#  $M_{W_{R}}$ dependence of leptogenesis in Minimal Left-Right Symmetric
Model with different strengths of Type-II seesaw mass.

Ankita Kakoti<EMAIL_ADDRESS>Mrinal Kumar Das<EMAIL_ADDRESS>Department of Physics, Tezpur University, Tezpur 784028, India

###### Abstract

Left Right Symmetric Model (LRSM) being an extension of the Standard model of
particle physics incorporates within itself Type-I and Type-II seesaw mass
terms naturally. Both the mass terms can have significant amount of
contribution to the resulting light neutrino mass within the model and hence
on the different phenomenology associated within. In this paper, we have
thoroughly analyzed and discussed the implications of specifying different
weightages to the type-I and type-II mass terms and also the study has been
carried out for different values of $M_{W_{R}}$ which is mass of the right-
handed gauge boson. This paper also gives a deeper insight into the new
physics contributions of Neutrinoless Double Beta Decay $(0\nu\beta\beta)$ and
their variations with the net baryon asymmetry arising out of the model.
Therefore, the main objective of the present paper rests on investigating the
implications of imposing different weightage to the type-I and type-II seesaw
terms and different values of $M_{W_{R}}$ on the new physics contributions of
$0\nu\beta\beta$ and net baryon asymmetry arising out as a result of resonant
leptogenesis. LRSM in this work has been realized using modular group of level
3, $\Gamma(3)$ which is isomorphic to non-abelian discrete symmetry group
$A_{4}$, the advantage being the non-requirement of flavons within the model
and hence maintaining the minimality of the model.

##  I Introduction

The establishment of the fact that neutrinos are massive and their mixing
between flavors by the neutrino oscillation experiments compels us to move
beyond the Standard paradigm. Although Standard Model (SM) has been enormously
successful in explaining the fundamental particles and their interactions,
however it fails to address some issues like the origin of tiny neutrino
masses, baryon asymmetry of the universe (BAU) [1], lepton flavor violation
(LFV) [2], origin of dark matter etc [3, 4].
In this context, we can state that the seesaw mechanism [5, 6, 7, 8, 9, 10,
11] has proven to be a very simple way to explain the tiny neutrino masses,
the two main componets of the mechanism being, the addition of right-handed
(RH) neutrinos to the SM and endowing the RH neutrinos with a Majorana mass
which breaks the accidental global (B-L) symmetry of the SM. However, in the
context of the electroweak gauge theory of the SM described by
$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$, there is no underlying principle
that explains this mechanism, but rather they are put in by hand. Coming to
the fact that there are two ultraviolet-complete theories where both the
components of seesaw arise in a natural manner and these are,

  * •

The left-right (LR) symmetric theories of weak interactions based on the gauge
group $SU(3)_{C}\otimes SU(2)_{L}\otimes SU(2)_{R}\otimes U(1)_{B-L}$ [7, 12,
13, 14, 15, 16, 17, 18].

  * •

$SO(10)$ grand-unified theory for all interations [19].

In this work, however our focus is on low-scale Left-Right symmetric model
(LRSM) where the seesaw scale can be in a few TeV range, such that it is
accessible to the LHC, while satisfying the observed charge-lepton and
neutrino mass spectra.
One imporatnt question here arises that why is the seesaw mechanism so
favourable for the explanation of tiny neutrino masses. This is because, when
we take seesaw mechanism into consideration, the same Yukawa that are
responsible for generating small neutrino masses can also resolve one of the
outstanding puzzles of cosmology, namely, the origin of matter-antimatter
asymmetry, via leptogenesis. Leptogenesis can be picturized by the out-of
equilibrium decays of the RH neutrinos via the modes $N\rightarrow L_{l}\phi$
and $N\rightarrow L_{l}^{c}\phi^{c}$, where, $L_{l}=(\nu_{L}l)_{L}^{T}$ are
the $SU(2)_{L}$ lepton doublets, $\phi$ is the Higgs doublet and the
superscript c denotes the CP conjugate. In the presence of CP violation in the
Yukawa sector, the decays mentioned above can lead to lepton asymmetry in the
early universe satisfying the three Sakharov conditions. This asymmetry then
undergoes thermodynamic evolution with the expansion of the universe and
different reactions present in the model have their impact on washing out part
of the asymmetry. The remaining final lepton asymmetry is converted to baryon
asymmetry via sphaleron processes before the electroweak phase transition.
For TeV scale seesaw models, we require resonant leptogenesis (RL) for the
generation of adequate lepton asymmetry, where atleast two of the heavy
Majorana neutrinos have a small difference comparable to their decay widths.
The TeV scale new particles in LRSM also leads to interesting collider
signatures.
In this work, we have also taken into account the study of neutrinoless double
beta decay (NDBD) [20], [21], because the possible observation of NDBD would
play an important role in understanding the origin of BAU as NDBD would imply
that lepton number is indeed not conserved, which is one of the essential
conditions for leptogenesis. In LRSM, there are several contributions to NDBD
that involve left-right sectors individually and also some contributions that
involves both the sector through left-right mixing accompanied by both light
and heavy neutrinos.Left-Right mixing is a ratio of the Dirac and Majorana
mass scales $[M_{D}M_{RR}^{-1}]$, appearing in the type-I seesaw mass formula
as we will discuss later. While taking to account large left-right mixing we
have to focus on the momentum dependent contributions of NDBD, that is, the
contribution that arises as a result of the mixed diagrams with simultaneous
mediation of $W_{L}$ and $W_{R}$ accompanied by light left-handed and heavy
right-handed neutrino known as $\lambda$ and $\eta$ contributions.
In this work, we have firstly realized the LRSM with the help of modular
$A_{4}$ symmetry and thus obtained the mass matrices in terms of the modular
forms $(Y_{1},Y_{2},Y_{3})$, and then we move forward to study the linkage
between the different contributions of NDBD and baryogenesis. We have
considered different strengths of the type-I and type-II seesaw masses with
differents values of the $SU(2)_{R}$ breaking scale, namely, type-II seesaw
mass being $30\%,50\%,70\%$ dominant for $M_{W_{R}}=3,5,10,18$ TeV. We have
not considered any particular texture for the Dirac and Majorana mass matrices
and in order to account for the neutrino mass of the order of sub eV, the
Dirac mass that we have determined in the scale of MeV, as the heavy right
handed neutrino is kept in the TeV scale. This leads to not so large left-
right mixing and hence we get non-negligible effects of the momentum dependent
contributions of NDBD. Although we have studied all the contributions of
NDBD,however to relate baryogenesis to the phenomena, we have considered only
the contributions appearing from the mediation of heavy right-handed neutrino
and the momentum dependent $\lambda$ and $\eta$ contributions.Since the
effective mass governing NDBD is dependent on the Majorana phases, $\alpha$
and $\beta$, it would be compelling to study if there is a link between NDBD
and BAU. So, this paper has been organized by firstly discussing about LRSM
and the origin of neutrino mass, the next section describing the phenomenon of
resonant leptogenesis and NDBD within LRSM, section (III) discusses the
numerical analysis and results and we conclude our work in section (IV). The
appendices discussed the minimisation of the Higgs potential for the model and
also introduces the concept of modular symmetry which has been used for the
realisation of the model.

