#  Common origin of $\theta_{13}$ and dark matter within the
flavor symmetric scoto-seesaw framework

Joy Ganguly<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Hyderabad, Telenagana, India  Department of BSH,
University of Engineering and Management, Kolkata, India  Janusz Gluza
<EMAIL_ADDRESS>Institute of Physics, University of Silesia, Katowice,
Poland  Biswajit Karmakar<EMAIL_ADDRESS>Institute of Physics,
University of Silesia, Katowice, Poland

###### Abstract

To understand the observed pattern of neutrino masses and mixing as well as to
account for the dark matter we propose a hybrid scoto-seesaw model based on
the $A_{4}$ discrete flavor symmetry. In this setup, the inclusion of at least
two heavy right-handed neutrinos is essential to employ the discrete flavor
symmetry which mimics once popular tribimaximal neutrino mixing at the leading
order via type-I seesaw. The scotogenic contribution then acts as a critical
deviation to reproduce the observed value of the reactor mixing angle
$\theta_{13}$ (within the trimaximal mixing scheme) and to accommodate
potential dark matter candidates, pointing towards a common origin of
$\theta_{13}$ and dark matter. The model predicts the atmospheric angle to be
in the upper octant, excludes some regions on the Dirac CP phase and restricts
the Majorana phases too. Further, normal and inverted mass hierarchies can be
distinguished for specific values of the relative phases associated with the
complex light neutrino mass matrix. Owing to the considered flavor symmetry, a
contribution coming from the scotogenic mechanism towards the lepton flavor
violating decay $\mu\rightarrow e\gamma$ vanishes and a lower limit on the
second right-handed neutrino mass can be obtained. Prediction for the
effective mass parameter appearing in the neutrinoless double beta decay falls
within the sensitivity of future experiments such as LEGEND-1k and nEXO.

##  I Introduction

The discovery of neutrino oscillation Pontecorvo:1967fh ; SNO:2001kpb ; Super-
Kamiokande:1998kpq ; Gonzalez-Garcia:2007dlo ; deSalas:2020pgw  suggests that
at least two neutrinos are massive but having very small masses with respect
to the charged leptons and quarks. This situation opens up a window for
interpretations which go beyond the Standard Model (SM) of particle physics.
On the other hand it is also well established that the neutrino flavor mixing
is significantly large compared to the quark mixing which also demands
extension of the SM (either by particle content or extension of the symmetry).
Another extraordinary problem in particle physics as of today is the nature of
dark matter (DM), whose relic abundance is precisely measured by the WMAP
WMAP:2012nax  and PLANCK Aghanim:2018eyx  satellite experiments and the
existence of such DM is strongly supported by the gravitational lensing,
galactic rotation curve and large scale structure of the Universe
Bertone:2004pz  as well. However, the SM of particle physics fails to provide
an appropriate candidate for DM. Now, from theoretical perspective, the origin
of tiny neutrino masses can be well understood within the framework of various
seesaw mechanisms. The simplest one is the type-I seesaw mechanism
Minkowski:1977sc ; Gell-Mann:1979vob ; Mohapatra:1979ia ; Schechter:1981cv ;
Schechter:1980gr  where usually three singlet right-handed neutrinos are added
to the SM; then three left-handed neutrinos can be massive. This “complete”
sequential seesaw scenario is denoted by ($\\#$$\nu_{L}$,$\\#$$N_{R}$)=(3,3)
where $\\#$ $\nu_{L}$, $\\#$$N_{R}$ denote the number of generations of left-
handed and right-handed neutrinos respectively. However, as first pointed out
in Schechter:1980gr , the number of right-handed neutrinos added to the SM is
not fixed as they do not carry any anomaly King:2015sfk . A simpler version of
the type-I seesaw is the minimal seesaw which further reduces the number of
free parameters Xing:2020ald ; King:1999mb ; King:2002nf . In the (3,2) seesaw
mechanism only two right-handed neutrinos are introduced to obtain viable
neutrino masses. However, the flavor structure of the relevant lepton mass
matrices still remains undetermined.

The flavor structures of the lepton mass matrices and hence the observed non-
trivial pattern of the lepton flavor mixing can be examined by incorporating
non-Abelian discrete flavor symmetries to the SM. For this purpose discrete
symmetry groups such as $A_{4},S_{4},A_{5},\Delta(27)$ are often used
Morisi:2012fg . For a general overview and implementation of discrete
symmetries in neutrino physics see Ishimori:2010au ; Altarelli:2010gt ;
Feruglio:2019ybq ; King:2013eh ; Petcov:2017ggy ; Xing:2020ijf ;
Chauhan:2022gkz . Such discrete flavor symmetries can explain various fixed
mixing schemes such as bi-maximal (BM), golden ratio (GR) and hexagonal (HG)
Vissani:1997pa ; Barger:1998ta ; Datta:2003qg ; Kajiyama:2007gx ;
Albright:2010ap  mixings, with the most popular one being the tri-bimaximal
(TBM) mixing Harrison:2002er ; Harrison:2002kp . In the TBM mixing scheme the
solar and atmospheric mixing angles take values $\sin^{2}\theta_{12}=1/3$ and
$\sin^{2}\theta_{23}=1/2$ respectively, whereas the reactor mixing angle is
fixed at $\sin^{2}\theta_{13}=0$. Therefore, a discrete flavor symmetric
construction with two right-handed neutrinos turns out to be a very economic
scenario to explain both neutrino masses and TBM mixing simultaneously
Chang:2004wy ; Park:2011zt ; Zhao:2011pv . Neutrino model building with two
right-handed neutrinos in various limiting cases can be found in King:1999cm .
Among various discrete groups employed in this purpose, $A_{4}$ turns out to
be the most popular one Ma:2001dn ; Ma:2004zv ; Ma:2002yp ; Altarelli:2005yp ;
Altarelli:2005yx ; Babu:2002dz  which was initially proposed as an underlying
family symmetry for quark sector. It is a discrete group of even permutations
of four objects with three inequivalent one-dimensional representations (1,
$1^{\prime}$ and $1^{\prime\prime}$ ) and a three-dimensional representation
(3). Interestingly, the three generations (or flavors) of right-handed charged
lepton singlets can fit into three inequivalent one-dimensional
representations. On the other hand, $A_{4}$ is the smallest group having a
three-dimensional irreducible representation, then three SM lepton doublets
can transform together as a triplet under $A_{4}$ Altarelli:2005yx ;
Altarelli:2005yp ; Ma:2002yp . So far, so good, however, in the last decade
the reactor neutrino mixing angle $\theta_{13}$ is decisively measured
Abe:2011fz ; An:2012eh ; Ahn:2012nd ; T2K:2013ppw ; MINOS:2013utc  to be
adequately large ($\sim 9^{\circ}$) and hence the era of fixed patterns (such
as BM, TBM, GR, HG) of the lepton mixing matrix is over. To generate non-zero
$\theta_{13}$ various approaches are considered either by additional
contributions to the neutrino sector or considering additional corrections
from the charged lepton sector or including corrections to vacuum alignments
of the flavons etc. Adhikary:2008au ; Brahmachari:2008fn ; King:2009qt ;
Branco:2009by ; AristizabalSierra:2009ex ; Morisi:2009sc ; Ahn:2011yj ;
Shimizu:2011xg ; Ganguly:2020riw ; Ganguly:2021nqx ; Ahn:2011if ;
Antusch:2011ic ; Borah:2019ldn ; Ding:2011gt ; King:2011ab ; Mukherjee:2015axj
; deMedeirosVarzielas:2010ppv ; Ahn:2012cga ; Ahn:2012tv ; BenTov:2012tg ;
Branco:2012vs ; Borah:2017qdu ; CarcamoHernandez:2013yiy ;
Bhattacharya:2016rqj ; Barry:2010zk ; Chen:2012st ; Karmakar:2016cvb ;
Zhao:2014yaa ; Antusch:2013wn ; Borah:2013upa ; Ding:2013bpa ; Ahn:2013mva ;
AristizabalSierra:2014zeq ; Vien:2014pta ; CarcamoHernandez:2015rmj ;
Holthausen:2012wz ; Pramanick:2015qga ; Kalita:2015jaa ; King:2013hj ;
Nomura:2016nfi ; Borah:2013jia ; Memenga:2013vc ; Karmakar:2014dva ;
Puyam:2022mej . As consequence various descendants of the fixed mixing schemes
have emerged. For example, even if the TBM mixing is obsolete now, there are
two of its successors which are still compatible with data. These mixing
schemes are known as trimaximal mixing (TM) mixings Albright:2010ap ;
Xing:2006ms ; King:2011ab ; Grimus:2008tt  which preserves the first (second)
column of the TBM mixing matrix and called ${\rm TM}_{1}$ (${\rm TM}_{2}$)
mixing, respectively.

In this work we consider a scotogenic contribution Ma:2006km  to the
underlying TBM mixing scheme establishing a common origin of the nonzero
$\theta_{13}$ and cosmological DM. There exists many radiative models which
account for the tiny neutrino masses Zee:1980ai ; Cheng:1980qt ;
Restrepo:2013aga ; Babu:1988ki ; Cai:2017jrq  and perhaps the simplest one is
the scotogenic model which also naturally accommodates potential candidate for
dark matter Ma:2006km . In the present article we explore the idea of
combining seesaw and scotogenic (termed as scoto-seesaw) model Rojas:2018wym
to explain neutrino mass and mixing in a consistent way with well motivated
$A_{4}$ discrete flavor symmetry. In our proposal the nature of cosmological
dark matter and reactor mixing angle $\theta_{13}$ share a unified origin. Our
full model comprises of (3,2) seesaw and then combine it with the scotogenic
mechanism which predict three neutrinos to be massive. The Yukawa structure of
the scotogenic contribution is formulated by adding flavon fields which
transforms non-trivially under the flavor symmetry. With this we show that our
model can successfully explain the lepton mixing with non-zero reactor angle
$\theta_{13}$ and includes leptonic CP violation. Due to the flavor symmetric
construction, the model is extremely predictive in nature and offers many
interesting results involving neutrino mass hierarchy, octant of the
atmospheric mixing angle $\theta_{23}$ and restricts the Dirac CP phase. In
addition to this, we also constrain the absolute neutrino masses and Majorana
CP phases. On the other hand, the model can be falsiﬁed by null results of
future neutrinoless double beta decay experiments. Interestingly, due to the
specific flavor structure of the Yukawa couplings as a consequence of the
flavor symmetry, the scotogenic part does not contribute in the lepton flavor
violating decays such as $\mu\rightarrow e\gamma$ and only right-handed
neutrinos contribute in such decays. We begin with a minimal type-I seesaw
assisted by the $A_{4}$ discrete flavor symmetry which helps in reproducing
the TBM mixing. The model also contains additional $Z_{N}$ symmetries to
forbid unwanted contributions and the complete discrete symmetries broken down
to the $\mathcal{Z}_{2}$ symmetry also ensures stability of the dark matter.
Thanks to the considered symmetry, the charged lepton mass matrix as well as
the heavy right-handed Majorana neutrino mass matrix are diagonal to start-
with. Therefore, the structure of the $3\times 2$ Dirac Yukawa matrix turns
out to be solely responsible for generating the TBM mixing with two right-
handed neutrinos. Now, inclusion of the scotogenic contribution to the
neutrino mass helps in reproducing the ${\rm TM}_{2}$ mixing and generates the
observed value of the reactor mixing angle $\theta_{13}$. It also naturally
incorporate dark matter candidates (three potential dark matter candidates
such as the dark fermion and real and imaginary components of the scalar field
involved in the scotogenic contribution) into the picture.

Rest of the paper is organized as follows. In Section II we first describe the
minimal scoto-seesaw model. Then in Section III we present the $A_{4}$ flavor
symmetric scoto-seesaw model with two right-handed neutrinos and describe the
construction of the model based on the symmetries of the framework. In Section
IV we present the correlation among the parameters involved in our analysis.
We carry out the complete analysis for various limiting and general cases and
present their predictions in Section V. Then in Section VI we mention various
phenomenological implications of undertaken analysis and finally conclude in
Section VII. We also included a short note on $A_{4}$ multiplication rules
involved in our analysis in the Appendix.

