aainstitutetext: Department of BSH, University of Engineering and Management,
Kolkata, Indiabbinstitutetext: Institute of Physics, University of Silesia,
Katowice, Polandccinstitutetext: Department of Physics and Institute of Basic
Science, Sungkyunkwan University, Suwon 16419, Korea

# Phenomenology of the flavor symmetric scoto-seesaw model with dark matter
and TM1 mixing

Joy Ganguly b Janusz Gluza b Biswajit Karmakar c Satyabrata Mahapatra
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We propose a hybrid scoto-seesaw model based on the $A_{4}$ non-Abelian
discrete flavor symmetry. Light neutrino masses come from the tree-level
type-I seesaw mechanism and from the one-loop scotogenic contribution
accommodating viable dark matter candidates responsible for observed relic
abundance of dark matter (DM). Respectively, both these contributions restore
the atmospheric and solar neutrino mass scales. With only one right-handed
neutrino, the model features specific predictions with the normal ordering of
light neutrino masses, the lightest neutrino being massless, and only one
relevant CP Majorana phase. The flavor symmetric setup helps us to realize the
TM1 mixing scheme with concrete correlations and constraints on the mixing
angles and associated CP phases. The framework predicts the atmospheric mixing
angle to be in the upper octant with specific ranges
$0.531\leq\sin^{2}\theta_{23}\leq 0.544$, $0.580\leq\sin^{2}\theta_{23}\leq
0.595$ and the Dirac CP phase is restricted within the range $\pm(1.44-1.12)$
radian. The Majorana phase is also tightly constrained with a range of
$0.82-0.95$ and $1.58-1.67$ radian, which is otherwise unconstrained from
neutrino oscillations. Strict predictions on the Majorana phases also yield an
accurate prediction for the effective mass parameter for neutrinoless double
beta within the range of $1.61-3.85$ meV. The model offers a rich
phenomenology regarding DM relic density and direct search constraints, and
the fermionic DM scenario has been discussed in detail, estimating its
possible connection with the neutrino sector. As an example of the model
studies at colliders, the SM Higgs in the diphoton decay channel is examined.
The model predicts strictly vanishing $\tau\to e\gamma$, $\tau\rightarrow 3e$
decays and testable signals by MEG-II and SINDRUM/Mu3e experiments for the
$\mu\to e\gamma$ and $\mu\to 3e$ decays, respectively.

##  1 Introduction

In the last few decades, several experiments around the globe have confirmed
the phenomenon of neutrino oscillation with incredible precision Fukuda:2001nk
; Ashie:2005ik ; Ahmad:2002jz ; Ahmad:2002ka ; Abe:2008aa ; Abe:2011sj ;
Abe:2011fz ; Abe:2013hdq . The immediate consequence of neutrino oscillation
is that at least two light neutrinos have nonzero mass. Furthermore, if we
combine this with a bound on the absolute neutrino masses coming from the end-
point spectrum of the tritium beta decay KATRIN:2021uub , as well as the
bounds from cosmological surveys Planck:2018vyg  and the neutrinoless double
beta decay KamLAND-Zen:2022tow , we conclude that the neutrino masses are in
the sub-eV scale. Despite these spectacular accomplishments, the origin of
tiny neutrino masses (compared to other Standard Model fermions) remains an
open question in particle physics. Over the years various ideas have been
proposed, and the most common schemes are seesaw mechanisms Minkowski:1977sc ;
GellMann:1980vs ; Mohapatra:1979ia ; Mohapatra:1986aw ; Magg:1980ut ;
Lazarides:1980nt ; Schechter:1980gr ; Mohapatra:1986bd  and radiative
generation of neutrino masses Cai:2017jrq . Nonzero neutrino masses can also
be realized within the framework of hybrid mass mechanisms where both seesaw
and radiative mass mechanisms contribute.

In addition to the tiny neutrino masses, we are yet to understand the observed
pattern of the lepton mixing comprehensively. In fact, two of the three mixing
angles, namely solar ($\theta_{12}$) and atmospheric ($\theta_{23}$), are
found to be large, while the reactor ($\theta_{13}$) mixing angle is
relatively small. Such a finding clearly shows the distinctive feature
associated with the lepton sector in contrast to the quark sector. The study
of the underlying principle behind this typical mixing is particularly
interesting with the precise measurement of the reactor mixing angle
$\theta_{13}$ Abe:2011fz ; An:2012eh ; Ahn:2012nd ; Abe:2013hdq . Neutrino
oscillation data also constraints the two mass squared differences (solar and
atmospheric) defined as $\Delta m_{21}^{2}=m_{2}^{2}-m_{1}^{2}$ and $|\Delta
m_{31}^{2}|=|m_{3}^{2}-m_{1}^{2}|$ where $m_{1},m_{2},m_{3}$ are the masses of
the three light neutrinos. The present global analysis from several
experimental data can be summarized as Esteban:2020cvm

$\displaystyle\Delta m^{2}_{21}=(6.82-8.03)\times 10^{-5}\hskip
2.84544pt\rm{eV}^{2},\hskip 14.22636pt|\Delta m^{2}_{31}|=(2.427-2.590)\times
10^{-3}\hskip 2.84544pt\rm{eV}^{2},$
$\displaystyle\sin^{2}\theta_{12}=0.270-0.341,\hskip
5.69046pt\sin^{2}\theta_{23}=0.408-0.603,\hskip
5.69046pt\sin^{2}\theta_{13}=0.02052-0.02398,$ (1)