##  II Left-Right Symmetric Model and Neutrino mass.

Left-Right Symmetric Model (LRSM) is described by the gauge group
$SU(3)_{C}\otimes SU(2)_{L}\otimes SU(2)_{R}\otimes U(1)_{B-L}$. As is evident
form the gauge group that unlike SM, both the left-handed and right handed
particles are taken as doublets in the said model. The scalar sector of the
model consists of one Higgs bidoublet $\phi(1,2,2,0)$ and two scalar triplets
given by $\Delta_{L}(1,3,1,2)$ and $\Delta_{R}(1,1,3,2)$. Because of the
presence of right-handed (RH) neutrinos and scalar triplets, LRSM incorporates
within itself $Type-I$ and $Type-II$ seesaw masses naturally.
In order for the fermions to attain mass, a Yukawa Lagrangian is necessary
which couples to the bidoublet $\phi$. The Yukawa Lagrangian for fermions
incorporating the bidoublet is given by,

$\mathcal{L_{D}}=\overline{l_{iL}}(Y_{ij}^{l}\phi+\widetilde{Y_{ij}^{l}}\widetilde{\phi})l_{jR}+h.c$
(1)

where, $l_{L}$ and $l_{R}$ are the left-handed and right-handed lepton fields,
$Q_{L}$ and $Q_{R}$ are the left-handed and right-handed quark fields. $Y^{l}$
being the Yukawa coupling corresponding to leptons and $Y^{q}$ being the
Yukawa coupling for the quarks. The Yukawa Lagrangian incorporating the scalar
triplets which play a role in providing Majorana mass to the neutrinos is
given by,

$\mathcal{L_{M}}=f_{L,ij}{\Psi_{L,i}}^{T}Ci\sigma_{2}\Delta_{L}\Psi_{L,j}+f_{R,ij}{\Psi_{R,i}}^{T}Ci\sigma_{2}\Delta_{R}\Psi_{R,j}+h.c$
(2)

$f_{L}$ and $f_{R}$ are the Majorana Yukawa couplings and are equal subjected
to discrete left-right symmetry. The family indices $i,j$ runs from $1$ to $3$
representing the three generations of the fermions. $C=i\gamma_{2}\gamma_{0}$
is the charge conjugation operator, where $\gamma_{\mu}$ are the Dirac
matrices and $\tilde{\phi}=\tau_{2}\phi^{*}\tau_{2}$.
The breaking pattern of the LRSM gauge group takes place in two steps. The
LRSM gauge group is first broken down to the Standard Model gauge group by the
vev of the scalar triplet $\Delta_{R}$, and then the Standard Model gauge
group is broken down to the electromagnetic gauge group i.e., $U(1)_{em}$ by
the vev of the bidoublet and a tiny vev of the scalar triplet $\Delta_{L}$.
So the resultant light neutrino mass of LRSM is expressed as a sum of the
type-I and type-II seesaw mass terms, given as,

$M_{\nu}=M_{\nu}^{I}+M_{\nu}^{II}$ (3)

where,

$M_{\nu}^{I}=M_{D}M_{R}^{-1}M_{D}^{T}$ (4)

is the type-I seesaw mass, and type-II seesaw mass is given by

$M_{\nu}^{II}=M_{LL}$ (5)

$M_{D}$ is the Dirac mass matrix and $M_{R}$ is the right-handed Majorana mass
matrix, where, $M_{R}=\sqrt{2}v_{R}f_{R}$ and $M_{L}=\sqrt{2}v_{L}f_{L}$.
$v_{R}$ and $v_{L}$ are the respective vev’s of $\Delta_{R}$ and $\Delta_{L}$.
The magnitudes of the VEVs follows the relation, $|v_{L}|^{2}<|k^{2}+k^{\prime
2}|<|v_{R}|^{2}$. In the context of LRSM however, both the type-I and type-II
mass terms can be expressed in terms of the heavy right-handed Majorana mass
matrix, so Eq.(3) will follow,

$M_{\nu}=M_{D}M_{R}^{-1}M_{D}^{T}+\gamma\Biggl{(}\frac{M_{W}}{v_{R}}\Biggl{)}^{2}M_{RR}$
(6)

where, $\gamma$ is a dimensionless parameter which is a function of various
couplings, appearing in the VEV of the triplet Higgs $\Delta_{L}$, i.e.,
$v_{L}=\gamma(\frac{v^{2}}{v_{R}})$ and here, $v=\sqrt{k^{2}+k^{\prime 2}}$,
and

$\gamma=\frac{\beta_{1}kk^{\prime}+\beta_{2}k^{2}+\beta_{3}k^{\prime
2}}{(2\rho_{1}-\rho_{3})(k^{2}+k^{\prime 2})}$ (7)

In our model, the dimensionless parameter $\gamma$ has been fine tuned to
$\gamma\approx 10^{-13}$ and $v_{R}$ is of the order of $TeV$. $\beta_{i}$ and
$\rho_{i}$ are dimensionless parameters occuring in the potential of the model
as described later in the appendix.
The light neutrino mass obtained can be expressed in terms of a matrix given
as,

$M_{\nu}=\begin{pmatrix}M_{LL}&M_{D}\\\ M_{D}^{T}&M_{RR}\end{pmatrix}$ (8)

This matrix is a $6\times 6$ matrix which can be diagonalized by a unitary
matrix as follows,

$\nu^{T}M_{\nu}\nu=\begin{pmatrix}\hat{M}_{\nu}&0\\\
0&\hat{M}_{RR}\end{pmatrix}$ (9)

where, $\nu$ represents the diagonalizing matrix of the full neutrino mass
matrix, $M_{\nu}$,$\hat{M_{\nu}}=diag(m_{1},m_{2},m_{3})$, with $m_{i}$ being
the light neutrino masses and $\hat{M_{RR}}=diag(M_{1},M_{2},M_{3})$, with
$M_{i}$ being the heavy right-handed neutrino masses.
The diagonalizing matrix can be represented as,

