# Retrieving texture zeros in 3+1 active-sterile neutrino framework under the
action of $A_{4}$ modular-invariants

Maibam Ricky Devi<EMAIL_ADDRESS>Department of Physics, Gauhati
University, Guwahati - 781014, Assam, India

###### Abstract

The flavour problem and the viability of texture zeros of the Majorana mass
matrix in the 3+1 active-sterile neutrino mixing scenario are investigated in
this work using a novel bottom-up technique where we leverage the pertinent
concepts of full modular group, modular invariants, and $A_{4}$ flavour
symmetry as theoretical tools for the explicit construction of neutrino
models. In this approach we treat each chiral field as modular forms in the
3+1 neutrino mixing which are constrained by the $A_{4}$ modular symmetry.
Using these techniques, we create straightforward predictive models that only
depends on a few parameters and simultaneously explains the observed pattern
of neutrino mixing without the need for fine-tuning, allowing us to perceive
the feasible zero textures of the Majorana mass matrix under the 3+1
framework.We discuss the implications of the allowed 3+1 zero textures by
analyzing the sterile neutrino parameters, providing insight into the flavor
problem and the viability of the Majorana mass matrix.

###### keywords:

$A_{4}$ flavour symmetry, 3+1 neutrino framework, neutrino mixing, modular
symmetry, active-sterile neutrino mixing.

##  1 Introduction

For decades, researchers have been intensely studying neutrino physics
following the groundbreaking discovery of neutrino oscillation by the Super-
Kamiokande experiment, which enabled the determination of the pattern of
neutrino mixing. Despite the progress made in recent years, there is still
much to be learned about neutrino flavor, prompting new efforts to find more
effective solutions to the flavour problem. The challenge lies in detecting
neutrinos, as they have very weak interactions with ordinary matter, and
accurately measuring the properties of neutrino oscillations. Through advances
in particle detectors and experimental techniques, researchers are continuing
to make strides in understanding the flavor problem in neutrino physics.

In the field of neutrino physics, physicists are currently grappling with two
important issues: the flavour problem in the Standard model and the search for
a fourth flavour neutrino, known as the sterile neutrino, and how it interacts
with other leptons and scalar fields. To address these issues, we have
developed a novel bottom-up approach that combines modular symmetries and
modular forms with the $A_{4}$ flavour symmetry [1]. By implementing this
$A_{4}$ modular symmetry, we can study the different texture zeros in the 3+1
neutrino mass matrices of active-sterile neutrino mixing. Incorporating the
$A_{4}$ modular symmetry into the allowed texture zeros of the 3+1 neutrino
mass matrix can be an extremely powerful tool in identifying the underlying
mixing pattern and intricate symmetries in neutrino. It provides a framework
for building elegant and efficient neutrino models with very few parameters
and rarely any need of fine-tuning. By utilizing modular symmetry, physicists
can predict the properties and behavior of neutrinos with a high degree of
accuracy, and modular forms offer a systematic and flexible way to build these
models.

One of the main advantages of using this modular invariance approach to build
neutrino models is that the modular symmetry acts as the source of flavour
symmetry. Instead of using the VEV of the scalar fields -flavons, to break
flavour groups in the conventional flavour space, we use the modulus $\tau$
for flavour symmetry breaking. This approach offers a new perspective on the
flavour problem and opens up new avenues for research in the field of neutrino
physics.

The use of different texture zeros in neutrino mass matrices has been
extensively studied as a potential solution to the flavor problem in the
Standard Model. In the 3+1 framework, the allowed texture-zero patterns of
lepton mass matrices have been elegantly explained and summarized in [2]. To
further understand the behaviour of the modular forms acting on the chiral
fields, we carefully examine which texture zeros( mentioned in [2] and shown
in B) would be viable under the action of $A_{4}$ modular invariance. We
systematically analyze these texture-zero patterns for different neutrino mass
models that can fit the experimental data that is based on the assumption that
the fermionic leptons and scalar fields transform as irreducible
representations of $A_{4}$ [3], [4], [5], [5], [6]. By exploring the allowed
texture zeros in neutrino mass matrices in active-sterile neutrino mixing, we
aim to provide a deeper understanding of the flavor problem in the Standard
Model. Our analysis takes into account the latest experimental data and
provides valuable insights into the behavior and interactions of the modular
forms for level N=3 which can be transformed to a finite flavor group $A_{4}$.

The concept of modular symmetry has gained significant attention in global
supersymmetry theories, particularly in addressing the flavor problem. To
tackle this pressing issue, researchers initially explored groups such as
$\Gamma_{2}\cong S_{3}$ [7, 8, 9, 10], $\Gamma_{3}\cong A_{4}$ [1, 11, 7, 12,
13, 14, 8, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35],, and $\Gamma_{4}\cong S_{4}$ [36, 37, 38, 39, 40, 41, 42, 24,
43]. As further research was conducted, other groups such as $\Gamma_{5}\cong
A_{5}$ [44, 45, 41], $\Gamma_{7}\cong PSL(2,Z_{7})$ [46],
$\Gamma^{\prime}_{3}\cong T^{\prime}$ [47, 48], $\Gamma^{\prime}_{4}\cong
S^{\prime}_{4}$ [49, 50] and $\Gamma^{\prime}_{5}\cong A^{\prime}_{5}$ [51,
52, 53] were also utilized to construct models that study quark and lepton
flavors and their unification. Yet, because to its infrequent fine tuning and
highly predictable findings on neutrino observables, this formalism has
recently attracted increased interest in the construction of neutrino models.

Several recent studies on modular forms include [54], [55], [56], [57], [58],
[59]. Theoretically, different phenomenological ansatz, such as texture
zeroes, hybrid textures, scaling matrices, and magic symmetry, to name a few,
have been proposed to increase the predictability and reduce the arbitrariness
of free parameters in the neutrino mass matrix. The realization of texture
zeroes, hybrid textures, broken scaling, broken $\mu-\tau$, and magic
symmetric neutrino models based on group symmetries has been explored to
understand the possible origin of such proposed ansatz and predictions on
neutrino masses and mixing [60], [61], [62], [63], [64], [65], [66], [67],
[68], [69], [70] and [71]. Several recent models of neutrinos based on $A_{4}$
and $A_{5^{\prime}}$ modular symmetry groups are discussed in the literature
[72], [73], [74], [75] and [76]. These models explore various topics such as
neutrino phenomenology, dark matter, leptogenesis, and lepton flavor
violations in different seesaw mechanisms. One can refer to the references
[77], [78], [79], [80], [81], [82],[83], [84], [85], [86], [87], [88], [89],
[90], [91] to gain insight on some highly sophisticated neutrino models based
on modular symmetry.

##  2 $A_{4}$ modular group of integral weight and level 3

We define the full modular group $SL_{2}(\mathbb{Z})$ (sometimes simply called
as the modular group) as a group of $2\times 2$ matrices with integer entries
and unit determinant. Mathematically it can be written in the set builder form
as (See ref Modular Forms: A Classical and Computational Introduction by Lloyd
J. P. Kilford)

$\displaystyle SL_{2}(\mathbb{Z}):=\Bigl{\\{}\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\Big{|}\quad a,b,c,d\in\mathbb{Z},$ (1) $\displaystyle
ad-bc=1\Bigr{\\}},$

The Special Linear group, $SL_{2}(\mathbb{Z})$ is usually denoted as $\Gamma$
, which is a homogeneous modular group. If N is a positive integer, then the
series of infinite normal subgroups of $\Gamma$ can be defined as principal
congruence subgroups of level N $(N\geq 2)$. Some congruence subgroups,
$\Gamma_{0}(N)$, $\Gamma_{1}(N)$, $\Gamma(N)$ of $SL_{2}(\mathbb{Z})$ can be
written as

$\displaystyle\Gamma_{0}(N):=\Bigl{\\{}\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in SL_{2}(\mathbb{Z})$ $\displaystyle:$ $\displaystyle
c\equiv 0$ $\displaystyle mod\;N\Bigr{\\}},$ (2)
$\displaystyle\Gamma_{1}(N):=\Bigl{\\{}\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in SL_{2}(\mathbb{Z}):$ $\displaystyle a\equiv d\equiv
1$ $\displaystyle mod\;N,$ (3) $\displaystyle c\equiv 0$ $\displaystyle
mod\;N\Bigr{\\}},$
$\displaystyle\Gamma(N):=\Bigl{\\{}\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in SL_{2}(\mathbb{Z}):$ $\displaystyle a\equiv d\equiv
1$ $\displaystyle mod\;N,$ (4) $\displaystyle b\equiv c\equiv 0$
$\displaystyle mod\;N\Bigr{\\}},$

If $\mathbb{H}$ represents the upper half of the complex plane such that
$\mathbb{H}=Im(\mathbb{\tau})>0$ with $\tau$ as the complex modulus, then
$\Gamma(N)$ acts on $\mathbb{H}$ using the the linear fractional
transformation given by

$\displaystyle\gamma=\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in\Gamma:~{}\tau\rightarrow\gamma\tau=\frac{a\tau+b}{c\tau+d}\;,$
(5)

The compactified space of the quotient group $\mathbb{H}/\Gamma(N)$ is
obtained by adjoing all the cusp forms at $i\infty$ or rational real numbers.
Apart from $\Gamma$ we have the infinite modular group $\overline{\Gamma}$
which have the form :

$\displaystyle\overline{\Gamma}=\Bigl{\\{}\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)/(\pm 1)\quad a,b,c,d\in\mathbb{Z},$ (6) $\displaystyle
ad-bc=1\Bigr{\\}},$

where $\mathbb{Z}$ refers to the set of integers. Thus from 1 we can imply
that $\overline{\Gamma}\equiv\Gamma/{\pm 1}$ which makes it an inhomogeneous
modular group. Two generators, designated as $S$ and $T$, meet the
equation$S^{2}=(ST)^{3}=\mathbbm{1}$ for the modular group. The matrices for
the generators $S$ and $T$ are

$\displaystyle S=\left(\begin{matrix}0&1\\\ -1&0\end{matrix}\right)\;,\quad
T=\left(\begin{matrix}1&1\\\ 0&1\end{matrix}\right)$ (7)

Under $S$ and $T$, the complex modulus $\tau$ transforms as

$\displaystyle S:~{}\tau\rightarrow-\frac{1}{\tau}\;,\quad
T:~{}\tau\rightarrow\tau+1\;,$ (8)

Figure 1:

respectively.