for normal ordering (NO) of light neutrino mass and similar constraints for
inverted ordering (IO) Esteban:2020cvm . In this regard, many conjectures have
been put forward. A particular pattern yielding
$\sin^{2}\theta_{23}=1/2,\sin^{2}\theta_{12}=1/3$ and $\theta_{13}=0$ known as
tri-bimaximal mixing (TBM) Harrison:2002er , received a lot of attention due
the proximity of $\theta_{23}$ and $\theta_{12}$ with experimental values.
Such a mixing pattern can also be elegantly generated using flavor symmetries.
Particularly, the use of non-Abelian discrete symmetries like
$S_{3},A_{4},S_{4},A_{5}$ is very well known Ma:2001dn ; Altarelli:2005yx  in
this context. For a detailed discussion on such frameworks see Ishimori:2010au
; Altarelli:2010gt ; King:2013eh ; Petcov:2017ggy ; Chauhan:2022gkz ;
Chauhan:2023faf  and references therein. Unsurprisingly, a deformation from
TBM mixing becomes inevitable after precisely measuring $\theta_{13}$.
Nevertheless, the TBM mixing scheme can still be considered as a leading-order
approximation, requiring adjustments such as accounting for non-zero
$\theta_{13}$ and the Dirac CP phase $\delta$. Possible simple deviations from
the TBM mixing are called trimaximal (TM1 and TM2) mixings where the first and
second columns of the TBM mixing respectively remain identical Albright:2008rp
; Bjorken:2005rm ; Albright:2010ap ; He:2006qd ; Luhn:2013lkn ;
deMedeirosVarzielas:2012apl ; Shimizu:2017fgu ; Thapa:2021ehj ;
Chakraborty:2020gqc . Such deviations can be elegantly achieved by considering
larger residual symmetry (compared to the TBM scenario) or introducing an
additional constituent that breaks the TBM structure Thapa:2021ehj ;
Chakraborty:2020gqc ; deAdelhartToorop:2011nfg ; deAdelhartToorop:2011re ;
Hernandez:2012sk ; Ding:2012xx ; Karmakar:2014dva ; Karmakar:2015jza ;
Karmakar:2016cvb ; Karmakar:2017sxt . Although the $3\sigma$ allowed ranges
for all three mixing angles can be explained by both TM1 and TM2 mixings, the
allowed value of the solar mixing angle $\theta_{12}$ (within these trimaximal
scenarios) slightly prefers the TM1 over the TM2 mixing scheme. For a detailed
discussion on the relative comparison of both mixings, see Chauhan:2023faf .

Apart from the neutrino masses and mixing, unraveling the nature of dark
matter (DM) remains a pressing challenge in contemporary particle physics.
While compelling astrophysical evidence, including observations like galaxy
rotation curves, gravitational lensing, and the cosmic large-scale structure,
substantiates the existence of DM Cheung:2013dua , the quest for a laboratory-
based confirmation persists. Satellite missions such as WMAP WMAP:2012nax  and
PLANCK Aghanim:2018eyx  have precisely determined that DM constitutes roughly
$26.8\%$ of the total energy content of the Universe. Expressing the
prevailing dark matter abundance through the density parameter $\Omega_{\rm
DM}$ and the normalized Hubble parameter $h$ (Hubble Parameter divided by
$100$ km $s^{-1}Mpc^{-1}$) yields $\Omega_{\rm DM}h^{2}=0.120\pm 0.001$ at a
$68\%$ confidence level. Still the intricacies of its properties beyond
gravitational interactions remain elusive. Among all proposed particle dark
matter, the most sought-after paradigm is the weakly interacting massive
particle (WIMP) paradigm, which suggests a dark matter particle with a mass
and interaction strength akin to the electroweak scale. Unfortunately, the
Standard Model of particle physics fails to comprehensively explain neutrino
masses, mixings and dark matter. Standing at this juncture, certainly, it is a
tempting challenge to find a common origin of these two seemingly uncorrelated
sectors, if any. Hence, we aim to go beyond the SM of particle physics to
explore scenarios that can accommodate a candidate of DM and explain non-zero
neutrino masses and mixings.

Neutrino oscillation data presented earlier, see Eq. (1), does not determine
the absolute scale or ordering of neutrino masses. The experiments have
measured the two mass-squared differences (solar and atmospheric) associated
with neutrino oscillations, and the ratio of the solar-to-atmospheric mass-
squared difference ($r$) is found to be Esteban:2020cvm

$\displaystyle r=\Delta m_{21}^{2}/|\Delta m_{31}^{2}|\sim 0.03.$ (2)

This may be an indication of the involvement of two different mass scales that
might originate from entirely separate mechanisms111Within the framework of
type-I seesaw mechanism, the hierarchy of the atmospheric and solar neutrino
mass scales can also be explained through the mechanisms of sequential
dominance King:1998jw ; King:1999cm ; King:1999mb ; King:2002nf ;
Antusch:2010tf ; Antusch:2004gf  and constrained sequential dominance
King:2005bj ; Antusch:2011ic  where one of the right-handed neutrinos is
considered to dominantly contribute the light neutrino masses.. Following this
ethos, the authors in Rojas:2018wym  showed a scotogenic extension of the
type-I seesaw scenario that can minimally explain the hierarchy of solar and
atmospheric neutrino mass scales. In this set-up, the Standard Model particle
content is extended by including one heavy isosinglet neutral lepton $N_{R}$
(for the type-I sector) along with a dark fermion $f$ and an inert scalar
doublet $\eta$ (for the scotogenic sector), both being odd under a dark
$Z_{2}$ symmetry. Here, the type-I contribution gives rise to the larger
atmospheric scale. In contrast, the one-loop scotogenic contribution turns out
to be the origin of the smaller solar mass scale mediated by a dark fermion,
also providing a potential dark matter candidate Leite:2023gzl ;
Mandal:2021yph . Unfortunately, such constructions turn out to be inadequate
in explaining the observed neutrino oscillation data associated with the
mixing and the Dirac CP phase mentioned above. This problem can be addressed
by augmenting non-Abelian discrete flavor symmetries Barreiros:2020gxu ;
Ganguly:2022qxj  mentioned above. Hence, in the work Ganguly:2022qxj , a
flavor symmetric scoto-seesaw (FSS) scenario is proposed to understand the
neutrino mass hierarchy, which also explains the TM2 mixing scheme. The whole
framework is embedded within widely used $A_{4}$ non-Abelian discrete flavor
symmetry, the smallest group with triplet irreducible representation providing
an opportunity to unify three generations of leptons having the remarkable
ability to produce realistic lepton mixing patterns. Moreover, such symmetry
can arise in many ways, starting from a continuous group Koide:2007sr ;
Adulpravitchai:2009kd ; Luhn:2011ip ; Merle:2011vy ; Rachlin:2017rvm ;
King:2018fke  or superstring theory in compactified extra dimensions
Altarelli:2005yp ; Altarelli:2005yx ; Burrows:2009pi ; King:2017guk ;
deAdelhartToorop:2011re ; Feruglio:2017spp ; deAnda:2018ecu ;
Novichkov:2018yse ; Criado:2018thu ; Kobayashi:2018wkl ; Penedo:2018nmg ;
Ding:2019xna . Earlier in Ref. Barreiros:2020gxu , the authors discussed a
scoto-seesaw scenario with $Z_{8}$ discrete symmetry where two right-handed
neutrinos were present. In a similar setup Ganguly:2022qxj  with two right-
handed neutrinos, we implemented a scoto-seesaw model within the $A_{4}$
flavor symmetric framework to realize the TM2 mixing.

