# Leptogenesis and dark matter in minimal inverse seesaw using $A_{4}$ modular
symmetry

Jotin Gogoi<EMAIL_ADDRESS>Lavina Sarma<EMAIL_ADDRESS>Mrinal
Kumar Das<EMAIL_ADDRESS>Department of Physics, Tezpur University,
Tezpur 784028, India

###### Abstract

In this paper we have studied neutrino masses and mixings by adding a scalar
triplet $\eta$ to the particle content of minimal Inverse seesaw. We have
realised this extension of minimal inverse seesaw by implementing an
isomorphic modular group $\Gamma(3)$ and a non-abelian discrete symmetry group
$A_{4}$. We have also used $Z_{3}$ symmetry group to restrain certain
interaction terms in the lagrangian of the model. We have studied baryon
asymmetry of the universe, neutrinoless double-beta decay and dark matter in
our work. In order to check the consistency of our model with various
experimental constraints, we have therefore calculated effective mass, relic
density and baryogenesis via leptogenesis. Interestingly, we have found our
model quite compatible with the experimental bounds and is also successful in
producing the neutrino masses and mixings in the 3$\sigma$ range.

###### pacs:

12.60.-i,14.60.Pq,14.60.St

##  I Introduction

With the discovery of the Higgs particle at the LHC in 2012, a very important
revolution was marked in the field of particle physics doi:10.1063/1.4727988 .
This discovery, apart from providing the missing piece to the Standard Model
(SM), paved the way for a better understanding of the fundamental particles
and through Higgs mechanism the process responsible for generating the masses
of quarks and leptons could be explained. But this grand success was not
devoid of shortcomings. There are many challenging issues in cosmology,
astrophysics, particle physics etc. which are still not addressed and the
answers to these phenomena are yet to be found. Some of these prominent
challenges are neutrino oscillations and generation of their tiny masses,
lepton number violating processes (LNV), baryon asymmetry of the universe
(BAU), dark matter, dark energy etc. The Standard Model of particle physics,
recognised as one of the most successful theories in the field, fails to
provide a proper explanation of these challenges. As a result, the concept of
Beyond the Standard Model (BSM) frameworks have come into the picture. And
people believe that a proper understanding with a suitable explanation of
these challenging issues can be found in this BSM paradigm.

One of the prime ingredients that motivates physicists to go for BSM theories
is associated with mass of the neutrinos. With the discovery of neutrino
oscillation Bellini:2013wra , it became obvious that neutrinos should have
tiny mass and mixing between them. The experimental findings made by Super-
Kamiokande KamLAND:2008dgz ; Super-Kamiokande:2001bfk  and Sudbury Neutrino
Observatories SNO:2002hgz  gave the final confirmation to this idea. But due
to the absence of right-handed neutrinos it remained massless within the
Standard Model. As a result, to generate this tiny mass of neutrinos an
extension of SM was inevitable. Consequently a new theory that could
incorporate and provide an elegant explanation to these questions came in the
form of Seesaw mechanism. A detailed review about seesaw mechanism can be
found in the literature Babu:1988qv ; Chang:1985en ; Mohapatra:1987fs . This
mechanism is broadly classified into Type I Albright:2004kb ; Mohapatra:2004vr
; King:2003jb , Type II Rodejohann:2004qh , Type III King:2014nza , Inverse
seesaw etc. In recent times, Inverse seesaw has become more popular in model
building. In this framework the mass of right-handed neutrinos can be brought
down to TeV scale. This peculiar property enhances the possibility of
detecting these particles at the LHC and future experiments. For a detailed
study on Inverse seesaw one can refer to Abada:2018qok ; Deppisch:2004fa ;
Dev:2009aw .

It is now an established fact that there is an asymmetry between matter and
anti-matter in the universe. Cosmological observations indicate that the
number of baryons in the universe is unequal to the number of anti-baryons.
This difference in number between baryons and anti-baryons is termed as Baryon
Asymmetry of the Universe (BAU) Barbieri:1999ma ; Kolb:1979ui ;
Pilaftsis:2003gt . At the beginning, as evident from various considerations,
the universe started with an equal number of both the types of particles i.e.
baryons and anti-baryons. As such the asymmetry observed in the universe
occured in much later times and must have been generated through a dynamical
process called baryogenesis. As per Planck data, the value of this asymmetry
is found to be ParticleDataGroup:2018ovx

$\eta_{B}=(6.04\pm 0.08)\times 10^{-10}$ (1)

As proposed by Sakharov, three important conditions are necessary for
baryogenesis: Baryon number violation (B), C and CP violation , interactions
out of thermal equilibrium Sakharov:1967dj . In the last couple of decades
several attempts have been made to address the phenomena of BAU. One of the
popular and successful theoretical process is baryogenesis via leptogenesis.
This mechanism was first proposed by Fukugita and Yanagida Fukugita:1986hr .
According to this mechanism, the L-violating out of equilibrium decays of
singlet neutrino creates an symmetry in the leptonic sector. This excess in
lepton number can be converted into the observed baryon asymmetry through B+L
violating sphaleron processes Trodden:2004mj ; Kolb:1990vq . In this regard,
Inverse seesaw contains gauge singlet right-handed neutrinos and sterile
fermions. As a result, the asymmetry generated by decay of one of the quasi-
Dirac pairs formed by these particles can be converted into baryon asymmetry
of the universe.

The presence of dark matter (DM), an inevitable mystery of the universe, has
been well-established through various observations in astrophysics and
cosmology. Some of the strong observations in this regard are galaxy cluster
observations by F. Zwicky Zwicky:1933gu , galaxy rotation curves Rubin:1970zza
, recent observation of the Bullet clusters Clowe:2006eq  and cosmological
data from the Planck collaboration Planck:2019nip . All of these remarks
suggest the existence of an unknown, non-luminous, non-baryonic dark matter
which constitutes about 26% of the energy density of the universe and is
approximately five times more than luminous matter. Currently the amount of
dark matter in the universe as found from the Planck data is Planck:2018vyg

$\Omega h^{2}=0.1199\pm 0.0027$ (2)

This is called the relic density of dark matter. The properties that a
candidate must have to qualify as dark matter candidate has been highlighted
in Taoso:2007qk ; Murayama:2007ek . Unfortunately, the SM particles do not
possess these required criteria and so none of them can be considered to be a
viable dark matter candidate. Therefore , from the particle physics view
point, one has to extend the SM particle content by incorporating new fields
to find a suitable candidate that could produce the correct relic abundance.