$\nu=\begin{pmatrix}U&S\\\
T&V\end{pmatrix}\approx\begin{pmatrix}1-\frac{1}{2}RR^{\dagger}&R\\\
-R^{\dagger}&1-\frac{1}{2}R^{\dagger}R\end{pmatrix}\begin{pmatrix}V_{\nu}&0\\\
0&V_{R}\end{pmatrix}$ (10)

where, $R$ describes the left-right mixing and is given by,

$R=M_{D}M_{RR}^{-1}+O(M_{D}^{3}(M_{RR}^{-1})).$ (11)

The matrices $U,V,S$ and $T$ are as follows,

$U=[1-\frac{1}{2}M_{D}M_{RR}^{-1}(M_{D}M_{RR}^{-1})^{\dagger}]V_{\nu}$ (12)
$V=[1-\frac{1}{2}(M_{D}M_{RR}^{-1})^{\dagger}M_{D}M_{RR}^{-1}]V_{\nu}$ (13)
$S=M_{D}M_{RR}^{-1}v_{R}f_{R}$ (14) $T=-(M_{D}M_{RR}^{-1})^{\dagger}V_{\nu}$
(15)

In this work, realization of LRSM has been done with the help of $A_{4}$
modular symmetry, which has been discussed thoroughly in the next section.

##  III Realization of LRSM with modular group $\Gamma(3)$ having weight 2 .

Symmetries have always played a very crucial role in the regime of particle
physics. LRSM has been realized with the help discrete flavor symmetries in
several earlier works like [22], [23],[24],[25]. In our present work, we have
realized the model using $A_{4}$ modular symmetry (non-SUSY case). The
advantage of using modular symmetry rather than flavor symmetry is the minimal
use of extra particles (flavons) and hence the model is minimal. The model
contains usual particle content of LRSM [24]. The lepton doublets transform as
triplets under $A_{4}$ and the bidoublet and scalar triplets transform as 1
under $A_{4}$ [26]. As we have considered modular symmetry, we assign modular
weights to the particles, keeping in mind that matter multiplets corresponding
to the model can have negative modular weights, but the modular forms cannot
be assigned negative weights. The assignment of these weights are done in such
a way that in the Lagrangian the sum of the modular weights in each term is
zero. Modular weights corresponding to each particle is shown in table 1. The
Yukawa Lagrangian for the leptonic and quark sector in LRSM is given by
Eq.(1),(2) and with a reference to that we can write the Yukawa Lagrangian of
our $A_{4}$ modular symmetric LRSM, for the lepton sector, by introducing
Yukawa coupling in the form of modular forms $Y$ is given as,

$\mathcal{L_{Y}}=\overline{l_{L}}\phi{l_{R}}Y+\overline{l_{L}}\tilde{\phi}{l_{R}}Y+{{l_{R}}^{T}}Ci{\tau_{2}}{\Delta_{R}}{l_{R}}{Y}+{{l_{L}}^{T}}Ci{\tau_{2}}{\Delta_{L}}{l_{L}}{Y}$
(16)

Gauge group | $l_{L}$ | $l_{R}$ | $\phi$ | $\Delta_{L}$ | $\Delta_{R}$
---|---|---|---|---|---
$SU(3)_{C}$ | 1 | 1 | 1 | 1 | 1
$SU(2)_{L}$ | 2 | 1 | 2 | 3 | 1
$SU(2)_{R}$ | 1 | 2 | 2 | 1 | 3
$U(1)_{B-L}$ | -1 | -1 | 0 | 2 | 2
$A_{4}$ | 3 | 3 | 1 | 1 | 1
$k_{I}$ | 0 | -2 | 0 | -2 | 2
Table 1: Charge assignments for the particle content of the model.

The Yukawa couplings $Y=(Y_{1},Y_{2},Y_{3})$ are expressed as modular forms of
level 3.

| Y (modular forms)
---|---
$A_{4}$ | $3$
$k_{I}$ | $2$
Table 2: Charge assignment and modular weight for the corresponding modular
Yukawa form for the model.

Table 2 shows the charge assignment for the modular form of level 3 and its
corresponding modular weight. In our work, we are concerned with the mass of
the neutrinos and as such, using $A_{4}$ modular symmetry and using the
multiplication rules for $A_{4}$ group, we construct mass matrices as given
below.
The Dirac mass matrix from the Yukawa Lagrangian has been determined in terms
of $(Y_{1},Y_{2},Y_{3})$ as,

$M_{D}=v\begin{pmatrix}2Y_{1}&-Y_{3}&-Y_{2}\\\ -Y_{2}&-Y_{1}&2Y_{3}\\\
-Y_{3}&2Y_{2}&-Y_{1}\end{pmatrix}$ (17)

where, $v$ is the VEV of the Higgs bidoublet $\phi$.
It is to be noted here that although we have determined the texture of the
Dirac mass matrix and also the right-handed mass matrix has been determined,
so it is a fact that the type-I seesaw mass texture can be determined. But,
the pivotal investigation in the present study revolves around the
substitution of different strengths to the type-II seesaw mass term and hence
from there, the value of the type-I seesaw mass has been calculated in order
to determine the light neutrino mass.
The right-handed Majorana mass matrix is given by,

$M_{R}=v_{R}\begin{pmatrix}2Y_{1}&-Y_{3}&-Y_{2}\\\ -Y_{3}&2Y_{2}&-Y_{1}\\\
-Y_{2}&-Y_{1}&2Y_{3}\end{pmatrix}$ (18)

where, $v_{R}$ is the VEV for the scalar triplet $\Delta_{R}$. As it is seen
that the Majorana mass matrix for our model is found to be symmetric in nature
as it should be. Under these assumptions for modular symmetric LRSM and the
basis that we have considered, our charged lepton mass matrix is also found to
be diagonal.
The type-II seesaw mass is given by,

$M_{\nu}^{II}=M_{LL}$ (19)

where,

$M_{\nu}^{II}=v_{L}\begin{pmatrix}2Y_{1}&-Y_{3}&-Y_{2}\\\
-Y_{3}&2Y_{2}&-Y_{1}\\\ -Y_{2}&-Y_{1}&2Y_{3}\end{pmatrix}$ (20)

where,$v_{L}$ is the vev of the scalar triplet $\Delta_{L}$.
The determination of the values of the modular forms $(Y_{1},Y_{2},Y_{3})$ has
been done using the relations as mentioned in the section of numerical
analysis and results.