The finite modular group, denoted as $\Gamma_{N}$, is the quotient group of
the modular group $\overline{\Gamma}$ by $\overline{\Gamma}(N)$. This group is
characterized by the condition that $T^{N}=\mathbbm{1}$ holds, except for the
relation $S^{2}=(ST)^{3}=\mathbbm{1}$. For specific values of $N$, namely
$N=2,3,4,5$, the finite modular group is isomorphic to certain well-known
groups. Specifically, we have $\Gamma_{2}\simeq S_{3}$, $\Gamma_{3}\simeq
A_{4}$, $\Gamma_{4}\simeq S_{4}$, and $\Gamma_{5}\simeq A_{5}$.

In this work we primarily use modular forms of level 3 that satisfies

$\displaystyle f(\gamma\tau)=(c\tau+d)^{2k}f(\tau)$
$\displaystyle\gamma=\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in\Gamma(3)$ (9)
$\displaystyle\Gamma(3):=\Bigl{\\{}\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in SL_{2}(\mathbb{Z}),$ (10)
$\displaystyle\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\equiv\left(\begin{matrix}1&0\\\
0&1\end{matrix}\right)(mod\;3)\Bigr{\\}},$

The fundamental domain of $\overline{\Gamma}(3)$, $\mathbb{D}$ is given by the
quotient space $\mathbb{H}/\Gamma(3)$ which is illustrated in Fig 1.

The fundamental domain of a congruence subgroup $\overline{\Gamma}(N)$ is
defined as $\mathbb{H}/\overline{\Gamma}(N)$, which is a quotient space. It’s
important to note that the fundamental domain $\mathcal{F}$ is congruent to
$\mathbb{H}/\Gamma$. This means that we can express
$\mathbb{H}/\overline{\Gamma}(N)$ as
$\mathbb{H}/\overline{\Gamma}(N)\equiv\mathbb{H}/\Gamma\times\Gamma/\overline{\Gamma}(N)\equiv\mathcal{F}\times\Gamma_{N}$
, although this expression is less rigorous. In simpler terms, we can obtain
the fundamental domain of $\overline{\Gamma}(N)$ by applying all elements of a
finite modular group $\Gamma_{N}$ to $\mathcal{F}$. We transform the modular
forms of level 3 undertaken in our theory based on the irreducible
representations of $A_{4}$ group.

In reference [1], the explicit construction of the basis of weight-2 modular
forms of $\Gamma(3)$ has been achieved. For those unfamiliar with the
Kronecker and Tensor Product of $A_{4}$, please refer to A. These modular
forms provide a triplet representation of $A_{4}$ and can be expressed using
the Dedekind eta function $\eta(\tau)$ and its derivative.

The Dedekind’s eta-function is a fundamental example of infinite product
representation which is also a holomorphic function defined in the complex
upper half-plane, denoted as $\eta(\tau)$. Thus it is clear from this
definition that $\eta(\tau)$ is non-vanishing in the complex upper half-plane
$\mathbb{H}$.

$\displaystyle\eta(\tau)=q^{1/24}\prod^{\infty}_{n=1}(1-q^{n})$ (11)

with $q=e^{i2\pi\tau}$. The Dedekind eta-function and its derivative are used
to find the three linearly independent modular forms of weight 2 and level 3
as [1]

$\displaystyle
Y^{(2)}_{1}(\tau)=\frac{i}{2\pi}\Bigl{[}\frac{\eta^{\prime}(\tau/3)}{\eta(\tau/3)}+\frac{\eta^{\prime}((\tau+1)/3)}{\eta((\tau+1)/3)}$
(12)
$\displaystyle+\frac{\eta^{\prime}((\tau+2)/3)}{\eta((\tau+2)/3)}-\frac{27\eta^{\prime}(3\tau)}{\eta(3\tau)}\Bigr{]}\;,$
$\displaystyle
Y^{(2)}_{2}(\tau)=-\frac{i}{\pi}\Bigl{[}\frac{\eta^{\prime}(\tau/3)}{\eta(\tau/3)}+\omega^{2}\frac{\eta^{\prime}((\tau+1)/3)}{\eta((\tau+1)/3)}$
$\displaystyle+\omega\frac{\eta^{\prime}((\tau+2)/3)}{\eta((\tau+2)/3)}\Bigr{]}\;,$
$\displaystyle
Y^{(2)}_{3}(\tau)=-\frac{i}{\pi}\Bigl{[}\frac{\eta^{\prime}(\tau/3)}{\eta(\tau/3)}+\omega\frac{\eta^{\prime}((\tau+1)/3)}{\eta((\tau+1)/3)}$
$\displaystyle+\omega^{2}\frac{\eta^{\prime}((\tau+2)/3)}{\eta((\tau+2)/3)}\Bigr{]}\;,$

where $\omega=e^{i2\pi/3}$. Eqns.(12) later satisfy the constraint: [1]

$\displaystyle Y^{(2)2}_{2}+2Y^{(2)}_{1}Y^{(2)}_{3}=0\;.$ (13)

These three modular forms can be organized into a triplet of $A_{4}$
transforming in the ${\bf 3}$ irreducible representation of $A_{4}$, denoted
as $Y^{(2)}_{i}(\tau)_{i=1,2,3}$ can be grouped together to transform as the
triplet representation of $A_{4}$ as

$\displaystyle Y^{(2)}_{\bf 3}(\tau)=\left(\begin{matrix}Y^{(2)}_{1}(\tau)\\\
Y^{(2)}_{2}(\tau)\\\ Y^{(2)}_{3}(\tau)\end{matrix}\right)\;,$ (14)

Following the q-expansion of $Y_{i}(\tau)$ by considering $q_{3}\equiv
e^{\dfrac{i2\pi\tau}{3}}$, Eqn. (12) takes the form:

$\displaystyle
Y^{(2)}_{\textbf{3}}(\tau)=\left(\begin{matrix}Y^{(2)}_{1}(\tau)\\\
Y^{(2)}_{2}(\tau)\\\ Y^{(2)}_{3}(\tau)\end{matrix}\right)$ (15)
$\displaystyle=\left(\begin{matrix}1+12q^{3}_{3}+36q^{6}_{3}+12+36q^{9}_{3}+...\\\
-6(q_{3}+7q^{4}_{3}+8q^{7}_{3}+...)\\\
-18(q^{2}_{3}+2q^{5}_{3}+5q^{8}_{3}+...)\end{matrix}\right)\;.$

It is possible to select a basis in the linear space of $M_{2k}(\Gamma(N))$
with weight 2k such that the transformation of the modular forms $Y(\tau)$ is
defined by the unitary representation $\rho$ of $\Gamma_{N}$. Modular
multiplets $Y(\tau)$ with a weight of 2k and a level of $N\geq 2$ yield non-
vanishing modular forms that are consistently of even weight [1]. This unique
property is particularly noteworthy as it allows for greater flexibility in
the analysis of modular forms. Furthermore, the function does not vanish for
odd weight modular forms, further enhancing the versatility of this approach.

$\displaystyle Y_{i}(\gamma\tau)=(c\tau+d)^{k}\rho(\gamma)_{ij}Y_{j}(\tau)\;,$
(16)

The earliest discussion of modular forms of general positive integer weights
(including positive even weights and odd weights) and finite modular groups
can be seen in the literature [47], which also contains the proof of Eqn.
(16), which is very helpful for understanding modular form multiplets. Modular
forms are crucial in the construction of models to explain fermion masses and
flavour mixing [1]. The linear space that is covered by modular forms of
weight $k$ has a dimension of $k+1$ for the case of level 3 that corresponds
to the group $\Gamma_{3}\simeq A_{4}$. Thus we can obtain only one trivial
modular form, which is constant and independent of the modulus $\tau$, for
$k=0$. A triplet of $A_{4}$ is formed for $k=2$ by three linearly independent
modular forms.