Another serious conundrum in particle physics is the nature of neutrinos
Barenboim:2002hx ; Czakon:1999cd ; Bilenky:2012qi : whether they are Dirac or
Majorana particle. A possible solution to this question lies in the discovery
of neutrinoless double beta decay (0$\nu\beta\beta$/NDBD). It is a SM
forbidden second order lepton number violating interaction, which if
discovered can shade light about the Majorana nature of the neutrinos. The
decay channel of this reaction can be expressed as:

$(A,Z)\rightarrow(A,Z+2)+2e^{-}$

Currently there are many ongoing experiments which actively look for this
decay. Some of these popular experiments are KamLANDZen KamLAND-Zen:2016pfg ,
GERDA GERDA:2020xhi , CUROE etc. These experiments use Xenon-136 and
Germanium-76 nuclei for possible detection of lepton number violating decays.
The recent results published from these experiments predict the allowed lower
bound for half-life ($T_{\frac{1}{2}}$) to be $\geq 10^{26}$ years. Moreover
the effective neutrino mass is found to be $\leq$ 0.165 eV.

Symmetry plays a very important role in model building in particle physics.
Numerous work that has been done based on discrete flavor symmetry can be
found in Borah:2017dmk ; King:2009ap ; King:2009db ; Gautam:2020wsd . In this
type of symmetry a large number of flavons, along with their VEV alignments,
are used in the models.The non-Abelian discrete symmetry groups like $A_{4}$,
$S_{4}$, $Z_{N}$ etc. play a very crucial role in developing these models. In
our work we have used modular symmetry in the framework of minimal Inverse
seesaw [ISS(2,3)]. In this type of symmetry the Yukawa couplings are not free.
Instead they are functions of a complex variable $\tau$, called the modulus
Feruglio:2017spp . As a result this helps in reducing the number of flavons
that are used in the model. In addition to this, one does not have to deal
with their VEV alignments. The modular group, $\Gamma(N)$ acts on the upper
half of the complex plane and transforms $\tau$ in the following way:

$\tau\rightarrow\frac{a\tau+b}{c\tau+d}$

where $a,b,c,d$ are the elements of a 2$\times$2 matrix, such that $ad-bc=1$.
The literatures Nomura:2019xsb ; Behera:2021eut ; Zhang:2021olk ;
Nomura:2020cog  highlight some of the recent works done in inverse seesaw
using modular symmetry. Some of the finite modular groups ( $N\leq 5$) are
isomorphic to non-Abelian discrete symmetry groups
($A_{4},S_{4},A^{\prime}_{5}$ etc). In our work we have used $\Gamma(3)$
modular group. This group has three Yukawa modular forms of weight 2. One can
obtain the higher weight modular forms by using these weight 2 modular
functions. Moreover their values can be found with the help of their
q-expansions Novichkov:2019sqv . A brief description on $\Gamma(3)$ is given
in Appendix A. The authors in Mukherjee:2015axj ; Verma:2021koo  have studied
dark matter phenomena using discrete flavor symmetry. But in our work, along
with dark matter, we have also studied leptogenesis and neutrinoless double
beta decay by using modular symmetry.

This paper is organised as follows: in section II.1 we discsuss the basic
structure of the model, charge assignments of particles under various groups
used in the work and then construct its lagrangian. In section III and IV we
give a brief description about neutrinoless double beta decay and leptogenesis
in the framework of ISS(2,3). We have given a brief explanation about dark
matter in section V. In section VI we provide the analysis and results of our
work. We have shown the different plots and their descriptions in this part of
the paper. Finally in section VII we conclude by giving an overview of our
entire work.

##  II The framework of inverse seesaw

Inverse seesaw is a popular BSM framework which is used in model building in
neutrino physics. It is an extension of the Standard Model with three right-
handed neutrinos and gauge singlet sterile fermions. Unlike the conventional
seesaw mechanism, the Majorana mass of right-handed neutrinos in Inverse
seesaw can be reduced to TeV scale. As a result, apart from generating tiny
neutrino masses, it also enhances the possibility of detecting these right-
handed particles in the ongoing and near future experiments.These
possibilities of inverse seesaw have been highlighted in Hirsch:2009mx ;
Gu:2010xc ; Nomura:2018cfu ; Awasthi:2013we ; Malinsky:2009df . For the basis
$n_{L}=(\nu_{L,\alpha},N^{c}_{R,i},S_{j})^{T}$, the Yukawa lagrangian for
neutrino in this framework can be written as:

$\mathcal{L}=-\frac{1}{2}n^{T}_{L}CMn_{L}+h.c.$ (3)

where $C\equiv\gamma^{2}\gamma^{0}$ is the charge conjugation matrix. The
component $\nu_{L,\alpha}$ for $\alpha=e,\mu,\tau$ are left-handed Standard
Model neutrinos. The complete mass matrix for neutral fermion arising from the
lagrangian can be written as Abada:2014vea :

$\mathcal{M}=\begin{pmatrix}0&M^{T}_{D}&0\\\ M_{D}&0&M_{NS}\\\
0&M^{T}_{NS}&M_{S}\end{pmatrix}_{9\times 9}$ (4)

The elements of the matrix in eq. (4) represent the different mass matrices
involved in ISS mechanism. The Dirac mass matrix ($M_{D}$) results from the
interaction between left and right-handed components of neutrinos. $M_{NS}$ is
the mixing matrix that occurs due to right-handed neutrinos and sterile
singlet fermions interactions. The interaction between the singlet sterile
fermions forms the Majorana mass term, $M_{S}$. The dimensions of the matrices
are related to the number of generations of the particles that have been
considered in the model. Accordingly in ISS the dimensions of these matrices
can be defined in the following way (# represents the number of generations of
particles.):

$\displaystyle Dimension\hskip 4.2679ptof\hskip
4.2679ptM_{D}=(\\#\nu_{L}\times\\#N_{R})$ (5) $\displaystyle Dimension\hskip
4.2679ptof\hskip 4.2679ptM_{NS}=(\\#N_{R}\times\\#S)$ $\displaystyle
Dimension\hskip 4.2679ptof\hskip 4.2679ptM_{S}=(\\#S\times\\#S)$

From eq. (5) one can find that the three matrices, $M_{D}$, $M_{NS}$ and
$M_{S}$, have the same dimension i.e. $3\times 3$. The effective neutrino mass
matrix for the active light neutrinos can be written as:

$m_{\nu}=M^{T}_{D}(M^{T}_{NS})^{-1}M_{S}M^{-1}_{NS}M_{D}$ (6)

The light neutrino mass matrix in eq. (6) can be diagonalised with the help of
an unitary matrix. The corresponding eigenvalues will be the mass of the
active neutrinos. In order to produce sub-eV Standard Model neutrinos, $M_{D}$
must be in electroweak range, $M_{NS}$ in the TeV range and $M_{S}$ must be in
the KeV range, respectively.