In the present work, we show that the scoto-seesaw mechanism can be embedded
within a flavor symmetric framework with the involvement of only one right-
handed neutrino, and the experimentally favored TM1 mixing scheme can be
realized. We will call the present model FSS1 from now on. The scotogenic
contribution contains neutral particles in both fermionic and scalar sectors.
Within this flavor symmetric framework, we perform a phenomenological study of
the fermionic dark matter and determine its correlation with the observed
neutrino mixing. The obtained magnitude and flavor structure of the scotogenic
contribution dictates the observed neutrino mixing pattern and facilitates us
in obtaining the correct dark matter relic density. The presence of flavor
symmetry makes an interesting prediction for the neutrino mass hierarchy,
determines the octant of the atmospheric mixing angle $\theta_{23}$ and
tightly constrains the TM1 prediction for the Dirac CP phase $\delta_{\rm
CP}$. Here, the type-I contribution produces a neutrino mass matrix of rank 1,
yielding only one massive light neutrino. Subsequently, the scotogenic
contribution generates another neutrino mass eigenstate, and together, we
obtain two massive neutrinos, which follow the normal ordering of light
neutrino mass spectrum. In Rojas:2018wym , the authors showed that the type-I
and scotogenic contributions could be the origin of the atmospheric and solar
mass scale. Here, we show that such a hierarchy can also be procured within a
flavor symmetric construction, explaining observed neutrino oscillation data.
Furthermore, in the present model, we can obtain the constraint on the
Majorana phase and predict the effective mass parameter appearing in the
neutrinoless double beta decay and the sum of the absolute neutrino masses.

One of the essential aspects of any theoretical model is its experimental
viability. For the version of the FSS model discussed here, we perform a
comprehensive phenomenological analysis involving the
$h\rightarrow\gamma\gamma$ decay, where $h$ is the SM Higgs boson. The signal
strength of the Higgs in the diphoton decay channel, $R_{\gamma\gamma}$, is
measured at the LHC, the value of which is around one ATLAS:2022tnm . The
additional contribution to the decay of $h\rightarrow\gamma\gamma$ in the FSS1
model is the charged scalar of the $\eta$ field. Our analysis shows that
$R_{\gamma\gamma}$ can be fitted in our model, which can constrain the mass of
the charged component of the $\eta$ field. Owing to the flavor structure of
this scoto-seesaw framework, we find that only the scotogenic part contributes
to the lepton flavor violating decays such as $\mu\rightarrow
e\gamma,\mu\rightarrow 3e$, whereas only the seesaw part contributes in the
decays such as $\tau\rightarrow\mu\gamma,\tau\rightarrow 3\mu$. However,
scotogenic and seesaw parts do not contribute to the $\tau\rightarrow e\gamma$
and $\tau\rightarrow 3e$ decays. All these phenomenological analyses for the
FSS1 framework serve as crucial tests of the model’s predictions and provide
valuable insights into its compatibility with experimental data.

The rest of the paper is organized as follows. In section 2, we briefly
introduce the minimal scoto-seesaw framework. In section 3 we present the
complete $A_{4}$ flavor symmetric scoto-seesaw scenario, and in section 4 we
analyze corresponding neutrino masses and mixing. We mention the low energy
scalar potential in section 5. In section 6, we discuss the detailed
phenomenology of fermionic dark matter and further phenomenological
implications for the Higgs to the diphoton decay and lepton flavor violation
in section 7 and section 8, respectively. Then, in section 9, we summarize the
phenomenological analysis. Finally, in section 10, we present the conclusion
and outlook of the FSS1 framework.

##  2 Minimal scoto-seesaw model

In this section, we present the minimal scoto-seesaw model which is introduced
in Rojas:2018wym . The minimal scoto-seesaw model consists of one222The number
of right-handed neutrinos added Schechter:1980gr  to the SM is not fixed as
they do not carry any anomaly King:2015sfk . right-handed neutrino $N_{R}$,
one singlet dark fermion $f$ and one extra scalar doublet $\eta_{R}$. In
addition to these particles, one $Z_{2}$ symmetry is introduced to stabilize
the dark matter. In this model333For various extensions of the minimal scoto-
seesaw scenario, see Refs. Barreiros:2020gxu ; Mandal:2021yph ;
Ganguly:2022qxj ; Leite:2023gzl ; Kumar:2023moh ; VanDong:2023xbd ., the usual
type-I seesaw mechanism with one right-handed neutrino $N_{R}$ is combined
with the scotogenic model with fermion $f$. The type-I seesaw generates the
atmospheric mass scale at the tree level, while the solar mass scale is
generated at the loop level in the scotogenic mechanism. As a result, the
hierarchy between solar mass scale and atmospheric mass scale is maintained.
The relevant Lagrangian in the model can be written as

$\displaystyle{\mathcal{L}}=-Y_{N}^{k}\bar{L}^{k}i\sigma_{2}H^{*}N_{R}+\frac{1}{2}M_{N}\bar{N}_{R}^{c}N_{R}+Y_{f}^{k}\bar{L}^{k}i\sigma_{2}\eta^{*}f+\frac{1}{2}M_{f}\bar{f}^{c}f+h.c..$
(3)

where $L^{k}$ are the lepton doublets. The scalars $H$ and $\eta$ are the
$SU(2)$ doublets defined in Eq. (35). $Y_{N}$ and $Y_{f}$ are complex $3\times
1$ Yukawa coupling matrices, and $M_{N,f}$ are the mass terms for $N_{R}$ and
$f$. The total neutrino mass reads Rojas:2018wym

$\displaystyle
M_{\nu}^{ij}=-\frac{v^{2}}{M_{N}}Y_{N}^{i}Y_{N}^{j}+{\mathcal{F}}(M_{\eta_{R}},M_{\eta_{I}},M_{f})M_{f}~{}Y_{f}^{i}Y_{f}^{j}.$
(4)

Here, the first term is due to the tree-level seesaw mechanism, while the
second term originates from the one-loop scotogenic contribution with