##  II Minimal scoto-seesaw model

In Ref. Rojas:2018wym , the (3,1) scenario of seesaw mechanism
Schechter:1980gr ; Schechter:1981bd ; Schechter:1981cv  and the scotogenic
model Ma:2006km  are combined to propose a minimal scoto-seesaw model. This
model consists of only one right-handed neutrino $N_{R}$, one singlet fermion
$f$, and one extra scalar doublet $\eta$. In addition to these particles, one
$Z_{2}$ symmetry is introduced, which is responsible for the stability of the
dark matter. All the standard model fields, $N_{R}$, are even under the
$Z_{2}$ symmetry, while the dark sector consists of one fermion $f$ and scalar
field $\eta$ which are odd under $Z_{2}$. This proposal generates the
atmospheric neutrino mass scale at the tree level with the conventional
$(3,1)$ seesaw term with $N_{R}$ and the solar neutrino mass scale is
generated at one-loop level, as a result the hierarchy between the atmospheric
and solar scale is maintained111 Earlier such hierarchy of the atmospheric and
solar neutrino mass scales (and associated mixing) was explained with type-I
seesaw mechanism where the right-handed neutrinos contribute hierarchically
and implemented within the frameworks of sequential dominance King:1998jw ;
King:1999cm ; King:1999mb ; King:2002nf ; Antusch:2010tf ; Antusch:2004gf  and
constrained sequential dominance King:2005bj ; Antusch:2011ic . With this
field content, the lepton Yukawa and mass terms can be written as

$\displaystyle{\mathcal{L}}=-Y_{N}^{k}\bar{L}^{k}i\sigma_{2}H^{*}N_{R}+\frac{1}{2}M_{R}\bar{N}_{R}^{c}N_{R}+Y_{f}^{k}\bar{L}^{k}i\sigma_{2}\eta^{*}f+\frac{1}{2}M_{f}\bar{f}^{c}f+h.c..$
(1)

where $L^{k}$ are the lepton doublets. The scalars $H=(H^{+},H^{0})^{T}$ and
$\eta=(\eta^{+},\eta^{0})$ are the $SU(2)$ doublets. $Y_{N}$ and $Y_{f}$ are
complex $3\times 1$ Yukawa coupling matrices, and $M_{R,f}$ are the mass
matrices for $N_{R}$ and $f$. The total neutrino mass reads Rojas:2018wym

$\displaystyle
M_{\nu}^{ij}=-\frac{v^{2}}{M_{N}}Y_{N}^{i}Y_{N}^{j}+{\mathcal{F}}(m_{\eta_{R}},m_{\eta_{I}},M_{f})Y_{f}^{i}Y_{f}^{j}.$
(2)

Here the first term is due to tree-level seesaw mechanism while the second
term originates from the scotogenic correction with

$\displaystyle\mathcal{F}(m_{\eta_{R}},m_{\eta_{I}},M_{f})=\frac{1}{32\pi^{2}}\Big{[}\frac{m_{\eta_{R}}^{2}\log\Big{(}M_{f}^{2}/m_{\eta_{R}}^{2}\Big{)}}{M_{f}^{2}-m_{\eta_{R}}^{2}}-\frac{m_{\eta_{I}}^{2}\log\Big{(}M_{f}^{2}/m_{\eta_{I}}^{2}\Big{)}}{M_{f}^{2}-m_{\eta_{I}}^{2}}\Big{]},$
(3)

where $m_{\eta_{R}}$ and $m_{\eta_{I}}$ are the masses of the neutral
component of $\eta$ Mandal:2021yph . However, this model predicts one massless
neutrino and demands extension to explain all the neutrino oscillation data.
Hence, to understand the observed pattern of neutrino masses and mixing, in
the next section we will present a modified scoto-seesaw model where two
right-handed neutrinos and $A_{4}$ discrete flavor symmetry will reproduce
non-zero $\theta_{13}$ mixing angle and will account for the dark matter
content. As mentioned earlier, the $A_{4}$ flavor symmetry is well motivated
reproducing the TBM mixing scheme. In Ref. Barreiros:2020gxu , the authors
already discussed the model with two right-handed neutrinos in the context of
the scoto-seesaw scenario but with the $Z_{8}$ symmetry. As we will see, the
set-up based on the $A_{4}$ flavor symmetry is substantially different in
construction from the $Z_{8}$ scenario, resulting in a completely different
texture of the Yukawa couplings and mass matrices. We will show that due to
the presence of the $A_{4}$ symmetry with diagonal structure of the charged
leptons and heavy right-handed neutrinos, the TBM mixing can be generated at
the leading order in the context of minimal type-I seesaw. Subsequently, the
scotogenic contribution acts as a crucial deviation from TBM mixing to
generate non-zero $\theta_{13}$ (reproducing the ${\rm TM}_{2}$ mixing scheme)
as well as providing essential dark matter candidates, thus unifying the
origin of $\theta_{13}$ and dark matter. The $A_{4}$ symmetry assists us to
obtain analytic expressions for neutrino masses and mixing angles as well as
yields interesting correlations among the oscillation parameters with
distinctive predictions; see section IV. In Barreiros:2020gxu , the CP
symmetry is spontaneously broken by the complex vacuum expectation value of
the singlet field whereas in our analysis the source of CP violation is due to
the complex couplings and relative values of CP phases determines the
hierarchy of the masses.

##  III Scoto-seesaw with flavor $A_{4}$ symmetry: the FSS model

The model which we propose is a hybrid scoto-seesaw framework with usual
scotogenic fermion $f$ and scalar doublet $\eta$, supported additionally by
the $A_{4}$ discrete flavor symmetry and two right-handed neutrinos
$N_{R_{1,2}}$. To obtain the flavor structure of the Yukawa couplings the
flavons $\phi_{s}$, $\phi_{a}$, $\phi_{T}$, $\xi$ are introduced. Inclusion of
flavon fields (SM gauge singlets) is a characteristic feature of models with
discrete flavor symmetries Ishimori:2010au ; Altarelli:2010gt ;
Feruglio:2019ybq ; King:2013eh ; Petcov:2017ggy ; Xing:2020ijf ;
Chauhan:2022gkz . In a similar manner, we also incorporate additional $Z_{N}$
discrete symmetries which forbid the exchange of flavon fields eliminating
unwanted terms Altarelli:2010gt ; Feruglio:2019ybq ; King:2013eh ;
Petcov:2017ggy ; Xing:2020ijf ; Chauhan:2022gkz ; Borah:2017dmk ;
Borah:2018nvu ; Borah:2018gjk . In what follows, we will call the whole
framework the Flavor-Scoto-Seesaw (FSS) model where each element of the FSS
model’s construction is well motivated towards understanding a common origin
of $\theta_{13}$ and DM. As we employ the $A_{4}$ discrete symmetry, compared
to Ref. Barreiros:2020gxu  and the $Z_{8}$ choice, we have a fuller symmetry
with larger particle content. This is the price we pay to predict the ${\rm
TM}_{2}$ structure of neutrino masses and mixing. The role of each of $Z_{N}$
auxiliary symmetries will be explained in details as we proceed.
Interestingly, the complete discrete symmetry is broken down to
$\mathcal{Z}_{2}$ (as one of the remnant subgroups) under which both $f$ and
$\eta$ are odd. The stability of the dark matter is ensured by this reserved
$\mathcal{Z}_{2}$ symmetry. In Table 1, we present transformation properties
of all the fields content of our model under the complete discrete $A_{4}$
flavor symmetry. The desired mass matrix structure will be obtained when the
flavons get vacuum expectation value (VEV) in a suitable direction.

Fields |  $e_{R}$, $\mu_{R}$, $\tau_{R}$ | $L_{\alpha}$ | $H$ | $N_{R_{1}}$ | $N_{R_{2}}$ | $f$ | $\eta$ | $\phi_{s}$ | $\phi_{a}$ | $\phi_{T}$ | $\xi$
---|---|---|---|---|---|---|---|---|---|---|---
$A_{4}$ |  $1$ , $1^{\prime\prime}$ , $1^{\prime}$ | $3$ | $1$ | $1$ | $1$ | $1$ | $1$ | $3$ | $3$ | 3 | $1^{\prime\prime}$
$Z_{4}$ | $-i$ | $-i$ | $1$ | $-1$ | $1$ | $1$ | $1$ | $i$ | $-i$ | $1$ | $-1$
$Z_{3}$ | $\omega$ | $\omega$ | $\omega^{2}$ | 1 | 1 | 1 | $\omega^{2}$ | 1 | 1 | $\omega$ | $1$
$Z_{2}$ | $-1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$
Table 1: Field contents and transformation under the symmetries of our model.
The second right-handed neutrino and the flavons field in second block of the
table are introduced to implement the $A_{4}$ symmetry.

The VEV alignment considered here is widely used Altarelli:2005yx ;
King:2005bj ; Zhao:2011pv  and can be realised in a natural way by analyzing
the complete scalar potential Altarelli:2010gt ; Karmakar:2016cvb ; He:2006dk
; Lin:2008aj ; King:2005bj . Here, the low energy scalar potential is
identical with the potential presented in Rojas:2018wym  and for brevity we
omit it here. With the fields content of Table 1, the charged lepton
Lagrangian can be described by

$\displaystyle\mathcal{L}_{l}=\frac{y_{e}}{\Lambda}(\bar{L}\phi_{T})He_{R}+\frac{y_{\mu}}{\Lambda}(\bar{L}\phi_{T})H\mu_{R}+\frac{y_{\tau}}{\Lambda}(\bar{L}\phi_{T})H\tau_{R}+h.c.,$
(4)

to the leading order, where $\Lambda$ is the cut-off scale of our model and
$y_{e}$, $y_{\mu}$ and $y_{\tau}$ are coupling constants. As the SM lepton
doublet $L$ transforms as a triplet under $A_{4}$, involvement of another
triplet is essential as seen in the above Lagrangian. Terms in the first
parenthesis of Eq. (4) represent the product of two $A_{4}$ triplets which
results a true singlet after contracting with $A_{4}$ singlets $e_{R}$,
$\mu_{R}$ and $\tau_{R}$ (charged as $1$, $1^{\prime\prime}$ and $1^{\prime}$
respectively). The multiplication rule of $A_{4}$ symmetry is summarized is
the Appendix and detailed discussion on $A_{4}$ symmetry can be found in
Altarelli:2010gt ; Ishimori:2010au . Now, when the flavon $\phi_{T}$ gets VEV
in the direction $\langle\phi_{T}\rangle=(v_{T},0,0)^{T}$ Altarelli:2005yx
and also the Higgs field gets VEV $\langle H\rangle=v$, the charged lepton
mass matrix will be a diagonal form

$\displaystyle M_{l}=\frac{v_{T}}{\Lambda}v\begin{pmatrix}y_{e}&0&0\\\
0&y_{\mu}&0\\\ 0&0&y_{\tau}\end{pmatrix}.$ (5)

The Lagrangian in the neutrino sector constitutes two parts: one is for type-I
seesaw contribution with two right-handed neutrinos $N_{R_{1}}$ and
$N_{R_{2}}$ and another is for scotogenic contribution with a scalar field
$\eta$ and a fermionic field $f$. The Lagrangian that generates neutrino mass
at a tree level by the type-I seesaw mechanism, in our model, can be written
as

$\displaystyle\mathcal{L}_{N}=\frac{y_{N_{1}}}{\Lambda}(\bar{L}\phi_{s})\tilde{H}N_{R_{1}}+\frac{y_{N_{2}}}{\Lambda}(\bar{L}\phi_{a})\tilde{H}N_{R_{2}}+\frac{1}{2}M_{N_{1}}\bar{N}_{R_{1}}^{c}N_{R_{1}}+\frac{1}{2}M_{N_{2}}\bar{N}_{R_{2}}^{c}N_{R_{2}}+h.c.,$
(6)

where $y_{N_{1,2}}$ are the corresponding couplings and $M_{N_{1,2}}$ are the
Majorana masses of right-handed neutrinos. To get the flavor structure, we
assume that the flavon fields get VEVs along
$\langle\phi_{s}\rangle=(0,v_{s},-v_{s})$,
$\langle\phi_{a}\rangle=(v_{a},v_{a},v_{a})$ Antusch:2011ic ; King:2005bj .
With these flavon vevs, the Dirac mass matrix will appear from the first two
terms of Eq. (6) while the Majorana matrix which follows from the next two
terms of the Lagrangian of Eq. (6) can be found as follows

$\displaystyle M_{D}=\frac{v}{\Lambda}\begin{pmatrix}0&y_{N_{2}}v_{a}\\\
-y_{N_{1}}v_{s}&y_{N_{2}}v_{a}\\\
y_{N_{1}}v_{s}&y_{N_{2}}v_{a}\end{pmatrix}=vY_{N},\quad
M_{R}=\begin{pmatrix}M_{N_{1}}&0\\\ 0&M_{N_{2}}\end{pmatrix}.$ (7)

The $Z_{4}$ symmetry in Table 1, under which $N_{R_{1}}$ is odd and
$N_{R_{2}}$ even and the $Z_{2}$ symmetry under which $N_{R_{1}}$ is even and
$N_{R_{2}}$ odd ensures the diagonal structure of $M_{R}$ as obtained in Eq.
7. The VEV alignment considered here is widely used in the context of form
dominance Chen:2009um , sequential dominance Antusch:2004gf , constrained
sequential dominance Antusch:2011ic  etc., to obtain the textures of the Dirac
and Majorana mass matrices. Now using type-I seesaw formula the light neutrino
mass matrix at the leading order can be written as

$\displaystyle M_{\nu}=-M_{D}M_{R}^{-1}M_{D}^{T}.$ (8)

With the structure of $M_{D}$ and $M_{R}$ obtained in Eq. (7), the light
neutrino mass matrix is given by

$\displaystyle({M_{\nu}})_{\rm TREE}=-\begin{pmatrix}B&B&B\\\ B&A+B&-A+B\\\
B&-A+B&A+B\end{pmatrix},\quad
A=\frac{v^{2}v_{s}^{2}y_{N_{1}}^{2}}{\Lambda^{2}M_{N_{1}}},\quad
B=\frac{v^{2}v_{a}^{2}y_{N_{2}}^{2}}{\Lambda^{2}M_{N_{2}}}.$ (9)