##  IV Resonant leptogenesis and neutrinoless double beta decay
$(0\nu\beta\beta)$ in TeV scale LRSM.

In TeV scale seesaw models, an adequate amount of lepton asymmetry can be
generated by the phenomenon of resonant leptogenesis (RL) [27, 28, 29, 30].
For RL, two of the RH Majorana neutrinos need to be degenerate which has been
found to be so in the present work. In the framework of TeV scale LRSM, the
presence of RH neutrinos and scalar triplets suggests their decays which give
rise to lepton asymmetry. The net lepton asymmetry will therefore be generated
due to the two seesaw terms. So, in this work, we investigate the contribution
of both the seesaw terms by towards the lepton asymmetry, assigning them
different weightages (e.g. 30%,50% and 70%).
For RL however there is a basic requirement that the a pair of Majorana
neutrinos must have a mass difference comparable to their decay widths which
will be illustrated later in the work.
The decay rates of the RH neutrinos are governed by the Yukawa couplings, and
is given by [31],

$\Gamma_{i}=(Y_{\nu}^{\dagger}Y_{\nu})_{ii}\frac{M_{i}}{8\pi}$ (21)

As already mentioned previously that one important condition for RL is that
the mass difference of the two heavy RH neutrinos must be comparable to their
decay width, i.e., $M_{i}-M_{j}=\Gamma$. In such a case, the CP asymmetry may
become very large. The CP violating asymmetry is thus given by [32],

$\epsilon_{i}=\frac{Im[(Y_{\nu}^{\dagger}Y_{\nu})_{ij}^{2}]}{(Y_{\nu}^{\dagger}Y_{\nu})_{11}(Y_{\nu}^{\dagger}Y_{\nu})_{22}}.\frac{(M_{i}-M_{j})^{2}M_{i}\Gamma_{j}}{(M_{i}^{2}-M_{j}^{2})+M_{i}^{2}\Gamma{j}^{2}}$
(22)

The variables $i,j$ run over 1 and 2 and $i\neq j$.
The CP asymmetries $\epsilon_{1}$ and $\epsilon_{2}$ can give rise to a net
lepton asymmetry, provided the expansion rate of the universe is larger than
$\Gamma_{1}$ and $\Gamma_{2}$. This can further be converted into baryon
asymmetry of the universe by $B+L$ violating sphaleron processes.
Now, due to the presence of several new heavy particles in LRSM, along with
the standard light neutrino contribution to $0\nu\beta\beta$, several new
physics contributions also come into the picture. There are total eight
contributions of the phenomenon to LRSM. They are mentioned as under,

  * •

Standard Model contribution to NDBD where the intermediate particles are the
$W_{L}$ bosons and light neutrinos, the process in which the amplitude depends
upon the leptonic mixing matrix elements and light neutrino masses.

  * •

Heavy right-handed neutrino contribution in which the mediator particles are
the $W_{L}$ bosons and the amplitude depends upon the mixing between light and
heavy neutrinos as well as the mass of the heavy neutrino.

  * •

Light neutrino contribution to NDBD where the intermediate particles are
$W_{R}$ bosons and the amplitude depends upon the mixing between light and
heavy neutrinos as well as mass of the right-handed gauge boson $W_{R}$.

  * •

Heavy right-handed neutrino contribution where the mediator particles are the
$W_{R}$ bosons. The amplitude of this process is dependent on the elements of
the right handed leptonic mixing matrix and mass of the right-handed gauge
boson, $W_{R}$ as well as the mass of the heavy right handed Majorana
neutrino.

  * •

Light neutrino contribution from the Feynman diagram mediated by both $W_{L}$
and $W_{R}$, and the amplitude of the process depends upon the mixing between
light and heavy neutrinos,leptonic mixing matrix elements, light neutrino
masses and the mass of the gauge bosons, $W_{L}$ and $W_{R}$ ($\lambda$
contribution).

  * •

Heavy neutrino contribution from the Feynman diagram mediated by both $W_{L}$
and $W_{R}$, and the amplitude of the process depends upon the right handed
leptonic mixing matrix elements, mixing between the light and heavy neutrinos,
also the mass of the gauge bosons, $W_{L}$ and $W_{R}$ and the mass of the
heavy right handed neutrino ($\eta$ contribution).

  * •

Scalar triplet contribution ($\Delta_{L}$) in which the mediator particles are
$W_{L}$ bosons, and the amplitude for the process depends upon the masses of
the $W_{L}$ bosons, left-handed triplet Higgs, as well as their coupling to
leptons.

  * •

Right-handed scalar triplet contribution ($\Delta_{R}$) contribution to NDBD
in which the mediator particles are $W_{R}$ bosons, and the amplitude for the
process depends upon the masses of the $W_{R}$ bosons, right-handed triplet
Higgs, $\Delta_{R}$ as well as their coupling to leptons.

In this work, we have considered the $\lambda$ and $\eta$ contributions of
$0\nu\beta\beta$. The standard light neutrino contribution and new physics
contributions by the heavy RH neutrino and scalar triplet contribution in
modular symmetric LRSM have already been done in one of our previous work
[33].

##  V Numerical Analysis and Results

The light neutrino mass in LRSM is given by,

$M_{\nu}=M_{\nu}^{I}+M_{\nu}^{II}$ (23)

where, the type-I seesaw mass term is,

$M_{\nu}^{I}=M_{D}M_{R}^{-1}M_{D}^{T}$ (24)

and,

$M_{\nu}^{II}=M_{LL}$ (25)

In this work, we aim to study the strengths of each seesaw terms that
contribute towards the light neutrino mass and hence the associated
phenomenology. So, we have considered,

$M_{\nu}^{II(diag)}=XM_{\nu}^{(diag)}$ (26)

where, the parameter X is introduced to describe the relative strength of the
type-II seesaw mass term. It can be assigned with any numerical value provided
the two seesaw mass terms gives rise to the correct light neutrino mass
matrix. In our case we have considered $X=0.3,0.5,0.7$ that is, contribution
from type-II seesaw term is considered to be $30\%,50\%,70\%$ respectively.
As we are using modular symmetry in LRSM, so our mass matrices are expressed
in terms of the modular forms $(Y_{1},Y_{2},Y_{3})$. So, the type-II mass
matrix is expressed as,

$M_{\nu}^{II}=v_{L}\begin{pmatrix}2Y_{1}&-Y_{3}&-Y_{2}\\\
-Y_{3}&2Y_{2}&-Y_{1}\\\ -Y_{2}&-Y_{1}&2Y_{3}\end{pmatrix}$ (27)

where, $v_{L}$ is of the order of $eV$.
Using (26), we can determine the values of the modular forms and hence further
we can calculate the values of the neutrino masses and take into consideration
effective masses of different contributions of $0\nu\beta\beta$ in the present
work. In LRSM, from type-II seesaw term, $M_{R}$ can be written as,