$\displaystyle\mathcal{F}(M_{\eta_{R}},M_{\eta_{I}},M_{f})=\frac{1}{32\pi^{2}}\Big{[}\frac{M_{\eta_{R}}^{2}\log\Big{(}M_{f}^{2}/M_{\eta_{R}}^{2}\Big{)}}{M_{f}^{2}-M_{\eta_{R}}^{2}}-\frac{M_{\eta_{I}}^{2}\log\Big{(}M_{f}^{2}/M_{\eta_{I}}^{2}\Big{)}}{M_{f}^{2}-M_{\eta_{I}}^{2}}\Big{]},$
(5)

where $M_{\eta_{R}}$ and $M_{\eta_{I}}$ are the masses of the neutral
component of $\eta$. Although the ratio of the above two contributions in Eq.
(4) can explain the hierarchy of the solar and atmospheric mass scales, it
fails to explain the observed neutrino mixing pattern. In this regard, the use
of non-Abelian discrete flavor symmetries is well motivated Altarelli:2010gt ;
King:2013eh ; Petcov:2017ggy ; Chauhan:2022gkz ; Chauhan:2023faf . In the
following sections, we discuss the phenomenological consequences of flavor
symmetric construction of the scoto-seesaw framework with only one right-
handed neutrino to explain the correct neutrino masses and mixing. We also
provide a detailed analysis of the fermionic dark matter relic abundance and
direct detection search constraint to determine the parameter space consistent
with neutrino oscillation data and predictions for Higgs to diphoton signal
strength and lepton flavor violating decays.

##  3 Scoto-seesaw with flavor symmetry: the FSS1 model

The model we are proposing is the flavor symmetric version of the scoto-seesaw
model described in the previous section with usual scotogenic fermion $f$ and
inert doublet $\eta$ in addition to one right-handed neutrino $N_{R}$. To
obtain the flavor structure, $A_{4}$ flavor discrete symmetry and flavons
$\phi_{s}$, $\phi_{a}$, $\phi_{T}$ and $\xi$ are introduced. To avoid unwanted
terms in the Lagrangian and get the correct flavor Yukawa structure,
additional $Z_{N}$ symmetries are introduced. The inclusion of flavon fields
and auxiliary symmetries are generic features of such flavor symmetric
constructions Ma:2001dn ; Ma:2002yp ; Altarelli:2005yp ; Altarelli:2005yx . A
remnant $Z_{2}$ symmetry of the $Z_{N}$ symmetries acts as a dark symmetry
that ensures the stability of dark matter under which only $f$ and $\eta$ are
odd. Similar types of flavored scoto-seesaw models were studied before in
Barreiros:2020gxu  and Ganguly:2022qxj  with $Z_{8}$ and $A_{4}$ discrete
symmetries, respectively. No simple analytic correlation can be obtained due
to the use of the $Z_{8}$ symmetry Barreiros:2020gxu  whereas the TM2 mixing
was reproduced in Ganguly:2022qxj  with the $A_{4}$ symmetry. In both cases,
two right-handed neutrinos are introduced in the seesaw contribution to get
the flavor structure and mixing. In the present work, we construct the
framework with only one right-handed neutrino and realize the experimentally
preferred TM1 mixing scheme compared to the TM2 mixing scheme (derived in
Ganguly:2022qxj ). The particle content of our model and charge assignment
under different symmetries are shown in Table 1. The role of each discrete
auxiliary symmetry will be described in detail as we proceed further.

Fields |  $e_{R}$, $\mu_{R}$, $\tau_{R}$ | $L_{\alpha}$ | $H$ | $N_{R}$ | $f$ | $\eta$ | $\phi_{s}$ | $\phi_{a}$ | $\phi_{T}$ | $\xi$
---|---|---|---|---|---|---|---|---|---|---
$A_{4}$ |  $1$ , $1^{\prime\prime}$ , $1^{\prime}$ | $3$ | $1$ | $1$ | $1$ | $1$ | $3$ | $3$ | 3 | $1^{\prime\prime}$
$Z_{4}$ | $i$ | $i$ | $1$ | $1$ | $1$ | $1$ | $i$ | $i$ | $1$ | $1$
$Z_{3}$ | $\omega^{2}$ | $\omega$ | $1$ | $\omega^{2}$ | 1 | $1$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $1$
$Z_{2}$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$
Table 1:  Fields content and transformations under the symmetries of the FSS1
model. Product rules of the $A_{4}$ singlets and triplets are given in
Appendix A. The flavon fields mentioned in the second block are essential to
implement the $A_{4}$ symmetry, and $\omega(=e^{2i\pi/3})$ is the cube root of
unity.

With the field content and charges assignment in Table 1, the charged lepton
Lagrangian can be written up to leading order as

$\displaystyle\mathcal{L}_{l}=\frac{y_{e}}{\Lambda}(\bar{L}\phi_{T})He_{R}+\frac{y_{\mu}}{\Lambda}(\bar{L}\phi_{T})H\mu_{R}+\frac{y_{\tau}}{\Lambda}(\bar{L}\phi_{T})H\tau_{R}+h.c.,$
(6)

where $\Lambda$ is the cut-off scale of our model. $y_{e}$, $y_{\mu}$ and
$y_{\tau}$ are the coupling constants. Now, when the flavon $\phi_{T}$ gets a
vacuum expectation value (VEV) in the direction
$\langle\phi_{T}\rangle=(v_{T},0,0)^{T}$ and subsequently the Higgs field also
get VEV as $\langle H\rangle=v$, where $v$ is the SM VEV, we get the charged
lepton mass matrix to be in the diagonal form as

$\displaystyle M_{l}=\frac{v_{T}}{\Lambda}v\begin{pmatrix}y_{e}&0&0\\\
0&y_{\mu}&0\\\ 0&0&y_{\tau}\end{pmatrix}.$ (7)

Now, the Lagrangian in the neutrino sector, which generates neutrino masses,
constitutes two parts: a type-I seesaw contribution with one right-handed
neutrino $N_{R}$ and another, one loop scotogenic part with the presence of
the dark fermion $f$ and scalar $\eta$. Following the symmetries and particle
content mentioned in table 1, the Lagrangian for the neutrino sector can be
written as

$\displaystyle\mathcal{L}=\frac{y_{N}}{\Lambda}(\bar{L}\phi_{s})\tilde{H}N_{R}+\frac{1}{2}M_{N}\bar{N}_{R}^{c}N_{R}+\frac{y_{s}}{\Lambda^{2}}(\bar{L}\phi_{a})\xi
i\sigma_{2}\eta^{*}f+\frac{1}{2}M_{f}\bar{f}^{c}f+h.c.,$ (8)

where $y_{N}$ and $y_{s}$ are the coupling constants and $M_{N}$ is the
Majorana mass of the right-handed neutrino $N_{R}$ while $M_{f}$ is the mass
of the fermion $f$. In the above Lagrangian, we have considered VEVs of the
flavons $\phi_{s}$, $\phi_{a}$ and $\xi$ in directions
$\langle\phi_{s}\rangle=(0,-v_{s},v_{s})$,
$\langle\phi_{a}\rangle=(2v_{a},v_{a},0)$ and $\langle{\xi}\rangle=v_{\xi}$,
respectively. A similar vacuum alignment can be found in the literature for
neutrino model building Antusch:2011ic ; Bjorkeroth:2014vha  which can be
realized inherently by analyzing the complete scalar potential
Altarelli:2010gt ; Karmakar:2016cvb ; He:2006dk ; Lin:2008aj ; King:2005bj ;
Holthausen:2011vd . The light neutrino mass matrix involving both type-I
seesaw and scotogenic contributions can be written as

