# Neutrinoless double beta decay and Sterile dark matter in extended left
right symmetric model

Bichitra Bijay Boruah<EMAIL_ADDRESS>Nayana Gautam<EMAIL_ADDRESS>Mrinal Kumar Das<EMAIL_ADDRESS>Department of Physics, Tezpur
University, Tezpur 784028, India

###### Abstract

We have studied a flavor symmetry based extended left-right symmetric
model(LRSM) with dominant type-II seesaw mechanism and have explored the
associated neutrino phenomenology. The particle content of the model includes
usual quarks, leptons along with additional sterile fermion per generation in
the fermion sector while the scalar content contains Higgs doublets and scalar
bidoublet. Realization of this extension of LRSM has been done by using
$A_{4}\times Z_{4}$ discrete symmetries. In this work, we have also included
the study of sterile neutrino dark matter(DM) phenomenology along with
neutrinoless double beta decay within the framework.

###### pacs:

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

##  I INTRODUCTION

Standard Model(SM) is the most economical and successful model of particle
physics, But it fails to explain various phenomena like lepton flavor
violation, small neutrino mass, baryon asymmetry of universe, dark matter etc.
One of the main drawback of the SM is that it explains the parity violation of
weak interaction adhocly by assuming that transformation of right handed
counter part of the particles to be transformed as singlet under $SU_{2}$
gauge group. Because of this, there has to be some framework where parity
violation can be explained in a natural way. Left right symmetric model is one
such framework where both right handed and left handed particle are treated on
equal footing and it is the most economical way to restore parity came a long
way in explaining neutrino mass, lepton number violation, dark matter, baryon
asymmetry of the universe thereby gaining popularity. LRSM Mohapatra:2006xy ;
Mohapatra:1986aw ; Mohapatra:1974gc ; Pati:1974yy ; Senjanovic:1975rk  is a
very simple extension of the standard model gauge group where parity
restoration is obtained at a high energy scale and the fermions are assigned
to the gauge group $SU(3)_{c}\times SU(2)_{L}\times SU(2)_{R}\times
U(1)_{B-L}$ which can be tested in present-day experiments. The usual type-
IMinkowski:1977sc ; Schechter:1980gr ; Mohapatra:1979ia ; Yanagida:1979as ;
Gell-Mann:1979vob  and type-IICheng:1980qt ; Magg:1980ut ; Lazarides:1980nt ;
Mohapatra:1980yp ; Wetterich:1981bx ; Antusch:2004xy  seesaw arises naturally
in LRSM. Several other problems like parity violation of weak interaction,
massless neutrinos, CP problems, hierarchy problems, etc can also be addressed
in the framework of LRSM. The seesaw scale is identified as the breaking of
the $SU(2)_{R}$ symmetry. In this model, the electric charge generator is
given by, $Q=T_{3L}+T_{3R}+\frac{B-L}{2}$, where $T_{3L}$ and $T_{3R}$ are the
generators of $SU(2)_{L}$ and $SU(2)_{R}$ and B-L being the baryon minus
lepton number charge operator. Mass of neutrino can be explained in LRSM with
canonical seesaw, inverse seesaw, linear seesaw, extended seesaw etc. In this
work LRSM is extended with a neutral fermions per generation, which is known
as extended seesaw. Within this extended seesaw type-II dominance is
considered for phenomenological analysis. With this extra singlet fermion the
mass matrix in type-II dominance can be given as-

$\mathcal{M}=\begin{pmatrix}M_{L}&0&M_{D}\\\ 0&0&M\\\ M_{D}^{T}&M^{T}&M_{R}\\\
\end{pmatrix}$ (1)

Where light neutrino mass is $m_{\nu}=\nu_{L}f_{L}$. A natural type-II seesaw
dominance allows large light-heavy neutrino mixing thus leading to many new
physics contributions to neutrinoless double beta decay along with constraints
on light neutrino mass. In one of the recent work Sahu:2022xkq  extended LRSM
is studied in details considering linear seesaw to show the phenomenological
importance in different LFV processes and unitarity violation etc using flavor
symmetry.

The nature of the neutrinos and absolute neutrino mass scale is still unknown,
whether they are four-component Dirac particles possessing a conserved lepton
number or two-component Majorana particles. This is directly related to the
issue of LN conservation. One such process of fundamental importance in
particle physics which arises in any BSM framework is neutrinoless double beta
decay(NDBD/0$\nu$$\beta$$\beta$). It is a second-order, and slow radioactive
process that transforms a nuclide of atomic number Z into its isobar with
atomic number Z+2,

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

which violates the lepton number(LN) conservation. The main aim for the search
of 0$\nu$$\beta$$\beta$ decay is the measurement of the effective Majorana
neutrino mass, which is a combination of the neutrino mass eigenstates and
neutrino mixing matrix terms.

There is no convincing experimental evidence of the decay that exists to date.
But the new generation of experiments are already running or about to run to
explore effective neutrino mass along with decay rates NDBD process. In
addition, from the lifetime of NDBD combined with sufficient knowledge of the
nuclear matrix elements (NME), one can set a constrain involving the neutrino
masses. The experiments that have improved the lower bound of the half-life of
the decay process include KamLANDZenKamLAND-Zen:2016pfg  and GERDAgerda  which
use Xenon-136 and Germanium-76 nuclei respectively. Incorporating the results
from the first and second phase of the experiment, KamLAND-Zen imposes the
best lower limit on the decay half-life using Xe-136 as $T_{1/2}^{0\nu}$$>$
$1.07\times 10^{26}$yr at $90$ percent CL and the corresponding upper limit of
effective Majorana mass in the range $(0.061-0.165)$eV. NDBD is extensively
studied within the framework of LRSM Mohapatra:1981pm ; Picciotto:1982qe ;
Hirsch:1996qw ; SuperNEMO:2010wnd ; Tello:2010am ; Chakrabortty:2012mh ;
Nemevsek:2012iq ; Patra:2012ur ; Barry:2013xxa ; BhupalDev:2013ntw ;
BhupalDev:2014qbx . New contribution to NDBD process in extended seesaw
mechanism also studied considering different type of dominance such as
inverse, linear etc in many literaturesSruthilaya:2017vui ; Senapati:2020alx ;
Deppisch:2014zta ; Parida:2018apw ; Awasthi:2013ff . In this work we aim to
discuss the extended contrbution to NDBD process using flavor symmetry.