$M_{R}=\frac{1}{\gamma}\Biggl{(}\frac{v_{R}}{M_{W_{L}}}\Biggl{)}^{2}M_{\nu}^{II}$
(28)

As already mentioned that $\gamma$ in this work has been fine tuned to
$10^{-13}$. We have considered four different values of the $SU(2)_{R}$
breaking scale $v_{R}$ for our further analysis, namely $3,5,10$ and $18$ TeV
respectively. These values will be useful for us to determine the common
parameter space of the phenomena we have considered like, baryon asymmetry of
universe (BAU), contributions of $0\nu\beta\beta$ etc. The mass of the left-
handed gauge boson is taken as, $M_{W_{L}}=80GeV$.
In the present work, we first determine the values of the leptonic CP
asymmetry $\epsilon_{1}$ and $\epsilon_{2}$, by using (22), where
$Y_{v}=\frac{M_{D}}{v}$, $M_{D}$ being the Dirac mass and $v$ is the vev of
the Higgs bidoublet and is in $GeV$. The decay rates in (22) can be calculated
using the relation (21).
The CP violating asymmetries $\epsilon_{1}$ and $\epsilon_{2}$ can give rise
to net lepton number asymmetry, provided the expansion rate of the universe is
larger than $\Gamma_{1}$ and $\Gamma_{2}$.The net baryon asymmetry is then
calculated using the following relation [34],

$\eta_{B}\approx-0.96\times 10^{-2}\sum_{i}(k_{i}\epsilon_{i})$ (29)

$k_{i}$ being the efficiency factors measuring the washout effects. Some
parameters are needed to defined as,

$K_{i}\equiv\frac{\Gamma_{i}}{H}$ (30)

Equation (30) is defined at temperature $T=M_{i}$. The Hubble’s constant is
given by, $H\equiv\frac{1.66\sqrt{g_{*}}T^{2}}{M_{Planck}}$, where,
$g_{*}=107$ and $M_{Planck}=1.2\times 10^{19}$ GeV is the Planck mass. decay
width is estimated using (21). The efficiency factors $k_{i}$ can be
calculated using the formula [35],

$k_{1}\equiv k_{2}\equiv\frac{1}{2}(\Sigma_{i}K_{i})^{-1.2}$ (31)

Equation (31) holds valid for two nearly degenerate heavy Majorana masses and
in most general cases $5\leq K_{i}\leq 100$. To calculate the baryon asymmetry
parameter, we have used formula (29). The results are shown as a variation of
the effective neutrino masses calculated for momentum dependent contributions
and heavy right handed neutrino contribution as described below. The
variations are shown for different values of $M_{W_{R}}$ and also variations
are shown for different strengths of the seesaw mass terms.
Talking about neutrinoless double beta decay $(0\nu\beta\beta)$, in this work,
we have taken into consideration the momentum dependent mechanisms namely,
$\lambda$ and $\eta$ contributions, which are given by the following formulae,

  * •

For $\lambda$ contribution, the dimensionless parameter $\eta_{\lambda}$ is
given by,

$|{\eta_{\lambda}}|=\Biggl{(}\frac{M_{W_{L}}}{M_{W_{R}}}\Biggl{)}|\Sigma_{i}U_{ei}T_{ei}^{*}|$
(32)

  * •

For $\eta$ contribution, the dimensionless parameter describing
$0\nu\beta\beta$ is given by,

$|\eta_{\eta}|=\tan\xi|\Sigma_{i}U_{ei}T_{ei}^{*}|$ (33)

In the above equations, $U_{ei}$ represents the first row of the neutrino
mixing matrix. $|\Sigma_{i}U_{ei}T_{ei}^{*}|$ can be simplified to the form
$-[M_{D}M_{RR}^{-1}]_{ee}$ as described in [36]. T is represented by the
equation (15) and,

$tan2\xi=-\frac{2k_{1}k_{2}}{v_{R}^{2}-v_{L}^{2}}$ (34)

The heavy right-handed (RH) neutrino contribution for $0\nu\beta\beta$ is
given by,

$M_{eff}^{N}=p^{2}\Biggl{(}\frac{M_{W_{L}}^{4}}{M_{W_{L}}^{4}}\Biggl{)}\frac{U_{Rei}^{2}}{M_{i}}$
(35)

where, $<p^{2}>=m_{e}m_{p}\frac{M_{N}}{M_{\nu}}$ is the typical momentum
exchange of the process. $m_{e}$ and $m_{p}$ are the masses of the electron
and proton repsectively and $M_{N}$ is the neuclear matrix element (NME)
corresponding to the RH neutrino exchange. The allowed value of $p$ is in the
range $(100-200)$ MeV. But for our analysis we have used $p=180MeV$ [37].
Figure 1,2 and 3 below shows the variation of baryon asymmetry parameter with
the effective mass for heavy right-handed neutrino contribution for right-
handed gauge boson masses 3,5,10 and 18TeV for type-II seesaw mass term
contributing $30\%,50\%$ and $70\%$ to light neutrino mass respectively.

Figure 1: Variation of $\eta_{B}$ with effective masses corresponding to heavy
right-handed neutrino contribution for $M_{W_{R}}$ = 3,5,10 and 18TeV
respectively for normal hierarchy(NH) when $X=0.3$.

Figure 2: Variation of $\eta_{B}$ with effective masses corresponding to heavy
right-handed neutrino contribution for $M_{W_{R}}$ = 3,5,10 and 18TeV
respectively for NH and IH when $X=0.5$.

Figure 3: Variation of $\eta_{B}$ with effective masses corresponding to heavy
right-handed neutrino contribution for $M_{W_{R}}$ = 3,5,10 and 18TeV
respectively for NH and IH when $X=0.7$.

The effective masses corresponding to the momentum dependent $\lambda$
contribution is given by,

$m_{eff}^{\lambda}=\frac{\eta_{\lambda}}{m_{e}}$ (36)

The variations of baryon asymmetry parameter with the effective masses
corresponding to the aforementioned contribution are shown in figures 4 to 6.

Figure 4:  Variation of effective mass with baryon asymmetry parameter for NH
and IH and X=0.3.

Figure 5:  Variation of effective mass with baryon asymmetry parameter for NH
and IH and X=0.5.

Figure 6:  Variation of effective mass with baryon asymmetry parameter for NH
and IH and X=0.7.

