathematica, and then find the best fit points by using the package MINUIT.

The modular symmetry is broken by the vacuum expectation value of the modulus
$\tau$ at high energy scale. We assume that modular invariance breaking scale
is around the GUT scale $2\times 10^{16}$ GeV. From the up type and down type
quark mass matrices $M_{u}$ and $M_{d}$ we can calculte the quark masses,
mixing angles and CP violation phase in terms of the input parameters of the
model. In order to find the point in parameter space which optimizes the
agreement between predictions and data, we generalize the numerical analysis
strategy to the quark sector, and we search the minimum of the $\chi^{2}$
contributions from quark mass ratios and CKM parameters. For the calculation
of $\chi^{2}_{\text{min}}$, we use the values of Yukawa couplings and the CKM
parameters calculated at the GUT scale from a minimal SUSY breaking scenario,
with SUSY breaking scale $M_{\text{SUSY}}=1$ TeV and $\tan\beta=7.5$,
$\bar{\eta}_{b}=0.09375$ [73],

$\displaystyle m_{u}/m_{c}=(1.9286\pm 0.6017)\times
10^{-3},~{}\quad~{}m_{c}/m_{t}=(2.7247\pm 0.1200)\times 10^{-3},$
$\displaystyle m_{d}/m_{s}=(5.0528\pm 0.6192)\times
10^{-2},~{}\quad~{}m_{s}/m_{b}=(1.7684\pm 0.0975)\times 10^{-2}\,,$
$\displaystyle m_{t}=89.5335~{}\text{GeV}\,,\quad
m_{b}=0.9336~{}\text{GeV}\,,\quad\delta^{q}_{CP}=69.213^{\circ}\pm
3.115^{\circ}\,,$ $\displaystyle\theta^{q}_{12}=0.22736\pm
0.00073,~{}~{}\theta^{q}_{13}=0.00338\pm 0.00012,~{}\theta^{q}_{23}=0.03888\pm
0.00062\,.$ (88)

where $\bar{\eta}_{b}$ denotes the contribution from SUSY threshold
corrections which mainly affects the bottom quark Yukawa coupling. After
examining many possible constructions, we succeeded in finding some models
which can accommodate the experimental data of quark masses and CKM matrix. In
the following, we present eight benchmark models with small number of free
parameters. The transformation properties of the quark fields under
$S^{\prime}_{4}$ and their modular weights are summarized in table 6.

| $u^{c}$ | $c^{c}$ | $t^{c}$ | $d^{c}$ | $s^{c}$ | $b^{c}$ | $Q_{1}$ | $Q_{2}$ | $Q_{3}$
---|---|---|---|---|---|---|---|---|---
Model I | $S^{\prime}_{4}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{\hat{2}}$ | $\mathbf{1^{\prime}}$ | $\mathbf{2}$ | $\mathbf{1}$
$k_{I}$ | $6-k_{Q_{3}}$ | $6-k_{Q_{3}}$ | $7-k_{Q_{3}}$ | $6-k_{Q_{3}}$ | $k_{Q_{3}}-2$ | $k_{Q_{3}}$
Model II | $S^{\prime}_{4}$ | $\mathbf{2}$ | $\mathbf{1^{\prime}}$ | $\mathbf{\hat{2}}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{1}$
$k_{I}$ | $6-k_{Q_{3}}$ | $6-k_{Q_{3}}$ | $7-k_{Q_{3}}$ | $6-k_{Q_{3}}$ | $k_{Q_{3}}-2$ | $k_{Q_{3}}$
Model III | $S^{\prime}_{4}$ | $\mathbf{\hat{2}}$ | $\mathbf{1^{\prime}}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{1}$
$k_{I}$ | $5-k_{Q_{3}}$ | $6-k_{Q_{3}}$ | $6-k_{Q_{3}}$ | $6-k_{Q_{3}}$ | $k_{Q_{3}}-2$ | $k_{Q_{3}}$
Model IV | $S^{\prime}_{4}$ | $\mathbf{2}$ | $\mathbf{\hat{1}^{\prime}}$ | $\mathbf{\hat{2}}$ | $\mathbf{1}$ | $\mathbf{3^{\prime}}$
$k_{I}$ | $2-k_{Q}$ | $5-k_{Q}$ | $5-k_{Q}$ | $4-k_{Q}$ | $k_{Q}$
Model V | $S^{\prime}_{4}$ | $\mathbf{\hat{2}}$ | $\mathbf{1^{\prime}}$ | $\mathbf{2}$ | $\mathbf{\hat{1}}$ | $\mathbf{3^{\prime}}$
$k_{I}$ | $3-k_{Q}$ | $6-k_{Q}$ | $4-k_{Q}$ | $5-k_{Q}$ | $k_{Q}$
Model VI | $S^{\prime}_{4}$ | $\mathbf{\hat{1}}$ | $\mathbf{1}$ |  $\mathbf{\hat{1}^{\prime}}$ | $\mathbf{\hat{1}}$ | $\mathbf{\hat{1}^{\prime}}$ | $\mathbf{\hat{1}}$ | $\mathbf{3}$
$k_{I}$ | $1-k_{Q}$ | $2-k_{Q}$ | $5-k_{Q}$ | $1-k_{Q}$ | $5-k_{Q}$ | $5-k_{Q}$ | $k_{Q}$
Model VII | $S^{\prime}_{4}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{\hat{1}}$ | $\mathbf{\hat{1}}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{3}$
$k_{I}$ | $2-k_{Q}$ | $4-k_{Q}$ | $5-k_{Q}$ | $1-k_{Q}$ | $2-k_{Q}$ | $6-k_{Q}$ | $k_{Q}$
Model VIII | $S^{\prime}_{4}$ | $\mathbf{\hat{1}}$ | $\mathbf{1}$ | $\mathbf{\hat{1}}$ | $\mathbf{\hat{1}}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{3}$
$k_{I}$ | $1-k_{Q}$ | $4-k_{Q}$ | $5-k_{Q}$ | $1-k_{Q}$ | $2-k_{Q}$ | $6-k_{Q}$ | $k_{Q}$
Table 6:  Transformation properties of the quark fields under the
$S^{\prime}_{4}$ modular symmetry and the modular weight assignments. The
Higgs fields $H_{u,d}$ are invariant under $S^{\prime}_{4}$ with vanishing
modular weight.