The dark matter(DM) and its nature are the most important and open questions
in particle physics community till date. Different astrophysical and
cosmological measurements confirm that some part of our present Universe is
made up of DM, which are basically non-luminous and non-baryonic. Cosmic
microwave background(CMB)power spectrum, cluster and galactic rotation curves
and gravitational lensing etc are some astrophysical measurements that
confirms the existence of the DM. Although the presence of the DM is confirmed
from above mentioned measurement, the nature and components of DM is still
unknown. Almost 26.8 % of the energy density of the universe is made up of DM
according to Planck data and the dark matter abundance is ade2016ade  :

$\Omega_{DM}h^{2}=0.1187\pm 0.0017$

Connecting BSM physics to DM with other unsolved problems like the smallness
of neutrino mass, BAU, etc has been a great challenge to the physics community
mass-mixingneutrino ; king2014neutrino . From the observational evidences till
date, it is clear that DM candidates must be massive, neutral to the
electromagnetic interactions, and stable. However, Standard Model fails to
provide a viable DM candidate. The particle physics community has begun to
study different possible BSM frameworks which can give rise to the correct DM
phenomenology. A plethora of DM candidates has been proposed within different
BSM frameworks. As proposed in several works, sterile neutrinos in keV scale
can be a well-motivated feasible dark matter candidate adhikari2017white ;
merle2017kev ; lucente2016implication ; admixturesterile ; kusenko2009sterile
; abazajian2017sterile ; adhikari2017white ; merle2017kev ;
lucente2016implication ; admixturesterile ; kusenko2009sterile ;
abazajian2017sterile ; upperlimit ; mixinglimit . Sterile neutrinos are right-
handed gauge singlet neutrinos that posses gravitational interactions only.
Even in the X-ray band, one of the most studied candidates of dark matter is
sterile neutrino, which can radiatively decay into an active neutrino and a
mono energetic photon in the process $N\longrightarrow\nu+\gamma$.

In this work, we have used $A_{4}\times Z_{4}$ discrete symmetries to realize
the extended seesaw with type-II dominance. We aim to study here the extension
of LRSM which offers the advantage of studying neutrino mass, new physics
contribution to NDBD process and DM phenomenology within one framework.
$A_{4}$ is the discrete group of even permutations of four objects is the
smallest non-Abelian group containing triplet irreducible representations.
More importantly, flavor symmetry has been used within the framework of LRSM
in very few works Rodejohann:2015hka ; Bonilla:2020hct ; Bazzocchi:2007au ;
Boruah:2021ktk . In some other important literatures extended LRSM is also
dicussed considering different dominant seesawSruthilaya:2017vui ;
Parida:2016hln ; Parida:2018apw ; Sahoo:2017cqg ; Chakraborty:2019uxk . We
extensively discussed different leading order contributions to NDBD process
and DM phenomenology in the considered framework. Discrete symmetry will
constrain different mass matrices arising from framework considered and which
will lead to phenomenological implications.

This paper is structured as follows. In section II, we briefly discuss the
left-right symmetric model with extended seesaw with type-II dominance. Flavor
symmetric model considering type-II dominance are discussed in section III. We
also discuss the different new physics contribution to the amplitude of the
decay process in section IV. In section V, we discuss light sterile neutrino
as a probable dark matter candidate. We present our numerical analysis and
results in section VI and then in Section VII, we conclude by giving a brief
overview of our work.

##  II LRSM with extended seesaw with type-II dominance:

In our work LRSM is extended with a neutral sterile fermion per generation
along with quark and leptons. Scalar sector of this extension consist of Higgs
bidoublet $\Phi$, two scalar triplets $\Delta_{L}$, $\Delta_{R}$ and two
doublets $H_{L}$,$H_{R}$. The B-L charge of Higgs bidoublet, scalar triplets
and doublets are 0,2 and -1 respectively. This kind of extension of LRSM is
known as extended LR model and naturally arising seesaw from it is called the
extended seesaw mechanism which is well explained in literaures such as
Hati:2018tge ; Sruthilaya:2017vui  . It is quite well known that type-II
dominance in such scenario allows large left right mixing which leads to
interesting phenomenology. The Quarks(Q) and leptons(l) (LH and RH) that
transform in L-R symmetric gauge group are given by,

$Q_{L}=\left[\begin{array}[]{cc}u^{{}^{\prime}}\\\
d^{{}^{\prime}}\end{array}\right]_{L},Q_{R}=\left[\begin{array}[]{cc}u^{{}^{\prime}}\\\
d^{{}^{\prime}}\end{array}\right]_{R},l_{L}=\left[\begin{array}[]{cc}\nu_{l}\\\
l\end{array}\right]_{L},l_{R}=\left[\begin{array}[]{cc}\nu_{l}\\\
l\end{array}\right]_{R}.$ (3)

where the Quarks are assigned with quantum numbers $(3,2,1,1/3)$ and
$(3,1,2,1/3)$ and leptons with $(1,2,1,-1)$ and $(1,1,2,-1)$ respectively
under $\rm SU(3)_{c}\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$. The
sterile fermion transformation under the relevant Gauge group is given as
$S_{L}(1,1,1,0)$ The scalar sector in this extension of LRSM consists of a bi-
doublet with quantum number $\rm\Phi(1,2,2,0)$ , the $SU(2)_{L,R}$ triplets,
$\Delta_{L}(1,3,1,2)$, $\Delta_{R}(1,1,3,2)$ and two $SU(2)_{L,R}$ doublets,
$H_{L}(1,2,1,-1)$, $H_{R}(1,1,2,-1)$ .

A natural type-II dominance in such case can be realised with the following
Yukawa interaction-

$\displaystyle-\mathcal{L_{Y}}=\bar{l_{L}}[Y_{1}\Phi+Y_{2}\tilde{\Phi}]l_{R}+f[\bar{l_{L}}^{c}l_{L}\Delta_{L}+\bar{l_{R}}^{c}l_{R}\Delta_{R}]+F(\bar{l_{R}})H_{R}S_{L}^{c}+h.c$
(4)

In terms of mass matrices we can write Eq(4) as-

$\displaystyle-\mathcal{L_{Y}}\supset
M_{D}\bar{\nu_{L}}N_{R}+M_{L}\bar{\nu_{L}}^{c}\nu_{L}+M_{R}\bar{\nu_{R}}^{c}\nu_{R}+M\bar{N_{R}S_{L}}+h.c$
(5)

where, $M_{D}=Y<\Phi>$ is the Dirac neutrino mass matrix which measures the
light and heavy neutrino mixing and Y stands for Yukawa coupling,
$M_{R}=f<\Delta_{R}>=f\nu_{R}$ ($M_{L}=f<\Delta_{L}>=f\nu_{L}$) is the
Majorana mass term for heavy(Light) neutrinos and $f$ is the Majorana left
right symmetric coupling along with $\nu_{R}$, $\nu_{L}$ corresponding to the
vacuum expectation values(VEV) of scalar triplets, finally $M=F<H_{R}>$ stands
for heavy neutrino-sterile mixing matrix. We have assumed the induced vev of
$<H_{L}>$ to be zero. Now in flavor basis the complete $9\times 9$ mass matrix
for neutral fermions can be represented as-