The effective mass for $\eta$ contribution to $0\nu\beta\beta$ is given by,

$m_{eff}^{\eta}=\frac{\eta_{\eta}}{m_{e}}$ (37)

Figure 7:  Variation of effective mass with baryon asymmetry parameter for NH
and IH and X=0.3.

Figure 8:  Variation of effective mass with baryon asymmetry parameter for NH
and IH and X=0.5.

Figure 9:  Variation of effective mass with baryon asymmetry parameter for NH
and IH and X=0.7.

Figures 7 to 9 shows the variation of baryon asymmetry parameter with the
effective masses for $\eta$ contribution of $0\nu\beta\beta$.

Figure 10:  Variation of effective mass with baryon asymmetry parameter for NH
and X=0.3,0.5 and 0.7.

Figure 10 shows the variation of baryon aymmetry parameter with the effective
mass corresponding to right-handed neutrino contribution for $1.5,2.2$ and
$2.5TeV$ for normal hierarchy.

Figure 11:  Variation of effective mass for heavy right-handed neutrino
contribution with baryon asymmetry parameter for IH and X=0.3,0.5 and 0.7.

Figure 11 shows the variation of baryon aymmetry parameter with the effective
mass corresponding to right-handed neutrino contribution for $1.5,2.2$ and
$2.5TeV$ for inverted hierarchy. It has been observed that the heavy right-
handed neutrino corresponding to $X=0.3$ and $X=0.5$ is very less in case of
$1.5,2.2$ and $2.5$ TeV. But when $X=0.7$ that is, when the type-II seesaw
mass term is dominant, there are some amount of data obtained within the
experimental limits. Same has been observed in case of lambda and eta
contributions of $0\nu\beta\beta$ as well and as such, we are showing only the
plots corresponding to $X=0.7$.

Figure 12:  Variation of effective mass for $\eta$ contribution with baryon
asymmetry parameter for NH and X=0.7.

Figure 12 shows the variation of effective mass corresponding to $\eta$
contribution with the baryon asymmetry parameter for the type-II seesaw mass
contributing $70\%$ towards the neutrino mass. The plots are for normal
hierarchy. The figure 13 shows the variations for inverted hierarchy.

Figure 13:  Variation of effective mass for $\eta$ contribution with baryon
asymmetry parameter for IH and X=0.7.

Variations of baryon asymmetry parameter with effective mass corresponding to
$\lambda$ contribution have also been shown in figures 14 and 15.

Figure 14:  Variation of effective mass for $\lambda$ contribution with baryon
asymmetry parameter for NH and X=0.7.

Figure 15:  Variation of effective mass for $\lambda$ contribution with baryon
asymmetry parameter for IH and X=0.7.

##  VI Discussion

The following points focusses on the complete study and analysis of the
present work and the results obtained hereafter,

  * •

Left-Right Symmetric model has been realized with the help of modular group of
level 3 and weight 2 ($\Gamma(3)$), isomorphic to the $A_{4}$ non-abelian
discrete symmetry group, as discussed in the appendix.

  * •

The aim of the study focusses on describing the variation of baryogenesis (via
leptogenesis) with variations in the value of the right-handed gauge boson
mass and strength of the type-II seesaw mass.

  * •

In several previous literatures, it was stated that while taking into
consideration resonant leptogenesis, the dominant contribution was given by
type-I seesaw mass, however, we have also taken into consideration the
strength of the type-II seesaw mass term and it has been found that the
analysis gives considerable results for the same.

  * •

We have considered strength of the type-II seesaw mass to be 30% , 50% and
70%, and masses of the right-handed gauge bosons are considered to be
$M_{W_{R}}=3,5,10,18$ TeV. Here, the condition $M_{N}\leq M_{W_{R}}$ has been
used.

  * •

From the present analysis, it can be stated that a lower bound of $M_{W_{R}}$
$>$ 1.5 TeV has been obatined for the case $g_{L}=g_{R}$, that is, when the
gauge couplings corresponding to left-handed and right-handed components are
equal. There may therefore be chances of obtaining a weaker constraint on the
mass of the right-handed gauge boson for successful leptogenesis in future
experiments.

  * •

The analysis shows that the weaker bounds on the $M_{W_{R}}$ mass is
applicable only in the case where the type-II term contributes dominantly.
This might be the result of the values of the Yukawa couplings and textures of
the mass matrices obtained as a result of implementing modular symmetry in
LRSM rather than flavor symmetry.

Coming to the fact that several previous works have been done taking into
consideration the flavor symmetric realization of LRSM as studied in [23, 25,
24, 38] etc. The authors in the mentioned literatures have also carried out
phenomenological studies related to the model. However, in the present work we
have not considered flavor symmetry but, we have taken into consideration
modular symmetry. The use of modular symmetry provides the advatage of minimal
construction of the model without the use of flavons. In the present work, we
have considered modular group of level 3, $\Gamma(3)$ and weight 2, which is
isomorphic to $A_{4}$ non-abelian discrete symmetry group. Here, it is to be
noted that the present model has been designed in the non-SUSY framework and
as such, Left-Right Symmetric Model does not restrict the use of infinite
number of modular forms, but as we are using $\Gamma(3)$ modular group having
weight 2, as stated in the appendix in table 3, we will have three number of
modular forms represented as $(Y_{1},Y_{2},Y_{3})$. As such, the mass matrices
corresponding to the present model have been expressed in terms of these three
modular forms, discussed in section (III) of the manuscript.
Some previous work as dicussed in [31, 39] have also taken into account the
study of several phenomenology like $0\nu\beta\beta$, Lepton Flavor Violation
(LFV) with changing values of $M_{W_{R}}$. In the present work, we have also
taken into account the different values of $M_{W_{R}}$ to study the relation
between $0\nu\beta\beta$ and leptogenesis, however it has been done taking
into account different strengths of the type-II seesaw mass term. One of the
most significant results of the present work can be stated to be the
observation of considerable amount of leptogenesis phenomena taking place when
the leading contribution of neutrino mass is given by the type-II seesaw mass
term ($70\%$). As stated by the authors in [31] that while taking into
consideration the phenomena of resonant leptogenesis, type-I seesaw term plays
the dominant role and the contribution of $\Delta_{L}$ is negligible. But, in
the present work, type-II seesaw mass term shows considerable contribution
towards the said phenomenology. The exact reason although not known, this may
be thought of as a result of emerging modular forms in the picture which
changes the structure of the mass matrices within the model.