①

Model I with gCP: 9 free real parameters including $\texttt{Re}(\tau)$ and
$\texttt{Im}(\tau)$
In this model, left-handed quarks $Q$ and right-handed up quarks
$u^{c},~{}c^{c},~{}t^{c}$ are assigned to a direct sum of doublet and singlet
$\mathbf{2}\oplus\mathbf{1}$ of $S^{\prime}_{4}$, the right-handed down quarks
$d^{c}$, $s^{c}$, $b^{c}$ are assigned to
$\mathbf{\hat{2}}\oplus\mathbf{1^{\prime}}$ of $S^{\prime}_{4}$. For
convenience, we use the subscript “$D$” to denote the doublet assignment, i.e.
$Q_{D}\equiv(Q_{1},~{}Q_{2})^{T}$, $u^{c}_{D}\equiv(u^{c},~{}d^{c})^{T}$,
$d^{c}_{D}\equiv(d^{c},~{}s^{c})^{T}$. The modular weights of these quark
superfields are set to

$k_{Q_{3}}=k_{Q_{D}}+2=6-k_{u^{c}_{D}}=6-k_{t^{c}}=7-k_{d^{c}_{D}}=6-k_{b^{c}}\,.$
(89)

Thus the modular invariant superpotentials for quark masses read as follows,

$\displaystyle\hskip-14.45377pt\mathcal{W}_{u}=\alpha_{u1}(u^{c}_{D}Q_{D})_{\mathbf{1}}Y^{(4)}_{\mathbf{1}}H_{u}+\alpha_{u2}(u^{c}_{D}Q_{D}Y^{(4)}_{\mathbf{2}})_{\mathbf{1}}H_{u}+\beta_{t}t^{c}(Q_{D}Y^{(4)}_{\mathbf{2}})_{\mathbf{1}}+\gamma_{u}Q_{3}(u^{c}_{D}Y^{(6)}_{\mathbf{2}})_{\mathbf{1}}H_{u}\,,$
$\displaystyle\hskip-14.45377pt\mathcal{W}_{d}=\alpha_{d}(d^{c}_{D}Q_{D}Y^{(5)}_{\mathbf{\hat{2}}})_{\mathbf{1}}H_{d}+\beta_{d}b^{c}(Q_{D}Y^{(4)}_{\mathbf{2}})_{\mathbf{1^{\prime}}}H_{d}+\gamma_{d}b^{c}Q_{3}Y^{(6)}_{\mathbf{1^{\prime}}}H_{d}\,.$
(90)

From the CG coefficients of $S^{\prime}_{4}$ group in Appendix A, we find the
up and down quark mass matrices are given by

$\displaystyle
M_{u}=\begin{pmatrix}\alpha_{u2}Y_{2}^{(4)}&\alpha_{u1}Y_{1}^{(4)}&\gamma_{u}Y_{4}^{(6)}\\\
\alpha_{u1}Y_{1}^{(4)}&\alpha_{u2}Y_{3}^{(4)}&\gamma_{u}Y_{3}^{(6)}\\\
\beta_{t}Y_{3}^{(4)}&\beta_{t}Y_{2}^{(4)}&0\\\ \end{pmatrix}v_{u}\,,\quad
M_{d}=\begin{pmatrix}\alpha_{d}Y_{1}^{(5)}&0&0\\\
0&-\alpha_{d}Y_{2}^{(5)}&0\\\
-\beta_{d}Y_{3}^{(4)}&\beta_{d}Y_{2}^{(4)}&\gamma_{d}Y_{1}^{(6)}\\\
\end{pmatrix}v_{d}\,,$ (91)

We see that this model makes use of five real positive parameters
$\alpha_{u1,d}$, $\beta_{t,d}$, $\gamma_{d}$ and two complex parameter
$\alpha_{u2},\gamma_{u}$ to describe quark masses and CKM matrix. If we impose
gCP symmetry on this model, $\alpha_{u2}$ and $\gamma_{u}$ restricted to be
real and they can be either positive or negative. A good agreement between
data and predictions is obtained for the following values of input parameters

$\begin{gathered}\langle\tau\rangle=-0.43847+0.90997i\,,\quad\alpha_{u2}/\alpha_{u1}=-1.8814\,,\quad\gamma_{u}/\alpha_{u1}=0.1846\,,\\\
\beta_{t}/\alpha_{u1}=719.01\,,\quad\beta_{d}/\alpha_{d}=23.3376\,,\quad\gamma_{d}/\alpha_{d}=0.0225\,,\\\
\alpha_{u1}v_{u}=0.000796~{}\text{GeV},\quad\alpha_{d}v_{d}=0.000254~{}\text{GeV}\,.\end{gathered}$
(92)

The quark mass ratios and mixing parameters are determined to be

$\begin{gathered}\theta^{q}_{12}=0.22732\,,\quad\theta^{q}_{13}=0.00338\,,\quad\theta^{q}_{23}=0.03880\,,\quad\delta^{q}_{CP}=68.095^{\circ}\,,\\\
m_{u}/m_{c}=0.001927\,,~{}m_{c}/m_{t}=0.002726\,,~{}m_{d}/m_{s}=0.060247\,,~{}m_{s}/m_{b}=0.017679\,.\end{gathered}$
(93)

②

Model II with gCP: 9 free real parameters including $\texttt{Re}(\tau)$ and
$\texttt{Im}(\tau)$
In this model, the representation assignments for quark fields are the same as
Model I except changing the right-handed quarks $t^{c},~{}b^{c}$ to
$\mathbf{1^{\prime}},~{}\mathbf{1}$ of $S^{\prime}_{4}$. The modular weights
of these quark superfields are set to

$k_{Q_{3}}=k_{Q_{D}}+2=6-k_{u^{c}_{D}}=6-k_{t^{c}}=7-k_{d^{c}_{D}}=6-k_{b^{c}}\,.$
(94)

Thus the modular invariant superpotentials for quark masses read as follows,

$\displaystyle\hskip-14.45377pt\mathcal{W}_{u}=\alpha_{u1}(u^{c}_{D}Q_{D})_{\mathbf{1}}Y^{(4)}_{\mathbf{1}}H_{u}+\alpha_{u2}(u^{c}_{D}Q_{D}Y^{(4)}_{\mathbf{2}})_{\mathbf{1}}H_{u}+\beta_{t}t^{c}(Q_{D}Y^{(4)}_{\mathbf{2}})_{\mathbf{1^{\prime}}}+\gamma_{u}Q_{3}(u^{c}_{D}Y^{(6)}_{\mathbf{2}})_{\mathbf{1}}H_{u}\,,$
$\displaystyle\hskip-14.45377pt\mathcal{W}_{d}=\alpha_{d}(d^{c}_{D}Q_{D}Y^{(5)}_{\mathbf{\hat{2}}})_{\mathbf{1}}H_{d}+\beta_{d}b^{c}(Q_{D}Y^{(4)}_{\mathbf{2}})_{\mathbf{1}}H_{d}+\gamma_{d}b^{c}Q_{3}Y^{(6)}_{\mathbf{1}}H_{d}\,.$
(95)

From the CG coefficients of $S^{\prime}_{4}$ group in Appendix A, we find the
up and down quark mass matrices are given by