$\mathcal{M}=\begin{pmatrix}M_{L}&0&M_{D}\\\ 0&0&M\\\ M_{D}^{T}&M^{T}&M_{R}\\\
\end{pmatrix}$ (6)

One can use standard seesaw mechanism to integrate out the heaviest neutrino,
Using Mass ordering $M_{R}>M>M_{D}\gg M_{L}$ one can obtain-

$\mathcal{M}^{{}^{\prime}}=\begin{pmatrix}M_{L}-M_{D}M_{R}^{-1}M_{D}^{T}&-M_{D}M_{R}^{-1}M_{D}^{T}\\\
MM_{R}^{-1}M_{D}^{T}&MM_{R}^{-1}M^{T}\\\ \end{pmatrix}$ (7)

In the Eq(7) which is the intermediate block diagonalised mass matrix, it is
seen that (2,2) element is larger than the entries in the limit
$M_{R}>M>M_{D}\gg M_{L}$. So using the same procedure we can integrate out
sterile neutrino mass matrix. Now we can write down light neutrino mass matrix
as-

$\displaystyle\begin{split}&m_{\nu}=[M_{L}-M_{D}M_{R}^{-1}M_{D}^{T}]-(-M_{D}M_{R}^{-1}M_{D}^{T})(MM_{R}^{-1}M_{D}^{T})(MM_{R}^{-1}M^{T})\\\
&=[M_{L}-M_{D}M_{R}^{-1}M_{D}^{T}]-M_{D}M_{R}^{-1}M_{D}^{T}\\\
&=M_{L}=m_{\nu}^{II}\end{split}$ (8)

After complete block diagonalisation which is extensively discussed in various
literatures, we can write t left handed neutrino, heavy right handed neutrino
and sterile neutrino mass matrices as-

$\displaystyle\begin{split}&m_{\nu}=M_{L}\\\
&M_{R}=\frac{\nu_{R}}{\nu_{L}}M_{L}\\\ &M_{S}=-MM_{R}^{-1}M^{T}\end{split}$
(9)

Again these mass matrices can be diagonalised by using respective $3\times 3$
unitary matrices as follows-

$\displaystyle\begin{split}&m_{\nu}^{diag}=U_{\nu}^{\dagger}m_{\nu}U_{\nu}^{*}=diag(m_{1},m_{2},m_{3})\\\
&M_{N}^{diag}=U_{N}^{\dagger}M_{N}U_{N}^{*}=diag(M_{N_{1}},M_{N_{2}},M_{N_{3}})\\\
&M_{S}^{diag}=U_{S}^{\dagger}M_{S}U_{S}^{*}=diag(M_{S_{1}},M_{S_{2}},M_{S_{3}})\end{split}$
(10)

Complete block diagonalization results-

$\displaystyle\begin{split}&\mathcal{M}_{diag}=V_{9\times
9}^{\dagger}\mathcal{M}V_{9\times 9}^{*}\\\
&=(\mathcal{W}.\mathcal{U})^{\dagger}\mathcal{M}(\mathcal{W}.\mathcal{U})\\\
&=diag(m_{1}.m_{2},m_{3,}M_{S_{1}},M_{S_{2}},M_{S_{3}},M_{N_{1}},M_{N_{2}},M_{N_{3}})\end{split}$
(11)

where, $\mathcal{W}$ is the bock diagonalized mixing matrix and $\mathcal{U}$
is the unitary mixing matrix. Thus the complete $9\times 9$ unitary mixing
matrix can be written as-

$V=\mathcal{W}.\mathcal{U}=\begin{pmatrix}U_{\nu}&M_{D}M^{-1}U_{S}&M_{D}M_{R}^{-1}U_{N}\\\
(M_{D}M^{-1})^{\dagger}U_{\nu}&U_{S}&MM_{R}^{-1}U_{N}\\\
\mathcal{O}&(MM_{R}^{-1})^{\dagger}U_{S}&U_{N}\\\ \end{pmatrix}$ (12)

##  III Flavor symmetric realization of extended LRSM with type-II dominance:

In this section we have realized the extended LRSM considering type-II
dominance case using $A_{4}\times Z_{4}$ flavor symmetry. Type-II dominance in
extended LRSM allows for large left right mixing. $A_{4}$ is a discrete group
of even permutations of four objects. It has three inequivalent one
dimensional representations 1, $1^{\prime}$ and$1^{\prime\prime}$ an
irreducible three dimensional representation 3. Product of the singlets and
triplets are given by-

$1\otimes 1=1$ $1^{\prime}\otimes 1^{\prime}=1^{\prime\prime}$
$1^{\prime}\otimes 1^{\prime\prime}=1$ $1^{\prime\prime}\otimes
1^{\prime\prime}=1^{\prime}$ $3\otimes 3=1\oplus 1^{\prime}\oplus
1^{\prime\prime}\oplus 3_{A}\oplus 3_{S}$ (13)

where subscripts A and S stands for “asymmetric” and “symmetric” respectively.
If we have two triplets ($a_{1}$, $a_{2}$, $a_{3}$) and ($b_{1}$, $b_{2}$,
$b_{3}$), their products are given by

$1\approx a_{1}b_{1}+a_{2}b_{3}+a_{3}b_{2}$ $1^{\prime}\approx
a_{3}b_{3}+a_{1}b_{2}+a_{2}b_{1}$ $1^{\prime\prime}\approx
a_{2}b_{2}+a_{3}b_{1}+a_{1}b_{3}$
$3_{S}\approx\left(\begin{array}[]{c}2a_{1}b_{1}-a_{2}b_{3}-a_{3}b_{2}\\\
2a_{3}b_{3}-a_{1}b_{2}-a_{2}b_{1}\\\
2a_{2}b_{2}-a_{1}b_{3}-a_{3}b_{1}\end{array}\right)$
$3_{A}\approx\left(\begin{array}[]{c}a_{2}b_{3}-a_{3}b_{2}\\\
a_{1}b_{2}-a_{2}b_{1}\\\ a_{3}b_{1}-a_{1}b_{3}\end{array}\right)$