The above mass matrix for light neutrinos obtained from type-I seesaw is
incapable to generate non-zero $\theta_{13}$ (charged lepton mass matrix being
diagonal). As neutrino oscillation data established adequately large
$\theta_{13}$, we include a scotogenic contribution to our model to explain
correct neutrino mixing which also naturally incorporates few potential DM
candidates. The scotogenic contribution in our model with the fermion $f$ and
scalar field $\eta$ can be written as

$\displaystyle\mathcal{L}_{S}=\frac{y_{s}}{\Lambda^{2}}(\bar{L}\phi_{s})\xi
i\sigma_{2}\eta^{*}f+\frac{1}{2}M_{f}\bar{f}^{c}f+h.c.,$ (10)

where $y_{s}$ is the coupling and $M_{f}$ is the mass of $f$. Owing to the
considered symmetry in Table 1, the leading order contribution
$\bar{L}i\sigma_{2}\eta^{*}f$ is disallowed, as it is not invariant under
$A_{4}$ (and $Z_{4}$, $Z_{2}$) symmetry. The SM lepton doublet being a triplet
under $A_{4}$, just like charged lepton sector, involvement of another $A_{4}$
triplet (here $\phi_{s}$) is essential. As a consequence of $Z_{4,2}$ symmetry
involvement of the $A_{4}$ flavons $\phi_{s}$ and $\xi$ is necessary in the
first term of Eq. (10). The VEV of $\phi_{s}$ (mentioned below) and the non-
trivial $A_{4}$ singlet $\xi$ (provides appropriate $A_{4}$ contraction)
crucially dictates the structure of the scotogenic contribution and helps in
breaking of TBM mixing Karmakar:2014dva ; Karmakar:2015jza ; Karmakar:2016cvb
; Bhattacharya:2016lts ; Bhattacharya:2016rqj . Therefore the contribution in
the effective neutrino mass matrix originated from the scotogenic radiative
corrections is given by Ma:2006km ; Rojas:2018wym ; Barreiros:2020gxu

$\displaystyle(M_{\nu})_{\rm
LOOP}=\mathcal{F}(m_{\eta_{R}},m_{\eta_{I}},M_{f})M_{f}Y_{f}^{i}Y_{f}^{j}.$
(11)

Once the flavons $\phi_{s}$ and $\xi$ acquire VEVs in the direction
$\langle\phi_{s}\rangle=(0,v_{s},-v_{s})$ and $\langle\xi\rangle=v_{\xi}$
respectively the associated couplings can be written as

$\displaystyle
Y_{f}^{1}=y_{s}\frac{v_{s}}{\Lambda}\frac{v_{\xi}}{\Lambda},\quad
Y_{f}^{2}=0,\quad
Y_{f}^{3}=-y_{s}\frac{v_{s}}{\Lambda}\frac{v_{\xi}}{\Lambda}.$ (12)

Therefore, the corresponding mass matrix takes the form

$\displaystyle(M_{\nu})_{\rm LOOP}=C\begin{pmatrix}1&0&-1\\\ 0&0&0\\\
-1&0&1\end{pmatrix},\quad
C={\mathcal{F}(m_{\eta_{R}},m_{\eta_{I}},M_{f})}y_{s}^{2}\frac{v_{s}^{2}v_{\xi}^{2}}{\Lambda^{4}}.$
(13)

Here ${\mathcal{F}(m_{\eta_{R}},m_{\eta_{I}},M_{f})}$ is the loop function
given in Eq. (3). Finally, combining the seesaw and scotogenic contributions,
the effective light neutrino mass matrix is the addition of the two mass
matrices given in Eq. (9) and Eq. (13) and reads

$\displaystyle M_{\nu}$ $\displaystyle=$ $\displaystyle(M_{\nu})_{\rm
TREE}+(M_{\nu})_{\rm LOOP}$ (14) $\displaystyle=$
$\displaystyle\begin{pmatrix}-B+C&-B&-B-C\\\ -B&-(A+B)&A-B\\\
-B-C&A-B&-A-B+C\end{pmatrix}.$

In the present context, neutrino masses are obtained through a combination of
type-I seesaw and scotogenic mechanisms. Now, there could also be operators
like $LHLH/\Lambda$, which can also contribute to the light neutrino mass. In
our model, this term is not invariant under the $Z_{4}$ symmetry mentioned in
Table 1. Any contributions coming from
$LHLH(\phi_{a},\phi_{s},\phi_{T},\xi)/\Lambda^{2}$ is also disallowed due to
the considered discrete symmetries $Z_{4},Z_{3}$ and $Z_{2}$. For the
scotogenic contribution, the coupling $\bar{L}i\sigma_{2}\eta^{*}f$ is allowed
only at $1/\Lambda^{2}$ level with the involvement of the flavons
$\phi_{s},\xi$, see Eq. (10). In this sector, any higher order contributions
in the bare mass term of $f$ can be absorbed in the leading order
contribution. On the other hand for charged lepton sector, the leading
contribution only appears at dimension-5 due to the considered $A_{4}$
symmetry. There are next-to-leading order corrections present in this sector
coming from $(\bar{L}\phi_{s}\phi_{a})H\alpha_{R}/\Lambda^{2}$, where
$\alpha_{R}$ is the associated right-handed charged lepton. Thanks to the VEV
alignment of the flavons $\phi_{s}$ and $\phi_{a}$ this term essentially
vanishes following the $A_{4}$ multiplication rules mentioned in the appendix.
For right-handed Majorana neutrinos, the nonvanishing next-to-leading order
corrections up-to $\mathcal{O}$ $(1/\Lambda^{2})$ in the mass matrix arise
from the following terms:

$\displaystyle\delta\mathcal{L}_{M_{R}}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda}\left(\bar{N}_{R_{1}}^{c}N_{R_{1}}+\bar{N}_{R_{2}}^{c}N_{R_{2}}\right)\left(\phi^{\dagger}_{a}\phi_{a}+\phi^{\dagger}_{s}\phi_{s}\right)$
(15) $\displaystyle+$
$\displaystyle\frac{1}{\Lambda}(\bar{N}_{R_{1}}^{c}N_{R_{2}}+\bar{N}_{R_{2}}^{c}N_{R_{1}})(\phi^{\dagger}_{s}\phi_{a}+\phi_{s}\phi^{\dagger}_{a}+\phi^{\dagger}_{s}\phi^{\dagger}_{a})+\frac{1}{\Lambda^{2}}(\bar{N}_{R_{1}}^{c}N_{R_{2}}+\bar{N}_{R_{2}}^{c}N_{R_{1}})\xi^{3}.$

Here in Eq. (15), the first term represents a correction to the diagonal entry
which can be absorbed in the leading order $M_{R}$. The second term represents
off-diagonal entries of the right-handed neutrino mass matrix at $\mathcal{O}$
$(1/\Lambda)$ which also vanishes due to specific VEV direction of $\phi_{s}$
and $\phi_{a}$. The last term in Eq. (15) represents off-diagonal entries at
$\mathcal{O}$ ($1/\Lambda^{2}$). Although this contribution is very small
compared to the leading order contribution, it can also be forbidden by
considering another $Z^{\prime}_{2}$ symmetry under which charged leptons, $f$
and the flavons $\phi_{T},\xi^{\prime}$ are odd (while all other particles are
even). The Dirac Yukawa coupling is allowed at dimension-5 as given in Eq.
(6). Here the next-to-leading order contribution at $\mathcal{O}$
$(1/\Lambda)$ can be written as
$(\bar{L}\phi^{\dagger}_{a}\phi_{T})\tilde{H}N_{R_{1}}/\Lambda^{2}$ and
$(\bar{L}\phi^{\dagger}_{s}\phi_{T})\tilde{H}N_{R_{2}}/\Lambda^{2}$
respectively. These terms are however forbidden owing to the $Z_{3}$ symmetry
mentioned in Table 1. Therefore, from Table 1, it is clear that along with the
$A_{4}$ symmetry the auxiliary discrete symmetries crucially dictate allowed
structures of the femionic mass matrices, and such symmetries are an integral
part of the flavor symmetric approach to understand neutrino mixing
Altarelli:2005yp ; Altarelli:2005yx ; Molinaro:2009lud ; Branco:2011iw ;
King:2011zj ; Altarelli:2010gt ; Ma:2001dn .

##  IV Neutrino masses and mixing in the FSS model

From the previous discussion, we find that the effective neutrino mass matrix
consists of two parts, one of them is coming from type-I seesaw mechanism
given by Eq. (9) and another one originates from the scotogenic contribution
given by Eq. (13). Now the mass matrix originated from type-I seesaw given in
Eq. (9) can be diagonalized by TBM mixing matrix ($U_{TB}$) via

$\displaystyle U_{TB}^{T}(M_{\nu})_{\rm TREE}U_{TB}=\begin{pmatrix}0&0&0\\\
0&-3B&0\\\ 0&0&-2A\end{pmatrix},$ (16)

where

$\displaystyle
U_{TB}=\begin{pmatrix}\sqrt{\frac{2}{3}}&\frac{1}{\sqrt{3}}&0\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{2}}\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\end{pmatrix}.$ (17)

Clearly, a pure type-I seesaw contribution in the present set-up predicts
$\theta_{13}=0$. However, thanks to the scotogenic contribution, we can obtain
a deviation from $\theta_{13}=0$ to be consistent with the observed
experimental value Abe:2011fz ; An:2012eh ; Ahn:2012nd . Therefore considering
the effective light neutrino mass matrix given in Eq. (14) and rotating it by
$U_{TB}$, $M_{\nu}$ takes the form

$\displaystyle M_{\nu}^{\prime}$ $\displaystyle=$ $\displaystyle
U_{TB}^{T}M_{\nu}U_{TB}$ (18)
$\displaystyle=\frac{1}{2}\begin{pmatrix}3C&0&-\sqrt{3}C\\\ 0&-6B&0\\\
-\sqrt{3}C&0&-4A+C\end{pmatrix},$

Here we find that the mass matrix in the tri-bimaximal basis is block
diagonalized. Therefore a further rotation by a unitary matrix $U_{13}$ in the
13 plane via $M_{\nu}^{diag}=U_{13}^{T}M_{\nu}^{\prime}U_{13}$ takes
$M^{\prime}_{\nu}$ to a diagonal one. This unitary matrix $U_{13}$ can be
parametrized as

$\displaystyle U_{13}=\begin{pmatrix}\cos\theta&0&\sin\theta e^{-i\phi}\\\
0&1&0\\\ -\sin\theta e^{i\phi}&0&\cos\theta\end{pmatrix},$ (19)

where $\theta$ is the rotation angle and $\phi$ is the associated phase
factor. The full diagonalization relation of the mass matrix $M_{\nu}$ can be
written as

$\displaystyle(U_{TB}U_{13})^{T}M_{\nu}U_{13}U_{TB}={\rm
diag}(m_{1}e^{i\gamma_{1}},m_{2}e^{i\gamma_{2}},m_{3}e^{i\gamma_{3}}),$ (20)

where $m_{1}$, $m_{2}$, $m_{3}$ are the real and positive mass eigenvalues and
$\gamma_{1}$, $\gamma_{2}$ and $\gamma_{3}$ are the phases extracted from the
corresponding complex eigenvalues. We are now in a position to evaluate the
neutrino mixing matrix $U_{\nu}$ such that $U_{\nu}^{T}M_{\nu}U_{\nu}={\rm
diag}(m_{1},m_{2},m_{3})$. Thus $U_{\nu}$ becomes $U_{\nu}=U_{TB}U_{13}U_{m}$,
where $U_{m}={\rm diag}(1,e^{i\alpha_{21}/2},e^{i\alpha_{31}/2})$ is the
Majorana phase matrix with $\alpha_{21}=\gamma_{1}-\gamma_{2}$ and
$\alpha_{31}=\gamma_{1}-\gamma_{3}$, one common phase being irrelevant. Using
Eq. (17) and Eq. (19), the $U_{\nu}$ mixing matrix in its explicit form can be
written as

$\displaystyle
U_{\nu}=\begin{pmatrix}\sqrt{\frac{2}{3}}\cos\theta&\frac{1}{\sqrt{3}}&\sqrt{\frac{2}{3}}e^{i\phi}\sin\theta\\\
-\frac{\cos\theta}{\sqrt{6}}+\frac{e^{i\phi\sin\theta}}{\sqrt{2}}&\frac{1}{\sqrt{3}}&-\frac{\cos\theta}{\sqrt{2}}-\frac{e^{i\phi\sin\theta}}{\sqrt{6}}\\\
-\frac{\cos\theta}{\sqrt{6}}-\frac{e^{i\phi\sin\theta}}{\sqrt{2}}&\frac{1}{\sqrt{3}}&\frac{\cos\theta}{\sqrt{2}}-\frac{e^{i\phi\sin\theta}}{\sqrt{6}}\end{pmatrix}U_{m}.$
(21)

Such deviation from the TBM mixing is well known and this particular pattern
of $U_{\nu}$ is called ${\rm TM}_{2}$ mixing as described earlier. This lepton
mixing matrix $U_{\nu}$ can now be compared with $U_{PMNS}$ which in its
standard parametrization is given by ParticleDataGroup:2020ssz

$\displaystyle
U_{PMNS}=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{\rm
CP}}\\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{\rm
CP}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{\rm CP}}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{\rm
CP}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{\rm
CP}}&c_{23}c_{13}\end{pmatrix}U_{m},$ (22)

where $\theta_{12},\theta_{13}$ and $\theta_{23}$ are three mixing angles,
$\delta_{\rm CP}$ is the CP violating Dirac phase and $\alpha_{21}$,
$\alpha_{31}$ are the Majorana phases.