$\displaystyle(M_{\nu})_{ij}$ $\displaystyle=$
$\displaystyle-\frac{v^{2}}{M_{N}}Y_{N}^{i}Y_{N}^{j}+\mathcal{F}(M_{\eta_{R}},M_{\eta_{I}},M_{f})M_{f}Y_{f}^{i}Y_{f}^{j}$
(9)

where the Yukawa couplings take the following form

$\displaystyle
Y_{N}=(Y_{N}^{e},Y_{N}^{\mu},Y_{N}^{\tau})^{T}=(0,y_{N}\frac{v_{s}}{\Lambda},-y_{N}\frac{v_{s}}{\Lambda})^{T},$
(10) $\displaystyle
Y_{F}=(Y_{F}^{e},Y_{F}^{\mu},Y_{F}^{\tau})^{T}=(y_{s}\frac{v_{\xi}}{\Lambda}\frac{v_{a}}{\Lambda},y_{s}\frac{v_{\xi}}{\Lambda}\frac{2v_{a}}{\Lambda},0)^{T}\equiv(\kappa,2\kappa,0)^{T}.$
(11)

Within this setup, the total effective light neutrino mass matrix of Eq. (9)
is the following

$\displaystyle M_{\nu}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}b&2b&0\\\ 2b&-a+4b&a\\\ 0&a&-a\end{pmatrix}$ (12)

with

$\displaystyle a$ $\displaystyle=$ $\displaystyle
y_{N}^{2}\frac{v^{2}}{M_{N}}\frac{v_{s}^{2}}{\Lambda^{2}},$ (13)
$\displaystyle b$ $\displaystyle=$ $\displaystyle
y_{s}^{2}\frac{v_{\xi}^{2}}{\Lambda^{2}}\frac{v_{a}^{2}}{\Lambda^{2}}\mathcal{F}(m_{\eta_{R}},m_{\eta_{I}},M_{f})M_{f}\equiv\kappa^{2}\mathcal{F}(M_{\eta_{R}},M_{\eta_{I}},M_{f})M_{f},$
(14)

where $\mathcal{F}$ is the loop function defined in Eq. (5). Clearly, from Eq.
(10) to Eq. (14), it is evident that the parameters $a$ and $b$ originate from
type I-seesaw and scotogenic contributions, respectively. In the next section,
we show how these parameters’ relative magnitude helps us explain the
hierarchy of the atmospheric and solar oscillation mass scales.

Though the neutrino mass matrix given in Eq. (12) is obtained through a
combination of type-I seesaw and scotogenic mechanisms, there can be
additional operators like $LHLH/\Lambda$, contributing to the light neutrino
masses. In our model, this higher dimensional term is not invariant explicitly
under the $Z_{4}$ symmetry given in Tab. 1. Also, terms like
$LHLH(\phi_{a},\phi_{s},\phi_{T},\xi)/\Lambda^{2}$ are disallowed due to the
considered discrete $Z_{N}$ symmetries. For the same $Z_{N}$ symmetries, the
scotogenic contribution ($\bar{L}i\sigma_{2}\eta^{*}f$) is only allowed at
$1/\Lambda^{2}$ with the involvement of flavons $\phi_{a}$ and $\xi$, which
are both odd under the $Z_{2}$ symmetry along with $f$ and $\eta$. Owing to
the $A_{4}$ symmetry, in the charged lepton sector, the leading order
contribution appears only at dimension-5. However, there could be a next-to-
leading correction at $\mathcal{O}(1/\Lambda^{2})$ via
$(\bar{L}\phi_{s}^{\dagger}\phi_{a})H\alpha_{R}/\Lambda^{2}$, where
$\alpha_{R}$ is the corresponding right-handed charged lepton. Interestingly,
such a contribution is disallowed due to the $Z_{2}$ symmetry given in Tab. 1.
As the right-handed Majorana neutrino present in our model is also a singlet
under $A_{4}$ symmetry, any higher-order correction can be absorbed in the
leading order contribution to $M_{N}$. For the same reason, we can also absorb
any higher order contribution to $M_{f}$ as it does not affect the flavor
structure of our model. Finally, the Dirac Yukawa coupling is allowed at
dimension-5 as given in Eq. (11). The next-to-leading order contribution at
$\mathcal{O}(1/\Lambda^{2})$ can be written as
$(\bar{L}\phi_{a}^{\dagger}\phi_{T})\tilde{H}N_{R}$ and
$(\bar{L}\phi_{s}^{\dagger}\phi_{T})\tilde{H}N_{R}$, which are not allowed due
to $Z_{2}$ and $Z_{4}$ symmetries.

##  4 Neutrino masses and mixings in the FSS1 model

The model we presented in the last section has two parts. One is coming from a
type-I seesaw with one right-handed neutrino $N_{R}$. Another part is the
scotogenic contribution with the dark fermion $f$. The full light neutrino
mass matrix is given in Eq. (12), and both contributions are essential in
explaining observed neutrino masses and mixing. To diagonalize the mass matrix
in Eq. (12), we first write the mass matrix in the TBM basis as

$\displaystyle
M_{\nu}^{\prime}=U_{TB}^{T}M_{\nu}U_{TB}=\begin{pmatrix}0&0&0\\\
0&3b&-\sqrt{6}b\\\ 0&-\sqrt{6}b&2(b-a)\end{pmatrix},$ (15)

where

$\displaystyle
U_{TB}=\begin{pmatrix}\sqrt{\frac{2}{3}}&\sqrt{\frac{1}{3}}&0\\\
-\sqrt{\frac{1}{6}}&\sqrt{\frac{1}{3}}&-\sqrt{\frac{1}{2}}\\\
-\sqrt{\frac{1}{6}}&\sqrt{\frac{1}{3}}&\sqrt{\frac{1}{2}}\end{pmatrix}.$ (16)