The particle content and their respective charge assignments of the model are
given in Table(1). One can get diagonal charged lepton mass matrix and Dirac
neutrino mass matrix by considering a suitable flavon with proper charge
assignments. Now, as we are interested in type-II dominant case within
extended LRSM, we have already discussed in the section(II) that mass matrices
of light neutrino, heavy neutrino and sterile neutrino are independent of
Dirac neutrino mass matrix (not mixing) allow us to assume flavor independent
structure of Dirac neutrino mass matrix. This is well motivated for the reason
that we can assume that this kind of extension originates from $SO(10)$ GUT
(grand unified theory) with proper choice of intermediate symmetry such as
Pat-Salaam symmetry. In this scenario, the charged lepton mass matrix and
Dirac neutrino mass matrix can be treated in equal footing and taken to be
equivalent to up type quark mass matrix. Main objective of this work to
constrain the mass matrix of light neutrino, heavy neutrino and sterile
neutrino with $A_{4}\times Z_{4}$ flavor symmetry study the impact of this
constrained mass matrices on mixing where Dirac neutrino mass matrix plays a
crucial role. The light neutrino mass formula is governed by type-II seesaw
mechanism, $m^{II}_{\nu}=M_{L}$ while type-I seesaw contribution
$m^{I}_{\nu}=-M_{D}M^{-1}_{R}M_{D}^{T}$ is exactly canceled out in the
complete diagonalization method as given in Sruthilaya:2017vui . Since light
neutrino mass formula is independent of Dirac neutrino mass matrix, any value
of $M_{D}$ is allowed consistent with GUT mass fitting without any fine tuning
of the Yukawa couplings. Further we study their impact on neutrinoless double
beta decay contributions and active sterile mixing with dark matter
phenomenology.

Field | $l_{L}$ | $l_{R}$ | $S_{L}$ | $\Phi$ | $\Delta_{L}$ | $\Delta_{R}$ | $H_{L}$ | $H_{R}$ | $\chi_{\nu}$ | $\epsilon^{{}^{\prime}}$ | $\rho$
---|---|---|---|---|---|---|---|---|---|---|---
$SU(2)_{L}$ | $2$ | $1$ | $1$ | $3$ | $3$ | $1$ | $2$ | $1$ | $1$ | $1$ | $1$
$SU(2)_{R}$ | $1$ | $2$ | $1$ | $1$ | $1$ | $3$ | $1$ | $2$ | $1$ | $1$ | $1$
$U(1)_{B-L}$ | $-1$ | $-1$ | $0$ | $2$ | $2$ | $2$ | $-1$ | $-1$ | $0$ | $0$ | $0$
$A_{4}$ | $3$ | $3$ | $3$ | $1$ | $1$ | $1$ | $3$ | $1$ | $3$ | $1$ | $1$
$Z_{4}$ | $-1$ | $-i$ | $-i$ | $1$ | $1$ | $1$ | $-i$ | $i$ | $1$ | $1$ | $i$
Table 1: Fields and their respective transformations under the symmetry group
of the model.

Now, the effective Lagrangian can be written as-

$\displaystyle\mathcal{L}\supset\bar{l}_{R}(f^{0}_{R}\epsilon^{{}^{\prime}}+f^{\nu}_{R}\chi^{\nu})\Delta_{R}l_{R}+{\bar{l}_{L}}(f^{0}_{L}\epsilon^{{}^{\prime}}+f^{\nu}_{L}\chi^{\nu})\Delta_{L}l_{L}+F\bar{l_{R}}H_{R}\rho
S_{L}^{c}+h.c$ (14)

The role of the flavon $\epsilon^{{}^{\prime}}$ is to break the $\mu-\tau$
symmetry of light and heavy neutrino mass matrices. In our work we take the
flavon alignment to be, $<\chi^{\nu}>\sim(1,1,1),<\rho>\sim v$.

The light neutrino mass matrix can be computed from-

$M_{L}=\nu_{L}f_{L}$ (15)

Where $f_{L}$ is the Majorana Yukawa coupling. Now, the light neutrino mass
matrix can be written as-

$M_{L}=\nu_{L}f^{0}_{L}\begin{pmatrix}0&0&1\\\ 0&1&0\\\ 1&0&0\\\
\end{pmatrix}+\nu_{L}f^{\nu}_{L}\begin{pmatrix}2\chi_{1}^{\nu}&-\chi_{3}^{\nu}&-\chi_{2}^{\nu}\\\
-\chi_{3}^{\nu}&2\chi_{2}^{\nu}&-\chi_{1}^{\nu}\\\
-\chi_{2}^{\nu}&-\chi_{1}^{\nu}&2\chi_{3}^{\nu}\\\ \end{pmatrix}$ (16)

Using the choosen flavon alignment we will get the heavy neutrino mass matrix
of form given below-

$M_{L}=a_{L}\begin{pmatrix}2z&-z&1-z\\\ -z&1+2z&-z\\\ 1-z&-z&2z\\\
\end{pmatrix}$ (17)

Where $a_{L}=\nu_{L}f^{0}_{L}$ and $z=\frac{f^{\nu}_{L}}{f^{0}_{L}}$. Beacuse
of the left-right symmetry, we can take $f_{R}=f_{L}$ and similarly we will
get the Majora neutrino mass matrix of form using (9)

$M_{R}=a_{R}\begin{pmatrix}2z&-z&1-z\\\ -z&1+2z&-z\\\ 1-z&-z&2z\\\
\end{pmatrix}$ (18)

Where $a_{R}=\nu_{R}f^{0}_{R}$ and $z=\frac{f^{\nu}_{R}}{f^{0}_{R}}$. So we
can say that the model parameter
$z=\frac{f^{\nu}_{R}}{f^{0}_{R}}=\frac{f^{\nu}_{L}}{f^{0}_{L}}$. $a_{L}$ and
$a_{R}$ are in the range of $\nu_{L}$ and $\nu_{R}$.

Now the sterile mass matrix is given by-

$M=m_{0}\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\\\ \end{pmatrix}$ (19)

where $m_{0}$ is the free parameter of the model containing $v$ which is the
vev of $\rho$. We want to work on a basis where sterile neutrino mass matrix
to be diagonal. To get the diagonal sterile mass matrix we change the basis by
a Unitary rotation given by-

$U=\begin{pmatrix}0&0&1\\\ 0&1&0\\\ 1&0&0\\\ \end{pmatrix}.$ (20)

After using this rotation one can get a diagonal sterile neutrino mass term as
At this basis light neutrino mass matrix is given as-

$M=m_{0}\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\ \end{pmatrix}$ (21)

At this basis light neutrino mass matrix is given as-

$M_{L}=a_{L}\begin{pmatrix}1-z&-z&2z\\\ -z&1+2z&-z\\\ 2z&-z&1-z\\\
\end{pmatrix}$ (22)

and Majorana neutrino mass matrix-

$M_{R}=a_{R}\begin{pmatrix}1-z&-z&2z\\\ -z&1+2z&-z\\\ 2z&-z&1-z\\\
\end{pmatrix}$ (23)