The parameters $A$, $B$ and $C$ appearing in Eq. (18) are in general complex
and without loss of generality we can write $A=|A|e^{i\phi_{A}}$,
$B=|B|e^{i\phi_{B}}$, $C=|C|e^{i\phi_{C}}$. Now, for calculation purpose, let
us define $\alpha=|A|/|C|$ and $\beta=|B|/|C|$, and phase differences
$\phi_{AC}=\phi_{A}-\phi_{C}$ and $\phi_{BC}=\phi_{B}-\phi_{C}$. As $U_{13}$
diagonalizes $M_{\nu}^{\prime}$ of Eq. (18), $\theta$ and $\phi$ can be
expressed in terms of model parameters as

$\displaystyle\tan\phi=\frac{\alpha\sin\phi_{AC}}{1-\alpha\cos\phi_{AC}},\quad\tan
2\theta=\frac{\sqrt{3}}{\cos\phi+2\alpha\cos(\phi_{AC}+\phi)}.$ (23)

Further comparing $U_{\nu}=U_{TB}U_{13}U_{m}$ as given in Eq. (21) with
$U_{PMNS}$ as in Eq. (22), we find the following relations for mixing angles
and $\delta_{\rm CP}$ as a function of $\theta$ and $\phi$ as Ma:2012ez ;
Hernandez:2012ra ; Tanimoto:2015nfa ; Shimizu:2014ria ; Karmakar:2014dva

$\displaystyle\sin\theta_{13}e^{-i\delta_{\rm
CP}}=\sqrt{\frac{2}{3}}e^{-i\phi}\sin\theta,\quad\tan^{2}\theta_{12}=\frac{1}{2-3\sin^{2}\theta_{13}},$
(24)
$\displaystyle\tan^{2}\theta_{23}=\frac{\Big{(}1+\frac{\sin\theta_{13}\cos\phi}{\sqrt{2-3\sin^{2}\theta_{13}}}\Big{)}^{2}+\frac{\sin^{2}\theta_{13}\sin^{2}\phi}{(2-3\sin^{2}\theta_{13})}}{\Big{(}1-\frac{\sin\theta_{13}\cos\phi}{\sqrt{2-3\sin^{2}\theta_{13}}}\Big{)}^{2}+\frac{\sin^{2}\theta_{13}\sin^{2}\phi}{(2-3\sin^{2}\theta_{13})}}.$
(25)

The above relations show that the mixing angles are correlated which is a
characteristic feature of the considered $A_{4}$ discrete flavor symmetry. For
$\sin\theta>0$, the relation of $\delta_{\rm CP}$ implies that $\delta_{\rm
CP}=\phi$ and for $\sin\theta<0$, the same relation implies that $\delta_{\rm
CP}=\phi\pm\pi$. Hence, for both cases, we have $\tan\delta_{\rm
CP}=\tan\phi$. Now, using Eq. (14), the complex mass eigenvalues are
calculated to be

$\displaystyle m_{1,3}^{c}$ $\displaystyle=$
$\displaystyle-A+C\pm\sqrt{A^{2}+AC+C^{2}},$ (26) $\displaystyle m_{2}^{c}$
$\displaystyle=$ $\displaystyle-3B.$ (27)

The real and positive eigenvalues can be written as

$\displaystyle
m_{1}=|C|\big{[}(1-\alpha\cos\phi_{AC}-P)^{2}+(Q+\alpha\sin\phi_{AC})^{2}\big{]}^{1/2},$
(28) $\displaystyle m_{2}=|C|3\beta,$ (29) $\displaystyle
m_{3}=|C|\big{[}(1-\alpha\cos\phi_{AC}+P)^{2}+(Q-\alpha\sin\phi_{AC})^{2}\big{]}^{1/2},$
(30)

where

$\displaystyle P^{2}=\frac{M\pm\sqrt{M^{2}+N^{2}}}{2},\quad
Q^{2}=\frac{-M\pm\sqrt{M^{2}+N^{2}}}{2},$ (31) $\displaystyle
M=1+\alpha\cos\phi_{AC}+\alpha^{2}\cos 2\phi_{AC},\quad
N=\alpha\sin\phi_{AC}+\alpha^{2}\sin 2\phi_{AC}.$ (32)

Following Eq. (26) and Eq. (27), the phase associated with complex mass
eigenvalues $m^{c}_{1,2,3}$ can be written as $\gamma_{i}=\phi_{C}+\phi_{i}$,
where $\phi_{i}$ are

$\displaystyle\phi_{1}=\tan^{-1}\Big{(}\frac{Q+\alpha\sin\phi_{AC}}{1-\alpha\cos\phi_{AC}-P}\Big{)},\quad\phi_{2}=\phi_{BC},\quad\phi_{3}=\tan^{-1}\Big{(}\frac{Q-\alpha\sin\phi_{AC}}{1-\alpha\cos\phi_{AC}+P}\Big{)}.$
(33)

The two Majorana phases in $U_{m}$ (see eq. (21)) therefore can be derived as

$\displaystyle\alpha_{21}$ $\displaystyle=$
$\displaystyle\tan^{-1}\Big{(}\frac{Q+\alpha\sin\phi_{AC}}{1-\alpha\cos\phi_{AC}-P}\Big{)}-\phi_{BC},$
(34) $\displaystyle\alpha_{31}$ $\displaystyle=$
$\displaystyle\tan^{-1}\Big{(}\frac{Q+\alpha\sin\phi_{AC}}{1-\alpha\cos\phi_{AC}-P}\Big{)}-\tan^{-1}\Big{(}\frac{Q-\alpha\sin\phi_{AC}}{1-\alpha\cos\phi_{AC}+P}\Big{)}.$
(35)

The overall phase factor $\phi_{C}$ appearing in $\gamma_{i}$ has no physical
significance in computing the Majorana phases. The mixing angles and the
phases depend on the parameters $\alpha,\phi_{AC},\phi_{BC}$ whereas the light
neutrino masses depend on these parameters as well as on $\beta$ and $|C|$ as
observed in Eq. (23) - Eq. (35). In the next section we constrain these
parameters using experimental data for neutrino mixing angles and masses.

parameters | best-fit |  $3\sigma$ range
---|---|---
$\dfrac{\Delta m_{21}^{2}}{10^{-5}{\rm eV}^{2}}$ | 7.50 | 6.94 - 8.14
$\dfrac{|\Delta m_{31}^{2}|}{10^{-3}{\rm eV}^{2}}$(NH) | 2.55 | 2.47 - 2.63
$\dfrac{|\Delta m_{31}^{2}|}{10^{-3}{\rm eV}^{2}}$(IH) | 2.45 | 2.37 - 2.53
$\sin^{2}\theta_{12}/10^{-1}$ | 3.18 | 2.71 - 3.69
$\sin^{2}\theta_{13}/10^{-2}$(NH) | 2.200 | 2.000 - 2.405
$\sin^{2}\theta_{13}/10^{-2}$(IH) | 2.225 | 2.018 - 2.424
$\sin^{2}\theta_{23}/10^{-1}$(NH) | 5.74 | 4.34 - 6.10
$\sin^{2}\theta_{23}/10^{-1}$(IH) | 5.78 | 4.33 - 6.08
Table 2: Global fits of three active-neutrino oscillation data taken for Ref.
deSalas:2020pgw  for NH and IH, used in our analysis.

##  V Numerical analysis of the FSS model

In order to constrain the parameters involved in our analysis, using Eq. (28)
- Eq. (30), we can define a ratio $r$ as

$\displaystyle r=\frac{\Delta m_{21}^{2}}{|\Delta m_{31}^{2}|},$ (36)

where $\Delta m_{21}^{2}=m_{2}^{2}-m_{1}^{2}$ and $|\Delta
m_{31}^{2}|=|m_{3}^{2}-m_{1}^{2}|$ are the solar and atmospheric mass squared
differences. From the expressions for the mixing angles (namely,
$\theta_{13}$, $\theta_{12}$ and $\theta_{23}$) as well as the absolute
neutrino masses ($m_{1,2,3}$), their sum ($\sum m_{i}$) and the ratio $r$
defined in Eq. (36) all depend on the variables $\alpha$, $\beta$, $\phi_{AC}$
and $\phi_{BC}$ as discussed in Section IV. Over the last two decades neutrino
oscillation parameters have been measured with incredible accuracy
deSalas:2020pgw ; Esteban:2020cvm ; Capozzi:2021fjo . Therefore using the
precisely determined neutrino oscillation data on $\theta_{13}$,
$\theta_{12}$, $\theta_{23}$, $r$, $\Delta m_{21}^{2}$ and $|\Delta
m_{31}^{2}|$ one can constrain four model parameters. Once the model
parameters are constrained, we can further compute the Dirac CP phase
$\delta_{\rm CP}$, Majorana phases $\alpha_{21}$, $\alpha_{31}$, sum of the
absolute masses of the three light neutrinos ($\sum m_{i}$) as well as the
effective mass parameter appearing in the neutrinoless double beta decay
($m_{\beta\beta}$). Table 2 summarizes the best fit and 3$\sigma$ ranges of
the neutrino oscillation data deSalas:2020pgw  for both normal and inverted
hierarchy of light neutrino masses which are used in the subsequent numerical
analysis.

Before we proceed further, let us point-out that the correlation between the
mixing angles $\theta_{13}$ and $\theta_{12}$ given in Eq. (24) is a feature
of the ${\rm TM}_{2}$ mixing mentioned above Shimizu:2014ria ;
Hernandez:2012ra . In Fig. 1, we have plotted this
$\sin^{2}\theta_{12}$$-$$\sin^{2}\theta_{13}$ correlation for ${\rm TM}_{2}$
mixing and we

Figure 1: $\sin^{2}\theta_{12}$ is plotted against $\sin^{2}\theta_{13}$ for
both NH and IH of neutrino masses. The vertical grid-lines are 3$\sigma$
allowed for $\sin^{2}\theta_{13}$ and horizontal grid-lines represent
corresponding restriction on $\sin^{2}\theta_{12}$ in our analysis.

find that $\sin^{2}\theta_{12}$ is restricted within a narrow range (between
0.3401 $\leq\sin^{2}\theta_{12}\leq$ 0.3415) corresponding to the $3\sigma$
ranges of $\sin^{2}\theta_{13}$.Now, in order to evaluate absolute neutrino
masses we also need to find the overall factor $|C|$ appearing in the mass
eigenvalues given in Eq. (28) - Eq. (30). Although this common factor $|C|$
cancels out when we calculate $r$ but $|C|$ can be calculated by fitting the
solar (or atmospheric) mass-squared differences knowing the model parameters.
After $|C|$ is evaluated, we can get the estimation of absolute neutrino
masses $m_{1,2,3}$ and their sum $\sum m_{i}$. Similarly, substituting the
estimations for $\alpha,\beta,\phi_{AC}$ and $\phi_{BC}$ in Eq. (34) and Eq.
(35) we can also quantify the Majorana phases. Knowing the neutrino mixing
angles, masses and associated CP phases, finally in our analysis we will have
a prediction on the effective neutrino mass parameter $m_{\beta\beta}$
characterizing the neutrinoless double beta decay. The effective mass
parameter $m_{\beta\beta}$ can be described as a function of the lightest
neutrino mass ($m_{1}$ for NH and $m_{3}$ for IH respectively) and can be
written as 222The expression for the effective mass parameter $m_{\beta\beta}$
can also be written in a symmetrical form where only two Majorana phases
Rodejohann:2011vc  appear instead of three phases appearing in the standard
PDG parametrization Zyla:2020zbs .