As evident from Eq. (15), a further rotation by $U_{23}$ (another unitary
matrix) in the 23 plane will diagonalize the light neutrino mass matrix via
$M_{\nu}^{\rm diag}=U_{23}^{T}M_{\nu}^{\prime}U_{23}$. The unitary rotation
matrix $U_{23}$ can be parameterized as

$\displaystyle U_{23}=\begin{pmatrix}1&0&0\\\ 0&\cos\theta&\sin\theta
e^{-i\psi}\\\ 0&-\sin\theta e^{i\psi}&\cos\theta\end{pmatrix}$ (17)

where $\theta$ and $\psi$ are the rotation angle and the associated phase
factor, respectively. So, the diagonalization of $M_{\nu}$ can be achieved
through

$\displaystyle(U_{TB}U_{23})^{T}M_{\nu}(U_{TB}U_{23})={\rm
diag}(m_{1}e^{i\gamma_{1}},m_{2}e^{i\gamma_{2}},m_{3}e^{i\gamma_{3}})$ (18)

where $m_{1,2,3}$ are the real and positive mass eigenvalues, and
$\gamma_{1,2,3}$ are the phases that are extracted from the corresponding
complex eigenvalues. In our framework, we have only one right-handed neutrino,
which, via type-I seesaw (the first term in Eq. (9)), yields a rank 1 mass
matrix which makes one light neutrino massive. Together with the scotogenic
contribution, we obtain a rank 2 mass matrix given in Eq. (12), generating two
massive neutrinos. Hence, within this flavor symmetric construction, one mass
eigenvalue (lightest) will be zero. So, we have $m_{1}=0$, which implies
$\gamma_{1}=0$. Now, we can get the form of neutrino mixing matrix $U_{\nu}$
such that $U_{\nu}^{T}M_{\nu}U_{\nu}={\rm diag}(0,m_{2},m_{3})$. Thus,
$U_{\nu}$ becomes $U_{\nu}=U_{TB}U_{23}U_{m}$, where $U_{m}={\rm
diag}(1,1,e^{i\alpha_{32}/2})$ is the Majorana phase matrix with
$\alpha_{32}=\gamma_{3}-\gamma_{2}$. Therefore, we have only one non-zero
phase in the Majorana phase matrix $U_{m}$ as the lightest neutrino is
massless. The explicit form of $U_{\nu}$ follows

$\displaystyle
U_{\nu}=\begin{pmatrix}\sqrt{\frac{2}{3}}&\frac{\cos\theta}{\sqrt{3}}&\frac{e^{-i\psi}\sin\theta}{\sqrt{3}}\\\
-\frac{1}{\sqrt{6}}&\frac{\cos\theta}{\sqrt{3}}+\frac{e^{i\psi}\sin\theta}{\sqrt{2}}&-\frac{\cos\theta}{\sqrt{2}}+\frac{e^{-i\psi}\sin\theta}{\sqrt{3}}\\\
-\frac{1}{\sqrt{6}}&\frac{\cos\theta}{\sqrt{3}}-\frac{e^{i\psi}\sin\theta}{\sqrt{2}}&\frac{\cos\theta}{\sqrt{2}}+\frac{e^{-i\psi}\sin\theta}{\sqrt{3}}\end{pmatrix}U_{m}.$
(19)

This form of $U_{\nu}$ is well known in the literature as a deviation from
$U_{\rm TBM}$ and is called the TM1 mixing pattern. The lepton mixing matrix
$U_{\nu}$ can now be compared with $U_{PMNS}$ which in its standard
parametrization is given by ParticleDataGroup:2022pth

$\displaystyle U_{\rm
PMNS}=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{\rm CP}}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{\rm
CP}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{\rm CP}}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{\rm
CP}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{\rm
CP}}&c_{23}c_{13}\end{pmatrix}U_{m},$ (20)

where $c_{ij}=\cos\theta_{ij}$, $s_{ij}=\sin\theta_{ij}$, $\delta_{\rm CP}$ is
the Dirac CP violating phase and $U_{m}$ is the Majorana phase matrix. We can
see that the total light neutrino mass matrix of Eq. (12) contains two
parameters $a$ and $b$ associated with the type-I seesaw and scotogenic
contributions, which can be complex in general. We can write these parameters
as $a=|a|e^{i\phi_{a}}$ and $b=|b|e^{i\phi_{b}}$ where $\phi_{a}$ and
$\phi_{b}$ are the associated phases. For calculational purpose, we define the
parameter $\alpha=|a|/|b|$ and the difference of phases by
$\phi_{ab}=\phi_{a}-\phi_{b}$. As $M_{\nu}^{\prime}$ is diagonalized by
$U_{23}$, the rotation angle $\theta$ and the phase $\psi$ appearing in Eq.
(17) can be expressed in terms of the model parameters as

$\displaystyle\tan\psi=\frac{2\alpha\sin\phi_{ab}}{5-2\alpha\cos\phi_{ab}},\quad\tan
2\theta=\frac{2\sqrt{6}}{\cos\psi+2\alpha\cos(\psi+\phi_{ab})}.$ (21)

As the charged lepton mass matrix is diagonal, to obtain the correlation among
the mixing angles and phases, we can compare $U_{\nu}=U_{TB}U_{23}U_{m}$ of
Eq. (19) with $U_{\rm PMNS}$ given in Eq. (20). These correlations can be
written as deMedeirosVarzielas:2012apl ; Shimizu:2017fgu ; Luhn:2013lkn

$\displaystyle\sin\theta_{13}e^{-i\delta_{\rm
CP}}=\frac{e^{-i\psi}\sin\theta}{\sqrt{3}},\quad\sin^{2}\theta_{12}=1-\frac{2}{3-\sin^{2}\theta},~{}~{}\sin^{2}\theta_{23}=\frac{1}{2}\Big{(}1-\frac{\sqrt{6}\sin
2\theta\cos\psi}{3-\sin^{2}\theta}\Big{)}.$ (22)

The above relations among the three mixing angles imply a mutual correlation.
These correlations are the unique feature of the considered $A_{4}$ flavor
symmetry, giving rise to the TM1 mixing scheme. More specifically, relations
in Eq. (22) are general for the TM1 mixing scheme Luhn:2013lkn ;
deMedeirosVarzielas:2012apl ; Shimizu:2017fgu  where the mixing angles
$\theta_{13}$, $\theta_{12}$ are being correlated to each other. The
correlation plot among these mixing angles can be found in Ref.
deMedeirosVarzielas:2012apl  where $\sin^{2}\theta_{12}$ is restricted to some
narrow range corresponding to the $3\sigma$ regions of $\sin^{2}\theta_{13}$.
Relations in Eq. (21) are unique for the considered FSS1 model. From Eqs.
(21)-(22), it is clear that the angle $\theta$ and the associated phase $\psi$
in $U_{23}$ can be linked with the parameters involved in $M_{\nu}$. Relations
in Eq. (22) imply that $\delta_{\rm CP}=\psi$ when $\sin\theta>0$, and
$\delta_{\rm CP}=\psi\pm\pi$ for $\sin\theta<0$ which can be written in a
compact form as $\tan\delta_{\rm CP}=\tan\psi$. Now, from Eq. (12), the
complex mass eigenvalues are calculated to be