###  II.1 The Model

In this work we have used minimal inverse seesaw [ISS(2,3)] framework.
Compared to ISS, this mechanism of minimal inverse seesaw constitutes of two
right-handed neutrinos ($N_{1},N_{2}$) and three singlet sterile fermions
($S_{1},S_{2},S_{3}$). As a result, the order of $M_{D}$, $M_{NS}$ changes to
$3\times 2$ and $2\times 3$; whereas for $M_{S}$ it remains unchanged. Along
with these particles we have used a flavon ($\phi$) whose VEV alignments
facilitate to get a diagonal charged lepton mass matrix. As such the role of
this flavon is restricted only to the charged lepton sector without affecting
the neutrino sector. We have extended the minimal inverse seesaw by a Higgs-
type scalar triplet $\eta=(\eta_{1},\eta_{2},\eta_{3})$ whose neutral
component is our dark matter candidate Boucenna:2011tj ; Hirsch:2010ru .
Proceeding ahead, we have used $A_{4}$ modular symmetry to construct the
desired lagrangian of the model. This group contains three Yukawa modular
forms ($y_{1},y_{2},y_{3}$) of weight 2 which have been considered as triplet
under $A_{4}$ charge assignment. The right-handed neutrinos ($N_{1}$ and
$N_{2}$) in the model are taken to be singlets under $A_{4}$ and they
transform as $1^{\prime}$ and $1^{\prime\prime}$, respectively; while the
sterile neutrinos ($S_{i}$) and lepton doublets (L) are considered as
triplets. The modular weight of the right-handed neutrinos is taken as -2
whereas for the lepton doublets (L), sterile neutrinos and $\eta$ it is taken
to be zero. In order to restrict certain interaction terms in the Lagrangian,
we have used $Z_{3}$ symmetry group. Moreover we introduce two weighton
fields, $\beta_{1}$ and $\beta_{2}$, which are associated with the right-
handed neutrinos. These charge assignments of the particles under various
groups in the model have been highlighted in table 4.

Fields | L | $N_{1}$ | $N_{2}$ | $S_{i}$ | H | $\phi$ | $\eta$ | Y
---|---|---|---|---|---|---|---|---
$A_{4}$ | 3 | $1^{\prime}$ | $1^{\prime\prime}$ | 3 | 1 | 1 | 3 | 3
$Z_{3}$ | $\omega^{2}$ | $\omega$ | $\omega$ | 1 | $\omega$ | $\omega$ | $\omega$ | $\omega^{2}$
$K_{I}$ | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 2
Table 1: Charge assignments of the particles under the various groups
considered in the model.

As we have considered a DM candidate in our model, it is necessary to
introduce a discrete symmetry $Z_{2}$ in order to maintain stability of the DM
candidate. The SM particles remain $Z_{2}$ even whereas the right handed
neutrino and the newly added field eta are odd under this symmetry. The scalar
potential of the Higgs sector can be found in Boucenna:2011tj . After
electroweak symmetry breaking, one of the $\eta$’s acquire VEV and their form
can be written as:

$\eta_{1}=\begin{pmatrix}\eta_{1}^{+}\\\
\frac{v_{\eta}+h_{1}+iA_{1}}{\sqrt{2}}\end{pmatrix},\hskip
14.22636pt\eta_{2}=\begin{pmatrix}\eta_{2}^{+}\\\
\frac{h_{2}+iA_{2}}{\sqrt{2}}\end{pmatrix},\hskip
14.22636pt\eta_{3}=\begin{pmatrix}\eta_{3}^{+}\\\
\frac{h_{3}+iA_{3}}{\sqrt{2}}\end{pmatrix}$ (7)

As mentioned in Hirsch:2010ru , the VEV alignment of $\eta$ can be written as
$\eta=v_{\eta}(1,0,0)$ and $\eta_{2},\eta_{3}$ will be the dark matter
candidates. Moreover we get a diagonal charge lepton mass matrix when the VEV
of $\phi$ is taken as $\phi=(v,0,0)$ Feruglio:2017spp . Based on the above
discussions and relevant charge assignmets, the Yukawa Lagrangian for the
neutrino sector can be written as:

$\mathcal{L}=N_{1}(LY)_{3}\eta+N_{2}(LY)_{3}\eta+\beta_{1}N_{1}(SY)_{1^{\prime\prime}}+\beta_{2}N_{2}(SY)_{1^{\prime}}+\mu_{0}(SS)_{1}$
(8)

The first two terms in the lagrangian denotes the interaction between left-
handed and right-handed neutrinos. The next two terms represents the
interaction between $N$’s and $S$’s while the last term denotes the
interaction between the sterile fermions. As a result, from eq. (8) we can
write down the corresponding neutrino mass matrices in the following way. The
Dirac mass matrix, which is a matrix of order 3$\times$2, for the neutrinos
can be written as:

$M_{D}=v_{\eta}\begin{pmatrix}-y_{1}&-y_{2}\\\ 2y_{2}&-y_{1}\\\
-y_{1}&2y_{3}\end{pmatrix}$ (9)

Similarly, following the $A_{4}$ multiplication rules, the Majorana mass
matrix for right-handed neutrino and sterile fermions, and the lepton number
violating mass term for the sterile fermions can be written as:

$M_{NS}=\begin{pmatrix}\beta_{1}y_{3}&\beta_{1}y_{2}&\beta_{1}y_{1}\\\
\beta_{2}y_{2}&\beta_{2}y_{1}&\beta_{2}y_{3}\end{pmatrix},\hskip
19.91692ptM_{S}=\mu_{0}\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix}$
(10)

Following the methods as mentioned in the literature, the full $8\times 8$
neutrino mass matrix for ISS(2,3) can be written as:

$\mathcal{M}=\begin{pmatrix}0&M^{T}_{D}&0\\\ M_{D}&0&M_{NS}\\\
0&M^{T}_{NS}&M_{S}\end{pmatrix}_{8\times 8}$ (11)

This $8\times 8$ matrix $M$ in eq (11) can be diagonalised with the help of an
Unitary matrix, $\mathcal{U}$ as

$\mathcal{U^{T}}M\mathcal{U}=M_{diag}=diag(m_{1},m_{2},m_{3},....,m_{8})$ (12)

where $m_{i}$’s in the above equation are masses of the particles of the
model. Since the matrix in eq (10) is a rectangular matrix, so we cannot find
the light neutrino mass matrix by the conventional approach Abada:2017ieq .
Because of this reason, the formula to find the effective light neutrino mass
matrix in eq, (6) changes and takes the form Abada:2014zra :

$m_{\nu}=M_{D}.d.M^{T}_{D}$ (13)

In the above equation, $d$ is a $2\times 2$ matrix which can be derived from
the $5\times 5$ heavy neutrino mass matrix $M_{H}$. The form of $d$ can be
obtained in the following way:

$M_{H}^{-1}=\begin{pmatrix}0&M_{NS}\\\
M_{NS}^{T}&M_{S}\end{pmatrix}^{-1}=\begin{pmatrix}d_{2\times 2}&.....\\\
......&.....\end{pmatrix}$ (14)

Thus we have constructed a model using $A_{4}$ modular symmetry in the
framework of ISS(2,3). We can diagonalise eq. (13) to obtain the masses of
light neutrinos which will facilitate to study the related neutrino
phenomenology.

##  III Leptogenesis

In order to validate our model considering the cosmological constraints, we
try to generate the observed baryon asymmetry of the universe through
leptogenesis. There are five heavy neutrinos in ISS(2,3). As mentioned above,
two of them are right handed neutrinos ($N_{1},N_{2}$) and the other three are
gauge singlet neutral sterile fermions ($S_{i}$). Four of these heavy
particles form two pairs, called quasi-Dirac pairs, and one of them gets
decoupled. Interestingly, the mass splitting between these pairs is comparable
to their decay width. The out-of-equilibrium decay of the lightest pair to any
lepton flavor creates an asymmetry in the leptonic sector. This asymmetry
created by decay of the lightest heavy neutrinos can be converted into baryon
asymmetry through sphaleron processes Awasthi:2013we ; Lindner:2014oea ;
Lucente:2018uaj . On the other hand, asymmetry generated by decay of the
heavier pair is washed out by lepton number violating scatterings of the
lightest pair, thereby, it does not contribute to the asymmetry produced.

###  III.1 Computation of CP asymmetry

To calculate the CP-asymmetry we need the mass matrix for the heavy neutrinos.
This matrix can be written as:

$M_{H}=\begin{pmatrix}0&M_{NS}\\\ M_{NS}^{T}&M_{S}\end{pmatrix}$ (15)

Now on diagonalising the above matrix, one can get masses of the five heavy
neutrinos. Also the mass splitting between the degenerate pairs is
proportional to $M_{S}$.

$M_{diag}=V^{T}M_{H}V=diag(m_{1},m_{2},m_{3},m_{4},m_{5})$ (16)

To diagonalise this $5\times 5$ matrix analytically is a challenging and
formidable task. So we opt for numerical diagonalisation so as to simplify our
analysis. Again for the calculation of CP-asymmetry, a particular basis is
preferred in which the matrix $M_{H}$ becomes diagonal. In this basis the
Lagrangian takes the form:

$\mathcal{L_{S}}=h_{i\alpha}N_{i}\eta
L_{\alpha}+M_{i}N_{i}^{T}C^{-1}N_{i}+h.c.$ (17)

where $h_{i\alpha}$ corresponds to the couplings in diagonal mass basis. These
are related to the couplings in the flavor basis represented by the following
relations Chakraborty:2021azg ; Agashe:2018cuf :

$\displaystyle h_{1\alpha}$ $\displaystyle=V_{11}^{*}\hskip
2.84526pty_{1\alpha}+V_{12}^{*}\hskip 2.84526pty_{2\alpha}$ (18)
$\displaystyle h_{2\alpha}$ $\displaystyle=V_{21}^{*}\hskip
2.84526pty_{1\alpha}+V_{22}^{*}\hskip 2.84526pty_{2\alpha}$ $\displaystyle
h_{3\alpha}$ $\displaystyle=V_{13}^{*}\hskip
2.84526pty_{1\alpha}+V_{23}^{*}\hskip 2.84526pty_{2\alpha}$ $\displaystyle
h_{4\alpha}$ $\displaystyle=V_{14}^{*}\hskip
2.84526pty_{1\alpha}+V_{24}^{*}\hskip 2.84526pty_{2\alpha}$

For the decay $N_{i}\rightarrow l_{\alpha}\phi\hskip
5.69054pt(\bar{l_{\alpha}}\phi)$, the formula for calculating the CP asymmetry
$\epsilon_{i}$ by summing over the SM flavor $\alpha$ is given by Covi:1996wh

$\epsilon_{i}=\frac{\sum_{\alpha}[\Gamma(\tilde{\psi}\rightarrow
l_{\alpha}\phi)-\Gamma(\tilde{\psi}\rightarrow\bar{l_{\alpha}}\phi^{\dagger})]}{\sum_{\alpha}[\Gamma(\tilde{\psi}\rightarrow
l_{\alpha}\phi)+\Gamma(\tilde{\psi}\rightarrow\bar{l_{\alpha}}\phi^{\dagger})]}=\frac{1}{8\pi}\sum_{i\neq
j}\frac{Im[(hh^{\dagger})^{2}_{ij}]}{(hh^{\dagger})_{ii}}f_{ij}$ (19)