Using the tensor product rules of $A_{4}$ as shown in A, we can construct the
higher weight modular forms ( $k\geq 4$ here) with the help of Eqs. (13) and
(14. In our models we use $k_{Y_{max}}=4,6,8,10$. For weight $k=4$, we have
two singlets ${\bf 1}$, ${\bf 1^{\prime}}$ and one triplet ${\bf 3}$ of
$A_{4}$ with a total of 5 linearly independent modular forms:

$\displaystyle Y^{(4)}_{\bf 1}=Y^{2}_{1}+2Y_{2}Y_{3}\;,\quad Y^{(4)}_{\bf
1^{\prime}}=Y^{2}_{3}+2Y_{1}Y_{2}\;,$ (17) $\displaystyle Y^{(4)}_{\bf
3}=\left(\begin{matrix}Y^{2}_{1}-Y_{2}Y_{3}\\\ Y^{2}_{3}-Y_{1}Y_{2}\\\
Y^{2}_{2}-Y_{1}Y_{3}\end{matrix}\right)$

Similarly when $k=6$, one has

$\displaystyle Y^{(6)}_{\bf
1}=Y^{3}_{1}+Y^{3}_{2}+Y^{3}_{3}-3Y_{1}Y_{2}Y_{3}\;,$ (18) $\displaystyle
Y^{(6)}_{{\bf 3},1}=(Y^{2}_{1}+2Y_{2}Y_{3})\left(\begin{matrix}Y_{1}\\\
Y_{2}\\\ Y_{3}\end{matrix}\right)\;,$ $\displaystyle Y^{(6)}_{{\bf
3},2}=(Y^{2}_{3}+2Y_{1}Y_{2})\left(\begin{matrix}Y_{3}\\\ Y_{1}\\\
Y_{2}\end{matrix}\right)$

the dimension is 7 where $Y^{(3)}_{3,3}=0$. Following the same procedure, for
$k=8$ , we have

$\displaystyle Y^{(8)}_{\bf 1}=(Y^{2}_{1}+2Y_{2}Y_{3})^{2}\;,$ (19)
$\displaystyle Y^{(8)}_{\bf
1^{\prime}}=(Y^{2}_{1}+2Y_{2}Y_{3})(Y^{2}_{3}+2Y_{1}Y_{2})\;,$ $\displaystyle
Y^{(8)}_{\bf 1^{\prime\prime}}=(Y^{2}_{3}+2Y_{1}Y_{2})^{2}\;,$ $\displaystyle
Y^{(8)}_{{\bf
3},1}=(Y^{2}_{1}+2Y_{2}Y_{3})\left(\begin{matrix}Y^{2}_{1}-Y_{2}Y_{3}\\\
Y^{2}_{3}-Y_{1}Y_{2}\\\ Y^{2}_{2}-Y_{1}Y_{3}\end{matrix}\right)\;,$
$\displaystyle Y^{(8)}_{{\bf
3},2}=(Y^{2}_{3}+2Y_{1}Y_{2})\left(\begin{matrix}Y^{2}_{2}-Y_{1}Y_{3}\\\
Y^{2}_{1}-Y_{2}Y_{3}\\\ Y^{2}_{3}-Y_{1}Y_{2}\end{matrix}\right)\;,$

a collective dimension of 9. For $k=10$, we get 11 linearly independent
modular forms which are assigned as $A_{4}$ multiplets as shown below

$\displaystyle Y^{(10)}_{\bf
1}=(Y^{2}_{1}+2Y_{2}Y_{3})(Y^{3}_{1}+Y^{3}_{2}+Y^{3}_{3}-3Y_{1}Y_{2}Y_{3})\;,$
(20) $\displaystyle Y^{(10)}_{\bf
1^{\prime}}=(Y^{2}_{3}+2Y_{1}Y_{2})(Y^{3}_{1}+Y^{3}_{2}+Y^{3}_{3}-3Y_{1}Y_{2}Y_{3})\;,$
$\displaystyle Y^{(10)}_{{\bf
3},1}=(Y^{2}_{1}+2Y_{2}Y_{3})^{2}\left(\begin{matrix}Y_{1}\\\ Y_{2}\\\
Y_{3}\end{matrix}\right),$ $\displaystyle Y^{(10)}_{{\bf
3},2}=(Y^{2}_{3}+2Y_{1}Y_{2})^{2}\left(\begin{matrix}Y_{2}\\\ Y_{3}\\\
Y_{1}\end{matrix}\right)\;,$ $\displaystyle Y^{(10)}_{{\bf
3},3}=(Y^{2}_{1}+2Y_{2}Y_{3})(Y^{2}_{3}+2Y_{1}Y_{2})\left(\begin{matrix}Y_{3}\\\
Y_{1}\\\ Y_{2}\end{matrix}\right)\;.$

These modular forms of higher weights are shown explicitly in [1] and [25].
The generic form of matter action in the Lagrangian $\mathcal{N}=1$ global
supersymmetry theory is [92, 93]

$\displaystyle{\mathcal{S}}=\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm
d}^{2}\overline{\theta}K(\phi_{i},\overline{\phi}_{i};\tau,\overline{\tau})$
(21) $\displaystyle+\int{\rm d}^{4}x{\rm d}^{2}\theta W(\phi_{i};\tau)+{\rm
h.c.}\;,$

Here $\theta$ and $\overline{\theta}$ represents the Grassmann variables,
$\phi_{i}$ is the collective matter chiral superfields involved in the theory
and K and W stands for the K$\ddot{\rm a}$hler potential and superpotential
respectively. the modulus $\tau$ as well as $\phi_{i}$ undergo transformation
as shown in Eq. (5) and Eq. (22) [92] for a finite modular group $\Gamma_{N}$
(to the limit of a field - independent phase $\alpha(\gamma)$ known as the
multiplier system) [93], [92]

$\displaystyle\gamma:~{}\phi_{i}\rightarrow(c\tau+d)^{-k_{i}}\rho_{I_{i}}(\gamma)\phi_{i}\;,$
(22)

From this, the superpotential $W(\phi_{i};\tau)$ can be expanded in powers of
the supermultiplets $\phi_{i}$ as

$\displaystyle
W(\phi_{i},\tau)=\sum(Y_{i_{1},\cdots,i_{n}}(\tau)\phi_{i_{1}}\cdots\phi_{i_{n}})_{\bf
1}\;,$ (23)

where ${\bf 1}$ denotes the sum of all possible field combinations and
singlets of the modular group $\Gamma_{N}$. Inorder for the superpotential to
under field transformation as the rule given in Eq. (22),
$Y^{(k_{Y})}_{I_{Y}}$ being the field couplings need to transform as modular
forms of weight $k_{Y}$ and level N such that
$k_{Y}=k_{i_{1}}+\cdots+k_{i_{n}}$ holds under $\Gamma_{N}$, i.e.,

$\displaystyle\gamma:~{}Y^{(k_{Y})}_{I_{Y}}(\tau)\rightarrow
Y^{(k_{Y})}_{I_{Y}}(\gamma\tau)=(c\tau+d)^{k_{Y}}$ (24)
$\displaystyle\times\rho_{I_{Y}}(\gamma)Y^{(k_{Y})}_{I_{Y}}(\tau)\;.$

The unitary representation of $\Gamma_{N}$, $\rho$ must satisfy
$\rho_{I_{Y}}\otimes\rho_{I_{1}}\otimes\cdots\otimes\rho_{I_{n}}$ such that it
involved atleast one singlet invariant under the finite modular group
$\Gamma_{N}$.

##  3 $A_{4}$ modular invariant zero-textures in 3+1 framework using minimal
seesaw mechanism

In the context of a diagonal charged lepton mass matrix $M_{l}$, there exist
fifteen distinct two-zero textures, with twenty out of 120 allowed three-zero
textures, fifteen out of 210 allowed four-zero textures, and six out of 246
allowed five-zero textures in the Majorana neutrino mass matrix $m_{\nu}$.
These findings are detailed in the reference. Additionally, B briefly outlines
the various zero textures that are not ruled out, each of which is labeled
with a specific alphabet and numeric designation where in the matrices of the
possible textures "$\times$" represents a non-zero entry.

In our work we have realized these possible textures using the non-Abelian
$A_{4}$ flavour symmetry. We have incorporated modular symmetry along with the
$A_{4}$ flavour symmetry to reduce the number of free parameters and to create
a more predictable models without fine tuning. The reason we chose $A_{4}$
flavour symmetry is because it has three irreducible singlet representations
and one triplet representation that can naturally accomodate its triplet
family of neutrinos and its three right-handed singlets. Here we generate the
neutrino masses using minimal extended seesaw mechanism. The neutral fermionic
mass matrix can be written in the basis $(L,N,S)$ as

$\displaystyle\mathbb{M}=\left(\begin{matrix}0&M_{D}&0\\\
M^{T}_{D}&M_{R}&M^{T}_{S}\\\ 0&M_{S}&0\end{matrix}\right)$ (25)

Table 1: A typical assignment of representations and weights for the chiral supermultiplets in Cases (a), (b), (c) and (d). | $L_{1}$ | $L_{2}$ | $L_{3}$ | $E^{c}_{1}$ | $E^{c}_{2}$ | $E^{c}_{3}$ | $H_{u}$ | $H_{d}$ | $N^{c}_{1}$ | $N^{c}_{2}$ | $S$ | $\phi$
---|---|---|---|---|---|---|---|---|---|---|---|---
$A_{4}$ | ${\bf 1}$ | ${\bf 1^{\prime\prime}}$ | ${\bf 1^{\prime}}$ | ${\bf 1}$ | ${\bf 1^{\prime}}$ | ${\bf 1^{\prime\prime}}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 1^{\prime}}$ | ${\bf 1^{\prime\prime}}$ | ${\bf 1^{\prime}}$ | 1
$-k$ | $k_{1}$ | $k_{2}$ | $k_{3}$ | $r_{1}$ | $r_{2}$ | $r_{3}$ | 0 | 0 | $l_{1}$ | $l_{2}$ | $s$ | $k_{\phi}$
Table 2: A typical assignment of representations and weights for the chiral supermultiplets in Cases (e) and (f). | $L_{1}$ | $L_{2}$ | $L_{3}$ | $E^{c}_{1}$ | $E^{c}_{2}$ | $E^{c}_{3}$ | $H_{u}$ | $H_{d}$ | $N^{c}_{1}$ | $N^{c}_{2}$ | $S$ | $\phi$
---|---|---|---|---|---|---|---|---|---|---|---|---
$A_{4}$ | ${\bf 1}$ | ${\bf 1^{\prime}}$ | ${\bf 1^{\prime\prime}}$ | ${\bf 1}$ | ${\bf 1^{\prime\prime}}$ | ${\bf 1^{\prime}}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 1^{\prime}}$ | ${\bf 1^{\prime\prime}}$ | $1^{\prime\prime}$
$-k$ | $k_{1}$ | $k_{2}$ | $k_{3}$ | $r_{1}$ | $r_{2}$ | $r_{3}$ | 0 | 0 | $l_{1}$ | $l_{2}$ | $s$ | $k_{\phi}$