$\displaystyle\text{NH:}\quad m_{\beta\beta}$ $\displaystyle=$
$\displaystyle\Big{|}m_{1}c_{12}^{2}c_{13}^{2}+\sqrt{m_{1}^{2}+\Delta
m_{21}^{2}}s_{12}^{2}c_{13}^{2}e^{i\alpha_{21}}+\sqrt{m_{1}^{2}+\Delta
m_{31}^{2}}s_{13}^{2}e^{i(\alpha_{31}-2\delta_{\rm CP})}\Big{|},\quad$ (37)
$\displaystyle\text{IH:}\quad m_{\beta\beta}$ $\displaystyle=$
$\displaystyle\Big{|}\sqrt{m_{3}^{2}+\Delta
m_{31}^{2}}c_{12}^{2}c_{13}^{2}+\sqrt{m_{3}^{2}+\Delta m_{31}^{2}+\Delta
m_{21}^{2}}s_{12}^{2}c_{13}^{2}e^{i\alpha_{21}}+m_{3}s_{13}^{2}e^{i(\alpha_{31}-2\delta_{\rm
CP})}\Big{|}.$ (38)

Now, for a better understanding of the behaviour of the parameters involved
and the model predictions, we can divide our numerical analysis into some
special cases by taking some particular values of the relative phases
$\phi_{AC}$ and $\phi_{BC}$. All mixing angles in Eq. (24), (25) and neutrino
mass eigenvalues in Eq. (28) - Eq. (30) and the Majorana phase $\alpha_{31}$
depend on one relative phase, namely, $\phi_{AC}$. In contrast, the other
Majorana phase $\alpha_{21}$ as well as $m_{\beta\beta}$ depend on both
$\phi_{AC}$ and $\phi_{BC}$. In the following, we choose five simple special
cases depending on the values of these phases, namely: $(i)$ Case I :
$\phi_{AC}=0$, $\phi_{BC}=0$ , $(ii)$ Case II $\phi_{AC}=0$, $(iii)$ Case III
$\phi_{AC}=\phi_{BC}$, $(iv)$ Case IV $\phi_{BC}=0$ and subsequently in $(v)$
Case V we present the general scenario where both $\phi_{AC}$ and $\phi_{BC}$
vary between $0-2\pi$. Below, we have explored these cases and as we proceed
it will be clear that some of these cases have potential to distinguish the
normal and inverted hierarchy of light neutrino masses and produce interesting
predictions on neutrino parameters.

###  V.1 Case I: ${\phi_{AC}=\phi_{BC}=0}$

Here we make the simplest choice for the relative phases, $i.e.,$
$\phi_{AC}=\phi_{BC}=0$. With this value, the Eq. (23) and (24) have the
simple form

$\displaystyle\tan
2\theta=\frac{\sqrt{3}}{1+2\alpha},\quad\sin\theta_{13}=\sqrt{\frac{2}{3}}|\sin\theta|.$
(39)

with $\tan\delta_{\rm CP}=0$.

Figure 2: Left Panel: Plot for $\sin\theta_{13}$ vs $\alpha$. Horizontal grid
lines are the $3\sigma$ allowed range for $\sin\theta_{13}$ (for NH and IH)
whereas the vertical grid lines represents corresponding allowed range for
$\alpha$ ( $1.6557-1.8391$ and $1.6465-1.8754$ for NH and IH respectively).
Right Panel: Contour plot for $r=0.03$ in $\alpha-\beta$ plane. The vertical
grid lines represent the allowed regions for $\alpha$ from the left panel and
the horizontal grid lines are the corresponding allowed regions for $\beta$.

Clearly, $\sin\theta_{13}$ only depends on $\alpha$ and in Fig. (2) left
panel, we have plotted $\sin\theta_{13}$ as a function of $\alpha$ using Eq.
(39). The $3\sigma$ allowed range for $\sin\theta_{13}$ (given by the area
between horizontal lines) restricts $\alpha$ within $1.6465-1.8754$
($1.6557-1.8391$) for IH (NH) given by the dashed (continuous) vertical lines.
With $\phi_{AC}=\phi_{BC}=0$ the real positive mass eigenvalues given in Eq.
(28) - Eq. (30) can be expressed as

$\displaystyle m_{1}$ $\displaystyle=$
$\displaystyle|C|(1-\alpha-\sqrt{1+\alpha+\alpha^{2}}),$ (40) $\displaystyle
m_{2}$ $\displaystyle=$ $\displaystyle|C|3\beta,$ (41) $\displaystyle m_{3}$
$\displaystyle=$ $\displaystyle|C|(1-\alpha+\sqrt{1+\alpha+\alpha^{2}}).$ (42)

With the above mass eigenvalues, one can write the ratio of solar to
atmospheric mass-squared differences as defined in Eq. (36) as

$\displaystyle
r=\pm\frac{9\beta^{2}-(1-\alpha-\sqrt{1+\alpha+\alpha^{2}})^{2}}{(1-\alpha+\sqrt{1+\alpha+\alpha^{2}})^{2}-(1-\alpha-\sqrt{1+\alpha+\alpha^{2}})^{2}},$
(43)

where the $\pm$ signs are for NH and IH respectively. When
${\phi_{AC}=\phi_{BC}=0}$, as a consequence of the considered discrete flavor
symmetry, NH of light neutrino masses can not be realised with Eq. (40) - Eq.
(43).

Figure 3: Left Panel: Absolute neutrino masses $m_{3}$ (orange line), $m_{2}$
(blue dashed line), $m_{3}$ (red line) and their sum $\sum m_{i}$ (green line)
plotted against $\alpha$. Right Panel : The prediction on effective mass
parameter $m_{\beta\beta}$. In both cases we have considered
$\phi_{AC}=\phi_{BC}=0$.

Hence only IH of light neutrino masses is allowed. From Eq. (43), we notice
that $r$ depends on both $\alpha$ and $\beta$ and in the right panel of Fig.
2, we have plotted this dependence in this $\alpha-\beta$ plane for the best
fit value of the ratio $r~{}(=0.03)$ deSalas:2020pgw  only for IH. For the
3$\sigma$ allowed range for $\alpha$ (fitting $\sin\theta_{13}$), obtained
from the left panel of Fig. 2, $\beta$ found to be in the range $0.997-1.14$
for IH. The contour plot for $r$ yields a one to one correspondence between
$\alpha$ and $\beta$ as evident form the right panel of Fig. 2. For example,
the best fit value of $\sin\theta_{13}$ and $r$ fixes
$\alpha:1.752,\beta:1.067$ for IH. With the known sets of $(\alpha,\beta)$
corresponding to 3$\sigma$ range of $\sin\theta_{13}$, we can calculate $|C|$
by fitting the best fit value of solar mass-squared difference (given in Table
2) as $|C|=(2.04-1.68)\times 10^{-2}$ eV. Then, for the allowed sets of
$(\alpha,\beta,|C|)$, we can estimate the absolute neutrino masses $m_{1,2,3}$
and their sum $\sum m_{i}$. In Fig. 3, we have plotted the individual neutrino
masses $m_{1,2,3}$ (by blue dashed, red and orange lines respectively) and
their sum $\sum m_{i}$ (green line) against $\alpha$ as obtained from left
panel of Fig. 2. Now for IH, cosmology sets an upper limit on the sum of the
masses of three light neutrinos as $\sum m_{i}\leq 0.15$ eV deSalas:2020pgw
as given by the horizontal cyan shaded region in the left panel of Fig. 3.
Thus all $(\alpha,\beta)$ found from the the right panel of Fig. 2 are not
allowed. The vertical dashed line in the left panel of Fig. 3 thus represents
a further restriction on $\alpha$ (and hence on $\beta$) and the lower bound
on $\alpha$ shifted from 1.6567 to 1.74. As a result the corresponding values
of $\beta$ and $|C|$ will also be shifted. The final allowed values of the
model parameters are summarized in Table. 3.

Parameters | Allowed ranges
---|---
$\alpha$ | 1.74-1.875
$\beta$ | 1.059-1.147
$|C|$ (eV) | (1.87-1.68)$\times 10^{-2}$
$\sum{m_{i}}$ (eV) | 0.1496-0.1408
$m_{\beta\beta}$ (eV) | 0.0568-0.0585
Table 3: The allowed ranges for $\alpha$, $\beta$, $|C|$, $\sum m_{i}$ and
$m_{\beta\beta}$ when $\phi_{AC}=\phi_{BC}=0$.

Substituting $\phi_{AC}=0$ in Eq. (33) we obtain $\phi_{1}=\phi_{3}=0$ and
with $\phi_{BC}=0$ we get $\phi_{2}=0$. Altogether substituting these in Eq.
(34) and Eq. (35), the Majorana phases are found to be zero. Thus for
vanishing values of the relative phase $\phi_{AC}=0$ and $\phi_{BC}=0$, the
Dirac and Majorana phases also vanishes . With these values of the phases and
known sets of $(\alpha,\beta,|C|)$ we can now finally estimate the effective
mass parameter $m_{\beta\beta}$ appearing in the neutrinoless double beta
decay. In the right panel of Fig. 3, we have plotted the prediction for
$m_{\beta\beta}$ as a function of $\alpha$ with $\phi_{AC}=\phi_{BC}=0$. The
finding for the sum of the absolute neutrino masses and the effective mass
parameter as obtained from Fig. 3 are also summarized in Table 3.

###  V.2 Case II : $\phi_{AC}=0$

In the second case, we consider $\phi_{AC}=0$ but do not make any choice of
$\phi_{BC}$. We already know that neutrino mixing angles and neutrino masses
both depend on the phase $\phi_{AC}$ alone which can be understood from the
general expressions of Eq. (23) - Eq. (25) and Eq. (28) - Eq. (30). The
Majorana phases $\alpha_{21}$ and hence neutrinoless double beta decay
effective mass parameter $m_{\beta\beta}$ depend on both $\phi_{AC}$ and
$\phi_{BC}$. Now, with the choice of $\phi_{AC}=0$, the simplified expressions
for $\theta$ and $\sin\theta_{13}$ are same as in Eq. (39). The real positive
mass eigenvalues and the ratio of the mass-squared differences $r$ will take
the form as in Eqs. (40) - (43) respectively. Hence, with these expressions,
the limits on $\alpha$, $\beta$, $|C|$, $m_{i}$ and $\Sigma m_{i}$ will same
as in Case I ($\phi_{AC}=\phi_{BC}=0$) which are summarized in Table 3. Thus
in this case too only IH of light neutrino masses is allowed. The only
difference between Case I and Case II is that in the later case $\phi_{BC}$ is
free. As $\phi_{BC}$ is only appearing in Majorana phases, which can
understood just by looking at the Eq. (33), the prediction on Majorana phases
will be different from the previous case and hence prediction on
$m_{\beta\beta}$ will change accordingly. For $\phi_{AC}=0$, $\phi_{1}$ and
$\phi_{3}$ are zero (obtained from Eq. (33)) as mentioned in Case I. Hence,
one of the Majorana phase $\alpha_{31}$ vanishes but due to non-zero
$\phi_{2}$, the other Majorana phase ($\alpha_{21}$) found to be
$\alpha_{21}=-\phi_{BC}$. Hence, we will calculate the neutrinoless double
beta decay effective mass parameter $m_{\beta\beta}$ using Eq. (37) by
choosing different values for $\phi_{BC}$.

$\phi_{BC}$ | ${\pi}/{6}$ | ${\pi}/{3}$ | ${\pi}/{2}$ | ${2\pi}/{3}$ | ${5\pi}/{6}$ | $\pi$
---|---|---|---|---|---|---
$m_{\beta\beta}$ (eV) | 0.057-0.055 | 0.051-0.050 | 0.043-0.042 | 0.034-0.033 | 0.024-0.023 | 0.019-0.018
Table 4: Prediction on $m_{\beta\beta}$ depending on different values of
$\phi_{BC}$ in the Case II with $\phi_{AC}=0$.

In Table 4 where we have summarized the ranges of $m_{\beta\beta}$ for
different values of $\phi_{BC}$ such as $\phi_{BC}={\pi}/{6}$, ${\pi}/{3}$,
${\pi}/{2}$, ${2\pi}/{3}$, ${5\pi}/{6}$ and $\pi$. Prediction on
$m_{\beta\beta}$ in this case for $\phi_{BC}=0-\pi$ recurs the same value
again for $\phi_{BC}=\pi-2\pi$. From Case I and Case II, one can say that only
IH is allowed with $\phi_{AC}=0$ (with $\phi_{BC}=0$ or arbitrary).

###  V.3 Case III: $\phi_{AC}=\phi_{BC}=\phi_{x}$

In this case we consider the scenario when the relative phases $\phi_{AC}$ and
$\phi_{BC}$ both are equal, say, $\phi_{AC}=\phi_{BC}=\phi_{x}$. Hence, the
general expressions for the rotation angle $\theta$ and associated phase
$\phi$ appearing in the unitary rotation matrix $U_{13}$ as given in Eq. (23)
can be rewritten as

$\displaystyle\tan\phi=\frac{\alpha\sin\phi_{x}}{1-\alpha\cos\phi_{x}},\quad\tan
2\theta=\frac{\sqrt{3}}{\cos\phi+2\alpha\cos(\phi_{x}+\phi)}.$ (44)

Thus we can substitute $\phi$ in the second equation above to evaluate
$\sin\theta_{13}$ using Eq. (24). Furthermore as $\tan\delta_{\rm
CP}=\tan\phi$, we find that a particular value of $\delta_{\rm CP}$,
$\sin\theta_{13}$ both depends on $\phi_{x}$ and $\alpha$. In Fig. 4 we
provide contour plots for $\sin\theta_{13}=0.148$ and $\delta_{\rm CP}=0.541$
( or $31^{\circ}$) denoted by purple and blue dotted lines respectively. The
intersection between $\sin\theta_{13}$ and $\delta_{\rm CP}$ contours indicate
the simultaneous satisfaction of them and Fig. 4 it is indicated by a black
dot with which a pair of $\cos\phi_{x}$ and $\alpha$ are attached. Similar
intersections of the green dot dashed lines ($\delta_{\rm CP}\leq 0.541$) with
the purple line represents other pairs of $\cos\phi_{x}$ and $\alpha$.