Using (9),(23) and (21) we can have the sterile neutrino mass matrix as-

$M_{S}=\frac{m_{0}^{2}}{(9z^{2}-1)a_{R}}\begin{pmatrix}2z+3z^{2}&-z+3z^{2}&-1-z+3z^{2}\\\
-z+3z^{2}&-1+2z+3z^{2}&-z+3z^{2}\\\ -1-z+3z^{2}&-z+3z^{2}&2z+3z^{2}\\\
\end{pmatrix}$ (24)

##  IV Neutrinoless Double beta deacy in extended LRSM:

Neutrinoless double beta decay has been studied extensively within the frame
work of LRSM in many literaratures Picciotto:1982qe ; Ge:2015yqa ;
Hirsch:1996qw ; Tello:2010am ; BhupalDev:2013ntw ; Chakrabortty:2012mh ;
BhupalDev:2014qbx ; Bambhaniya:2015ipg . In this section, we will discuss
different new physics contributions to neutrinoless double beta decay process
in extended LRSM scenerio. In our framework, the light neutrino mass arises
from extended type-II dominant seesaw mechanism. The type-II seesaw dominance
not only provides mass relation between light and heavy neutrinos but also
allows large Dirac neutrino mass and thereby gives large light-heavy neutrino
mixing. This light-heavy neutrino mixing plays an important role in giving
sizable contributions to neutrinoless double beta decay. In this kind of
scenario mass of right handed boson ($W_{R}$) is kept very high scale, so the
contribution to the NDBD process due to purely right handed current is
suppressed as the contribution from purely right handed current is
proportional to $\frac{1}{M_{W_{R}}^{4}}$. Similarly, $\lambda$ and $\eta$
diagram gets supressed beacuse of the same argumentSruthilaya:2017vui .

The charge current interaction for leptons and quarks which mainly governs the
NDBD process are given as follows,

$\mathcal{L}^{q}_{cc}=\frac{g_{L}}{\sqrt{2}}\bar{d}\gamma^{\mu}P_{L}W^{-}_{L\mu}+\frac{g_{R}}{\sqrt{2}}\bar{d}\gamma^{\mu}P_{R}W^{-}_{R\mu}+h.c$
(25)

$\mathcal{L}^{lep}_{cc}=\frac{g_{L}}{\sqrt{2}}\sum_{\alpha=e,\mu,\tau}\bar{l_{\alpha}}\gamma^{\mu}P_{L}\nu_{\alpha}W^{-}_{L\mu}+\frac{g_{R}}{\sqrt{2}}\sum_{\alpha=e,\mu,\tau}\bar{l_{\alpha}}\gamma^{\mu}P_{R}\nu_{\alpha}W^{-}_{R\mu}+h.c.$
(26)

Mass eigenstates of neutral lepton are related to light neutrino flavor states
(e,$\mu$,$\tau$) as-

$\nu_{\alpha}=U_{\alpha_{i}}\nu_{i}+Z_{\alpha_{i}}N_{j}+Y_{\alpha_{i}}S_{k}$
(27)

Where, $U_{\alpha_{i}}$, $Z_{\alpha_{j}}$ and $Y_{\alpha_{k}}$ are the mixing
matrices of light neutrino, heavy neutrino and sterile neutrino.

The leading order contribution which contributes to the NDBD process arises
from purely left handed current due to the exchange light neutrino($\nu$),
heavy right handed neutrino ($N_{R}$) and sterile neutrino ($S_{L}$).The
Feynmann amplitude of these contributions are proportional to-

$\mathcal{A}_{\nu}^{LL}\propto
G_{F}^{2}\sum_{i=1,2,3}\frac{U_{ei}^{2}}{p^{2}}$ (28)
$\mathcal{A}_{N}^{LL}\propto
G_{F}^{2}\sum_{j=1,2,3}\big{(}-\frac{Z_{ej}^{2}}{M_{N_{j}}}\big{)}$ (29)
$\mathcal{A}_{S}^{LL}\propto
G_{F}^{2}\sum_{k=1,2,3}\big{(}-\frac{Y_{ek}^{2}}{M_{S_{k}}}\big{)}$ (30)

Where $G_{F}$ is the Fermi constant. The corresponding effective Majorana mass
parameters, which are bsically the measure of lepton number violation are
given as-

$|m_{ee}^{\nu}|=\sum_{i=1,2,3}U_{ei}^{2}m_{\nu i},$ (31)

$|m_{ee}^{N}|=<p^{2}>\sum_{j=1,2,3}\frac{Z_{ej}^{2}}{M_{Nj}}$ (32)

$|m_{ee}^{S}|=<p^{2}>\sum_{k=1,2,3}\frac{Y_{ek}^{2}}{M_{Sk}}$ (33)

where $<p^{2}>$ is the virtual neutrino momentum of order nuclear Fermi scale.

##  V Light sterile neutrino in extended LRSM :

The extended left right symmetric model discussed above can lead to a sterile
neutrino of mass at eV-GeV range depending on the parameter $m_{0}$. Thus it
is possible to achieve a sterile neutrino in keV range within the framework
which is considered as a viable dark matter candidate in present scenario. The
sterile neutrino can be produced from active-sterile oscillation by Dodelson-
Widrow (DW) mechanism dodelson1994sterile . One can obtain the mass as well as
the mixing of the extra sterile neutrino with the active neutrinos using (12).
It is evident from the expressions that both are functions of p and one can
obtain the desired mass range of sterile neutrino by fine tuning $m_{0}$.

There are many cosmological and astrophysical constraints on sterile neutrino
dark matter Adamson:2020jvo ; ng2019new ; heeck2013exotic ; upper . Sterile
neutrinos can be produced from Standard model plasma through their mixing with
active neutrinos in the early universe. Since sterile neutrino is a fermionic
dark matter candidate, lower bounds exist on its mass known as Tremaine-Gunn
bound. Again, the upper limit on mass can be obtained from X-ray constraints.
Direct and indirect detection of dark matter also impose significant bounds on
sterile neutrino which can be seen in horiuchi2014sterile ;
boyarsky2009realistic ; baur2017constraints ; campos2016testing  .