The Dirac mass matrix with basis $(L,N)$ is given as

$M_{D}=\left(\begin{matrix}a_{11}&a_{12}\\\ a_{21}&a_{22}\\\
a_{31}&a_{32}\end{matrix}\right)$ (26)

where $L$ is the left-handed lepton doublet with three generations, $E^{c}$ is
the right-handed charged leptons of three families, $N^{c}$ is the two right-
handed neutrinos, and S is the sterile neutrino. $H_{u,d}$ defines the Higgs
doublets with zero weight. The rest of the two matrices can be written as

$M_{R}=\left(\begin{matrix}0&M_{N}\\\ M_{N}&0\end{matrix}\right)$ (27)
$M_{S}=\left(\begin{matrix}\rho_{1}&\rho_{2}\end{matrix}\right)$ (28)

If the condition $M_{R}\gg M_{S}>M_{D}$ then for minimal seesaw mechanism, the
effective $4\times 4$ active-sterile neutrino mass matrix is given by

$m_{\nu}=-\left(\begin{matrix}M_{D}M^{-1}_{R}M^{T}_{D}&M_{D}M^{-1}_{R}M^{T}_{S}\\\
M_{S}(M^{-1}_{R})^{T}M^{T}_{D}&M_{S}M^{-1}_{R}M^{T}_{S}\end{matrix}\right)$
(29)

We have to take care of the fact that atleast of the neutrino mass eigenvalue
is zero. Eqn. (29) thus takes the form:

$m_{\nu}=\left(\begin{matrix}p_{1}&q&r&s\\\ q&p_{2}&m&t\\\ r&m&p_{3}&w\\\
s&t&w&p_{4}\end{matrix}\right)$ (30)

where $p_{1}=\frac{2a_{11}a_{12}}{M_{N}}$,
$p_{2}=\frac{2a_{12}a_{22}}{M_{N}}$, $p_{3}=\frac{2a_{31}a_{32}}{M_{N}}$,
$p_{4}=\frac{2\rho_{1}\rho_{2}}{M_{N}}$,
$q=\frac{a_{12}^{2}}{M_{N}}+\frac{a_{11}a_{22}}{M_{N}}$,
$r=\frac{a_{12}a_{31}}{M_{N}}+\frac{a_{11}a_{32}}{M_{N}}$,
$s=\frac{a_{12}\rho_{1}}{M_{N}}+\frac{a_{11}\rho_{2}}{M_{N}}$,
$m=\frac{a_{22}a_{31}}{M_{N}}+\frac{a_{12}a_{32}}{M_{N}}$,
$t=\frac{a_{22}\rho_{1}}{M_{N}}+\frac{a_{12}\rho_{2}}{M_{N}}$,
$w=\frac{a_{32}\rho_{1}}{M_{N}}+\frac{a_{31}\rho_{2}}{M_{N}}$.

To simplify our analysis, we have included a table (LABEL:cont) that displays
the $A_{4}$ representations and modular weights assigned to the chiral
supermultiplets. In our models, we have established fixed textures for $M_{R}$
and $M_{S}$ that are invariant under the $A_{4}$ modular symmetry. However,
the textures of $M_{D}$ do transform along with the assigned weights and
$A_{4}$ representations. We then apply minimal seesaw mechanism to find only
the allowed textures out of the possible textures as shown in B under the
$A_{4}$ modular symmetry. It is interesting to find that out of all the
possible cases only 6 textures (3 two-zeros+ 1 three-zeros + 2 four-zeros) are
allowed and the rest are ruled out which is summarized in Table (3). These
models are shown below where $\alpha$, $\beta$, $\gamma$, $\lambda_{ij}$ (i=
1,2,3; j=1,2), $\Omega$, $\rho$ are arbritary coefficients with $\langle
H_{u}\rangle=\upsilon_{u}$ and there is absence of non-trivial modular forms
in the charged lepton sector. For cases (a), (b), (c), and (d), we are
following Table (1) and for cases (e) and (f) we are following Table (2) where
the particle content of the various fields under $A_{4}$ modular
transformation is given.

Case No. | Constraints | $M_{D}$ | $m_{\nu}$ | Allowed textures
---|---|---|---|---
(a) | $a_{11}=a_{22}=0$ | $\left(\begin{matrix}0&\times\\\ \times&0\\\ \times&\times\end{matrix}\right)$ | $\left(\begin{matrix}0&\times&\times&\times\\\ \times&0&\times&\times\\\ \times&\times&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ |  $E_{1}$ of two-zeros
(b) | $a_{11}=a_{32}=0$ | $\left(\begin{matrix}0&\times\\\ \times&\times\\\ \times&0\end{matrix}\right)$ | $\left(\begin{matrix}0&\times&\times&\times\\\ \times&\times&\times&\times\\\ \times&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ |  $E_{2}$ of two-zeros
(c) | $a_{22}=a_{31}=0$ | $\left(\begin{matrix}\times&\times\\\ \times&0\\\ 0&\times\end{matrix}\right)$ | $\left(\begin{matrix}\times&\times&\times&\times\\\ \times&0&\times&\times\\\ \times&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ |  $C$ of two-zeros
(d) | $a_{22}=a_{32}=0$ | $\left(\begin{matrix}\times&\times\\\ \times&0\\\ \times&0\end{matrix}\right)$ | $\left(\begin{matrix}\times&\times&\times&\times\\\ \times&0&0&\times\\\ \times&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ |  $C_{3}$ of three-zeros
(e) | $a_{12}=a_{21}=a_{31}=0$ | $\left(\begin{matrix}\times&0\\\ 0&\times\\\ 0&\times\end{matrix}\right)$ | $\left(\begin{matrix}0&\times&\times&\times\\\ \times&0&0&\times\\\ \times&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ |  $A_{3}$ of four-zeros
(f) | $a_{12}=a_{21}=a_{32}=0$ | $\left(\begin{matrix}\times&0\\\ 0&\times\\\ \times&0\end{matrix}\right)$ | $\left(\begin{matrix}0&\times&0&\times\\\ \times&0&\times&\times\\\ 0&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ |  $A_{2}$ of four-zeros

Table 3: Possible textures in $m_{\nu}$ in the 3+1 scenario for different
cases of $M_{D}$ and $m_{\nu}$

(a) ${\bf E_{1}}$ of Two-zero textures

$\displaystyle k_{1}=-2\;,\;k_{2}=-4\;,\;k_{3}=-2\;,\quad$ (31) $\displaystyle
r_{1}=2\;,\;r_{2}=4\;,\;r_{3}=2\;,\quad$ $\displaystyle
l_{1}=-2\;,\;l_{2}=-6\;,\;s=-2\;,k_{\phi}=2.$

In this model of texture $E_{1}$ of two-zeros, we utilize the multiplets of
modular forms, namely $Y^{(4)}_{1^{\prime}}$, $Y^{(6)}_{1}$, $Y^{(8)}_{1}$,
and $Y^{(8)}_{1^{\prime}}$, which were derived earlier. This results in a
maximum value of $k_{Y{\rm max}}=8$. The superpotential can be explicitly
expressed as follows:

$\displaystyle W=$ $\displaystyle(\alpha L_{1}H_{d}E^{c}_{1}+\beta
L_{2}H_{d}E^{c}_{2}+\gamma L_{3}H_{d}E^{c}_{3})$
$\displaystyle+(\lambda_{12}Y^{(8)}_{\bf
1^{\prime}}L_{1}H_{u}N^{c}_{2}+\lambda_{21}Y^{(6)}_{\bf 1}L_{2}H_{u}N^{c}_{1}$
$\displaystyle+\lambda_{31}Y^{(4)}_{\bf
1^{\prime}}L_{3}H_{u}N^{c}_{1}+\lambda_{32}Y^{(8)}_{\bf
1}L_{3}H_{u}N^{c}_{2})$ $\displaystyle+\Omega Y^{(6)}_{\bf
1}\phi(N^{c}_{1}N^{c}_{2}+N^{c}_{2}N^{c}_{1})$
$\displaystyle+\rho(Y^{(4)}_{\textbf{1}^{\prime}}SN^{c}_{1}+Y^{(8)}_{\textbf{1}}SN^{c}_{2})\;,$

Thus from Eqns. (3) and (30) we get

$m_{\nu}=\left(\begin{matrix}0&\frac{a_{12}^{2}}{M_{N}}+\frac{a_{11}a_{22}}{M_{N}}&\frac{a_{12}a_{31}}{M_{N}}+\frac{a_{11}a_{32}}{M_{N}}&\frac{a_{12}\rho_{1}}{M_{N}}+\frac{a_{11}\rho_{2}}{M_{N}}\\\
\frac{a_{12}^{2}}{M_{N}}+\frac{a_{11}a_{22}}{M_{N}}&0&\frac{a_{22}a_{31}}{M_{N}}+\frac{a_{12}a_{32}}{M_{N}}&\frac{a_{22}\rho_{1}}{M_{N}}+\frac{a_{12}\rho_{2}}{M_{N}}\\\
\frac{a_{12}a_{31}}{M_{N}}+\frac{a_{11}a_{32}}{M_{N}}&\frac{a_{22}a_{31}}{M_{N}}+\frac{a_{12}a_{32}}{M_{N}}&\frac{2a_{31}a_{32}}{M_{N}}&\frac{a_{32}\rho_{1}}{M_{N}}+\frac{a_{31}\rho_{2}}{M_{N}}\\\
\frac{a_{12}\rho_{1}}{M_{N}}+\frac{a_{11}\rho_{2}}{M_{N}}&\frac{a_{22}\rho_{1}}{M_{N}}+\frac{a_{12}\rho_{2}}{M_{N}}&\frac{a_{32}\rho_{1}}{M_{N}}+\frac{a_{31}\rho_{2}}{M_{N}}&\frac{2\rho_{1}\rho_{2}}{M_{N}}\end{matrix}\right)$

(33)

where, $a_{12}=\lambda_{12}Y^{(8)}_{1^{\prime}}\upsilon_{u}$,
$a_{21}=\lambda_{21}Y^{(6)}_{1}\upsilon_{u}$,
$a_{31}=\lambda_{31}Y^{(4)}_{1^{\prime}}\upsilon_{u}$,
$a_{32}=\lambda_{32}Y^{(8)}_{1}\upsilon_{u}$, $M_{N}=\Omega Y^{(6)}_{1}\phi$ ,
$\rho_{1}=\rho Y^{(4)}_{1^{\prime}}$ and $\rho_{2}=\rho Y^{(8)}_{1}$.