$\displaystyle
M_{u}=\begin{pmatrix}\alpha_{u2}Y_{2}^{(4)}&\alpha_{u1}Y_{1}^{(4)}&\gamma_{u}Y_{4}^{(6)}\\\
\alpha_{u1}Y_{1}^{(4)}&\alpha_{u2}Y_{3}^{(4)}&\gamma_{u}Y_{3}^{(6)}\\\
-\beta_{t}Y_{3}^{(4)}&\beta_{t}Y_{2}^{(4)}&0\\\ \end{pmatrix}v_{u}\,,\quad
M_{d}=\begin{pmatrix}\alpha_{d}Y_{1}^{(5)}&0&0\\\
0&-\alpha_{d}Y_{2}^{(5)}&0\\\
\beta_{d}Y_{3}^{(4)}&\beta_{d}Y_{2}^{(4)}&\gamma_{d}Y_{2}^{(6)}\\\
\end{pmatrix}v_{d}\,,$ (96)

We see that this model make use of five real parameters $\alpha_{u1,d}$,
$\beta_{t,d}$, $\gamma_{d}$ and two complex parameter $\alpha_{u2},\gamma_{u}$
to describe quark masses and CKM matrix. If we impose gCP symmetry on this
model, $\alpha_{u1},\gamma_{u}$ are either positive or negative. A good
agreement between data and predictions is obtained for the following values of
input parameters

$\begin{gathered}\langle\tau\rangle=0.48945+0.94228i\,,\quad\alpha_{u2}/\alpha_{u1}=-2.13638\,,\quad\gamma_{u}/\alpha_{u1}=0.21630\,,\\\
\beta_{t}/\alpha_{u1}=814.674\,,\quad\beta_{d}/\alpha_{d}=22.74651\,,\quad\gamma_{d}/\alpha_{d}=0.01134\,,\\\
\alpha_{u1}v_{u}=0.000751~{}\text{GeV},\quad\alpha_{d}v_{d}=0.000278~{}\text{GeV}\,.\end{gathered}$
(97)

The quark mass ratios and mixing parameters are determined to be

$\begin{gathered}\theta^{q}_{12}=0.22731\,,\quad\theta^{q}_{13}=0.00338\,,\quad\theta^{q}_{23}=0.03876\,,\quad\delta^{q}_{CP}=67.916^{\circ}\,,\\\
m_{u}/m_{c}=0.001956\,,~{}m_{c}/m_{t}=0.002724\,,~{}m_{d}/m_{s}=0.060138\,,~{}m_{s}/m_{b}=0.017783\,.\end{gathered}$
(98)

③

Model III with gCP: 9 free real parameters including $\texttt{Re}(\tau)$ and
$\texttt{Im}(\tau)$
In this model, the Irrep assignments are same as Model I except that the
assignments of the right-handed up quarks and down quarks are interchanged.
The modular weights of these quark superfields are set to

$k_{Q_{3}}=k_{Q_{D}}+2=5-k_{u^{c}_{D}}=6-k_{t^{c}}=6-k_{d^{c}_{D}}=6-k_{b^{c}}$
(99)

Thus the modular invariant superpotentials for quark masses read as follows,

$\displaystyle\hskip-14.45377pt\mathcal{W}_{u}=\alpha_{u}(d^{c}_{D}Q_{D}Y^{(3)}_{\mathbf{\hat{1}^{\prime}}})_{\mathbf{1}}H_{u}+\beta_{u}t^{c}(Q_{D}Y^{(4)}_{\mathbf{2}})_{\mathbf{1^{\prime}}}H_{u}+\gamma_{u}t^{c}Q_{3}Y^{(6)}_{\mathbf{1^{\prime}}}H_{u}\,,$
(100)
$\displaystyle\hskip-14.45377pt\mathcal{W}_{d}=\alpha_{d1}(u^{c}_{D}Q_{D})_{\mathbf{1}}Y^{(4)}_{\mathbf{1}}H_{u}+\alpha_{d2}(u^{c}_{D}Q_{D}Y^{(4)}_{\mathbf{2}})_{\mathbf{1}}H_{d}+\beta_{d}b^{c}(Q_{D}Y^{(4)}_{\mathbf{2}})_{\mathbf{1}}+\gamma_{d}Q_{3}(u^{c}_{D}Y^{(6)}_{\mathbf{2}})_{\mathbf{1}}H_{d}\,.$

From the CG coefficients of $S^{\prime}_{4}$ group in Appendix A, we find the
up and down quark mass matrices are given by

$\displaystyle\quad M_{u}=\begin{pmatrix}0&\alpha_{u}Y_{1}^{(3)}&0\\\
\alpha_{u}Y_{1}^{(3)}&0&0\\\
-\beta_{u}Y_{3}^{(4)}&\beta_{u}Y_{2}^{(4)}&\gamma_{u}Y_{1}^{(6)}\\\
\end{pmatrix}v_{u}\,,M_{d}=\begin{pmatrix}\alpha_{d2}Y_{2}^{(4)}&\alpha_{d1}Y_{1}^{(4)}&\gamma_{d}Y_{4}^{(6)}\\\
\alpha_{d1}Y_{1}^{(4)}&\alpha_{d2}Y_{3}^{(4)}&\gamma_{d}Y_{3}^{(6)}\\\
\beta_{d}Y_{3}^{(4)}&\beta_{d}Y_{2}^{(4)}&0\\\ \end{pmatrix}v_{d}\,,$ (101)

We see that this model make use of five real parameters $\alpha_{u,d1}$,
$\beta_{u,d}$, $\gamma_{u}$ and two complex parameter $\alpha_{d2},\gamma_{d}$
to describe quark masses and CKM matrix. If we impose gCP symmetry on this
model, $\alpha_{d2},\gamma_{d}$ are either positive or negative. A good
agreement between data and predictions is obtained for the following values of
input parameters

$\begin{gathered}\langle\tau\rangle=0.45221+0.92620i\,,\quad\beta_{u}/\gamma_{u}=124.4933\,,\quad\gamma_{u}/\beta_{u}=0.04754\,,\\\
\alpha_{d2}/\alpha_{d1}=3.14749\,,\quad\gamma_{d}/\alpha_{d1}=-0.30559\,,\quad\beta_{d}/\alpha_{d1}=183.16024\,,\\\
\alpha_{u}v_{u}=0.00476~{}\text{GeV},\quad\alpha_{d1}v_{d}=3.3723\times
10^{-5}~{}\text{GeV}\,.\end{gathered}$ (102)

The quark mass ratios and mixing parameters are determined to be

$\begin{gathered}\theta^{q}_{12}=0.22736\,,\quad\theta^{q}_{13}=0.00333\,,\quad\theta^{q}_{23}=0.03888\,,\quad\delta^{q}_{CP}/^{\circ}=69.214\,,\\\
m_{u}/m_{c}=0.003322\,,~{}m_{c}/m_{t}=0.002725\,,~{}m_{d}/m_{s}=0.05056\,,~{}m_{s}/m_{b}=0.017689\,.\end{gathered}$
(103)