For the case of resonant leptogenesis, the self energy correction,
$f_{ij}=\frac{(M_{i}^{2}-M_{j}^{2})M_{i}M_{j}}{(M_{i}^{2}-M_{j}^{2})^{2}+(M_{i}\Gamma_{i}+M_{j}\Gamma_{j})^{2}}$.
$M_{i}$ and $M_{j}$ are the real and positive eigenvalues of the heavy
neutrino mass matrix. $\Gamma_{i}$ is the decay width of one of the quasi-
Dirac pair which is expressed as
$\Gamma_{i}=\frac{M_{i}}{8\pi}(hh)^{\dagger}_{ii}$. Thus the explicit form of
CP parameter for the decay of a quasi-Dirac pair, say $(N_{1},S_{1})$, can be
expressed as:

$\displaystyle\epsilon_{1}$
$\displaystyle=\frac{1}{8\pi(hh^{\dagger})_{11}}\textnormal{Im}[(hh^{\dagger})^{2}_{12}f_{12}+(hh^{\dagger})^{2}_{13}f_{13}+(hh^{\dagger})^{2}_{14}f_{14}]$
(20) $\displaystyle\epsilon_{2}$
$\displaystyle=\frac{1}{8\pi(hh^{\dagger})_{22}}\textnormal{Im}[(hh^{\dagger})^{2}_{21}f_{21}+(hh^{\dagger})^{2}_{23}f_{23}+(hh^{\dagger})^{2}_{24}f_{24}]$

As already mentioned, the asymmetry produced by decay of the heavier pair is
washed out. Thus the wash out parameter for such decays, in terms of the
Hubble parameter H, is written as:

$K_{i}=\frac{\Gamma_{i}}{H}=\frac{M_{i}}{8\pi}(hh)^{\dagger}_{ii}\times\frac{M_{pl}}{1.66\sqrt{g*}M_{i}^{2}}$
(21)

In the above equation, $M_{pl}$ is the Planck mass and $g*$ denotes the
effective number of relativistic degrees of freedom. The final expression for
BAU can be written as:

$Y_{B}=10^{-2}\sum\kappa_{i}\epsilon_{i}$ (22)

where $\kappa_{i}$ is the dilution factor responsible for washout out of the
asymmetry associated with the heavy pair. $\epsilon_{i}$ is the CP asymmetry
generated in the leptonic sector. The expressions for $\kappa_{i}$ depends on
the values of washout factor in eq (21). These relations can be summarised as
Pilaftsis:2005rv :

$\displaystyle-\kappa$ $\displaystyle\approx\sqrt{0.1K}\hskip
5.69054ptexp[\frac{-4}{3(0.1K)^{0.25}}],\hskip 17.07164pt\textnormal{for}K\geq
10^{6}$ (23) $\displaystyle\approx\frac{0.3}{K(\textnormal{ln}K)^{0.6}},\hskip
71.13188pt\textnormal{for}\hskip 5.69054pt10\leq K\leq 10^{6}$
$\displaystyle\approx\frac{1}{2\sqrt{K^{2}+9}},\hskip
71.13188pt\textnormal{for}\hskip 5.69054pt0\leq K\leq 10$

Thus, we use eq (22) to calculate the value of BAU in the framework of
ISS(2,3). In our work we have computed the values of this asymmetry for both
the hierarchies. We have highlighted our findings in the numerical analysis
section of the paper. We find that for both the hierarchies there are a large
number of points that satisfy the Planck value of BAU.

##  IV Neutrinoless double beta decay

Neutrinoless double beta decay $(0\beta\beta\nu/\textnormal{NDBD})$ is an
important phenomena in particle physics. It is a lepton number violating decay
which if discovered can give an answer to many of the open questions in
particle physics. In our work, we have studied the effective electron neutrino
Majorana mass which characterizes $0\beta\beta\nu$. As mentioned in
Blennow:2010th , the experiments like KamLAND-ZenKamLAND-Zen:2016pfg ,
GERDAGERDA:2020xhi , CUORECUORE:2019yfd  etc. provide a stringent bound on
$m_{ee}$. In absence of any sterile neutrino, this effective mass is expressed
as given below:

$m_{ee}=\big{|}\sum^{3}_{i}U_{ei}^{2}m_{i}\big{|}$ (24)

The presence of sterile neutrinos in ISS(2,3) changes the expression for
effective mass in eq (24). As a result, the modified expression become
Gogoi:2022jwf ; Abada:2018qok :

$m_{ee}=\big{|}\sum^{3}_{i=1}U_{ei}^{2}m_{i}\big{|}+\big{|}\sum_{j=1}^{5}U_{ej}^{2}\frac{M_{j}}{k^{2}+M_{j}^{2}}\big{|}\big{<}k\big{>}\big{|}^{2}\big{|}$
(25)

where $U_{ej}$ represents the coupling of the heavy neutrinos to the electron
neutrino. $M_{j}$ is the mass of the heavy neutrinos. The parameter $k$ is
known as the virtuality momentum and its value is
$\big{|}\big{<}k\big{>}\big{|}$$\approx$ 190 MeV.

##  V Dark Matter

At the beginning of the universe the particles present in the thermal pool
were in thermal equilibrium with each other. This implies that the rate at
which lighter particles combined to form heavy particles and vice-versa was
the same. During the course of evolution, the conditions that were required to
maintain this equilibrium state were disturbed. As a result, after a certain
temperature the density of some particle species became too low. Once this is
achieved, the abundance of those particle remains the same and their density
becomes constant. This phase of the particle species is called freeze-out and
the density hereafter is referred to as relic density. For a particle $\chi$,
which was in thermal equilibrium, the relic density can be obtained from the
Boltzman equations Griest:1990kh ; Gondolo:1990dk :

$\frac{dn_{\chi}}{dt}+3\mathcal{H}n_{\chi}=-\textless\sigma
v\textgreater\left(n_{\chi}^{2}-\left(n_{\chi}^{eqb}\right)^{2}\right)$ (26)

In the above equation $n_{\chi}$ is the number density of the dark matter
particle whereas $n_{\chi}^{eqb}$ is the density of $\chi$ when it was in
equilibrium with the thermal bath. Here $\mathcal{H}$ represents the Hubble
constant and $\textless\sigma v\textgreater$ is the thermally averaged
annihilation cross-section of the dark matter candidate. For the interactions
in eq. (8), the cross-section formula can be writtern as Bai:2013iqa :