(b) ${\bf E_{2}}$ of Two-zero textures

$\displaystyle k_{1}=-4\;,\;k_{2}=-4\;,k_{3}=2\;,\quad$ (34) $\displaystyle
r_{1}=4\;,r_{2}=4\;,r_{3}=-2\;,\quad$ $\displaystyle
l_{1}=-6\;,\;l_{2}=-4\;,\;s=-2\;,k_{\phi}=6.$

In this model of texture $E_{2}$ of two-zeros, we utilize the multiplets of
modular forms, namely $Y^{(4)}_{1}$, $Y^{(4)}_{1^{\prime}}$, $Y^{(6)}_{1}$,
$Y^{(8)}_{1^{\prime}}$, $Y^{(8)}_{1^{\prime\prime}}$, and $Y^{(10)}_{1}$,
which were derived earlier. This results in a maximum value of $k_{Y{\rm
max}}=10$. The superpotential can be explicitly expressed as follows:

$\displaystyle W=$ $\displaystyle(\alpha L_{1}H_{d}E^{c}_{1}+\beta
L_{2}H_{d}E^{c}_{2}+\gamma L_{3}H_{d}E^{c}_{3})$
$\displaystyle+(\lambda_{12}Y^{(8)}_{\bf
1^{\prime}}L_{1}H_{u}N^{c}_{2}+\lambda_{21}Y^{(10)}_{\bf
1}L_{2}H_{u}N^{c}_{1}$ $\displaystyle+\lambda_{22}Y^{(8)}_{\bf
1^{\prime\prime}}L_{2}H_{u}N^{c}_{2}+\lambda_{31}Y^{(4)}_{\bf
1^{\prime}}L_{3}H_{u}N^{c}_{1})$ $\displaystyle+\Omega Y^{(4)}_{\bf
1}\phi(N^{c}_{1}N^{c}_{2}+N^{c}_{2}N^{c}_{1})$
$\displaystyle+\rho(Y^{(8)}_{\textbf{1}^{\prime}}SN^{c}_{1}+Y^{(6)}_{\textbf{1}}SN^{c}_{2})\;,$

$m_{\nu}=\left(\begin{matrix}0&\frac{a_{12}^{2}}{M_{N}}+\frac{a_{11}a_{22}}{M_{N}}&\frac{a_{12}a_{31}}{M_{N}}+\frac{a_{11}a_{32}}{M_{N}}&\frac{a_{12}\rho_{1}}{M_{N}}+\frac{a_{11}\rho_{2}}{M_{N}}\\\
\frac{a_{12}^{2}}{M_{N}}+\frac{a_{11}a_{22}}{M_{N}}&\frac{2a_{12}a_{22}}{M_{N}}&\frac{a_{22}a_{31}}{M_{N}}+\frac{a_{12}a_{32}}{M_{N}}&\frac{a_{22}\rho_{1}}{M_{N}}+\frac{a_{12}\rho_{2}}{M_{N}}\\\
\frac{a_{12}a_{31}}{M_{N}}+\frac{a_{11}a_{32}}{M_{N}}&\frac{a_{22}a_{31}}{M_{N}}+\frac{a_{12}a_{32}}{M_{N}}&0&\frac{a_{32}\rho_{1}}{M_{N}}+\frac{a_{31}\rho_{2}}{M_{N}}\\\
\frac{a_{12}\rho_{1}}{M_{N}}+\frac{a_{11}\rho_{2}}{M_{N}}&\frac{a_{22}\rho_{1}}{M_{N}}+\frac{a_{12}\rho_{2}}{M_{N}}&\frac{a_{32}\rho_{1}}{M_{N}}+\frac{a_{31}\rho_{2}}{M_{N}}&\frac{2\rho_{1}\rho_{2}}{M_{N}}\end{matrix}\right)$

(36)

where, $a_{12}=\lambda_{12}Y^{(8)}_{1^{\prime}}\upsilon_{u}$,
$a_{21}=\lambda_{21}Y^{(10)}_{1}\upsilon_{u}$,
$a_{22}=\lambda_{22}Y^{(8)}_{1^{\prime\prime}}\upsilon_{u}$,
$a_{31}=\lambda_{31}Y^{(4)}_{1^{\prime}}\upsilon_{u}$, $M_{N}=\Omega
Y^{(4)}_{1}\phi$, $\rho_{1}=\rho Y^{(8)}_{1^{\prime}}$ and $\rho_{2}=\rho
Y^{(6)}_{1}$.

(c) ${\bf C}$ of Two-zero textures

$\displaystyle k_{1}=-4\;,\;k_{2}=-4\;,k_{3}=-2\;,\quad$ (37) $\displaystyle
r_{1}=4\;,r_{2}=4\;,r_{3}=2\;,\quad$ $\displaystyle
l_{1}=-4\;,\;l_{2}=-6\;,\;s=-4\;,k_{\phi}=6.$

In this model of texture $C$ of two-zeros, we utilize the multiplets of
modular forms, namely $Y^{(4)}_{1}$, $Y^{(8)}_{1}$, $Y^{(8)}_{1^{\prime}}$,
$Y^{(8)}_{1^{\prime\prime}}$, $Y^{(10)}_{1}$, and $Y^{(10)}_{1^{\prime}}$
which were derived earlier. This results in a maximum value of $k_{Y{\rm
max}}=10$. The superpotential can be explicitly expressed as follows:

$\displaystyle W=$ $\displaystyle(\alpha L_{1}H_{d}E^{c}_{1}+\beta
L_{2}H_{d}E^{c}_{2}+\gamma L_{3}H_{d}E^{c}_{3})$
$\displaystyle+(\lambda_{11}Y^{(8)}_{\bf
1^{\prime\prime}}L_{1}H_{u}N^{c}_{1}+\lambda_{12}Y^{(10)}_{\bf
1^{\prime}}L_{1}H_{u}N^{c}_{2}$ $\displaystyle+\lambda_{21}Y^{(8)}_{\bf
1}L_{2}H_{u}N^{c}_{1}+\lambda_{32}Y^{(8)}_{\bf 1}L_{3}H_{u}N^{c}_{2})$
$\displaystyle+\Omega Y^{(4)}_{\bf
1}\phi(N^{c}_{1}N^{c}_{2}+N^{c}_{2}N^{c}_{1})$
$\displaystyle+\rho(Y^{(8)}_{\textbf{1}^{\prime}}SN^{c}_{1}+Y^{(10)}_{\textbf{1}}SN^{c}_{2})\;,$

$m_{\nu}=\left(\begin{matrix}\frac{2a_{11}a_{12}}{M_{N}}&\frac{a_{12}^{2}}{M_{N}}+\frac{a_{11}a_{22}}{M_{N}}&\frac{a_{12}a_{31}}{M_{N}}+\frac{a_{11}a_{32}}{M_{N}}&\frac{a_{12}\rho_{1}}{M_{N}}+\frac{a_{11}\rho_{2}}{M_{N}}\\\
\frac{a_{12}^{2}}{M_{N}}+\frac{a_{11}a_{22}}{M_{N}}&0&\frac{a_{22}a_{31}}{M_{N}}+\frac{a_{12}a_{32}}{M_{N}}&\frac{a_{22}\rho_{1}}{M_{N}}+\frac{a_{12}\rho_{2}}{M_{N}}\\\
\frac{a_{12}a_{31}}{M_{N}}+\frac{a_{11}a_{32}}{M_{N}}&\frac{a_{22}a_{31}}{M_{N}}+\frac{a_{12}a_{32}}{M_{N}}&0&\frac{a_{32}\rho_{1}}{M_{N}}+\frac{a_{31}\rho_{2}}{M_{N}}\\\
\frac{a_{12}\rho_{1}}{M_{N}}+\frac{a_{11}\rho_{2}}{M_{N}}&\frac{a_{22}\rho_{1}}{M_{N}}+\frac{a_{12}\rho_{2}}{M_{N}}&\frac{a_{32}\rho_{1}}{M_{N}}+\frac{a_{31}\rho_{2}}{M_{N}}&\frac{2\rho_{1}\rho_{2}}{M_{N}}\end{matrix}\right)$

(39)

where, $a_{11}=\lambda_{11}Y^{(8)}_{1^{\prime\prime}}\upsilon_{u}$,
$a_{12}=\lambda_{12}Y^{(10)}_{1^{\prime}}\upsilon_{u}$,
$a_{21}=\lambda_{21}Y^{(8)}_{1}\upsilon_{u}$,
$a_{32}=\lambda_{32}Y^{(8)}_{1}\upsilon_{u}$, $M_{N}=\Omega Y^{(4)}_{1}\phi$,
$\rho_{1}=\rho Y^{(8)}_{1^{\prime}}$ and $\rho_{2}=\rho Y^{(10)}_{1}$.