Any stable neutrino state with a non-vanishing mixing to the active neutrinos
will be produced through active-sterile neutrino conversion. Thus the
abundance is generated through the mixing between sterile and the active
neutrinos. The mechanism of non resonant production of sterile neutrinos is
known as Dodelson-Widrow (DW) mechanism. The resulting relic abundance can be
expressed as:

$\Omega_{DM}h^{2}=1.1\times 10^{7}\sum C_{\alpha}(m_{s})|\epsilon_{\alpha
s}|^{2}{\left(\frac{m_{s}}{keV}\right)}^{2},\alpha=e,\mu,\tau$ (34)

where $sin^{2}2\theta=4\sum|\epsilon_{\alpha s}|^{2}$ with $|\epsilon_{\alpha
s}|$is the active-sterile leptonic mixing matrix element and $m_{s}$
represents the mass of the lightest sterile fermion.

The lightest sterile neutrino present in the model is not completely stable
and can decay into an active neutrino through their mixing. The radiative
decay of sterile neutrino induced at one loop level result in photon producing
monochromatic X-rays. Thus there exists X-ray bounds on sterile neutrino dark
matter. In LRSM, along with the bounds on bound on the mixing angle, x-ray
bound leads to some constraints on the properties of the bosonic sector of the
theory because of mixing of the right WR gauge bosons $W_{R}$ with the SM
$W_{L}$ Bezrukov:2009th . However, in our model, the mixing can be neglected.
The total decay width of the process $N\longrightarrow\gamma\nu$ in presence
of gauge boson mixing can be written as Bezrukov:2009th ,

$\Gamma_{N\longrightarrow\gamma\nu}=\frac{G_{F}^{2}\alpha
m_{s}^{3}}{64\pi^{4}}\sum\big{|}4m_{l\alpha}(V_{R})_{\alpha
1}.\zeta-\frac{3}{2}\theta_{\alpha 1}m_{s}\big{|}^{2}$ (35)

Since in our model, gauge boson mixing is negligible i.e $\zeta=0$, the
expression for decay width will be

$\Gamma_{N\longrightarrow\gamma\nu}=\frac{9G_{F}^{2}\alpha
m_{s}^{5}}{1024\pi^{4}}sin^{2}2\theta,\alpha=e,\mu,\tau$ (36)

The non observation of X-ray lines from clusters provides upper limits to the
active-sterile mixing angle as well as the sterile-neutrino mass. We have
implemented the bounds from X-ray in our analysis. An important constraint on
sterile neutrino dark matter is Ly-$\alpha$ bound. This bound provides
stronger bounds on the velocity distribution of the DM particles from the
effect of their free streaming on the large scale structure formation. This
constraint can be converted into a bound for the mass of the DM particle which
can be seen in ng2019new . The constraint is strongly model dependent and the
bounds are governed by the production mechanism of the DM candidate. In this
work, We have adopted the bounds considering $XQ-100$ Ly-$\alpha$ data
baur2017constraints .

As seen from the above equations, the decay rate and as well as the relic
abundance depend on mixing and mass of the DM candidate. Hence, the same set
of model parameters which are supposed to produce correct neutrino
phenomenology can also be used to evaluate the relic abundance and the decay
rate of the sterile neutrino.

##  VI Numerical results and analysis:

In this section, we will study different new physics contribution to NDBD
process in this particular framework. Here, we are interested to study
different new physics contribution to the NDBD process as mentioned earlier.
As the purely right handed contribution gets suppressed and dominant
contribution comes from purely left handed current, we have to put Dirac
neutrino mass matrix as input parameter. In our framework, we are assuming
that TeV scale LRSM is originating from $SO(10)$ GUT or its sub group Pati
Salam symmetry. In this limit we can take the charge lepton mass matrix and
Dirac neutrino mass matrix on equal footing which is equivalent to up type
quark mass matrix. The Dirac neutrino mass matrix is given as-

$M_{D}=V_{CKM}M_{U}V_{CKM}^{T}\\\
=\begin{pmatrix}0.067-0.004i&0.302-0.022i&0.550-0.530i\\\
0.302-0.022i&1.480&6.534-0.001i\\\ 0.550-0.530i&6.534-0.001i&159.72\\\
\end{pmatrix}GeV$ (37)

The $V_{CKM}$ and up type mass matrix is given as:

$V_{CKM}\\\ =\begin{pmatrix}0.97427&0.22534&0.00351-0.0033i\\\
-0.2252+0.0001i&0.97344&0.0412\\\ 0.00876-0.0032i&-0.0404-0.0007i&0.99912\\\
\end{pmatrix}GeV$ (38) $M_{U}=diag(2.3MeV,1.275GeV,173.210GeV)$ (39)

The sterile and right handed neutrino mixing is coming from the model and it
is a diagonal one given in (21).Which is-

$M=m_{0}\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\ \end{pmatrix}$ (40)

The parameter $m_{o}$ is chosen to be free and we assume two different values,
50 GeV and 150 GeV for numerical analysis. The complete diagonalization of
mixing matrix give of extended LRSM with additional singlet fermion is
discussed earlier and detailed analysis is given inHati:2018tge . Mass of
light neutrino, heavy neutrino and sterile neutrino in terms of oscillation
parameter can be written as:

$m_{\nu}=M_{L}=U_{PMNS}m^{diag}_{\nu}U^{T}_{PMNS}$ (41)
$M_{N}=M_{R}=\frac{\nu_{R}}{\nu_{L}}U_{PMNS}m^{diag}_{\nu}U^{T}_{PMNS}$ (42)
$M_{S}=-m^{2}_{o}\frac{\nu_{L}}{\nu_{R}}U^{*}_{PMNS}(m^{diag}_{\nu})^{-1}U^{\dagger}_{PMNS}$
(43)

$U_{PMNS}$ is the diagonalizing matrix of the light neutrino mass matrix,
$m_{\nu}$ such that,

$m_{\nu}=U_{PMNS}M_{\nu}^{(diag)}U_{PMNS}^{T}$ (44)

where, $M_{\nu}^{(diag)}=diag(m_{1},m_{2},m_{3})$ and,

$\rm
U_{\text{PMNS}}=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13}\end{array}\right)U_{\text{Maj}}$
(45)

is the PMNS (Pontecorvo-Maki-Nakagawa-Sakata) matrix and
$c_{ij}=\cos{\theta_{ij}},\;s_{ij}=\sin{\theta_{ij}}$ and $\delta$ is the
leptonic Dirac CP phase. The diagonal matrix $\rm
U_{\text{Maj}}=\text{diag}(1,e^{i\alpha},e^{i(\beta+\delta)})$ contains the
Majorana CP phases $\alpha,\beta$.

The diagonal mass matrix of the light neutrinos can be written as,

$M_{\nu}^{(diag)}=\text{diag}(m_{1},\sqrt{m^{2}_{1}+\Delta
m_{21}^{2}},\sqrt{m_{1}^{2}+\Delta m_{31}^{2}})$ (46)

for normal hierarchy and

$M_{\nu}^{(diag)}=\text{diag}(\sqrt{m_{3}^{2}+\Delta m_{23}^{2}-\Delta
m_{21}^{2}},\sqrt{m_{3}^{2}+\Delta m_{23}^{2}},m_{3})$ (47)

for inverted hierarchy.