$\textless\sigma
v\textgreater=\frac{v^{2}y^{4}m_{\chi}^{2}}{48\pi(m_{\chi}^{2}+m_{\psi}^{2})^{2}}$
(27)

where the parameters $m_{\chi},m_{\psi}\hskip 5.69054pt\textnormal{and}\hskip
5.69054pty$ represent the mass of the relic particle, mass of the Majorana
fermion and the interaction between the dark matter and fermions in the model.
$v$ in the above expression represents the relative velocity of the relic
particles whose value at the time of freeze-out is taken to be $0.3c$. The
solution of eq. (26) in terms of reduced Hubble constant ($h$), as found in
Jungman:1995df , can be written as:

$\Omega_{\chi}h^{2}=\frac{3\times
10^{-27}\textnormal{cm}^{3}\textnormal{s}^{-1}}{\textless\sigma
v\textgreater}$ (28)

The above expression of $\Omega_{\chi}h^{2}$ gives the relic density of dark
matter particle. It is found that the self annihilation between dark matter
and next to lightest neutral component of $\eta$ contributes to the
annihilation cross-section. For low mass region i.e. $m_{DM}\textless M_{W}$,
the cross-section for the self annihilation of either $\eta_{2}$ or $\eta_{3}$
into Standard Model particles via the Higgs boson can be written as
Bell:2013wua :

$\sigma_{xx}=\frac{|Y_{f}|^{2}|\lambda_{x}|^{2}}{16\pi
s}\frac{(s-4m_{f}^{2})^{\frac{3}{2}}}{\sqrt{s-4m_{x}^{2}}((s-m_{h}^{2})^{2}+m_{h}^{2}\Gamma_{h}^{2})}$
(29)

where $x$ is the dark matter particle ($\eta_{2},\eta_{3}$) and $\lambda_{x}$
is the coupling of $x$ with the SM Higgs boson $h$. In eq. (29) $Y_{f}$
represents the Yukawa couplings of the fermions. $\Gamma_{h}$ represents the
decay width of the SM Higgs and $m_{h}$ is equal to 125 GeV. And $s$ in the
expression represents the thermally averaged center of mass squared energy and
is given by $s=4m_{x}^{2}+m_{x}^{2}v^{2}$, $m_{x}$ is mass of the relic. In
our work the neutral component of the scalar triplet $\eta$ is the dark matter
candidate and its mass should be less than the mass of $W$ boson, $M_{W}$. The
results that we have found in this work and their respective analysis are
shown in the next section.

##  VI Numerical Analysis

In this section we discuss the processes that we followed to arrive at the
results of our work. In our calculations we have used the latest nu-fit data
for the oscillation parameters. This set of data has been shown in the table
below:

Oscillation parameters | Normal Ordering | Inverted Ordering
---|---|---
$\textnormal{sin}^{2}\theta_{12}$ | [0.269,0.343] | [0.269,0.343]
$\textnormal{sin}^{2}\theta_{23}$ | [0.407,0.618] | [0.411,0.621]
$\textnormal{sin}^{2}\theta_{13}$ | [0.02034,0.02430] | [0.020530,0.02436]
$\Delta m^{2}_{21}/10^{-5}\textnormal{eV}^{2}$ | [6.28,8.04] | [6.82,8.04]
$\Delta m^{2}_{31}/10^{-3}\textnormal{eV}^{2}$ | [2.431,2.598] | [2.412,2.583]
Table 2: The latest $3\sigma$ nu-fit values of oscillation parameters
Esteban:2020cvm .

In order to find the model parameters, we diagonalise the light neutrino mass
matrix using the standard relation
$m_{\nu}=U^{T}\textnormal{diag}(m_{1},m_{2},m_{3})U$, where $U$ is the PMNS
matrix. In normal hierarchy the diagonal matrix can be written as
$\textnormal{diag}(0,\sqrt{m_{1}^{2}+\Delta
m^{2}_{solar}},\sqrt{m_{1}^{2}+\Delta m^{2}_{atm}})$, whereas in the inverted
ordering it takes the form $\textnormal{diag}(\sqrt{m_{3}^{2}+\Delta
m^{2}_{atm}}),\sqrt{\Delta m_{atm}^{2}+\Delta m^{2}_{solar}},0)$ Nath:2016mts
. Through this process we get the masses of light active neutrinos. Moreover,
the solar and atmospheric mass squared differences, along with the sum of
neutrino masses, provide additional constraints on the model. In order to
evaluate the values of the mixing angles from the model we use the following
relations Nomura:2019xsb :

$\displaystyle\textnormal{sin}^{2}\theta_{13}=|U_{e3}|^{2},\hskip
17.07164pt\textnormal{sin}^{2}\theta_{23}=\frac{|U_{\mu
3}|^{2}}{1-|U_{e3}|^{2}},\hskip
17.07164pt\textnormal{sin}^{2}\theta_{12}=\frac{|U_{e2}|^{2}}{1-|U_{e3}|^{2}},$
(30)

The real and imaginary parts of the complex modulus, $\tau$ is found to lie
within the fundamental domain Feruglio:2017spp . Here in this work the ranges
for the real and imaginary parts are found to be:
Re($\tau$)$\rightarrow$[-0.9, 0.9] and Im($\tau$)$\rightarrow$[0.3, 6]. With
these values of $\tau$ we find the Yukawa modular forms for both normal and
inverted hierarchies. These values of the couplings have been highlighted in
table 3. For further calculations we have taken $\Lambda$ in the range [10,20]
KeV and $v_{\eta}$ is considered in between ($30-50$) GeV. In a similar way,
we have taken the values of $\beta_{1}$ and $\beta_{2}$ in the ranges
[$10^{5},10^{7}$] GeV and [$10^{2},10^{3}$] GeV, respectively, so as to obtain
the values of different parameters in the allowed ranges. In the next part of
this section we will show the different plots and then discuss the respective
results that are obtained in this work.

Yukawa couplings | Normal Hierarchy | Inverted Hierarchy
---|---|---
|$y_{1}$| | 0.93 - 2 | 0.94 - 2.5
|$y_{2}$| | 0.5 - 1 | 0.01 - 0.6
|$y_{3}$| | 0.1 - 0.7 | 0.01 - 0.5
Table 3: Values of the Yukawa modular forms.