④

Model IV with gCP: 9 free real parameters including $\texttt{Re}(\tau)$ and
$\texttt{Im}(\tau)$
In this model, we assume the left-handed quarks $Q$ transform as triplet
$\mathbf{3^{\prime}}$ of $S^{\prime}_{4}$, the right-handed up quark fields
$u^{c}$, $c^{c}$, $t^{c}$ transform as a direct sum of doublet and singlet
$\mathbf{2}\oplus\mathbf{\hat{1}^{\prime}}$,, the the right-handed down quark
fields $d^{c}$, $s^{c}$, $b^{c}$ transform as
$\mathbf{\hat{2}}\oplus\mathbf{1}$ of $S^{\prime}_{4}$. The modular weights of
these quark superfields are set to

$k_{Q}=2-k_{u_{D}^{c}}=5-k_{t^{c}}=5-k_{d_{D}^{c}}=4-k_{b^{c}}\,.$ (104)

Thus the modular invariant superpotentials for quark masses read as follows,

$\displaystyle\hskip-14.45377pt\mathcal{W}_{u}=\alpha_{u}(u^{c}_{D}QY_{\mathbf{3}}^{(2)})_{\mathbf{1}}H_{u}+\beta_{u}t^{c}(QY_{\mathbf{\hat{3}^{\prime}},I}^{(5)})_{\mathbf{\hat{1}}}H_{u}+\gamma_{u}t^{c}(QY_{\mathbf{\hat{3}^{\prime}},II}^{(5)})_{\mathbf{\hat{1}}}H_{u}\,,$
$\displaystyle\hskip-14.45377pt\mathcal{W}_{d}=\alpha_{d}(d^{c}_{D}QY_{\mathbf{\hat{3}}}^{(5)})_{\mathbf{1}}H_{d}+\beta_{d}(d^{c}_{D}QY_{\mathbf{\hat{3}^{\prime}},I}^{(5)})_{\mathbf{1}}H_{d}+\gamma_{d}(d^{c}_{D}QY_{\mathbf{\hat{3}^{\prime}},II}^{(5)})_{\mathbf{1}}H_{d}+\delta_{d}b^{c}(QY_{\mathbf{3}^{\prime}}^{(4)})_{\mathbf{1}}H_{d}\,.$
(105)

From the CG coefficients of $S^{\prime}_{4}$ group in Appendix A, we find the
up and down quark mass matrices are given by

$\displaystyle
M_{u}=\begin{pmatrix}-\alpha_{u}Y_{4}^{(2)}&-\alpha_{u}Y_{3}^{(2)}&-\alpha_{u}Y_{5}^{(2)}\\\
\alpha_{u}Y_{5}^{(2)}&\alpha_{u}Y_{4}^{(2)}&\alpha_{u}Y_{3}^{(2)}\\\
\beta_{u}Y_{6}^{(5)}+\gamma_{u}Y_{9}^{(5)}&\beta_{u}Y_{8}^{(5)}+\gamma_{u}Y_{11}^{(5)}&\beta_{u}Y_{7}^{(5)}+\gamma_{u}Y_{10}^{(5)}\\\
\end{pmatrix}v_{u}\,,$ $\displaystyle
M_{d}=\begin{pmatrix}\alpha_{d}Y_{4}^{(5)}-\beta_{d}Y_{7}^{(5)}-\gamma_{d}Y_{10}^{(5)}&\alpha_{d}Y_{3}^{(5)}-\beta_{d}Y_{6}^{(5)}-\gamma_{d}Y_{9}^{(5)}&\alpha_{d}Y_{5}^{(5)}-\beta_{d}Y_{8}^{(5)}-\gamma_{d}Y_{11}^{(5)}\\\
\alpha_{d}Y_{5}^{(5)}+\beta_{d}Y_{8}^{(5)}+\gamma_{d}Y_{11}^{(5)}&\alpha_{d}Y_{4}^{(5)}+\beta_{d}Y_{7}^{(5)}+\gamma_{d}Y_{10}^{(5)}&\alpha_{d}Y_{3}^{(5)}+\beta_{d}Y_{6}^{(5)}+\gamma_{d}Y_{9}^{(5)}\\\
\delta_{d}Y_{7}^{(4)}&\delta_{d}Y_{9}^{(4)}&\delta_{d}Y_{8}^{(4)}\\\
\end{pmatrix}v_{d}\,.$

It can been seen that there are four real parameters $\alpha_{u,d}$,
$\beta_{u}$, $\delta_{d}$ and three complex parameter $\gamma_{u,d},\beta_{d}$
to describe quark masses and CKM matrix. If we impose gCP symmetry on this
model, $\gamma_{u,d},\beta_{d}$ are either positive or negative. A good
agreement between data and predictions is obtained for the following values of
input parameters

$\begin{gathered}\langle\tau\rangle=-0.33817+1.47789i\,,\quad\beta_{u}/\alpha_{u}=0.19597\,,\quad\gamma_{u}/\alpha_{u}=19.87307\,,\\\
\delta_{d}/\alpha_{d}=1.32375\,,\quad\beta_{d}/\alpha_{d}=1.76097\,,\quad\gamma_{d}/\alpha_{d}=0.10850\,,\\\
\alpha_{u}v_{u}=0.01653~{}\text{GeV},\quad\alpha_{d}v_{d}=0.00117~{}\text{GeV}\,.\end{gathered}$
(107)

The quark mass ratios and mixing parameters are determined to be

$\begin{gathered}\theta^{q}_{12}=0.22747\,,\quad\theta^{q}_{13}=0.00338\,,\quad\theta^{q}_{23}=0.03849\,,\quad\delta^{q}_{CP}/^{\circ}=70.166\,,\\\
m_{u}/m_{c}=0.001570\,,\quad m_{c}/m_{t}=0.002735\,,\quad
m_{d}/m_{s}=0.050425\,,\quad m_{s}/m_{b}=0.018259\,,\end{gathered}$ (108)

which are compatible with experimental data at $1\sigma$ level.