##  VII Conclusion

Neutrino oscillations paved the gateway for physics beyond the Standard Model.
In the present work, we have realized LRSM with the help of modular $A_{4}$
symmetry. Using modular symmetry provides the advantage of using no extra
particles called ’flavons’. The Yukawa couplings are represented as modular
forms expressed as expansions of $q$. The values of the Yukawa couplings
$(Y_{1},Y_{2},Y_{3})$ are calculated using the relation
$M_{\nu}^{II(diag)}=XM_{\nu}^{(diag)}$. The mass matrix for the type-II seesaw
dominance has been determined using the multiplication rules for $A_{4}$ group
stated in the Appendix. The Majorana mass matrix is found to be symmetric and
under the considered basis, the charged lepton mass matrix is also diagonal.
To summarize our work, some results are stated as under,

  * •

In the present work we have considered the arbitrary constant that can be
multiplied with the Yukawa term to be one. This is to reduce the number of
free parameters in the model and to determine the data more accurately.

  * •

The model has been realized using $A_{4}$ modular symmetry, the advantage
being no use of extra particles or flavons. We can also use other symmetry
groups depending upon the level of modular form used and as such, the number
of modular forms will also change, as per the table 3 in the manuscript. As we
have used $A_{4}$ modular symmetry, there are three modular forms
$(Y_{1},Y_{2},Y_{3})$ and hence it became easier for us to construct the
$3\times 3$ mass matrices in terms of the aforementioned modular forms which
are consistent with the neutrino data.

  * •

The Yukawa couplings determined for the cases when $X=0.3,0.5,0.7$ has been
found to satisfy the condition that its modulus is greater than unity.

  * •

Effective masses corresponding to heavy right-handed neutrino contribution,
and momentum dependent contributions of $0\nu\beta\beta$ are determined and
their variations with the baryon asymmetry parameter $\eta_{B}$ has been
studied. The results are found to satisfy the experimental constraints, for
both normal and inverted hierarchies.

The effective masses for the $0\nu\beta\beta$ contributions are calculated and
it has been found that the values for the effective mass corresponding to each
contribution is well within the experimental bounds, which in fact makes us
clearly state that the building of the model with modular symmetry is
advantageous to that of flavor symmetries.In this model, we have not used any
extra particles and the analysis has been done taking into consideration the
calculated and computed values for the model parameters and the results are
found to be satisfactory, so it can be stated that the Left-Right Symmetric
Model can be constructed with modular symmetry while satisfying the
experimental bounds on the desired parameters.

## Appendix A : Minimisation of the potential term.

Let us consider the Higgs potential of our model that has quadratic and
quartic coupling terms given by [40],

$V_{\phi,\Delta_{L},\Delta_{R}}=-\mu_{ij}^{2}Tr[\phi_{i}^{\dagger}\phi_{j}]+\lambda_{ijkl}Tr[\phi_{i}^{\dagger}\phi_{j}]Tr[\phi_{k}^{\dagger}\phi_{l}]+\lambda_{ijkl}^{{}^{\prime}}Tr[\phi_{i}^{\dagger}\phi_{j}\phi_{k}^{\dagger}\phi_{l}]-\mu_{ij}^{2}Tr[\Delta_{L}^{\dagger}\Delta_{L}+\Delta_{R}^{\dagger}\Delta_{R}]$
$\rho_{1}[(Tr[\Delta_{L}^{\dagger}\Delta_{L}])^{2}+(Tr[\Delta_{L}^{\dagger}\Delta_{L}])^{2}]+\rho_{2}(Tr[\Delta_{L}^{\dagger}\Delta_{L}\Delta_{L}^{\dagger}\Delta_{L}]+Tr[\Delta_{R}^{\dagger}\Delta_{R}\Delta_{R}^{\dagger}\Delta_{R}])+\rho_{3}(Tr[\Delta_{L}^{\dagger}\Delta_{L}]Tr[\Delta_{R}^{\dagger}\Delta_{R}])+$
$\alpha_{ij}Tr[\phi_{i}^{\dagger}\phi_{j}](Tr[\Delta_{L}^{\dagger}\Delta_{L}]+Tr[\Delta_{R}^{\dagger}\Delta_{R}])+\beta_{ij}(Tr[\Delta_{L}^{\dagger}\Delta_{L}\phi_{i}\phi_{j}^{\dagger}]+(Tr[\Delta_{R}^{\dagger}\Delta_{R}\phi_{i}^{\dagger}\phi_{j}])+\gamma_{ij}(Tr[\Delta_{L}^{\dagger}\phi_{i}\Delta_{R}\phi_{j}^{\dagger}]+h.c)$
(38)

where, i,j,k,l runs from 1 to 2 with $\phi_{1}=\phi$ and
$\phi_{2}=\tilde{\phi}$. As mentioned above after SSB, the scalar sector
obtains VEV. So after the substitution of the respective VEVs and determining
the traces, so after simplification the potential can be written as,

$V=-\mu^{2}(v_{L}^{2}+v_{R}^{2})+\frac{\rho}{4}(v_{L}^{4}+v_{R}^{4})+\frac{\rho^{\prime}}{2}+\frac{\alpha}{2}(v_{L}^{2}+v_{R}^{2})k^{2}+\gamma
v_{L}v_{R}k^{2}$ (39)

where, we have used the approximation $k^{\prime}<<k$, and
$\rho^{\prime}=2\rho_{3}$. Our minimization conditions are, $\frac{\delta
V}{\delta v_{L}}=\frac{\delta V}{\delta v_{R}}=\frac{\delta V}{\delta
k}=\frac{\delta V}{\delta k^{\prime}}=0$

Therefore, we get,

$\frac{\delta V}{\delta v_{L}}=-2\mu^{2}v_{L}+\rho
v_{L}^{3}+\rho^{\prime}v_{L}k^{2}+\gamma v_{R}k^{2}$ (40)

Here, it is evident that the Majorana mass of the left-handed neutrino
$M_{LL}$ is dependent on the vev $v_{L}$ as already defined above. Again, we
have

$\frac{\delta V}{\delta v_{R}}=-2\mu^{2}v_{R}+\rho
v_{R}^{3}+\rho^{\prime}v_{R}k^{2}+\gamma v_{L}k^{2}$ (41)

So, the right handed Majorana mass $M_{RR}$ is dependent on the vev $v_{R}$.
Similarly, the calculations for the same can be carried out and it can be
found out the Dirac mass term $M_{D}$ can be expressed in terms of the vev for
the Higgs bidoublet as also defined previously.
Now, we are to determine a relation between the VEVs for the scalars and so
after using the minimization conditions and simplifying the equations, we come
to a relation given by,