In fig.(1) and (2), we show the relation between the Yukawa modular forms and
real part of the modulus, $\tau$. In case of normal hierarchy, the values of
Yukawa coupling ($y_{2}$) is found to lie mainly around the region (0.1-0.3)
and for $y_{3}$ the parameter space is about 0.1. Some patterns of clusters
can be found between the parameters in these zones. These patterns are more
prominent in the case of inverted hierarchy. In the case of inverted
hierarchy, however, the pattern is almost similar to NH. Fig. (3) shows the
relation between the imaginary part of $\tau$ and the Yukawa modular forms. It
is evident from the graphs that most of the values of the couplings lie in
between (0.9-2.5) of Im($\tau$).

Figure 1: Corelation between Re($\tau$) and Yukawa modular form $y_{2}$ for
both normal and inverted hierarchies.

Next we discuss the relation between the mixing angles and modular forms used
in this work. Fig. (4), (5), and (6) highlight these relations for both the
normal and inverted hierarchies. As evident from these graphs, there are
sufficient parameter spaces that lie within the allowed ranges. In fig. (4) we
can see the relation with respect to $\textnormal{sin}^{2}\theta_{13}$.

Figure 2: The above figure shows the relation between Re($\tau$) and $y_{3}$
for both the hierarchies.

Figure 3: Corelation between Im($\tau$) and the Yukawa modular forms ($y_{2}$
and $y_{3}$).

There are three prominent regions of the Yukawa couplings that give the mixing
angle in the allowed range. These ranges can be identified as:
$0.99\leq|y_{1}|\leq 1.2$, $0.78\leq|y_{2}|\leq 0.87$ and $0.29\leq|y_{3}|\leq
0.35$. Similarly we can find the ranges for the other graphs also. From fig.
(5) we can see that the allowed range of sin${}^{2}\theta_{12}$ is prominent
for the values of Yukawa couplings in the range: $0.99\leq|y_{1}|\leq 1.2$,
$0.25\leq|y_{2}|\leq 0.35$ and $0.05\leq|y_{3}|\leq 0.1$. A few values of
$y_{3}$ are also found in the upper part of the graphs. In case of atmospheric
mixing angle (sin${}^{2}\theta_{23}$), the ranges of the Yukawa couplings from
fig. (6) can be found as: $0.98\leq|y_{1}|\leq 1.2$, $0.5\leq|y_{2}|\leq 0.7$
and $0.15\leq|y_{3}|\leq 0.25$. Moreover there are few values of $y_{2}$ and
$y_{3}$ which are randomly spread in the upper and middle portion of the
graphs. This shows that there are certain regions of Yukawa modular forms
which are more favorable at producing the mixing angles. In the following
figures, the vertical lines represent the 3$\sigma$ bounds of the mixing
angles for both the normal and inverted hierarchies. The points in the region
within these lines represent the allowed ranges of the parameters.

Figure 4: Corelation between sin${}^{2}\theta_{13}$ and yukawa couplings
($y_{1},y_{2},y_{3}$).

Figure 5: Corelation between sin${}^{2}\theta_{12}$ and yukawa couplings
($y_{1},y_{2},y_{3}$).

Figure 6: Corelation between sin${}^{2}\theta_{23}$ and yukawa couplings
($y_{1},y_{2},y_{3}$).

Fig. (7) shows the relation between sum of neutrino mass ($\sum m_{\nu}$) and
effecive mass ($m_{eff}$). The recent cosmological findings from KamLANDZen
provide the upper bound of $\sum m_{\nu}$ $\leq$ 0.12 eV, whereas the allowed
range for the effective neutrino mass for NDBD is found to be $\leq$ 0.165 eV.
The horizontal line in the figure represents the upper bound of effective
electron neutrino mass, whereas the vertical line represents the upper bound
for sum of neutrino mass. It is quite evident from both the graphs that there
are sufficient parameter space within the allowed range for both the normal
and inverted hierarchies.

In fig. (8) we show the relation between BAU and lightest right-handed
neutrino mass, $M_{1}$. We performed the calculations for both the
hierarchies. From these graphs it is clear that for both the type of
hierarchies there are sufficient values of BAU which satisfy the Planck limit.
This value of BAU for normal ordering is mainly concentrated in between
(100-1000) GeV of $M_{1}$. Beyond this mass range the values of asymmetry is
very less. But for inverted ordering this mass range is found to lie in
between (100-5000) GeV. From this discussion it is clear that the values of
baryon asymmetry of the universe can be obtained for both the hierarchies from
this model. Also in fig. (9), we show the relation between BAU and sum of
neutrino masses.

Figure 7: The above figures illustrate the corelation between $\sum m_{\nu}$
and $m_{eff}$. The horizontal and vertical lines represent the upper bounds of
the parameters.

Figure 8: The figures above show the corelation between BAU and $M_{1}$ for
both the hierarchies. In these figures, the horizontal lines represent the
Planck value of BAU.

Figure 9: These figures show the corelation between BAU and sum of neutrino
masees $\sum m_{\nu}$. The horizontal and vertical lines represent the Planck
value of BAU and upper bound of $\sum m_{\nu}$, respectively.

Finally in fig. (10) and (11) we show the variation of dark matter relic
density and thermally averaged annihilation cross-section with respect to
lightest heavy neutrino $M_{1}$. It can be seen that for $M_{1}$ in between
(100-450) GeV, the values of relic density is more prominent for both the
hierarchies. However, in case of inverted hierarchy, there are certain areas
of mass that produce the observed relic density. As for the thermally averaged
cross-section, we see that for almost the entire mass range of the $M_{1}$,
the values of the scattering cross section is consistent with the experimental
upper bounds Fermi-LAT:2010qeq .

Figure 10: The above figure shows the variation between DM relic density and
lightest right-handed neutrino $M_{1}$. The horizontal line represents the
current dark matter abundance in the universe.

Figure 11: The above figure shows the corelation between cross-section
$\textless\sigma v\textgreater$ and $M_{1}$.