⑤

Model V with gCP: 9 free real parameters including $\texttt{Re}(\tau)$ and
$\texttt{Im}(\tau)$
In this model, we assume the left-handed quarks $Q$ transform as triplet
$\mathbf{3^{\prime}}$ of $S^{\prime}_{4}$, the right-handed up quark fields
$u^{c}$, $c^{c}$, $t^{c}$ transform as a direct sum of doublet and singlet
$\mathbf{\hat{2}}\oplus\mathbf{1^{\prime}}$,, the the right-handed down quark
fields $d^{c}$, $s^{c}$, $b^{c}$ transform as
$\mathbf{2}\oplus\mathbf{\hat{1}}$ of $S^{\prime}_{4}$. The modular weights of
these quark superfields are set to

$k_{Q}=3-k_{u_{D}^{c}}=6-k_{t^{c}}=4-k_{d_{D}^{c}}=5-k_{b^{c}}\,.$ (109)

Thus the modular invariant superpotentials for quark masses read as follows,

$\displaystyle\hskip-14.45377pt\mathcal{W}_{u}=\alpha_{u}(u^{c}_{D}QY_{\mathbf{\hat{3}}}^{(3)})_{\mathbf{1}}H_{u}+\beta_{u}(u^{c}_{D}QY_{\mathbf{\hat{3}^{\prime}}}^{(3)})_{\mathbf{1}}H_{u}+\gamma_{u}t^{c}(QY_{\mathbf{3},I}^{(6)})_{\mathbf{1^{\prime}}}H_{u}+\delta_{u}t^{c}(QY_{\mathbf{3},II}^{(6)})_{\mathbf{1^{\prime}}}H_{u}\,,$
$\displaystyle\hskip-14.45377pt\mathcal{W}_{d}=\alpha_{d}(d^{c}_{D}QY_{\mathbf{3}}^{(4)})_{\mathbf{1}}H_{d}+\beta_{d}(d^{c}_{D}QY_{\mathbf{3}^{\prime}}^{(4)})_{\mathbf{1}}H_{d}+\gamma_{d}b^{c}(QY_{\mathbf{\hat{3}}}^{(5)})_{\mathbf{\hat{1}^{\prime}}}H_{d}\,.$
(110)

From the CG coefficients of $S^{\prime}_{4}$ group in Appendix A, we find the
up and down quark mass matrices are given by

$\displaystyle
M_{u}=\begin{pmatrix}\alpha_{u}Y_{3}^{(3)}-\beta_{u}Y_{6}^{(3)}&\alpha_{u}Y_{2}^{(3)}-\beta_{u}Y_{5}^{(3)}&\alpha_{u}Y_{4}^{(3)}-\beta_{u}Y_{7}^{(3)}\\\
\beta_{u}Y_{7}^{(3)}+\alpha_{u}Y_{4}^{(3)}&\beta_{u}Y_{6}^{(3)}+\alpha_{u}Y_{3}^{(3)}&\beta_{u}Y_{5}^{(3)}+\alpha_{u}Y_{2}^{(3)}\\\
\gamma_{u}Y_{5}^{(6)}+\delta_{u}Y_{8}^{(6)}&\gamma_{u}Y_{7}^{(6)}+\delta_{u}Y_{10}^{(6)}&\gamma_{u}Y_{6}^{(6)}+\delta_{u}Y_{9}^{(6)}\\\
\end{pmatrix}v_{u}\,,$ $\displaystyle
M_{d}=\begin{pmatrix}\beta_{d}Y_{8}^{(4)}-\alpha_{d}Y_{5}^{(4)}&\beta_{d}Y_{7}^{(4)}-\alpha_{d}Y_{4}^{(4)}&\beta_{d}Y_{9}^{(4)}-\alpha_{d}Y_{6}^{(4)}\\\
\beta_{d}Y_{9}^{(4)}+\alpha_{d}Y_{6}^{(4)}&\beta_{d}Y_{8}^{(4)}+\alpha_{d}Y_{5}^{(4)}&\beta_{d}Y_{7}^{(4)}+\alpha_{d}Y_{4}^{(4)}\\\
\gamma_{d}Y_{3}^{(5)}&\gamma_{d}Y_{5}^{(5)}&\gamma_{d}Y_{4}^{(5)}\\\
\end{pmatrix}v_{d}\,.$ (111)

It can been seen that there are four real parameters $\alpha_{u,d}$,
$\gamma_{u,d}$ and three complex parameter $\beta_{u,d},\delta_{u}$ to
describe quark masses and CKM matrix. If we impose gCP symmetry on this model,
$\beta_{u,d},\delta_{u}$ are either positive or negative. A good agreement
between data and predictions is obtained for the following values of input
parameters

$\begin{gathered}\langle\tau\rangle=-0.43820+2.04453i\,,\quad\gamma_{u}/\alpha_{u}=103.47956\,,\quad\beta_{u}/\alpha_{u}=-0.01046\,,\\\
\delta_{u}/\alpha_{u}=-51.88721\,,\quad\gamma_{d}/\alpha_{d}=362.65478\,,\quad\beta_{d}/\alpha_{d}=-189.08346\,,\\\
\alpha_{u}v_{u}=0.00450~{}\text{GeV},\quad\alpha_{d}v_{d}=5.6122\times
10^{-6}~{}\text{GeV}\,.\end{gathered}$ (112)

The quark mass ratios and mixing parameters are determined to be

$\begin{gathered}\theta^{q}_{12}=0.22736\,,\quad\theta^{q}_{13}=0.00338\,,\quad\theta^{q}_{23}=0.03888\,,\quad\delta^{q}_{CP}/^{\circ}=69.197\,,\\\
m_{u}/m_{c}=0.001928\,,\quad m_{c}/m_{t}=0.002724\,,\quad
m_{d}/m_{s}=0.050546\,,\quad m_{s}/m_{b}=0.017684\,.\end{gathered}$ (113)

which are compatible with experimental data at $1\sigma$ level.

⑥

Model VI without gCP: 10 free real parameters including $\texttt{Re}(\tau)$
and $\texttt{Im}(\tau)$

In this model, we assume the left-handed quarks $Q$ transform as triplet
$\mathbf{3}$ of $S^{\prime}_{4}$, the right-handed up quark fields $u^{c}$,
$c^{c}$, $t^{c}$ transform as $\mathbf{\hat{1}}$, $\mathbf{1}$,
$\mathbf{\hat{1}^{\prime}}$, the the right-handed down quark fields $d^{c}$,
$s^{c}$, $b^{c}$ transform as
$\mathbf{\hat{1}},\,\mathbf{\hat{1}^{\prime}},\,\mathbf{\hat{1}}$ of
$\Gamma^{\prime}_{4}$. The modular weights of these quark superfields are set
to

$k_{Q}=1-k_{u^{c}}=2-k_{c^{c}}=5-k_{t^{c}}=1-k_{d^{c}}=5-k_{s^{c}}=5-k_{b^{c}}\,.$
(114)