$v_{L}v_{R}=\frac{\gamma}{\xi}k$ (42)

where, $\xi=\rho-\rho^{\prime}$.
The neutrino mass for LRSM is given as a summation of the type-I and type-II
term as already mentioned above. So, in the approximation that
$k^{\prime}<<k$, and if we consider that our Yukawa coupling $Y^{l}$
corresponding to the neutrino masses is $y_{D}$ and the coupling
$\widetilde{Y^{l}}$ for the charged fermion masses is denoted by $y_{L}$, so
considering $y_{D}k>>y_{l}k^{\prime}$ we can write,

$M_{\nu}=\frac{k^{2}}{v_{R}}y_{D}f_{R}^{-1}y_{D}^{T}+f_{L}v_{L}$ (43)

Since, for due to left-right symmetry, we can consider $f_{L}=f_{R}=f$, so the
above equation can be written as,

$M_{\nu}=\frac{k^{2}}{v_{R}}y_{D}f^{-1}y_{D}^{T}+fv_{L}$ (44)

So, from this equation we can come to a relation given by,

$M_{\nu}=(f\frac{\gamma}{\xi}+y_{D}f^{-1}y_{D}^{T})\frac{k^{2}}{v_{R}}$ (45)

Here, we can consider two situations, namely

  * •

If $f(\frac{\gamma}{\xi})<<y_{D}f^{-1}y_{D}^{T}$, the light neutrino mass is
given by the type-I term $M_{D}M_{RR}^{-1}M_{D}^{T}$. That is, here type-I is
dominant and the light neutrino mass is from the suppression of heavy
$\nu_{R}$.

  * •

If $f(\frac{\gamma}{\xi})>>y_{D}f^{-1}y_{D}^{T}$, the light neutrino mass is
given by the type-II term $fv_{L}$. That is, in this case type-II mass term is
dominant and the light neutrino mass is because of the tiny value of
$\nu_{L}$.

## Appendix B : Properties of $A_{4}$ discrete symmetry group.

$A_{4}$ is a non-abelian discrete symmetry group which represents even
permuatations of four objects. It has four irreducible representations, three
out of which are singlets $(1,1^{\prime},1^{\prime\prime})$ and one triplet
$3$ ($3_{A}$ represents the anti-symmetric part and $3_{S}$ the symmetric
part). Products of the singlets and triplets are given by,

$1\otimes 1=1$ $1^{\prime}\otimes 1^{\prime}=1^{\prime\prime}$
$1^{\prime}\otimes 1^{\prime\prime}=1$ $1^{\prime\prime}\otimes
1^{\prime\prime}=1^{\prime}$ $3\otimes 3=1\oplus 1^{\prime}\oplus
1^{\prime\prime}\oplus 3_{A}\oplus 3_{S}$

If we have two triplets under $A_{4}$ say, $(a_{1},a_{2},a_{3})$ and
$(b_{1},b_{2},b_{3})$ , then their multiplication rules are given by,

$1\approx a_{1}b_{1}+a_{2}b_{3}+a_{3}b_{2}$ $1^{\prime}\approx
a_{3}b_{3}+a_{1}b_{2}+a_{2}b_{1}$ $1^{\prime\prime}\approx
a_{2}b_{2}+a_{3}b_{1}+a_{1}b_{3}$
$3_{S}\approx\begin{pmatrix}2a_{1}b_{1}-a_{2}b_{3}-a_{3}b_{2}\\\
2a_{3}b_{3}-a_{1}b_{2}-a_{2}b_{1}\\\
2a_{2}b_{2}-a_{1}b_{3}-a_{3}b_{1}\end{pmatrix}$
$3_{A}\approx\begin{pmatrix}a_{2}b_{3}-a_{3}b_{2}\\\ a_{1}b_{2}-a_{2}b_{1}\\\
a_{3}b_{1}-a_{1}b_{3}\end{pmatrix}$

## Appendix C : Modular Symmetry.

Modular symmetry has gained much importance in aspects of model building [41],
[42]. This is because it can minimize the extra particle called ’flavons’
while analyzing a model with respect to a particular symmetry group. An
element $q$ of the modular group acts on a complex variable $\tau$ which
belongs to the upper-half of the complex plane given as [42] [43]

$q\tau=\frac{a\tau+b}{c\tau+d}$ (46)

where $a,b,c,d$ are integers and $ad-bc=1$, Im$\tau$$>$0\.
The modular group is isomorphic to the projective special linear group
PSL(2,Z) = SL(2,Z)/$Z_{2}$ where, SL(2,Z) is the special linear group of
integer $2\times 2$ matrices having determinant unity and $Z_{2}=({I,-I})$ is
the centre, $I$ being the identity element. The modular group can be
represented in terms of two generators $S$ and $T$ which satisfies
$S^{2}=(ST)^{3}=I$. $S$ and $T$ satisfies the following matrix
representations:

$S=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$ (47)

$T=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$ (48)

corresponding to the transformations,

$S:\tau\rightarrow-\frac{1}{\tau};T:\tau\rightarrow\tau+1$ (49)

Finite modular groups (N $\leq$ 5) are isomorphic to non-abelian discrete
groups, for example, $\Gamma(3)\approx A_{4}$, $\Gamma(2)\approx S_{3}$,
$\Gamma(4)\approx S_{4}$. While using modular symmetry, the Yukawa couplings
can be expressed in terms of modular forms, and the number of modular forms
present depends upon the level and weight of the modular form. For a modular
form of level N and weight 2k, the table below shows the number of modular
forms associated within and the non-abelian discrete symmetry group to which
it is isomorphic [43].

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 3: No. of modular forms corresponding to modular weight 2k.

In our work, we will be using modular form of level 3, that is, $\Gamma(3)$
which is isomorphic to $A_{4}$ discrete symmetry group. The weight of the
modular form is taken to be 2, and hence it will have three modular forms
$(Y_{1},Y_{2},Y_{3})$ which can be expressed as expansions of q given by,

$Y_{1}=1+12q+36q^{2}+12q^{3}+84q^{4}+72q^{5}+36q^{6}+96q^{7}+180q^{8}+12q^{9}+216q^{10}$
(50)
$Y_{2}=-6q^{1/3}(1+7q+8q^{2}+18q^{3}+14q^{4}+31q^{5}+20q^{6}+36q^{7}+31q^{8}+56q^{9})$
(51)
$Y_{3}=-18q^{2/3}(1+2q+5q^{2}+4q^{3}+8q^{4}+6q^{5}+14q^{6}+8q^{7}+14q^{8}+10q^{9})$
(52)

where, $q=\exp(2\pi i\tau)$.

## Acknowledgements

Ankita Kakoti acknowledges Department of Science and Technology (DST), India
(grant DST/INSPIRE Fellowship/2019/IF190900) for the financial assistantship.

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