$\displaystyle m_{1}^{c}$ $\displaystyle=$ $\displaystyle 0,$ (23)
$\displaystyle m_{2}^{c}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}-2a+5b-\sqrt{4a^{2}+4ab+25b^{2}}\Big{)},$ (24)
$\displaystyle m_{3}^{c}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}-2a+5b+\sqrt{4a^{2}+4ab+25b^{2}}\Big{)}.$ (25)

The real and positive mass eigenvalues are calculated as

$\displaystyle m_{1}$ $\displaystyle=$ $\displaystyle 0,$ (26) $\displaystyle
m_{2}$ $\displaystyle=$
$\displaystyle\frac{|b|}{2}\big{[}(5-2\alpha\cos\phi_{ab}-P)^{2}+(Q+2\alpha\sin\phi_{ab})^{2}\big{]}^{1/2},$
(27) $\displaystyle m_{3}$ $\displaystyle=$
$\displaystyle\frac{|b|}{2}\big{[}(5-2\alpha\cos\phi_{ab}+P)^{2}+(Q-2\alpha\sin\phi_{ab})^{2}\big{]}^{1/2}.$
(28)

where

$\displaystyle P^{2}=\frac{M\pm\sqrt{M^{2}+N^{2}}}{2},\quad
Q^{2}=\frac{-M\pm\sqrt{M^{2}+N^{2}}}{2},$ (29) $\displaystyle
M=25+4\alpha\cos\phi_{ab}+4\alpha^{2}\cos 2\phi_{ab},\quad
N=4\alpha\sin\phi_{ab}+4\alpha^{2}\sin 2\phi_{ab}.$ (30)

Now, from Eq.(23) to Eq. (25), we get the phases associated with the complex
eigenvalues $m^{c}_{1,2,3}$. These phases can be written as
$\gamma_{i}=\phi_{b}+\phi_{i}$, $i=2,3$. $i=1$ is excluded here as the
lightest mass eigenvalue is zero, the phase associated with $m_{1}^{c}$ is
$\gamma_{1}=0$. Now, $\phi_{2,3}$ in our model can be written as

$\displaystyle\phi_{2}=\tan^{-1}\Big{(}\frac{Q+2\alpha\sin\phi_{ab}}{5-2\alpha\cos\phi_{ab}-P}\Big{)},\quad\phi_{3}=\tan^{-1}\Big{(}\frac{Q-2\alpha\sin\phi_{ab}}{5-2\alpha\cos\phi_{ab}+P}\Big{)}$
(31)

Using the above relations, we can calculate the Majorana phase $\alpha_{32}$
in $U_{m}$, which can be written as

$\displaystyle\alpha_{32}=\tan^{-1}\Big{(}\frac{Q-2\alpha\sin\phi_{ab}}{5-2\alpha\cos\phi_{ab}+P}\Big{)}-\tan^{-1}\Big{(}\frac{Q+2\alpha\sin\phi_{ab}}{5-2\alpha\cos\phi_{ab}-P}\Big{)}.$
(32)

The phase $\phi_{b}$ is irrelevant while calculating the Majorana phase as it
is the difference between $\gamma_{3}$ and $\gamma_{2}$. Finally, the Jarlskog
invariant $J_{\rm CP}$ Jarlskog:1985ht ; Krastev:1988yu

$\displaystyle J_{\rm
CP}={\mathcal{I}}(U_{11}U_{22}U_{12}^{*}U_{21}^{*})=s_{12}c_{12}s_{13}c_{13}^{2}s_{23}c_{23}\sin\delta_{\rm
CP}$ (33)

will be used to quantify the $CP$ violation in the FSS1 model. From Eqs.
(21)-(32), we observe that the mixing angles and all the phases depend on
parameters $\alpha$ and $\phi_{ab}$ while the light neutrino masses depend on
these parameters as well as on $|b|$. Now, we will estimate these model
parameters ($\alpha$, $|b|$ and $\phi_{ab}$ ) using neutrino oscillation data
on neutrino mixing angles and mass squared differences. With measured values
deSalas:2020pgw ; Esteban:2020cvm ; Capozzi:2021fjo  of mixing angles
$\theta_{13}$, $\theta_{12}$ and $\theta_{23}$, mass-squared differences
$\Delta m_{21}^{2}$, $|\Delta m_{31}^{2}|$ (mentioned in Eq. (1), taken from
Esteban:2020cvm ) and the ratio $r$ defined in Eq. (2), we first estimate
$\alpha$ and the phase $\phi_{ab}$ using the $3\sigma$ range of neutrino
oscillation data. The allowed ranges for $\alpha$ and $\phi_{ab}$ are plotted
in the left panel of Fig 1 in the $\alpha-\phi_{ab}$ plane. Here, we find that
the allowed ranges of $\alpha$ vary between $4.82-5.27$ whereas two distinct
regions of $\phi_{ab}$ are allowed between $4.72-4.76$ and $5.03-5.06$ radian.

Figure 1: The allowed regions for $\alpha$ and $\phi_{ab}$ (left panel), and
$|a|$ and $|b|$ (right panel) for the $3\sigma$ ranges of neutrino oscillation
data Esteban:2020cvm .

As mentioned earlier, the effective light neutrino mass matrix in the FSS1
model has rank 2 due to the considered $A_{4}$ symmetry. Hence we obtain two
massive light neutrinos as given in Eqs. (26)-(28), predicting only the normal
ordering (NO) of light neutrino masses. To obtain the absolute values of
$m_{2}$ and $m_{3}$, we need to find the overall factor $|b|$ appearing in
Eqs. (27) and (28). Though the factor $|b|$ cancels out while calculating $r$,
it can be calculated by fitting solar or atmospheric mass-squared differences
after knowing $\alpha$ and $\phi_{ab}$ from the left panel of Fig. 1. After
evaluating $|b|$, $|a|$ can be easily estimated using the relation
$|a|=\alpha|b|$. Hence, in the right panel of Fig. 1, we have plotted the
allowed region in the $|a|-|b|$ plane for the $3\sigma$ range of neutrino
oscillation data. Corresponding to two distinct regions of $\phi_{ab}$ in the
left panel there also exist two distinct regions of the parameters $|a|$ as
shown in the right panel of Fig. 1. Now, from Eqs. (27) and (28), we find that
the light neutrino masses are functions of both $a$ and $b$, whose origin lies
in the type-I seesaw and the scotogenic contributions, respectively. Since
$m_{1}=0$ in the FSS1 framework, $m_{2}$ and $m_{3}$ are proportional to the
solar and atmospheric mass-squared differences.