Thus the modular invariant superpotentials for quark masses read as follows,

$\displaystyle\hskip-14.45377pt\mathcal{W}_{u}=\alpha_{u}u^{c}(QY^{(1)}_{\mathbf{\hat{3}^{\prime}}})_{\mathbf{\hat{1}^{\prime}}}H_{u}+\beta_{u}c^{c}(QY^{(2)}_{\mathbf{3}})_{\mathbf{1}}H_{u}+\gamma_{u}t^{c}(QY^{(5)}_{\mathbf{\hat{3}}})_{\mathbf{\hat{1}}}H_{u}\,,$
$\displaystyle\hskip-14.45377pt\mathcal{W}_{d}=\alpha_{d}d^{c}(QY^{(1)}_{\mathbf{\hat{3}^{\prime}}})_{\mathbf{\hat{1}^{\prime}}}H_{d}+\beta_{d}s^{c}(QY^{(5)}_{\mathbf{\hat{3}}})_{\mathbf{\hat{1}}}H_{d}+\gamma_{d1}b^{c}(QY^{(5)}_{\mathbf{\hat{3}^{\prime}},I})_{\mathbf{\hat{1}^{\prime}}}H_{d}+\gamma_{d2}b^{c}(QY^{(5)}_{\mathbf{\hat{3}^{\prime}},II})_{\mathbf{\hat{1}^{\prime}}}H_{d}\,.$
(115)

The phases of $\alpha_{u}$, $\beta_{u}$, $\gamma_{u}$, $\alpha_{d}$,
$\beta_{d}$ and $\gamma_{d1}$ can be absorbed into the quark fields while the
phase of $\gamma_{d2}$ can not be removed by field redefinition. From the CG
coefficients of $S^{\prime}_{4}$ group in Appendix A, we find the up and down
quark mass matrices are given by

$\displaystyle
M_{u}=\begin{pmatrix}\alpha_{u}Y^{(1)}_{1}&\alpha_{u}Y^{(1)}_{3}&\alpha_{u}Y^{(1)}_{2}\\\
\beta_{u}Y^{(2)}_{3}&\beta_{u}Y^{(2)}_{5}&\beta_{u}Y^{(2)}_{4}\\\
\gamma_{u}Y^{(5)}_{3}&\gamma_{u}Y^{(5)}_{5}&\gamma_{u}Y^{(5)}_{4}\\\
\end{pmatrix}v_{u}\,,$ $\displaystyle
M_{d}=\begin{pmatrix}\alpha_{d}Y^{(1)}_{1}&\alpha_{d}Y^{(1)}_{3}&\alpha_{d}Y^{(1)}_{2}\\\
\beta_{d}Y^{(5)}_{3}&\beta_{d}Y^{(5)}_{5}&\beta_{d}Y^{(5)}_{4}\\\
\gamma_{d1}Y^{(5)}_{6}+\gamma_{d2}Y^{(5)}_{9}&\gamma_{d1}Y^{(5)}_{8}+\gamma_{d2}Y^{(5)}_{11}&\gamma_{d1}Y^{(5)}_{7}+\gamma_{d2}Y^{(5)}_{10}\\\
\end{pmatrix}v_{d}\,,$ (116)

We see that this model make uses of six real parameters $\alpha_{u,d}$,
$\beta_{u,d}$, $\gamma_{u,d1}$ and one complex parameter $\gamma_{d2}$ to
describe quark masses and CKM matrix. A good agreement between data and
predictions is obtained for the following values of input parameters

$\begin{gathered}\langle\tau\rangle=-0.4999+0.8958i\,,\quad\beta_{u}/\alpha_{u}=62.5142\,,\quad\gamma_{u}/\alpha_{u}=0.00104\,,\\\
\beta_{d}/\alpha_{d}=0.7378\,,\quad\gamma_{d1}/\alpha_{d}=1.4946\,,\quad\gamma_{d2}/\alpha_{d}=-0.1958-0.2762i\,,\\\
\alpha_{u}v_{u}=0.0799~{}\text{GeV},\quad\alpha_{d}v_{d}=0.0009~{}\text{GeV}\,.\end{gathered}$
(117)

The quark mass ratios and mixing parameters are determined to be

$\begin{gathered}\theta^{q}_{12}=0.22731\,,\quad\theta^{q}_{13}=0.00298\,,\quad\theta^{q}_{23}=0.04873\,,\quad\delta^{q}_{CP}=67.1762^{\circ}\,,\\\
m_{u}/m_{c}=0.00204\,,\quad m_{c}/m_{t}=0.00268\,,\quad
m_{d}/m_{s}=0.05182\,,\quad m_{s}/m_{b}=0.01309\,,\end{gathered}$ (118)

which are compatible with the experimental data in Eq. (88) except that
$\theta^{q}_{23}$ is somewhat larger. Notice that the top and bottom quark
masses can be reproduced by adjusting the parameters $\alpha_{u}$ and
$\alpha_{d}$.

⑦

Model VII with gCP: 10 free real parameters including $\texttt{Re}(\tau)$ and
$\texttt{Im}(\tau)$
The left-handed quarks $Q$ are assigned to triplet $\mathbf{3}$ of
$S^{\prime}_{4}$, $u^{c}$, $c^{c}$ and $t^{c}$ transform as $\mathbf{1}$,
$\mathbf{1}$ and $\mathbf{\hat{1}}$ respectively under $S^{\prime}_{4}$, down
type quarks $d^{c},\,s^{c},\,b^{c}$ transform as
$\mathbf{\hat{1}},\,\mathbf{1},\,\mathbf{1}$ respectively. Note that $u^{c}$
and $c^{c}$ are distinguished by their different modular weight, and similarly
for $s^{c}$ and $b^{c}$. We choose of the weights of quark fields to fulfill
$k_{Q}=2-k_{u^{c}}=4-k_{c^{c}}=5-k_{t^{c}}=1-k_{d^{c}}=2-k_{s^{c}}=6-k_{b^{c}}$.
The superpotentials of the quark sector are given by,

$\displaystyle\mathcal{W}_{u}=\alpha_{u}u^{c}(QY^{(2)}_{\mathbf{3}})_{\mathbf{\hat{1}}}H_{u}+\beta_{u}c^{c}(QY^{(4)}_{\mathbf{3}})_{\mathbf{1}}H_{u}+\gamma_{u1}t^{c}(QY^{(5)}_{\mathbf{\hat{3}^{\prime}},I})_{\mathbf{\hat{1}^{\prime}}}H_{u}+\gamma_{u2}t^{c}(QY^{(5)}_{\mathbf{\hat{3}^{\prime}},II})_{\mathbf{\hat{1}^{\prime}}}H_{u}\,,$
$\displaystyle\mathcal{W}_{d}=\alpha_{d}d^{c}(QY^{(1)}_{\mathbf{\hat{3}^{\prime}}})_{\mathbf{\hat{1}^{\prime}}}H_{d}+\beta_{d}s^{c}(QY^{(2)}_{\mathbf{3}})_{\mathbf{1}}H_{d}+\gamma_{d1}b^{c}(QY^{(6)}_{\mathbf{3},I})_{\mathbf{1}}H_{d}+\gamma_{d2}b^{c}(QY^{(6)}_{\mathbf{3},II})_{\mathbf{1}}H_{d}\,,$
(119)