Figure 4: The contour plot for $\delta_{\rm CP}=0.78$ (blue dots) and best fit
of $\sin\theta_{13}$ (red) for NH of neutrino masses. The region between red
dots represent the $3\sigma$ region of $\sin\theta_{13}$ value of neutrino
oscillation data for NH of neutrino masses. The intersection point represents
the value of $\alpha$ and $\cos\phi_{x}$.

In this case the mass eigenvalues $m_{1}$, $m_{2}$ and $m_{3}$ can be written
as

$\displaystyle m_{1}$ $\displaystyle=$
$\displaystyle|C|\big{[}(1-\alpha\cos\phi_{x}-P)^{2}+(Q+\alpha\sin\phi_{x})^{2}\big{]}^{1/2},$
(45) $\displaystyle m_{2}$ $\displaystyle=$ $\displaystyle|C|3\beta,$ (46)
$\displaystyle m_{3}$ $\displaystyle=$
$\displaystyle|C|\big{[}(1-\alpha\cos\phi_{x}+P)^{2}+(Q-\alpha\sin\phi_{x})^{2}\big{]}^{1/2},$
(47)

and using these equations we can also evaluate the ratio of the mass-squared
differences $r$ using Eq. (36). One can then compute $\beta$ with $r=0.03$ and
the corresponding value for $|C|$ for each pair of $\alpha$ and $\cos\phi_{x}$
obtained from intersecting points in Fig. 4. With this flavor structure of the
mass eigenvalues given in Eq. (45) - Eq. (47), inverted hierarchy of neutrino
mass is not possible with $\phi_{AC}=\phi_{BC}$. For each set of
$\alpha,\beta,\phi_{x}$ and $|C|$ estimated above, we can predict the sum of
the absolute neutrino mass $\sum m_{i}$ and in Table 5 we present few such
representative values. From Eq. (33) - Eq. (35), we find that the Majorana
phases $\alpha_{21}$ and $\alpha_{31}$ are defined as
$\alpha_{21}=\phi_{1}-\phi_{2}$ and $\alpha_{31}=\phi_{1}-\phi_{3}$ where
$\phi_{1}$, $\phi_{2}$ and $\phi_{3}$ in the present case can be written as

$\displaystyle\phi_{1}=\tan^{-1}\Big{(}\frac{Q+\alpha\sin\phi_{x}}{1-\alpha\cos\phi_{x}-P}\Big{)},\quad\phi_{2}=\phi_{x},\quad\phi_{3}=\tan^{-1}\Big{(}\frac{Q-\alpha\sin\phi_{x}}{1-\alpha\cos\phi_{x}+P}\Big{)}.$
(48)

Substituting $\alpha$ and $\cos\phi_{x}$ obtained from Fig. 4 in the above
equations we can compute the Majorana phases and estimate the effective mass
parameter appearing in the neutrinoless double beta decay as defined in Eq.
(37).

$\delta_{\rm CP}$ | $\alpha$ | $\cos\phi_{x}$ | $\beta$ |  $|C|$ (eV) |  $\sum m_{i}$ (eV) |  $m_{\beta\beta}$ (eV) |  $m_{\beta\beta}\big{|}_{\phi_{BC}=0}$ (eV)
---|---|---|---|---|---|---|---
0.087 | 2.765 | -0.9848 | 0.5721 | 0.0084 | 0.0773 | 0.00589 | 0.0128
0.174 | 2.765 | -0.9659 | 0.5895 | 0.00858 | 0.0792 | 0.0077 | 0.0148
0.262 | 2.765 | -0.9327 | 0.6356 | 0.008956 | 0.0804 | 0.0100 | 0.0141
0.349 | 2.759 | -0.8853 | 0.7089 | 0.0096 | 0.0922 | 0.0144 | 0.0108
0.436 | 2.754 | -0.8332 | 0.7849 | 0.0104 | 0.1026 | 0.0188 | 0.0188
0.523 | 2.754 | -0.7669 | 0.872 | 0.0117 | 0.1181 | 0.0252 | 0.0259
0.541 | 2.754 | -0.748 | 0.894 | 0.0122 | 0.123 | 0.0257 | 0.0251
Table 5: The allowed ranges of parameters satisfying neutrino data for various
values of $\delta_{\rm CP}$ with $\phi_{AC}=\phi_{BC}$. The last column
represents prediction for $m_{\beta\beta}$ for $\phi_{AC}=0\div 2\pi$ and
$\phi_{BC}=0$.

These predictions for $m_{\beta\beta}$ are listed in the last column of Table
5. From the results obtained form Fig. 4 and Table 5, we find that all values
for $\delta_{\rm CP}$ are not compatible with the cosmological bound on the
sum of neutrino masses $\sum m_{i}$ (satisfying correct neutrino oscillation
data). The region $\delta_{\rm CP}\leq 0.54~{}(31^{\circ})$ given by the cyan
shaded region is disallowed due the cosmological upper bound $\sum m_{i}\leq
0.12$ eV for NH deSalas:2020pgw . Thus the green dot dashed region in Fig. 4
with $0<\delta_{\rm CP}\leq\pi/6$ satisfy both neutrino oscillation data and
the cosmological bound on the sum of neutrino masses. In Table 5 we have
summarized constraints on $\alpha$, $\beta$, $\cos\phi_{x}$ and $|C|$ for a
specific value of $\delta_{\rm CP}$ as well as predictions on $\sum m_{i}$ and
$m_{\beta\beta}$. Here we also find that predictions for the parameters given
in Table 5 repeats again for $\pi<\delta_{\rm CP}\leq 7\pi/6$. From the
results summarized in the Table 5 one can find that with increase of ($\beta$,
$\cos\phi_{x}$, $|C|$), $\alpha$ decreases whereas both $\sum m_{i}$,
$m_{\beta\beta}$ increases with the increase of $\delta_{\rm CP}$.

###  V.4 Case IV: $\phi_{BC}=0$

In this case we consider $\phi_{BC}=0$ while $\phi_{AC}$ is varied arbitrarily
between $0-2\pi$. This scenario is very similar to the previous case. with
$\phi_{AC}=\phi_{BC}=\phi_{x}$. From Eq. (33)$-$(35) we find that
$\phi_{BC}=0$ only effects the Majorana phases and hence the effective mass
parameter appearing in the neutrinoless double beta decay. The conclusion
drawn in Case III for absolute values of light neutrino masses and their
hierarchy will be identical. For simplicity and resembles with previous case
we consider $\phi_{AC}=\phi_{x}$. With this the expressions for rotation angle
$\theta$ and the phase $\phi$ appearing in $U_{13}$ is already given in Eq.
(44). Using Eq. (24) we also find that $\delta_{\rm CP}$ and $\sin\theta_{13}$
are both function of $\alpha$ and $\phi_{x}$. For particular value of
$\delta_{\rm CP}$ we can again compute the pairs of $\alpha$ and $\phi_{x}$
from the intersections of contour plots for the chosen value of $\delta_{\rm
CP}$ and best-fit value of $\sin\theta_{13}$ as given in Fig. 4. Similar to
Case III, by fitting $r=0.03$, we can get $\beta$ (and subsequently $|C|$),
where the expression of $r$ in Eq. (36) involves neutrino masses those are
given in Eq. (45) - Eq. (47). Therefore the results of Table 5 will be same
for this case up-to the sixth coloumn for $\sum m_{i}$. The change will occur
in the seventh column of Table 5 as here in Case IV, we have $\phi_{BC}=0$.
Although, with this choice, the expression for $\phi_{1}$ and $\phi_{3}$ will
be same as Eq. (48) but unlike previous case we have $\phi_{2}=0$ here. The
two Majorana phases follow the relation $\alpha_{21}=\phi_{1}-\phi_{2}$,
$\alpha_{31}=\phi_{1}-\phi_{3}$ hence $\alpha_{31}$ will be identical and
$\alpha_{21}$ will be different from Case III. As a result, $m_{\beta\beta}$
will be different compared to the previous scenario. Hence in last column of
Table 5 we append the prediction of $m_{\beta\beta}$ for few allowed values
for $\delta_{\rm CP}$. Note that here also the allowed ranges of the Dirac CP
phase are $0<\delta_{\rm CP}\leq\pi/6$ and $\pi<\delta_{\rm CP}\leq 7\pi/6$
respectively which satisfy both neutrino oscillation data and the cosmological
bound on the sum of absolute neutrino masses. A point to remember is that this
case only satisfy NH of neutrino mass and IH is not allowed.

Figure 5: The allowed regions for $\alpha$ and $\beta$ for the $3\sigma$
ranges for $\sin^{2}\theta_{13}$ and $r$ for NH (left panel, light red shaded
region) and IH (right panel, blue shaded region) of neutrino masses where
$\phi_{AC}$ and $\phi_{BC}$ are varied from $0$ to $2\pi$.

###  V.5 General case

In the above cases, we have analyzed neutrino mixing for various limiting
values for the relative phases associated in our study. Now, we will carry out
a full numerical analysis for the most general case where we vary both
$\phi_{AC}$ and $\phi_{BC}$ to their entire range from $0$ to $2\pi$. Then,
Eq. (23) - Eq. (25) will be used for the calculation of mixing angles. On the
other hand, the ratio of the mass-squared differences $r$ defined in Eq. (36)
can also be calculated using the general expression for the mass eigenvalues
of Eq. (28) - Eq. (30).

Figure 6: The allowed ranges for $\phi_{AC}$ and $\phi_{BC}$ for the correct
value of $\sin^{2}\theta_{13}$ and $r$ along with for both NH (left panel) and
IH (right panel) of neutrino masses.

As explained earlier, in order to evaluate the absolute neutrino masses, we
also need to evaluate the the common factor $|C|$ associated with each mass
eigenvalues. Here we obtain $|C|$ by fitting the solar mass-squared difference
taken from deSalas:2020pgw . In our analysis for this most general case, we
have also included the bound coming from cosmological observations on the sum
of absolute neutrino masses as $\sum m_{i}<0.12$ eV for NH and $\sum
m_{i}<0.15$ eV for IH deSalas:2020pgw . Using the $3\sigma$ allowed range for
the neutrino oscillation data deSalas:2020pgw  given in Table 2, we vary both
$\phi_{AC}$ and $\phi_{BC}$ between 0 to $2\pi$. In Fig. 5 we have plotted the
allowed region in $\alpha-\beta$ plane for NH (left panel, light red shaded
region) and IH (right panel, blue shaded region) respectively. Here we find
the for NH (IH), the allowed ranges for $\alpha$ varies between
$2.33\leq\alpha\leq 2.9$ ($1.65\leq\alpha\leq 2.8$). On the other hand,
allowed ranges for $\beta$ is restricted by $0.3\leq\beta\leq 1$ for NH
whereas for IH we have $1.9\geq\beta\geq 1$. This implies $\beta$ values less
than 1 is favoured for NH whereas values greater than 1 is favoured for IH.
Similarly, in Fig. 6 we have plotted the the allowed region in
$\phi_{AC}-\phi_{BC}$ plane for NH (left panel, light red shaded region) an IH
(right panel, blue shaded region) respectively. Here $\phi_{BC}$ between
$0-2\pi$ is compatible for both of the hierarchies, however, two distinct
regions for $\phi_{AC}$ are allowed, namely, $2.61\leq\phi_{AC}\leq 5.38$ for
NH and $\phi_{AC}\leq 2$ $\&$ $6\leq\phi_{AC}\leq 2\pi$ for IH respectively.
Clearly the full range of $\phi_{BC}$ is allowed because it is not sensitive
to low energy masses, mixing and only appears in one of the Majorana phase
$\alpha_{21}$. Thus the values of $\phi_{AC}$ crucially dictates the hierarchy
of light neutrino masses. As an artifact of the considered flavor symmetry,
combining the results from Fig. 5 and 6, we can conclude that with $\beta\leq
1$ and $2.61\leq\phi_{AC}\leq 5.38$ one can reproduce NH whereas to obtain IH
we need $\beta\geq 1$ and $0\leq\phi_{AC}\leq 2$ (or $6\leq\phi_{AC}\leq
2\pi$).

Figure 7:  $\sin^{2}\theta_{23}-\delta_{\rm CP}$ correlation for NH (left
panel) and IH (right panel). The cyan patch represents the disallowed region
coming from the constraints on light neutrino masses.

Now with the allowed values for $\alpha,\beta,\phi_{AC}$ and $\phi_{BC}$
obtained from Fig. 5 and 6, we are now equipped to study the correlation
between neutrino mixing parameters and predictions associated with the phases
and masses. Due to the presence of the $A_{4}$ discrete flavor symmetry, this
model yields interesting correlation among the observables appearing in
neutrino mixing. Following Eq. (23) - Eq. (25), we find one such important
correlation between the atmospheric mixing angle $\theta_{23}$ and Dirac CP
phase $\delta_{\rm CP}$. This correlation is very crucial because still there
are some unsettled issues with the measurement of these two oscillation
parameter such as $(a)$ octant of $\theta_{23}$ , i.e.,
$\theta_{23}<45^{\circ}$ (lower octant, LO) or $\theta_{23}>45^{\circ}$
(higher octant, HO) and $(b)$ magnitude of Dirac CP phase $\delta_{\rm CP}$.
The $\delta_{\rm CP}-\theta_{23}$ correlation obtained here is plotted in Fig.
7 and given by light red (blue) shaded region for NH (IH) in the left (right)
panel and shades some light on the above mentioned unsettled issues. It is
evident from Fig. 7 that for both hierarchies only higher octant of
$\theta_{23}$ is favoured ($i.e.~{}\theta_{23}\geq 45^{\circ}$) in our
analysis. Furthermore, the cyan patch in both of the panels represents the
disallowed region for $\delta_{\rm CP}$ in order to satisfy the limits on
light neutrino masses deSalas:2020pgw . From Fig. 7 the allowed regions for
Dirac CP phase $\delta_{\rm CP}$ are given by $-1.57\leq\delta_{\rm CP}\leq
1.37$ and $1.4\leq\delta_{\rm CP}\leq 1.57$ for NH, whereas for IH the
predictions are $-1.57\leq\delta_{\rm CP}\leq 0.5$ and $0.86\leq\delta_{\rm
CP}\leq 1.57$. The disallowed region for $\delta_{\rm CP}$ is small in IH of
neutrino masses compared to the NH as cosmology puts a tighter constraint on
NH compared to IH deSalas:2020pgw .