In this work, light neutrino mass directly comes from type-II dominance. Here
we want to see the different contribution to the NDBD process. First of all
all, we will see the case of exchange of light neutrino. The dimensionless
parameter which is relevant to lepton number violating process in this process
is given by

$\eta_{\nu}=\frac{1}{m_{e}}\sum_{i=1}^{3}U^{2}_{ei}m_{i}=\frac{m^{\nu}_{ee}}{m_{e}}$
(48)

where $m_{e}$ is the mass of electron. And the effective mass parameter is
given as

$|m_{ee}^{\nu}|=\sum_{i=1,2,3}U_{ei}^{2}m_{\nu i},$ (49)

$m_{\nu}^{eff}=m_{1}c_{12}^{2}c_{13}^{2}+m_{2}s_{12}^{2}c_{13}^{2}e^{2i\alpha}+m_{3}s_{13}^{2}e^{2i\beta}$
(50)

with $c_{ij}=\cos{\theta_{ij}},\;s_{ij}=\sin{\theta_{ij}}$ are respective
oscillation angle and $\alpha$ and $\beta$ are the Majorana phase.

Now, we consider the non standard contribution to the NDBD process due to
purely left handed current because of the exchange of right handed neutrino.
This results in dimensionless parameter which can given as-

$\eta_{N}=-m_{P}\sum_{j=1}^{3}\frac{Z^{2}_{ej}}{M_{i}j}$ (51)

where $m_{p}$ is the mass of proton. And the effective Majorana mass parameter
is-

$|m_{ee}^{N}|=<p^{2}>\sum_{j=1,2,3}\frac{Z_{ej}^{2}}{M_{Nj}}$ (52)

where $<P^{2}>=\mid m_{e}m_{p}\mathcal{M}_{N}/\mathcal{M}_{\nu}\mid$ is the
virtual neutrino momentum.

Now for sterile neutrino contribution, the dimensionless parameter arising
from exchange of sterile neutrino can be given as-

$\eta_{S}=-m_{P}\sum_{k=1}^{3}\frac{Y^{2}_{ek}}{M_{S_{k}}}$ (53)

Effective Majorana mass parameter is -

$|m_{ee}^{S}|=<p^{2}>\sum_{k=1,2,3}\frac{Y_{ek}^{2}}{M_{S_{k}}}$ (54)

Now the combined contribution of all the process discussed above can be
written as-

$\mid m^{tot}_{ee}\mid=\sum_{i=1,2,3}U_{ei}^{2}m_{\nu
i}+<p^{2}>\sum_{j=1,2,3}\frac{Z_{ej}^{2}}{M_{Nj}}+<p^{2}>\sum_{k=1,2,3}\frac{Y_{ek}^{2}}{M_{S_{k}}}$
(55)

We have computed the light neutrino mass matrix from the model described in
the sectionIII given in equation (22). As discussed before, the structures of
mass matrices of heavy neutrino $M_{R}$ and sterile neutrino $M_{S}$ involved
in analysis are constructed using the discrete flavor symmetry $A_{4}\times
Z_{4}$. The light neutrino mass matrix arising from the model is consistent
with non-zero $\theta_{13}$ as $A_{4}$ product rules lead to the light
neutrino mass matrix in which the $\mu-\tau$ symmetry is explicitly broken.
The $3\sigma$ ranges of the mass squared differences and mixing angles from
global analysis of oscillation data are outlined as in the table2.

Oscillation parameters | 3$\sigma$(NO) | 3$\sigma$(IO)
---|---|---
$\frac{\Delta m_{21}^{2}}{10^{-5}eV^{2}}$ | 6.82 - 8.04 | 6.82 - 8.04
$\frac{\Delta m_{31}^{2}}{10^{-3}eV^{2}}$ | 2.431 - 2.60 | 2.31-2.51
$sin^{2}\theta_{12}$ | 0.269 - 0.343 | 0.269 - 0.343
$sin^{2}\theta_{23}$ | 0.407 - 0.612 | 0.411 - 0.621
$sin^{2}\theta_{13}$ | 0.02034 - 0.02430 | 0.02053 - 0.02436
$\frac{\delta}{\pi}$ | 0.87 - 1.94 | 1.12- 1.94
Table 2: Latest gobal fit neutrino oscillation data for both mass ordering in
3-$\sigma$ range Esteban:2020cvm

Figure 1:  Light neutrino contribution to neutrinoless double beta decay
process(left) and non standard contribution due to exchange of heavy
neutrino(right) considering both NH and IH case. The band of black and the
blue solid horizontal line represents the KamLAND-Zen bound on the effective
mass and Legend-1k reach. The red solid vertical line represents the Planck
bound on the sum of the absolute neutrino mass respectively.

Using these values we solve for parameters of the model. From these we have
evaluated the corresponding mixing matrices which are denoted as
$U_{ei}$,$Z_{ej}$ and $Y_{ek}$. Then we calculated the effective mass for both
the cases. During our calculation the effective mass assumes different values
depending on whether the neutrino mass states follows normal hierarchy (NH) or
inverted hierarchy (IH). The standard neutrino contribution to NDBD process as
function of light neutrino mass is shown in the left side of the Fig1. It is
seen from the figure that the light neutrino contribution to neutrinoless
double beta decay (NDBD) can saturate the bound imposed by KamLAND-ZEN and
GERDA. We have also incorporate the future sensitivity of experiment like
Legend-1k in our analysis. We have seen that for NH the contribution is well
within the upper bound provided by mentioned experiments. In case of IH the
contribution is well within the bound provided by KamLAND-ZEN and GERDA but
some points are above the upper bound of future sensitivity of Legend-1k
reach. The right handed neutrino contribution to the NDBD process is depicted
in right side of the Fig1. From this figure we see that lightest neutrino mass
for NH case is between the order of $10^{-2}$ to $10^{-3}$ ev and for IH can
it is below $10^{-2}$ ev with effective mass ranging from $10^{-5}$ to 1 eV.
For sterile contribution we have already mentioned that we will estimate this
for two emperical values of $m_{o}$, 50 GeV and 150 GeV.

Figure 2:  Sterile neutrino contribution to neutrinoless double beta decay
process for $m_{o}=50$ GeV (left) and $m_{o}=150$ GeV(right) considering both
NH and IH case. The band of black and the blue solid horizontal line
represents the KamLAND-Zen bound on the effective mass and Legend-1k reach.
The red solid vertical line represents the Planck bound on the sum of the
absolute neutrino mass respectively.