(d) ${\bf C_{3}}$ of Three-zero textures

$\displaystyle k_{1}=-3\;,\;k_{2}=-1\;,k_{3}=-1\;,\quad$ (40) $\displaystyle
r_{1}=3\;,r_{2}=1\;,r_{3}=1\;,\quad$ $\displaystyle
l_{1}=-5\;,\;l_{2}=-1\;,\;s=-5\;,k_{\phi}=-2.$

In this model of texture $C_{3}$ of three-zeros, we utilize the multiplets of
modular forms, namely $Y^{(4)}_{1^{\prime}}$, $Y^{(6)}_{1}$,$Y^{(8)}_{1}$,
$Y^{(8)}_{1^{\prime}}$, $Y^{(8)}_{1^{\prime\prime}}$, and
$Y^{(10)}_{1^{\prime}}$ which were derived earlier. This results in a maximum
value of $k_{Y{\rm max}}=10$. The superpotential can be explicitly expressed
as follows:

$\displaystyle W=$ $\displaystyle(\alpha L_{1}H_{d}E^{c}_{1}+\beta
L_{2}H_{d}E^{c}_{2}+\gamma L_{3}H_{d}E^{c}_{3})$
$\displaystyle+(\lambda_{11}Y^{(8)}_{\bf
1^{\prime\prime}}L_{1}H_{u}N^{c}_{1}+\lambda_{12}Y^{(4)}_{\bf
1^{\prime}}L_{1}H_{u}N^{c}_{2}$ $\displaystyle+\lambda_{21}Y^{(6)}_{\bf
1}L_{2}H_{u}N^{c}_{1}+\lambda_{31}Y^{(8)}_{\bf
1^{\prime}}L_{3}H_{u}N^{c}_{1})$ $\displaystyle+\Omega Y^{(8)}_{\bf
1}\phi(N^{c}_{1}N^{c}_{2}+N^{c}_{2}N^{c}_{1})$
$\displaystyle+\rho(Y^{(10)}_{\textbf{1}^{\prime}}SN^{c}_{1}+Y^{(6)}_{\textbf{1}}SN^{c}_{2})\;,$

$m_{\nu}=\left(\begin{matrix}\frac{2a_{11}a_{12}}{M_{N}}&\frac{a_{12}^{2}}{M_{N}}+\frac{a_{11}a_{22}}{M_{N}}&\frac{a_{12}a_{31}}{M_{N}}+\frac{a_{11}a_{32}}{M_{N}}&\frac{a_{12}\rho_{1}}{M_{N}}+\frac{a_{11}\rho_{2}}{M_{N}}\\\
\frac{a_{12}^{2}}{M_{N}}+\frac{a_{11}a_{22}}{M_{N}}&0&0&\frac{a_{22}\rho_{1}}{M_{N}}+\frac{a_{12}\rho_{2}}{M_{N}}\\\
\frac{a_{12}a_{31}}{M_{N}}+\frac{a_{11}a_{32}}{M_{N}}&0&0&\frac{a_{32}\rho_{1}}{M_{N}}+\frac{a_{31}\rho_{2}}{M_{N}}\\\
\frac{a_{12}\rho_{1}}{M_{N}}+\frac{a_{11}\rho_{2}}{M_{N}}&\frac{a_{22}\rho_{1}}{M_{N}}+\frac{a_{12}\rho_{2}}{M_{N}}&\frac{a_{32}\rho_{1}}{M_{N}}+\frac{a_{31}\rho_{2}}{M_{N}}&\frac{2\rho_{1}\rho_{2}}{M_{N}}\end{matrix}\right)$

(42)

where, $a_{11}=\lambda_{11}Y^{(8)}_{1^{\prime\prime}}\upsilon_{u}$,
$a_{12}=\lambda_{12}Y^{(4)}_{1^{\prime}}\upsilon_{u}$,
$a_{21}=\lambda_{21}Y^{(6)}_{1}\upsilon_{u}$,
$a_{31}=\lambda_{31}Y^{(8)}_{1^{\prime}}\upsilon_{u}$, $M_{N}=\Omega
Y^{(8)}_{1}\phi$, $\rho_{1}=\rho Y^{(10)}_{1^{\prime}}$ and $\rho_{2}=\rho
Y^{(6)}_{1}$.

(e) ${\bf A_{3}}$ of Four-zero textures

$\displaystyle k_{1}=-1\;,\;k_{2}=-3\;,k_{3}=1\;,\quad$ (43) $\displaystyle
r_{1}=1\;,\;r_{2}=3\;,r_{3}=-1\;,\quad$ $\displaystyle
l_{1}=-3\;,\;l_{2}=-5\;,\;s=-1,k_{\phi}=4\;,\;\;$

In this model of texture $A_{3}$ of four-zeros, we utilize the multiplets of
modular forms, namely $Y^{(4)}_{1}$, $Y^{(4)}_{1^{\prime}}$, $Y^{(6)}_{1}$,
and $Y^{(8)}_{1^{\prime}}$, which were derived earlier. This results in a
maximum value of $k_{Y{\rm max}}=8$. The superpotential can be explicitly
expressed as follows:

$\displaystyle W=$ $\displaystyle(\alpha L_{1}H_{d}E^{c}_{1}+\beta
L_{2}H_{d}E^{c}_{2}+\gamma L_{3}H_{d}E^{c}_{3})$
$\displaystyle+(\lambda_{11}Y^{(4)}_{\bf
1}L_{1}H_{u}N^{c}_{1}+\lambda_{22}Y^{(8)}_{\bf 1^{\prime}}L_{2}H_{u}N^{c}_{2}$
$\displaystyle+\lambda_{32}Y^{(4)}_{\bf 1}L_{3}H_{u}N^{c}_{2})$
$\displaystyle+\Omega Y^{(4)}_{\bf
1}\phi(N^{c}_{1}N^{c}_{2}+N^{c}_{2}N^{c}_{1})$
$\displaystyle+\rho(Y^{(4)}_{\textbf{1}^{\prime}}SN^{c}_{1}+Y^{(6)}_{\textbf{1}}SN^{c}_{2})\;,$

$m_{\nu}=\left(\begin{matrix}0&\frac{a_{12}^{2}}{M_{N}}+\frac{a_{11}a_{22}}{M_{N}}&\frac{a_{12}a_{31}}{M_{N}}+\frac{a_{11}a_{32}}{M_{N}}&\frac{a_{12}\rho_{1}}{M_{N}}+\frac{a_{11}\rho_{2}}{M_{N}}\\\
\frac{a_{12}^{2}}{M_{N}}+\frac{a_{11}a_{22}}{M_{N}}&0&0&\frac{a_{22}\rho_{1}}{M_{N}}+\frac{a_{12}\rho_{2}}{M_{N}}\\\
\frac{a_{12}a_{31}}{M_{N}}+\frac{a_{11}a_{32}}{M_{N}}&0&0&\frac{a_{32}\rho_{1}}{M_{N}}+\frac{a_{31}\rho_{2}}{M_{N}}\\\
\frac{a_{12}\rho_{1}}{M_{N}}+\frac{a_{11}\rho_{2}}{M_{N}}&\frac{a_{22}\rho_{1}}{M_{N}}+\frac{a_{12}\rho_{2}}{M_{N}}&\frac{a_{32}\rho_{1}}{M_{N}}+\frac{a_{31}\rho_{2}}{M_{N}}&\frac{2\rho_{1}\rho_{2}}{M_{N}}\end{matrix}\right)$

(45)

where, $a_{11}=\lambda_{11}Y^{(4)}_{1}\upsilon_{u}$,
$a_{22}=\lambda_{22}Y^{(8)}_{1^{\prime}}\upsilon_{u}$,
$a_{32}=\lambda_{32}Y^{(4)}_{1}\upsilon_{u}$, $M_{N}=\Omega Y^{(4)}_{1}\phi$,
$\rho_{1}=\rho Y^{(4)}_{1^{\prime}}$ and $\rho_{2}=\rho Y^{(6)}_{1}$.