which lead to the quark mass matrices,

$\displaystyle
M_{u}=\begin{pmatrix}\alpha_{u}Y^{(2)}_{3}&\alpha_{u}Y^{(2)}_{5}&\alpha_{u}Y^{(2)}_{4}\\\
\beta_{u}Y^{(4)}_{4}&\beta_{u}Y^{(4)}_{6}&\beta_{u}Y^{(4)}_{5}\\\
\gamma_{u1}Y^{(5)}_{6}+\gamma_{u2}Y^{(5)}_{9}~{}&\gamma_{u1}Y^{(5)}_{8}+\gamma_{u2}Y^{(5)}_{11}~{}&\gamma_{u1}Y^{(5)}_{7}+\gamma_{u2}Y^{(5)}_{10}\\\
\end{pmatrix}v_{u}\,,$ $\displaystyle
M_{d}=\begin{pmatrix}\alpha_{d}Y^{(1)}_{1}&\alpha_{d}Y^{(1)}_{3}&\alpha_{d}Y^{(1)}_{2}\\\
\beta_{d}Y^{(2)}_{3}&\beta_{d}Y^{(2)}_{5}&\beta_{d}Y^{(2)}_{4}\\\
\gamma_{d1}Y^{(6)}_{5}+\gamma_{d2}Y^{(6)}_{8}~{}&\gamma_{d1}Y^{(6)}_{7}+\gamma_{d2}Y^{(6)}_{10}~{}&\gamma_{d1}Y^{(6)}_{6}+\gamma_{d2}Y^{(6)}_{9}\\\
\end{pmatrix}v_{d}\,,$ (120)

The parameters $\alpha_{u,d}$, $\beta_{u,d}$ and $\gamma_{u1,d1}$ can be made
real and positive by field redefinition while $\gamma_{u2}$ and $\gamma_{d2}$
are complex. If we impose CP as symmetry on model, all couplings are
constrained to be real, and $\gamma_{u2}$ and $\gamma_{d2}$ are either
positive or negative. The best fit values of input parameters and the
predictions for quark mass ratios and CKM mixing parameters are

$\displaystyle\langle\tau\rangle=-0.4362+1.8184i\,,\quad\beta_{u}/\alpha_{u}=9104.86\,,\quad\gamma_{u1}/\alpha_{u}=19.7442\,,\quad\gamma_{u2}/\alpha_{u}=-19.9232\,,$
$\displaystyle\beta_{d}/\alpha_{d}=0.02443\,,\quad\gamma_{d1}/\alpha_{d}=0.24788\,,\quad\gamma_{d2}/\alpha_{d}=-0.00214\,,$
$\displaystyle\alpha_{u}v_{u}=0.00009~{}\text{GeV},\quad\alpha_{d}v_{d}=0.00672~{}\text{GeV}\,,$
$\displaystyle\theta^{q}_{12}=0.22736\,,\quad\theta^{q}_{13}=0.00337\,,\quad\theta^{q}_{23}=0.03923\,,\quad\delta^{q}_{CP}=69.15^{\circ}\,,$
$\displaystyle
m_{u}/m_{c}=0.001812\,,~{}~{}m_{c}/m_{t}=0.002714\,,~{}~{}m_{d}/m_{s}=0.050526\,,~{}~{}m_{s}/m_{b}=0.017687\,.$
(121)

It is remarkable that all observables are within the $1\sigma$ experimental
ranges.

⑧

Model VIII with gCP: 10 free real parameters including $\texttt{Re}(\tau)$ and
$\texttt{Im}(\tau)$
This model is different from Model VII in the assignment of $u^{c}$ which is
assigned to $\mathbf{\hat{1}}$ with weight $k_{u^{c}}=1-k_{Q}$. Thus the
superpotential in the quark sector reads as,

$\displaystyle\mathcal{W}_{u}=\alpha_{u}u^{c}(QY^{(1)}_{\mathbf{\hat{3}^{\prime}}})_{\mathbf{\hat{1}}}H_{u}+\beta_{u}c^{c}(QY^{(4)}_{\mathbf{3}})_{\mathbf{1}}H_{u}+\gamma_{u1}t^{c}(QY^{(5)}_{\mathbf{\hat{3}^{\prime}},I})_{\mathbf{\hat{1}^{\prime}}}H_{u}+\gamma_{u2}t^{c}(QY^{(5)}_{\mathbf{\hat{3}^{\prime}},II})_{\mathbf{\hat{1}^{\prime}}}H_{u}\,,$
$\displaystyle\mathcal{W}_{d}=\alpha_{d}d^{c}(QY^{(1)}_{\mathbf{\hat{3}^{\prime}}})_{\mathbf{\hat{1}^{\prime}}}H_{d}+\beta_{d}s^{c}(QY^{(2)}_{\mathbf{3}})_{\mathbf{1}}H_{d}+\gamma_{d1}b^{c}(QY^{(6)}_{\mathbf{3},I})_{\mathbf{1}}H_{d}+\gamma_{d2}b^{c}(QY^{(6)}_{\mathbf{3},II})_{\mathbf{1}}H_{d}\,,$
(122)

where all the couplings are enforced to be real by the generalized CP
symmetry. The resulting quark mass matrices are different from those of Eq.
(96) in the first row of $M_{u}$,

$\displaystyle
M_{u}=\begin{pmatrix}\alpha_{u}Y^{(1)}_{1}&\alpha_{u}Y^{(1)}_{3}&\alpha_{u}Y^{(1)}_{2}\\\
\beta_{u}Y^{(4)}_{4}&\beta_{u}Y^{(4)}_{6}&\beta_{u}Y^{(4)}_{5}\\\
\gamma_{u1}Y^{(5)}_{6}+\gamma_{u2}Y^{(5)}_{9}~{}&\gamma_{u1}Y^{(5)}_{8}+\gamma_{u2}Y^{(5)}_{11}~{}&\gamma_{u1}Y^{(5)}_{7}+\gamma_{u2}Y^{(5)}_{10}\\\
\end{pmatrix}v_{u}\,,$ $\displaystyle
M_{d}=\begin{pmatrix}\alpha_{d}Y^{(1)}_{1}&\alpha_{d}Y^{(1)}_{3}&\alpha_{d}Y^{(1)}_{2}\\\
\beta_{d}Y^{(2)}_{3}&\beta_{d}Y^{(2)}_{5}&\beta_{d}Y^{(2)}_{4}\\\
\gamma_{d1}Y^{(6)}_{5}+\gamma_{d2}Y^{(6)}_{8}~{}&\gamma_{d1}Y^{(6)}_{7}+\gamma_{d2}Y^{(6)}_{10}~{}&\gamma_{d1}Y^{(6)}_{6}+\gamma_{d2}Y^{(6)}_{9}\\\
\end{pmatrix}v_{d}\,,$ (123)

The numerical minimisation of the $\chi^{2}$ function gives the best fit point
of the model,