##  VII Conclusion

In this work, we have extended the minimal inverse seesaw by adding an extra
Higgs-type scalar triplet ($\eta$) to its particle content. As mentioned in
the earlier sections, this is a very popular mechanism in model building
bescause of its ability to reduce the mass scale of right-handed neutrinos to
TeV. We have used $A_{4}$ modular symmetry and $Z_{3}$ symmetry groups in our
work. In $A_{4}$ modular group there are three Yukawa modular forms which are
functions of the complex modulus $\tau$. These modular forms play a vital role
in our calculations. In this paper, we have studied neutrino masses and
mixings, neutrinoless double beta decay, baryogenesis via leptogenesis and
dark matter. We have considered the results and bounds related to these
phenomena which are published by different ongoing experiments and
observations. We find that the values of the parameters calculated from the
model are within the allowed ranges. Moreover, we have performed these
calculations for both the normal and inverted hierarchies. In case of the
Yukawa couplings, most of the values are mainly found to lie in the region
(0.01 - 2). Also there is a sufficient parameter space within the allowed
range for sum of neutrino masses and effective electron neutrino mass for
NDBD.

Furthermore, on studying BAU in our model, we have obtained a satisfying
parameter space corresponding to $M_{1}$ which abide by the Planck limit for
both NO/IO. Again, we have extended our investigation to dark matter sector as
well by calculating the relic abundance and scattering cross section of the DM
candidate. From plots Fig. 10 and 11, we have obtained a certain parameter
space for DM mass which generates the observed relic abundance. Also, we have
sufficient points which showcase very small scattering cross section. Thus,
from the study of various phenomena that we have carried out in this work, we
can have a conclusive idea that this model is a viable one.

##  Appendix A Modular Symmetry

In recent times modular symmetry has become quite popular in the study of
neutrino phenomenology. The modular group $\Gamma(N)$ ($N=1,2,3..$) can be
defined in the following way:

$\Gamma(N)=\left\\{\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
SL(2,Z),\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=\begin{pmatrix}1&0\\\
0&1\end{pmatrix}(\textnormal{mod N})\right\\}$ (31)

such that $ad-bc=1.$ These groups act on the upper half of the complex plane,
(Im$(\tau)\textgreater 0$) and transforms the complex variable $\tau$ linearly
as:

$\tau\rightarrow\frac{a\tau+b}{c\tau+d}$

The matrix form of the two generators of modular symmetry are:

$\displaystyle S=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},$
$\displaystyle\hskip 56.9055ptT=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}.$ (32)

These operators act on $\tau$ and transform them in the following ways:

$\displaystyle S$ $\displaystyle\xrightarrow{\tau}-\frac{1}{\tau},$
$\displaystyle T$ $\displaystyle\xrightarrow{\tau}1+\tau.$ (33)

The finite modular groups $(N\leq 5)$ and non-abelian discrete groups are
isomorphic to each other. As a result $\Gamma_{2}\approx S_{3}$,
$\Gamma_{3}\approx A_{4}$,$\Gamma_{4}\approx S_{4}$,$\Gamma_{5}\approx
A^{\prime}_{5}.$ For a group of level $N$, the number of modular forms varies
with respect to their weights. The table below shows how to find out the
number of modular forms that a particular group with a particular level can
have.

$N$ | No. of modular forms | $\Gamma(N)$
---|---|---
2 | $k+1$ | $S_{3}$
3 | $2k+1$ | $A_{4}$
4 | $4k+1$ | $S_{4}$
5 | $10k+1$ | $A_{5}$
6 | $12k$ |
7 | $28k-2$ |
Table 4: No. of modular forms of weight $2k$.

###  A.1 $\Gamma(3)$ modular group

It is a level three modular group which is isomorphic to discrete symmetry
group $A_{4}$. The three modular forms of weight 2 present in this group can
be expressed as:

$\displaystyle
y_{1}(\tau)=\frac{i}{2\pi}[\frac{\eta^{\prime}(\frac{\tau}{3})}{\eta(\frac{\tau}{3})}+\frac{\eta^{\prime}(\frac{\tau+1}{3})}{\eta(\frac{\tau+1}{3})}+\frac{\eta^{\prime}(\frac{\tau+2}{3})}{\eta(\frac{\tau+2}{3})}-27\frac{\eta^{\prime}(3\tau)}{\eta(3\tau)}]$
(34) $\displaystyle
y_{2}(\tau)=\frac{-i}{\pi}[\frac{\eta^{\prime}(\frac{\tau}{3})}{\eta(\frac{\tau}{3})}+\omega^{2}\frac{\eta^{\prime}(\frac{\tau+1}{3})}{\eta(\frac{\tau+1}{3})}+\omega\frac{\eta^{\prime}(\frac{\tau+2}{3})}{\eta(\frac{\tau+2}{3})}]$
$\displaystyle
y_{3}(\tau)=\frac{-i}{\pi}[\frac{\eta^{\prime}(\frac{\tau}{3})}{\eta(\frac{\tau}{3})}+\omega\frac{\eta^{\prime}(\frac{\tau+1}{3})}{\eta(\frac{\tau+1}{3})}+\omega^{2}\frac{\eta^{\prime}(\frac{\tau+2}{3})}{\eta(\frac{\tau+2}{3})}]$

where $\eta(\tau)$ is the Dedekind eta-function and is defined in the
following way:

$\eta(\tau)=q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n}),\hskip
28.45274ptq=e^{2\pi i\tau}.$ (35)

The eta functions satisfy the equations

$\eta(\tau+1)=\exp^{i\pi/12}\eta(\tau),\hskip
22.76219pt\eta(-1/\tau)=\sqrt{-i\tau}\eta(\tau)$ (36)

Another way of expanding these modular forms is the q-expansions, where q is
expressed in terms of $\tau$ in the following way $q=\exp(2i\pi\tau)$. These
expansions take the form:

$\displaystyle y_{1}(\tau)=1+12q+36q^{2}+12q^{3}+.........$ (37)
$\displaystyle y_{2}(\tau)=-6q^{1/3}(1+7q+8q^{2}+.......)$ $\displaystyle
y_{3}(\tau)=-18q^{2/3}(1+2q+5q^{2}+......)$

Moreover with the help of these lower weight modular forms we can construct
the higher weight modular functions. For a detailed study one can refer to
Zhang:2019ngf ; Novichkov:2019sqv .

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