(f) ${\bf A_{2}}$ of Four-zero textures

$\displaystyle k_{1}=1\;,\;k_{2}=-1\;,k_{3}=1\;,\quad$ (46) $\displaystyle
r_{1}=-1\;,r_{2}=1\;,r_{3}=-1\;,\quad$ $\displaystyle
l_{1}=-5\;,\;l_{2}=-3\;,\;s=-3\;,k_{\phi}=4.$

In this model of texture $A_{2}$ of four-zeros, we utilize the multiplets of
modular forms, namely $Y^{(4)}_{1}$, $Y^{(4)}_{1^{\prime}}$, $Y^{(6)}_{1}$,
$Y^{(8)}_{1^{\prime}}$, and $Y^{(8)}_{1^{\prime\prime}}$, which were derived
earlier. This results in a maximum value of $k_{Y{\rm max}}=8$. The
superpotential can be explicitly expressed as follows:

$\displaystyle W=$ $\displaystyle(\alpha L_{1}H_{d}E^{c}_{1}+\beta
L_{2}H_{d}E^{c}_{2}+\gamma L_{3}H_{d}E^{c}_{3})$
$\displaystyle+(\lambda_{11}Y^{(4)}_{\bf
1}L_{1}H_{u}N^{c}_{1}+\lambda_{22}Y^{(4)}_{\bf 1^{\prime}}L_{2}H_{u}N^{c}_{2}$
$\displaystyle+\lambda_{31}Y^{(4)}_{\bf 1^{\prime}}L_{3}H_{u}N^{c}_{1})$
$\displaystyle+\Omega Y^{(4)}_{\bf
1}\phi(N^{c}_{1}N^{c}_{2}+N^{c}_{2}N^{c}_{1})$
$\displaystyle+\rho(Y^{(8)}_{\textbf{1}^{\prime}}SN^{c}_{1}+Y^{(6)}_{\textbf{1}}SN^{c}_{2})\;,$

$m_{\nu}=\left(\begin{matrix}0&\frac{a_{12}^{2}}{M_{N}}+\frac{a_{11}a_{22}}{M_{N}}&0&\frac{a_{12}\rho_{1}}{M_{N}}+\frac{a_{11}\rho_{2}}{M_{N}}\\\
\frac{a_{12}^{2}}{M_{N}}+\frac{a_{11}a_{22}}{M_{N}}&0&\frac{a_{22}a_{31}}{M_{N}}+\frac{a_{12}a_{32}}{M_{N}}&\frac{a_{22}\rho_{1}}{M_{N}}+\frac{a_{12}\rho_{2}}{M_{N}}\\\
0&\frac{a_{22}a_{31}}{M_{N}}+\frac{a_{12}a_{32}}{M_{N}}&0&\frac{a_{32}\rho_{1}}{M_{N}}+\frac{a_{31}\rho_{2}}{M_{N}}\\\
\frac{a_{12}\rho_{1}}{M_{N}}+\frac{a_{11}\rho_{2}}{M_{N}}&\frac{a_{22}\rho_{1}}{M_{N}}+\frac{a_{12}\rho_{2}}{M_{N}}&\frac{a_{32}\rho_{1}}{M_{N}}+\frac{a_{31}\rho_{2}}{M_{N}}&\frac{2\rho_{1}\rho_{2}}{M_{N}}\end{matrix}\right)$

(48)

where, $a_{11}=\lambda_{11}Y^{(4)}_{1}\upsilon_{u}$,
$a_{22}=\lambda_{22}Y^{(4)}_{1^{\prime}}\upsilon_{u}$,
$a_{31}=\lambda_{31}Y^{(4)}_{1^{\prime}}\upsilon_{u}$, $M_{N}=\Omega
Y^{(4)}_{1}\phi$, $\rho_{1}=\rho Y^{(8)}_{1^{\prime}}$ and $\rho_{2}=\rho
Y^{(6)}_{1}$.

Figure 2:  Figure 3:  Figure 4:  Figure 5:

##  4 Discussion and Summary

Our research involves the implementation of the $A_{4}$ symmetry group and
modular symmetry to identify the most promising textures from a range of
possibilities, as outlined in [2]. To construct neutrino models, we assign
$A_{4}$ representations to each chiral supermultiplet which are even weighted.
This approach ensures that the Lagrangians remain invariant under modular
transformation, satisfying the relation of $k_{Y}=k_{i}+k_{r}+...+k_{l}$. The
maximal weight $k_{Y_{max}}$ for the neutrino models of the allowed cases
varies from 4 to 10.

Our findings indicate that by fixing the texture of $M_{R}$ and $M_{S}$
matrices and varying only the texture of $M_{D}$, we can identify six allowed
zero-texture of the active-sterile neutrino mixing matrix. These include three
two-zero textures, one three-zero texture, and two four-zero textures. We have
exclusively used here the minimal seesaw mechanism to derive the $4\times 4$
active-sterile neutrino mixing matrix.Our models have proven to be highly
predictive and consistent with experimental values of active-sterile neutrino
parameters within a 3$\sigma$ range. We have utilized the 3$\sigma$ values of
both active and sterile neutrino oscillations, as shown in Table (4), using
input variables such as $\Delta M^{2}_{41}$, $\theta_{14}$, $\theta_{24}$, and
$\theta_{34}$ from recent publications [94], [95], [96], to determine unknown
active and sterile neutrino parameters, such as $m_{lightest}$, $m_{4}$
(sterile neutrino mass), and Dirac CP violating phases $\delta_{14}$,
$\delta_{24}$, $\delta_{13}$, as well as Majorana CPV phases $\alpha$ and
$\beta$.

Our models have produced allowed textures that are highly consistent with
experimental values, as mentioned in Table (4). For instance, if we consider
the texture $E_{1}$ texture of the allowed two-zero textures of the active-
sterile mixing matrix then it is found that the range of $|U_{41}|^{2}$,
$|U_{24}|^{2}$, $|U_{34}|^{2}$ are within the 3$\sigma$ values of
$|U_{41}|^{2}$, $|U_{24}|^{2}$, $|U_{34}|^{2}$ as obtained in [94] for
Inverted Hierarchy (IH). This is shown briefly in Table (5). The correlation
plots between $m_{4}$ vs $\Delta M^{2}_{41}$, $\theta_{34}/^{o}$ vs
$|U_{43}|$, $\theta_{24}/^{o}$ vs $|U_{42}|$, and $\Sigma m_{i(i=1,2,3,4)}$ vs
$|U_{42}|/|U_{41}|$, for $E_{1}$ two-zero texture is shown in Fig (2), Fig(3),
Fig(4) and Fig(5)respectively.

Parameters | Normal hierarchy | Inverted hierarchy
---|---|---
$\Delta m^{2}_{21}$ $(10^{-5}eV^{2})$ | 6.95 $\rightarrow$ 8.14 (7.50) | 6.94 $\rightarrow$ 8.14 (7.50)
$|\Delta m^{2}_{31}|$ $(10^{-3}eV^{2})$ | 2.47 $\rightarrow$ 2.63 (2.55) | 2.37 $\rightarrow$ 2.53 (2.45)
$Sin^{2}\theta^{2}_{12}/10^{-1}$ | 2.71 $\rightarrow$ 3.69 (3.18) | 2.71 $\rightarrow$ 3.69 (3.18)
$Sin^{2}\theta^{2}_{23}/10^{-1}$ | 4.34 $\rightarrow$ 6.10 (5.74) | 4.33 $\rightarrow$ 6.08 (5.78)
$Sin^{2}\theta^{2}_{13}/10^{-2}$ | 2.000 $\rightarrow$ 2.405 (2.200) | 2.018 $\rightarrow$ 2.424 (2.225)
$\delta/\pi$ | 0.71 $\rightarrow$ 1.99 (1.08) | 1.11 $\rightarrow$ 1.96 (1.58)
$\Delta m^{2}_{41}/eV^{2}$ | 0.33 $\rightarrow$ 148 | 0.33 $\rightarrow$ 148
$\theta_{14}/^{o}$ | 6.1 $\rightarrow$ 15.3 | 6.1 $\rightarrow$ 15.3
$\theta_{24}/^{o}$ | 2.8 $\rightarrow$ 12.5 | 2.8 $\rightarrow$ 12.5
$\theta_{34}/^{o}$ | 0 $\rightarrow$ 30 | 0 $\rightarrow$ 30
$|U_{14}|^{2}$ | 0.012 $\rightarrow$ 0.047 | 0.012 $\rightarrow$ 0.047
$|U_{24}|^{2}$ | 0.005 $\rightarrow$ 0.03 | 0.005 $\rightarrow$ 0.03
$|U_{34}|^{2}$ | 0 $\rightarrow$ 0.16 | 0 $\rightarrow$ 0.16

Table 4: $3\sigma$ parameter values of active-sterile neutrinos [94]

Parameter values for $E_{1}$ two zero texture in IH
---
Parameters | Model predicted value | Experimental value ($3\sigma$)
$|U_{14}|^{2}$ | 0.01202 $\rightarrow$ 0.0469956 | 0.012 $\rightarrow$ 0.047
$|U_{24}|^{2}$ | 0.00500654 $\rightarrow$ 0.0299983 | 0.005 $\rightarrow$ 0.03
$|U_{34}|^{2}$ | 0.000185605 $\rightarrow$ 0.159948 | 0 $\rightarrow$ 0.16

Table 5: $3\sigma$ parameter values of $|U_{14}|^{2}$, $|U_{24}|^{2}$ and
$|U_{34}|^{2}$ as predicted for $E_{1}$ two-zero texture model and latest
$3\sigma$ value obtained from experiments.

Overall, our research provides valuable insights into the use of symmetry
groups and modular symmetry to construct neutrino models. By identifying the
most viable textures of active-sterile neutrino, we can better understand the
behavior of active-sterile neutrino mixing matrices and their implication in
Dark Matter, Baryon Asymmetry of the Universe etc.

Acknowledgements

The author expresses sincere gratitude to Xiang-Gan Liu from UC Irvine in USA,
Prof. Ferruccio Feruglio from INFN in Italy, Shivajee Gupta from IITGN, Monal
Kashav from Central University of Himachal Pradesh and Prof. Ramakrishnan
Balakrishnan from the Indian Statistical Institute (TASU-NE) for their
invaluable guidance, important inputs and insightful discussions on modular
forms. Their expertise has been instrumental in shaping the author’s
understanding of this complex topic. Additionally, the author would like to
extend thanks to Prof. Rupam Barman from the Department of Mathematics at IIT
Guwahati for granting permission to attend the NCWM-ATM Schools on Jacobi
Forms. The lectures delivered during the workshop have been an immense help in
comprehending the intricate concepts of Modular and Jacobi Forms. Overall, the
author is deeply appreciative of the support and knowledge provided by these
esteemed individuals, which has undoubtedly contributed to the success of this
research work.