$\displaystyle\langle\tau\rangle=0.0617+1.5127i\,,~{}~{}\beta_{u}/\alpha_{u}=16.5002\,,~{}~{}\gamma_{u1}/\alpha_{u}=3919.03\,,~{}~{}\gamma_{u2}/\alpha_{u}=-1945.11\,,$
$\displaystyle\beta_{d}/\alpha_{d}=91.1983\,,\quad\gamma_{d1}/\alpha_{d}=0.3027\,,\quad\gamma_{d2}/\alpha_{d}=-1.997\,,$
$\displaystyle\alpha_{u}v_{u}=0.00019~{}\text{GeV},\quad\alpha_{d}v_{d}=0.00035~{}\text{GeV}\,,$
$\displaystyle\theta^{q}_{12}=0.22737\,,\quad\theta^{q}_{13}=0.00338\,,\quad\theta^{q}_{23}=0.03889\,,\quad\delta^{q}_{CP}/^{\circ}=69.21\,,$
$\displaystyle
m_{u}/m_{c}=0.001964\,,~{}~{}m_{c}/m_{t}=0.002729\,,~{}~{}m_{d}/m_{s}=0.05037\,,~{}~{}m_{s}/m_{b}=0.01767\,,$
(124)

which are compatible with experimental data at $1\sigma$ level as well.

##  7 Toward quark-lepton unification

As shown in sections 5 and 6, the charged lepton masses and the neutrino
oscillation data can be explained very well in the $S^{\prime}_{4}$ modular
symmetry models, and the $S^{\prime}_{4}$ modular symmetry can also help to
understand the quark mass hierarchies and CKM mixing matrix. In this section,
we shall investigate the quark-lepton unified models which can explain the
experimental data of quarks and leptons simultaneously for certain common
value of modulus $\tau$. Such kind of models at level $N=3$ has been studied
[33, 18, 21, 35] in the literature. It is generally not an easy task to
construct a quark-lepton unified model. After trying many possibilities, we
find out a realistic quark-lepton unified model with small number of free
parameters, the quark sector is described by Model VIII of section 6, we
extend the lepton models by introducing higher weight modular forms, and the
generalized CP symmetry in combination with $S^{\prime}_{4}$ modular symmetry
is imposed to match the quark sector.

We assign the three generations of left-handed lepton doublets $L$ and of
right-handed neutrinos $N^{c}$ to two triplets $\mathbf{3}$ of
$S^{\prime}_{4}$, while the right-handed charged leptons $E_{1}^{c}$,
$E_{2}^{c}$ and $E_{3}^{c}$ transform as $\mathbf{\hat{1}}$,
$\mathbf{\hat{1}^{\prime}}$ and $\mathbf{1}$ respectively. We choose the
modular weights of lepton fields as $k_{N^{c}}=2$, $k_{L}=-2$, $k_{e^{c}}=3$,
$k_{\mu^{c}}=5$ and $k_{\tau^{c}}=6$. Then the modular invariant
superpotential of the lepton sector is given by

$\displaystyle\mathcal{W}_{e}=\alpha_{e}(E^{c}_{1}LY^{(1)}_{\mathbf{\hat{3}^{\prime}}})_{\mathbf{1}}H_{d}+\beta_{e}(E^{c}_{2}LY^{(3)}_{\mathbf{\hat{3}}})_{\mathbf{1}}H_{d}+\gamma_{e}(E^{c}_{3}LY^{(4)}_{\mathbf{3}})_{\mathbf{1}}H_{d}\,,$
$\displaystyle\mathcal{W}_{\nu}=g(N^{c}L)_{\mathbf{1}}H_{u}+\Lambda_{1}(N^{c}N^{c})_{\mathbf{1}}Y^{(4)}_{\mathbf{1}}+\Lambda_{2}((N^{c}N^{c})_{\mathbf{2}}Y^{(4)}_{\mathbf{2}})_{\mathbf{1}}+\Lambda_{3}((N^{c}N^{c})_{\mathbf{3^{\prime}}}Y^{(4)}_{\mathbf{3^{\prime}}})_{\mathbf{1}}\,,$
(125)

where all couplings $\alpha_{e}$, $\beta_{e}$, $\gamma_{e}$, $g_{1}$ and
$\Lambda_{1,2,3}$ are real because of the generalized CP invariance. We can
read out the lepton mass matrices as follows,

$\displaystyle
M_{e}=\begin{pmatrix}\alpha_{e}Y^{(1)}_{1}~{}&\alpha_{e}Y^{(1)}_{3}~{}&\alpha_{e}Y^{(1)}_{2}\\\
\beta_{\mu}Y^{(3)}_{2}~{}&\beta_{\mu}Y^{(3)}_{4}~{}&\beta_{\mu}Y^{(3)}_{3}\\\
\gamma_{\tau}Y^{(4)}_{4}~{}&\gamma_{\tau}Y^{(4)}_{6}~{}&\gamma_{\tau}Y^{(4)}_{5}\end{pmatrix}v_{d}\,,~{}~{}~{}M_{D}=g\begin{pmatrix}1~{}&0~{}&0\\\
0~{}&0~{}&1\\\ 0~{}&1~{}&0\end{pmatrix}v_{u}\,,$ $\displaystyle
M_{N}=\begin{pmatrix}\Lambda_{1}Y^{(4)}_{1}+2\Lambda_{3}Y^{(4)}_{7}~{}&\Lambda_{2}Y^{(4)}_{2}-\Lambda_{3}Y^{(4)}_{9}~{}&\Lambda_{2}Y^{(4)}_{3}-\Lambda_{3}Y^{(4)}_{8}\\\
\Lambda_{2}Y^{(4)}_{2}-\Lambda_{3}Y^{(4)}_{9}~{}&\Lambda_{2}Y^{(4)}_{3}+2\Lambda_{3}Y^{(4)}_{8}~{}&\Lambda_{1}Y^{(4)}_{1}-\Lambda_{3}Y^{(4)}_{7}\\\
\Lambda_{2}Y^{(4)}_{3}-\Lambda_{3}Y^{(4)}_{8}~{}&\Lambda_{1}Y^{(4)}_{1}-\Lambda_{3}Y^{(4)}_{7}~{}&\Lambda_{2}Y^{(4)}_{2}+2\Lambda_{3}Y^{(4)}_{9}\end{pmatrix}\,.$
(126)

The up type and down type quark mass matrices are given in Eq. (④ ‣ 6). It is
notable that the model has less free parameters than the number of observable
quantities including quark and lepton masses and mixing parameters. We perform
a comprehensive numerical scan over the parameter space, we find that good
agreement with experimental data can be achieved for the following value of
$\tau$ common to quark and lepton sectors,

$\langle\tau\rangle=0.0581+1.5088i\,,$ (127)

which is mainly determined by the quark masses and CKM mixing parameters.