Figure 8:  Sum of absolute neutrino masses $\sum m_{i}$ plotted against
lightest neutrino mass for both NH (left panel, light red shaded region) and
IH (right panel, blue shaded region). The area between the dashed lines
represents $3\sigma$ allowed range and the cyan patch represents the area
excluded by cosmology.

Again, with the permitted values of $\alpha$, $\beta$, $\phi_{AC}$ and
$\phi_{BC}$, we have the predictions on light neutrino masses which can be
understood from the correlation plot of $\sum m_{i}$ vs lightest neutrino
($m_{1}$ for NH and $m_{3}$ for IH) mass for both hierarchies as given in Fig.
8. Here also the allowed regions are given by light red (blue) shaded region
for NH (IH) in the left (right) panel. The horizontal cyan region in each
plots represents disallowed regions mentioned earlier in this subsection.
Clearly, this framework predicts that the lightest neutrino mass can take
smaller values for NH ($m_{\rm lightest}\geq 0.0012$ eV), compared to the IH
scenario ($m_{\rm lightest}\geq 0.014$ eV). The Majorana phases in our
analysis, for the most general case, can be evaluated using the expressions
given in Eq. (34) and (35) with the allowed regions of $\alpha$, $\phi_{AC}$,
$\phi_{BC}$ given in Fig. 5 and 6. Thus we can constrain the Majorana phases
using the low energy neutrino oscillation data. In Fig. 9, present a
correlation plot in the $\alpha_{21}-\alpha_{31}$ plane for NH (left panel,
light red shaded region) and IH (right panel, blue shaded region) respectively
with $3\sigma$ allowed ranges of neutrino oscillation data deSalas:2020pgw .

Figure 9: The correlation plot between two Majorana phases $\alpha_{21}$ and
$\alpha_{31}$ for NH (left panel) and IH (right panel) for $3\sigma$ allowed
range of neutrino oscillation data deSalas:2020pgw . Figure 10: Neutrinoless
double beta decay effective mass parameter $m_{\beta\beta}$ is plotted as a
function of lightest neutrino mass for the case of both NH (light red shaded
region) and IH (blue shaded region). Green and magenta regions are three
neutrino allowed regions when all the parameters are varied within their
3$\sigma$ range. The gray shaded region, brown dashed and black dotted-dash
lines stand for experimental limits by KamLAND-Zen+GERDA, LEGEND-1k and nEXO
respectively. The vertical cyan area represents disallowed region by
cosmology.

Finally, with the estimation for neutrino masses and phases in hand we are now
able to plot the effective mass parameter characterizing neutrinoless double
beta decay ($m_{\beta\beta}$) given in Eq. (37) and Eq. (38). In Fig. 10, we
have plotted $m_{\beta\beta}$ against lightest neutrino mass for both NH
($m_{1}$) and IH ($m_{3}$) respectively by light red and blue shaded regions
respectively. The predictions for $m_{\beta\beta}$ are $1-30$ meV for NH and
16-60 meV for IH. Here the green and magenta shaded regions represents
$3\sigma$ allowed regions for the $m_{\beta\beta}$ predictions for NH and IH
respectively. The vertical cyan shaded regions represent the cosmological
upper-limit on the sum of absolute neutrino masses ($\sum m_{i}$). The gray
shaded region represent the upper limit for $m_{\beta\beta}$ by combined
analysis of KamLAND-Zen KamLAND-Zen:2016pfg  and GERDA GERDA:2018pmc
experiments and predictions for $m_{\beta\beta}$ in our model fall within this
upper limit.

Cases |  NH |  IH | $\delta_{\rm CP}$ | $\alpha_{21}$ | $\alpha_{31}$ |  $\sum m_{i}$(eV) |  $m_{\beta\beta}$(eV)
---|---|---|---|---|---|---|---
Case I | ✗ | ✓ | 0,$\pi$ | 0 | 0 | (0.1408, 0.1496) | (0.057,0.059)
Case II | ✗ | ✓ | 0,$\pi$ | (0,2$\pi$) | 0 | (0.1408-0.1496) | (0.019-0.054)
Case III | ✓ | ✗ | (0-0.78) | (-2.362,-1.26) | (0.61,1.20) | (0.0773,0.12) | (0.0059,0.026)
(3.14,3.66)
Case IV | ✓ | ✗ | (0,0.78) | (0.60,1.84) | (0.61,1.20) | (0.0773,0.12) | (0.013,0.025)
(3.14,3.66)
General Case | ✓ | ✗ | (-1.5,0.6) | (-3,3) | (-2.85,1.13) | (0.06,0.12) | (0.001,0.03)
(1.38,1.57)
General Case | ✗ | ✓ | (-1.5,0.6) | (-3,3) | (-1.5,-0.01) | (0.115,0.15) | (0.016,0.06)
(0.8,1.56) | (1.33,1.7)
Table 6: Summary of the different cases of our numerical analysis. The ✓and ✗
symbols stand for allowed and disallowed regimes. The numbers in the
parenthesis represent the allowed ranges in each scenario.

In this plot the brown dashed and black dotted-dash lines stand for future
sensitivities of the LEGEND-1k LEGEND:2021bnm  and nEXO nEXO:2021ujk
experiments respectively. Thus these near future experiments have potential to
almost entirely falsify the IH prediction and probe a major part prediction
for $m_{\beta\beta}$ for NH of light neutrino mass. Guided by the symmetry
construction, the model also sets a lower limit on the effective mass
parameter as $m_{\beta\beta}\geq 1$ meV for NH.

To sum up the full numerical analysis, we present Table 6, where we give a
summary of all the results including both special cases and general cases. In
our analysis, we divided our special case into four categorise depending on
the values of the input relative phases $\phi_{AC}$ and $\phi_{BC}$. In Case
I, where $\phi_{AC}=\phi_{BC}=0$, only IH of neutrino masses are allowed and
$\delta_{CP}$ = $0$ or $\pi$, and two Majorana phases are coming out to be
zero. In case II, where $\phi_{AC}=0$ but $\phi_{BC}$, IH is predicted and the
main difference with occurs in the prediction of Majorana phases. As a result,
the prediction on $m_{\beta\beta}$ is different from Case I. In both Case III
and Case IV where $\phi_{AC}=\phi_{BC}\neq 0$ and $\phi_{AC}\neq
0,\phi_{BC}=0$, respectively, NH of neutrino masses are predicted. The allowed
regions of $\delta_{\rm CP}$ are given in Table 6 which are same for both
these cases. The prediction on the Majorana phase $\alpha_{31}$ is same for
both of these cases whereas the prediction on the $\alpha_{21}$ and hence
$m_{\beta\beta}$ are different in both cases. Finally, as a most general case
study, in Case V we vary $\phi_{AC}$ and $\phi_{BC}$ arbitrarily within its
full range. The analysis constrains $\delta_{\rm CP}$ into two particular
regions. We also find the values of the parameters $\beta$ and the phase
$\phi_{AC}$ plays a crucial role in determining the neutrino mass hierarchy
with distinct limits on neutrino masses for each hierarchies.

##  VI Phenomenological implications for the FSS model

Owing to the flavor symmetry of the model, charged lepton sector’s Yukawa
couplings are diagonal so the flavors are flavors are conserved. But there are
sources of the lepton flavor violation arising outside the charged lepton
sector from both the Yukawa couplings $y_{N}$ and $y_{s}$ associated with the
seesaw and scotogenic contributions, respectively. These Yukawa interactions
lead to lepton flavor violating processes such as $\mu\rightarrow e\gamma$,
$\mu\rightarrow 3e$ etc. For related studies on lepton flavor violation in a
pure scotogenic model, can be see Toma:2013zsa ; Vicente:2014wga . In the
minimal scoto-seesaw model Rojas:2018wym , it was shown that the scotogenic
part dominantly contribute in the lepton flavor violating decays compared to
the seesaw contribution. Due to the proposed symmetry configuration, our
analysis significantly differs from this observation. In our framework, the
branching ratios of $\mu\rightarrow e\gamma$ and $\mu\rightarrow 3e$ for
scotogenic contribution can be respectively written as Rojas:2018wym

$\displaystyle{\rm Br}(\mu\rightarrow e\gamma)$ $\displaystyle\approx$
$\displaystyle\frac{\alpha
m_{\mu}}{(8\pi)^{4}\Gamma_{t}}\Big{(}\frac{m_{\mu}}{m_{\eta}}\Big{)}^{4}(Y_{f}^{1}Y_{f}^{2}\epsilon^{4})^{2}\Bigg{(}F\Bigg{(}\frac{M_{f}^{2}}{m_{\eta}}\Bigg{)}\Bigg{)}^{2},$
(49) $\displaystyle{\rm Br}(\mu\rightarrow 3e)$ $\displaystyle\approx$
$\displaystyle\frac{\alpha^{2}m_{\mu}}{(8\pi)^{5}\Gamma_{t}}\Big{(}\frac{m_{\mu}}{m_{\eta}}\Big{)}^{4}(Y_{f}^{1}Y_{f}^{2}\epsilon^{4})^{2}G\Big{(}\frac{m_{\mu}}{m_{e}}\Big{)}\Bigg{(}F\Bigg{(}\frac{M_{f}^{2}}{m_{\eta}}\Bigg{)}\Bigg{)}^{2}.$
(50)

Here $\Gamma_{t}$ is the total decay width of the muon,
$\alpha={e^{2}}/{4\pi}$ is the fine structure constant and
$\epsilon=v_{f}/\Lambda$ where we have considered the flavon VEVs to be equal,
$i.e$, $v_{\xi}=v_{s,a}=v_{f}$ for simplicity. The expression for the
functions $F$ and $G$ are given by

$\displaystyle F(x)$ $\displaystyle=$
$\displaystyle\frac{2-9x+18x^{2}-11x^{3}+6x^{3}{\rm log}(x)}{6(1-x)^{4}},$
(51) $\displaystyle G(x)$ $\displaystyle=$ $\displaystyle\frac{16}{3}{\rm
log}(x)-\frac{22}{3}.$ (52)

In our discussion, the considered discrete symmetries dictates the structure
of the associated Yukawa couplings. Due to the specific VEV alignment of the
$A_{4}$ triplet flavon $\phi_{s}$ and its contraction with the non-trivial
$A_{4}$ singlet $\xi$ (and following the multiplication rules given in the
appendix), we find $Y_{f}^{2}=0$ as given in Eq. (12). Therefore owing to the
$A_{4}$ symmetry, the scotogenic part alone yields a vanishing contribution in
the lepton flavor violating decays in our construction.

Figure 11: Left Panel : $\beta^{2}|C|^{2}$ vs $|C|$ for 3$\sigma$ allowed
regions of neutrino oscillation data obtained form Fig. 5 and Fig. 6. Right
Panel: Branching ratio of $\mu\rightarrow e\gamma$ is plotted against the mass
of the right-handed neutrino $N_{2}$. The horizontal shaded region and dashed
line represent current and future experimental upper limits. In both panels
light red and blue shaded patch represent regions for NH and IH respectively.