##  Appendix A The Tensor Product and Clebsch Gordon coefficient of $A_{4}$
symmetry

$A_{4}$ group [3], [4], [5], [5], [6] is an alternating group (of even
permutation) of object 4. It has two generators S and T which represents
rotation and translation. The tensor product and Clebsch Gordon coefficient
for $A_{4}$ symmmetry is shown below:

The $A_{4}$ group has a total of 12 elements that originates from 4 conjugacy
classes. It has three singlets and one triplet irreducible representation

The product rules of $A_{4}$ symmetry is given by

$\displaystyle{\bf 1^{\prime}}\otimes{\bf 1^{\prime}}={\bf
1^{\prime\prime}}\;,\quad{\bf 1^{\prime\prime}}\otimes{\bf
1^{\prime\prime}}={\bf 1^{\prime}}\;,\quad$ $\displaystyle{\bf
1^{\prime}}\otimes{\bf 1^{\prime\prime}}={\bf 1^{\prime\prime}}\otimes{\bf
1^{\prime}}={\bf 1},\;\;\quad\;\;$ (49) $\displaystyle{\bf 3}\otimes{\bf
3}={\bf 1}\oplus{\bf 1^{\prime}}\oplus{\bf 1^{\prime\prime}}\oplus{\bf
3_{s}}\oplus{\bf 3_{a}}\;\quad.$

##  Appendix B The allowed textures of 3+1 Majorana mass matrix under $A_{4}$
modular group

$A_{1}$ | $A_{2}$ |  |
---|---|---|---
$\left(\begin{matrix}0&0&\times&\times\\\ 0&\times&\times&\times\\\ \times&\times&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&\times&0&\times\\\ \times&\times&\times&\times\\\ 0&\times&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ |  |
$B_{1}$ | $B_{2}$ | $B_{3}$ | $B_{4}$
$\left(\begin{matrix}\times&\times&0&\times\\\ \times&0&\times&\times\\\ 0&\times&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&0&\times&\times\\\ 0&\times&\times&\times\\\ \times&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&0&\times&\times\\\ 0&0&\times&\times\\\ \times&\times&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&\times&0&\times\\\ \times&\times&\times&\times\\\ 0&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$
C |  |  |
$\left(\begin{matrix}\times&\times&\times&\times\\\ \times&0&\times&\times\\\ \times&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ |  |  |
$D_{1}$ | $D_{2}$ |  |
$\left(\begin{matrix}\times&\times&\times&\times\\\ \times&0&0&\times\\\ \times&0&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&\times&\times&\times\\\ \times&\times&0&\times\\\ \times&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ |  |
$E_{1}$ | $E_{2}$ | $E_{3}$ |
$\left(\begin{matrix}0&\times&\times&\times\\\ \times&0&\times&\times\\\ \times&\times&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&\times&\times&\times\\\ \times&\times&\times&\times\\\ \times&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&\times&\times&\times\\\ \times&\times&0&\times\\\ \times&0&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ |
$F_{1}$ | $F_{2}$ | $F_{3}$ |
$\left(\begin{matrix}\times&0&0&\times\\\ 0&\times&\times&\times\\\ 0&\times&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&0&\times&\times\\\ 0&\times&0&\times\\\ \times&0&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&\times&0&\times\\\ \times&\times&0&\times\\\ 0&0&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ |

Table 6: Possible two-zero textures in $m_{\nu}$ in the 3+1 scenario

$A$ |  |
---|---|---
$\left(\begin{matrix}0&\times&\times&\times\\\ \times&0&\times&\times\\\ \times&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ |  |
$B_{1}$ | $B_{2}$ | $B_{3}$
$\left(\begin{matrix}0&\times&0&\times\\\ \times&0&\times&\times\\\ 0&\times&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&\times&\times&\times\\\ \times&0&0&\times\\\ \times&0&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&0&\times&\times\\\ 0&\times&\times&\times\\\ \times&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$
$B_{4}$ | $B_{5}$ | $B_{6}$
$\left(\begin{matrix}0&\times&\times&\times\\\ \times&\times&0&\times\\\ \times&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&0&\times&\times\\\ 0&0&\times&\times\\\ \times&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&\times&0&\times\\\ \times&0&\times&\times\\\ 0&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$
$C_{1}$ | $C_{2}$ | $C_{3}$
$\left(\begin{matrix}0&0&\times&\times\\\ 0&0&\times&\times\\\ \times&\times&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&\times&0&\times\\\ \times&\times&\times&\times\\\ 0&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&\times&\times&\times\\\ \times&0&0&\times\\\ \times&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$
$D_{1}$ | $D_{2}$ | $D_{3}$
$\left(\begin{matrix}0&0&\times&\times\\\ 0&\times&0&\times\\\ \times&0&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&\times&0&\times\\\ \times&\times&0&\times\\\ 0&0&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&0&0&\times\\\ 0&0&\times&\times\\\ 0&\times&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$
$D_{4}$ | $D_{5}$ | $D_{6}$
$\left(\begin{matrix}\times&\times&0&\times\\\ \times&0&0&\times\\\ 0&0&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&0&0&\times\\\ 0&\times&\times&\times\\\ 0&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&0&\times&\times\\\ 0&\times&0&\times\\\ \times&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$
$E_{1}$ | $E_{2}$ | $E_{3}$
$\left(\begin{matrix}0&0&0&\times\\\ 0&\times&\times&\times\\\ 0&\times&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&0&\times&\times\\\ 0&0&0&\times\\\ \times&0&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&\times&0&\times\\\ \times&\times&0&\times\\\ 0&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$

Table 7: Possible three-zero textures in $m_{\nu}$ in the 3+1 scenario

$A_{1}$ | $A_{2}$ | $A_{3}$
---|---|---
$\left(\begin{matrix}0&0&\times&\times\\\ 0&0&\times&\times\\\ \times&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&\times&0&\times\\\ \times&0&\times&\times\\\ 0&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&\times&\times&\times\\\ \times&0&0&\times\\\ \times&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$
$A_{4}$ | $A_{5}$ | $A_{6}$
$\left(\begin{matrix}0&0&0&\times\\\ 0&0&\times&\times\\\ 0&\times&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&0&\times&\times\\\ 0&0&0&\times\\\ \times&0&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&\times&0&\times\\\ \times&0&0&\times\\\ 0&0&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$
$A_{7}$ | $A_{8}$ | $A_{9}$
$\left(\begin{matrix}0&0&0&\times\\\ 0&\times&\times&\times\\\ 0&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&0&\times&\times\\\ 0&\times&0&\times\\\ \times&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&\times&0&\times\\\ \times&\times&0&\times\\\ 0&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$
$A_{10}$ | $B_{1}$ | $B_{2}$
$\left(\begin{matrix}0&0&0&\times\\\ 0&\times&0&\times\\\ 0&0&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&0&\times&\times\\\ 0&0&0&\times\\\ \times&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&\times&0&\times\\\ \times&0&0&\times\\\ 0&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$
$B_{3}$ | $B_{4}$ | $B_{5}$
$\left(\begin{matrix}\times&0&0&\times\\\ 0&0&\times&\times\\\ 0&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&0&0&\times\\\ 0&0&0&\times\\\ 0&0&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&0&0&\times\\\ 0&\times&0&\times\\\ 0&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$

Table 8: Possible four-zero textures in $m_{\nu}$ in the 3+1 scenario

$A$ | $B$ | $C$
---|---|---
$\left(\begin{matrix}0&0&0&\times\\\ 0&0&0&\times\\\ 0&0&\times&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&0&0&\times\\\ 0&0&\times&\times\\\ 0&\times&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&0&0&\times\\\ 0&\times&0&\times\\\ 0&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$
$D$ | $E$ | $F$
$\left(\begin{matrix}0&0&\times&\times\\\ 0&0&0&\times\\\ \times&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}0&\times&0&\times\\\ \times&0&0&\times\\\ 0&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$ | $\left(\begin{matrix}\times&0&0&\times\\\ 0&0&0&\times\\\ 0&0&0&\times\\\ \times&\times&\times&\times\\\ \end{matrix}\right)$

Table 9: Possible five-zero textures in $m_{\nu}$ in the 3+1 scenario

This section provides a concise summary of the permitted zero-texture 3+1
neutrino mass matrices, which have been extensively reviewed and elaborately
explained in the reference [2]. When considering the matrix of active-sterile
neutrino mixing with the sterile neutrino as the fourth generation, there are
a total of 45 allowed two-zero textures in $m_{\nu}$. However, only 15 of
these two-zero textures are actually possible. Interestingly, these 15 allowed
textures for active-sterile neutrino mixing are the same as the 15 allowed
textures of the neutrino mass matrix with only three generations. To provide
further clarity, we have included a Table (6) displaying the 15 permitted two-
zero textures.

When dealing with three-zero textures, there are 120 possible textures to
consider. However, only 20 of these textures are allowed, as the other 100
have been discarded. These 20 textures can be found in Table (7). It’s worth
noting that the $C_{3}$ texture is only allowed in a very small region of the
IH parameter space as mentioned in the reference [2].

Moving on to four-zero textures, there are a total of 210 textures to analyze.
Unfortunately, 195 of these textures have been discarded, leaving only 15
allowed textures. These 15 textures can be found in Table (8).

When it comes to five-zero textures, there are 252 possible options. However,
only six of these textures are allowed. It’s important to note that none of
our allowed $m_{\nu}$ textures fall under the category of five-zero textures.
The possible textures for five-zero can be found in Table (9).

In summary, while there are many possible textures to consider, only few of
them are allowed. By analyzing these allowed textures with mathematical tools
such as modular and flavour symmetries, we can gain a better understanding of
3+1 neutrino mixing matrix.

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