Effective mass parameter due to the exchange of sterile neutrino considering
$m_{o}=50$ GeV as a function of lightest neutrino mass is shown in left side
of the Fig2. It is seen that lightest neutrino mass ranging from $10^{-4}$ to
$10^{-1}$ eV gives effective mass ranging from $10^{-4}$ to $10^{-8}$ eV for
both NH and IH case which is well within the experimental limit. Again for
$m_{o}=150$ GeV, the plot of effective mass parameter with lightest neutrino
mass for sterile contribution is shown in the right side of the Fig2. In this
case lightest neutrino mass ranging from $10^{-4}$ to $10^{-1}$ eV gives
effective mass ranging from $10^{-6}$ to $10^{-10}$ eV for both NH and IH
case. So it can be inferred that on increasing the value of mass parameter of
sterile neutrino the contribution from sterile neutrino to the NDBD process
become smaller. The total contribution to the NDBD process considering
standard, right handed neutrino and sterile neutrino for the case $m_{o}=50$
GeV and $m_{o}=150$GeV with respect to lightest neutrino mass is shown in left
and right side of Fig3 respectively. From these plots it can be said that
there is not much difference in total contribution due to the change in the
value of $m_{o}$. For both the cases the total effective mass ranges from
$10^{-4}$ to $10^{-1}$ eV for both NH and IH which is well within the bound
provided by KamLAND-ZEN and GERDA. But specifically for NH in both the cases
many of data points are slightly above the future sensitivity of Legend -1k
reach.

Figure 3:  Total contribution to neutrinoless double beta decay process for
$m_{o}=50$ GeV (left) and $m_{o}=150$ GeV(right) considering both NH and IH
case. The band of black and the blue solid horizontal line represents the
KamLAND-Zen bound on the effective mass and Legend-1k reach. The red solid
vertical line represents the Planck bound on the sum of the absolute neutrino
mass respectively.

We also study sterile neutrino dark matter phenomenology in the framework. The
same set of the evaluated model parameters are used to calculate the mass and
mixing of the sterile neutrinos present in the model. We have considered the
lightest sterile neutrino as potential dark matter candidate. We have obtained
the relic abundance and the decay of the dark matter particle using Eq. 34 and
Eq. 36 respectively. As mentioned above, we perform the dark matter study
using two different values of the parameter $m_{0}$ i.e. 50 GeV and 150 GeV.
The obtained results are shown from fig 4 to fig 9.

Figure 4:  Mass mixing parameter space for the DM candidate for $m_{o}=50$ GeV
(left) and $m_{o}=150$ GeV(right)for NH. The cosmological bounds from
Ly-$\alpha$ and X-rays have been implemented.

Figure 5:  Mass mixing parameter space for the DM candidate for $m_{o}=50$ GeV
(left) and $m_{o}=150$ GeV(right)for IH. The cosmological bounds from
Ly-$\alpha$ and X-rays have been implemented.

Fig 4 and 5 depict the mass mixing parameter of the sterile neutrino dark
matter. After implementing the current cosmological bounds,the allowed
parameter space lie within $10-25$ keV for both the cases in NH. In case of
IH, the allowed parameter space is $10-24$ keV for $m_{o}=50$ keV. It has been
observed that the value of $m_{o}$ has no significant effects on the mass and
mixing limit. However, one can obtain more allowed data points for $m_{o}=150$
GeV as can be seen from these figures.

Figure 6: Decay rate of the DM candidate for $m_{o}=50$ GeV (left) and
$m_{o}=150$ GeV(right)for NH. The cosmological bounds from Ly-$\alpha$ and
X-rays have been implemented.

Figure 7: Decay rate of the DM candidate for $m_{o}=50$ GeV (left) and
$m_{o}=150$ GeV(right)for IH. The cosmological bounds from Ly-$\alpha$ and
X-rays have been implemented.

The decay rate of the DM candidate are shown in the above figures fig 6 and
fig 7 for NH and IH respectively. For $m_{o}=50$ GeV, the decay rate for the
allowed mass range is around $10^{-26}$ s-1 and that in case of $m_{o}=150$
GeV is around $10^{-26}$ s-1 in case of NH. However, the decay rates are
slightly larger in case of IH for both the cases.

Figure 8: Sterile neutrino contribution to the total DM abundance for
$m_{o}=50$ GeV (left) and $m_{o}=150$ GeV(right)for IH. The cosmological
bounds from Ly-$\alpha$ and X-rays have been implemented.

Figure 9: Sterile neutrino contribution to the total DM abundance for
$m_{o}=50$ GeV (left) and $m_{o}=150$ GeV(right)for IH. The cosmological
bounds from Ly-$\alpha$ and X-rays have been implemented.

Fig 8 and fig 9 show the relic abundance of the proposed DM candidate. The
sterile neutrino DM can contribute to 17 $\%$ to almost 100 $\%$ of the total
DM relic abundance in all the cases.

Figure 10: Correlation between DM mass and effective mass for $m_{o}=50$ GeV
(left) and $m_{o}=150$ GeV(right)for NH.

Figure 11: Correlation between DM mass and effective mass for $m_{o}=50$ GeV
(left) and $m_{o}=150$ GeV(right)for IH.

Fig 10 and fig 11 represent the correlation of the dark matter property with
the active neutrino phenomenology. It has been observed that for the allowed
mass range of the DM i.e. for (10-25) keV, the effective mass range lies well
within the experimental limit in all the cases. Thus the model can propose a
viable DM candidate along with rich neutrino phenomenology.

##  VII Conclusion

In this work, we extend the generic left right symmetric model with an extra
singlet fermion per generation. We have realized this extension with $A_{4}$
and $Z_{4}$ flavor symmetry considering type-II dominance case. Because of the
extension there will be new physics contribution to NDBD process and type-II
dominance will constrain some of the contribution. We have computed all the
mass matrices of light neutrino, heavy neutrino and sterile neutrino using
flavor symmetry which will constrain the model. Type-II dominance gives raise
to large left-right mixing which is discussed in details our work. By
estimating the contribution coming from light neutrino, heavy neutrino and
sterile neutrino, it is seen that all the contribution coming from these
exchange are well within the experimental bound. The singlet fermion in keV
range can be a viable DM candidate in the model. We have extensively studied
the sterile neutrino dark matter phenomenology. It has been found that the
allowed mass range lies within $10-25$ keV in this model. We have found that
the DM candidate can provide significant contribution to the total relic
abundance. The implications of the model on low energy processes as well as
baryon asymmetry of the Universe can also be addressed which we leave for our
future study.

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