Now, for the type-I seesaw contribution in the lepton flavor violating decays,
most stringent constrain comes from the $\mu\rightarrow e\gamma$ decay. The
branching ratio for such decay in our framework can be written as
Ilakovac:1994kj ; Tommasini:1995ii ; Dinh:2012bp ; Bambhaniya:2016rbb ;
Ghosh:2017fmr

$\displaystyle{\rm Br}(\mu\rightarrow e\gamma)$ $\displaystyle\approx$
$\displaystyle\frac{3\alpha
v^{4}}{8\pi}\Bigg{|}\sum_{i=1}^{2}\mathcal{K}_{ei}\mathcal{K}^{\dagger}_{i\mu}f\Bigg{(}\frac{M^{2}_{N_{i}}}{M_{W}^{2}}\Bigg{)}\Bigg{|}^{2},$
(53) $\displaystyle=$ $\displaystyle\sum_{i=1}^{2}\frac{3\alpha v^{4}}{8\pi
M_{N_{i}}^{4}}\Bigg{|}(Y_{N})_{ei}(Y_{N}^{\dagger})_{i\mu}f\Bigg{(}\frac{M^{2}_{N_{i}}}{M_{W}^{2}}\Bigg{)}\Bigg{|}^{2}$

where $\mathcal{K}=Y^{\dagger}_{N}(M^{-1}_{R})^{*}$ and $Y_{N}=M_{D}/v$ as
obtained from Eq. (7). Similar to the scotogenic contribution, the VEV
alignment of the $A_{4}$ flavon $\phi_{s}$ once again plays a crucial role in
obtaining the estimation for the branching ration for the $\mu\rightarrow
e\gamma$ decay. The VEV configuration of $\phi_{s}$ is such that it give rise
to $(Y_{N})_{e1}=0$ (see Eq. (7)) and contribution associated with $N_{1}$
essentially vanishes. Therefore only surviving contribution in the branching
ratio originates from ${N_{2}}$ and found to be

$\displaystyle{\rm Br}(\mu\rightarrow e\gamma)=\frac{3\alpha v^{4}}{8\pi
M_{N_{2}}^{4}}\Bigg{|}(y_{N_{2}}y_{N_{2}}^{*}\epsilon^{2}f\Bigg{(}\frac{M^{2}_{N_{2}}}{M_{W}^{2}}\Bigg{)}\Bigg{|}^{2}=\frac{3\alpha}{8\pi
M_{N_{2}}^{2}}\beta^{2}|C|^{2}\Bigg{(}f\Bigg{(}\frac{M^{2}_{N_{2}}}{M_{W}^{2}}\Bigg{)}\Bigg{)}^{2}.$
(54)

where we have used $B=v^{2}y^{2}_{N_{2}}\epsilon^{2}/M_{N_{2}}$ from Eq. (9)
and the definition $\beta=|B|/|C|$ to obtain Eq. (54). The loop function
$f(x)$ in Eq. (54) can be written as

$\displaystyle f(x)=\frac{x(2x^{3}+3x^{2}-6x-6x^{2}{\rm
log}(x)+1)}{2(1-x)^{4}}.$ (55)

From Eq. (54) we find that the contribution in ${\rm Br}(\mu\rightarrow
e\gamma)$ coming from seesaw mechanism depends on the mass of the heavy right-
handed neutrino $N_{2}$, $\beta$ and $|C|$. The parameters $\beta$ and $|C|$
are already fixed by neutrino data as discussed in Section V. As ${\rm
Br}(\mu\rightarrow e\gamma)\propto\beta^{2}|C|^{2}$, in Fig. 11 left panel we
have showed the variation of $\beta^{2}|C|^{2}$ with $|C|$ for the most
general case of our analysis. In this plot, the light red and blue shaded
region represent estimation for $\beta^{2}|C|^{2}$ as a function of $|C|$ for
NH and IH of light neutrino mass respectively. Furthermore, Fig. 11 left panel
also depicts that $\beta^{2}|C|^{2}$ acquires higher values for IH compared to
NH. This is because for NH $\beta$ is smaller ($\beta\leq 1$) compared to IH
($\beta\geq 1$) for similar values of $|C|$. With this when we plot ${\rm
Br}(\mu\rightarrow e\gamma)$ for seesaw contribution in our scenario as a
function of $M_{N_{2}}$ in the right panel of Fig. 11. Here the light red and
blue shaded region represent prediction for ${\rm Br}(\mu\rightarrow e\gamma)$
for NH and IH of light neutrino mass. As the $\beta^{2}|C|^{2}$ takes higher
values for IH compared to NH, the branching ratio for the $\mu\rightarrow
e\gamma$ decay is higher for IH compared to NH as seen in this figure. The
horizontal orange shaded region represent current experimental limit ${\rm
Br}(\mu\rightarrow e\gamma)\leq 4.2\times 10^{-13}$ MEG:2016leq  and the
purple dashed stands for the future reach ${\rm Br}(\mu\rightarrow
e\gamma)\leq 6\times 10^{-14}$ MEGII:2018kmf  which puts a lower limit on the
mass of $N_{2}$ in the range $0.2~{}{\rm TeV}\leq M_{N_{2}}\leq 1.6$ TeV
($1.7~{}{\rm TeV}\leq M_{N_{2}}\leq 3$ TeV) for NH (IH) of light neutrino
mass.

The scotogenic contribution in our analysis offers us the opportunity to
explain the nature of dark matter. In this model there is an remnant dark
$\mathcal{Z}_{2}$ symmetry that ensures the stability of a lightest dark
particle and there exists three feasible dark matter candidates. These are,
namely, the fermionic dark matter $f$ and scalar dark matter which are real
and imaginary components of $\eta$, given by $\eta_{R}$ and $\eta_{I}$
respectively. When the lightest dark particle originates from $\eta$, it has a
resemblance with the inert Higgs doublet model Dolle:2009fn . Considering
$\eta_{R}$ as the DM candidate, there exists several annihilation and co-
annihilation channels in this model which involve annihilation to quarks and
leptons, SM gauge bosons and the Higgs boson such as
$\eta_{R}\eta_{R}\rightarrow ZZ,~{}W^{+}W^{-},~{}hh,~{}q\bar{q}$ etc.
Collectively they all contribute in the relic abundance of $\eta_{R}$. Present
dark matter abundance often expressed in terms of the relic density parameter
$\Omega^{2}h$ and reported to be $\Omega^{2}h=0.120\pm 0.001$ at $68\%$ CL
Planck:2018vyg . In minimal scoto-seesaw framework Mandal:2021yph , it has
been argued that correct relic density can be obtained for three different
mass ranges for $\eta_{R}$. These are respectively $m_{\eta_{R}}<50$ GeV, $70$
GeV $<m_{\eta_{R}}<100$ GeV and $m_{\eta_{R}}>550$ GeV. The dark matter mass
in the range $m_{\eta_{R}}<50$ is disallowed as it is in conflict with the LHC
Higgs invisible decay limit CMS:2022qva . The intermediate region $70$ GeV
$<m_{\eta_{R}}<100$ GeV is not completely ruled out by LHC and LEP data and
dark matter mass in the range $m_{\eta_{R}}>550$ GeV is not affected by the
collider constraints. The phenomenology of fermionic dark matter is worth
exploring and beyond the scope of current study.

##  VII Conclusions

We have proposed the flavor-scoto-seesaw (FSS) model that establishes a common
origin of the nonzero $\theta_{13}$ and cosmological dark matter. The
framework is based on the $A_{4}$ flavor symmetry where both type-I seesaw and
scotogenic mechanisms contribute to the effective light neutrino mass. FSS
explains observed neutrino masses and mixing angles, provides rich
phenomenology and accommodates potential dark matter candidates. Guided by the
discrete symmetry, we show that the minimal type-I seesaw first reproduces the
widely popular TBM mixing, a first order approximation of the lepton mixing
matrix. Subsequently, the scotogenic contribution acts as a requisite
deviation to the TBM mixing and addresses the issue of the nature of dark
matter. We also demonstrate that the neutrino mixing pattern exhibits ${\rm
TM}_{2}$ mixing scheme (a viable descendent of the TBM mixing pattern) within
this scoto-seesaw scenario.
The model which we construct here is highly predictive in nature. Using the
current experimental observation on neutrino oscillation and other
cosmological limits, we found that the allowed parameter space in the FSS
model restricts some of the key observables associated with neutrinos (the
atmospheric mixing angle, Dirac and Majorana CP phases, effective mass
parameter appearing in the neutrinoless double beta decay ) and crucially
dictates the lepton flavor violating decays. To understand the behaviours of
the parameters involved (namely, $\alpha,\beta,\phi_{AC},\phi_{BC}$), we
divide numerical analysis into a few specific cases based on the choice of the
associated phases ($\phi_{AC},\phi_{BC}$) and then carried out the complete
general numerical analysis. These limiting cases can easily distinguish
between light neutrino masses’ normal and inverted hierarchy. For example,
when the relative phase between the seesaw contribution associated with right-
handed neutrino $N_{1}$ and the scotogenic contribution is considered zero
($i.e.$, $\phi_{AC}=0$ for Case I and II), only inverted hierarchy is allowed.
On the other hand when $\phi_{AC}\neq 0$ (Case III and IV), only normal
hierarchy is allowed. Considered discrete flavor symmetries play an
instrumental role in producing such distinctive constraints. Subsequently, for
the most general case, we carried out the numerical analysis for all possible
choices of the parameters involved and found that normal hierarchy can be
realised only with $\beta\leq 1$ and $2.61\leq\phi_{AC}\leq 5.38$ whereas to
realise inverted hierarchy we need $\beta\geq 1$ and $0\leq\phi_{AC}\leq 2$
(or $6\leq\phi_{AC}\leq 2\pi$). This analysis also predicts the atmospheric
mixing angle lying in the upper octant, in good agreement with the latest
global ﬁt neutrino oscillation data and restricts Dirac CP phase $\delta_{\rm
CP}$ within $-1.57\leq\delta_{\rm CP}\leq 1.37$ (and $1.4\leq\delta_{\rm
CP}\leq 1.57$) for normal hierarchy and $-1.57\leq\delta_{\rm CP}\leq 0.5$
(and $0.86\leq\delta_{\rm CP}\leq 1.57$) for inverted hierarchy. Along with
the Dirac CP phase $\delta_{\rm CP}$, the Majorana phases also get restricted
in our analysis. Furthermore, we obtain a lower limit on the lightest neutrino
mass as $m_{\rm lightest}\geq 0.0012$ eV for normal hierarchy and $m_{\rm
lightest}\geq 0.014$ eV for inverted hierarchy. We have also estimated the
prediction for the effective mass parameter $m_{\beta\beta}$ characterizing
the neutrinoless double beta decay and found it to be in the range $1-30$ meV
for normal hierarchy and $16-60$ meV for inverted hierarchy, respectively.
These values are within the reach of future neutrinoless double beta decay
experiments. In the end, we also comment on the phenomenological implications,
such as lepton flavor violation and the prospects of dark matter candidates in
such a scenario. As a consequence of the flavor structure, the scotogenic part
of the model does not contribute to the lepton flavor violating decays, and a
lower limit on the mass of the heavy right-handed neutrinos can be obtained
from the constraints on the branching ratios involving such decays. A detailed
discussion on the phenomenological aspects in this direction is dedicated for
future investigations.

## Appendix: $A_{4}$ symmetry

$A_{4}$ is a discrete group of even permutation of four objects333For a
detailed discussion on $A_{4}$ see Ref. Altarelli:2010gt ; Ishimori:2010au .
Geometrically, it is an invariance group of a tetrahedron. It has 12 elements
which can be generated by two basic objects $S$ and $T$ which obey the
following relation

$\displaystyle S^{2}=T^{3}=(ST)^{3}=1$ (56)

The $A_{4}$ group has three one-dimensional irreducible representations
$1$,$1^{\prime}$ and $1^{\prime\prime}$ and one three dimensional irreducible
representation 3. Products of the singlets and triplets are given by

$\displaystyle 1$ $\displaystyle\otimes$ $\displaystyle
1=1;~{}1^{\prime}\otimes 1^{\prime\prime}=1,$ (57) $\displaystyle 1^{\prime}$
$\displaystyle\otimes$ $\displaystyle
1^{\prime}=1^{\prime\prime};~{}1^{\prime\prime}\otimes
1^{\prime\prime}=1^{\prime},$ (58) $\displaystyle 3$ $\displaystyle\otimes$
$\displaystyle 3=1\oplus 1^{\prime}\oplus 1^{\prime\prime}\oplus 3_{s}\oplus
3_{a},$ (59)

where the subscripts $``s"$ and $``a"$ denote symmetric and antisymmetric part
respectively. Writing two triplets as $(x_{1},x_{2},x_{3})$ and
$(y_{1},y_{2},y_{3})$ respectively, their products are given by

$\displaystyle 1$ $\displaystyle\sim$ $\displaystyle
x_{1}y_{1}+x_{2}y_{3}+x_{3}y_{2},$ (60) $\displaystyle 1^{\prime}$
$\displaystyle\sim$ $\displaystyle x_{3}y_{3}+x_{1}y_{2}+x_{2}y_{1},$ (61)
$\displaystyle 1^{\prime\prime}$ $\displaystyle\sim$ $\displaystyle
x_{2}y_{2}+x_{1}y_{3}+x_{3}y_{1},$ (62) $\displaystyle 3_{s}$
$\displaystyle\sim$
$\displaystyle\begin{pmatrix}2x_{1}y_{1}-x_{2}y_{3}-x_{3}y_{2}\\\
2x_{3}y_{3}-x_{1}y_{2}-x_{2}y_{1}\\\
2x_{2}y_{2}-x_{1}y_{3}-x_{3}y_{1}\end{pmatrix},$ (63) $\displaystyle 3_{a}$
$\displaystyle\sim$ $\displaystyle\begin{pmatrix}x_{2}y_{3}-x_{3}y_{2}\\\
x_{1}y_{2}-x_{2}y_{1}\\\ x_{3}y_{1}-x_{1}y_{3}\end{pmatrix}.$ (64)

###### Acknowledgements.

This work has been supported in part by the Polish National Science Center
(NCN) under grant 2020/37/B/ST2/02371 and the Freedom of Research (Swoboda
Badań) initiative of the University of Silesia in Katowice. BK would like to
thank José W. F. Valle for useful discussions.

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