Figure 2: $r$ vs $|b|$ with variation of $|a|$.

Hence, in Fig 2, we have plotted variation of $|b|$ with respect to $|a|$
(represented by the color variation from blue to red) to reproduce correct
$r$. This plot shows that the hierarchy between $|a|$ and $|b|$ essentially
explains the observed value of the ratio of the solar to atmospheric mass-
squared differences $r$, where $|a|$ is the dominant contribution originated
from the type-I seesaw.

Figure 3: $\sin^{2}\theta_{23}-\delta_{\rm CP}$ correlation plot (denoted by
red dots) is presented in the left panel. The grey-shaded region represents
the generic prediction for the TM1 mixing scheme. In the right panel,
$\delta_{\rm CP}$ is plotted against the relative (between $a$ and $b$) phase
factor $\phi_{ab}$.

With the allowed values of $\alpha$, $\phi_{ab}$, $|a|$ and $|b|$ obtained
from Fig. 1, we are in a position to study the correlations among neutrino
mixing angles, phases, and masses. Due to the presence of the $A_{4}$ discrete
flavor symmetry we have realized the TM1 mixing scheme yielding interesting
correlations among the observables appearing in the neutrino mixing. It is
well known that there are still some unsettled issues in the measurement of
$\theta_{23}$ and $\delta_{\rm CP}$ deSalas:2020pgw ; Esteban:2020cvm ;
Capozzi:2021fjo . These are (i) octant of $\theta_{23}$ ( i.e., whether
$\theta_{23}<45^{o}$ or $\theta_{23}>45^{o}$) and (ii) the precise measurement
of $\delta_{\rm CP}$. Following Eqs. (21) and (22), we find a correlation
between the atmospheric mixing angle $\theta_{23}$ and the Dirac CP phase
$\delta_{\rm CP}$ for TM1 mixing scheme. Together with Eq. (21) and Fig. 1
within the FSS1 framework, the predictions regarding $\theta_{23}$ and
$\delta_{\rm CP}$ for the TM1 scheme gets constrained further as plotted in
the left panel of Fig. 3. Here, the gray-shaded region represents the TM1
prediction in the $\theta_{23}-\delta_{\rm CP}$ plane where the red-shaded
region is the prediction for the FSS1 framework. We find that our model
prefers the higher octant of $\theta_{23}$ for narrow regions of $\delta_{\rm
CP}$. The allowed regions of $\sin^{2}\theta_{23}$ are $0.531-0.544$ and
$0.580-0.595$ whereas the allowed regions of $\delta_{\rm CP}$ are
$\pm(1.44-1.12)$ rad. Here, the relative phase between type-I and scotogenic
contributions (denoted by $\phi_{ab}$) is the source of CP violation in the
lepton sector, see Eq. (21) and subsequent discussion. Hence, in the right
panel of Fig. 3, we have plotted the dependence of $\delta_{\rm CP}$ on
$\phi_{ab}$ (the relative phase between $a$ and $b$) denoted by the red shaded
regions.

Figure 4: Correlation plot between the Dirac CP phase $\delta_{CP}$ and the
Majorana phase $\alpha_{32}$ (left panel) & Dependence of the Jarlskog
invariant $J_{\rm CP}$ on $\phi_{ab}$ (right panel).

It is established that the Majorana phases cannot be constrained from neutrino
oscillation data directly, as they do not appear in the neutrino oscillation
probability Bilenky:1980cx ; Langacker:1986jv ; Gluza:2001de . In the FSS1
framework, the Majorana phase $\alpha_{23}$ can be constrained using Eq. (32)
with the allowed range for $\alpha$ and $\phi_{ab}$. Hence, in the left panel
of Fig. 4, we showed the correlation among the CP phases in
$\alpha_{23}-\delta_{CP}$ plane, and the Majorana phase is found to be within
the range $0.82-0.95$ and $1.58-1.67$ radian. Estimating the Majorana phase
will play a crucial role in predicting the effective mass parameter appearing
in the neutrinoless double decay Bilenky:1980cx . Now, following Eq. (33), we
have plotted the Jarlskog invariant $J_{\rm CP}$ as a function of $\phi_{ab}$
in the right panel of Fig. 4. Here the magnitude of $J_{\rm CP}$ is found to
be within the range $0.0290-0.0313$ and $0.0318-0.0344$. Finally, with the
allowed parameter space obtained in Fig. 1, we make predictions for the light
neutrino masses ($m_{2},m_{3}$), their sum ($\sum m_{i}$) and the effective
mass parameter appearing in the neutrinoless double decay ($m_{\beta\beta}$)
as summarized in Table 2.

$m_{2}$ | $m_{3}$ | $\sum m_{i}$ | $m_{\beta\beta}$
---|---|---|---
$8.3-9.0$ | $49.7-51.3$ | $58.0-60.3$ | ${1.61-3.85}$
Table 2: Predictions for $m_{2}$, $m_{3}$, $\sum m_{i}$ and $m_{\beta\beta}$
(all in meV) for the FSS1 framework.

The prediction for $\sum m_{i}$ is consistent with cosmological observation
Planck:2018vyg  whereas prediction for $m_{\beta\beta}$ falls below the upper
limit provided by the next generation double beta decay experiment nEXO
nEXO:2021ujk .

##  5 Scalar Potential

The FSS1 model considered here consists of two $SU(2)$ doublet scalars $H$ and
$\eta$. To obtain the flavor structure of the leptons we have four flavons
$\phi_{s},\phi_{a},\phi_{T}$ and $\xi$ as mentioned in Tab 1. These $SU(2)$
singlet flavons are considered to be very heavy compared to $H$ and $\eta$ and
hence remain decoupled from the low energy phenomenology of scalars. The low
energy scalar potential of the model can be written as

$\displaystyle V$ $\displaystyle=$
$\displaystyle-\mu_{1}^{2}(H^{\dagger}H)+\mu_{2}^{2}(\eta^{\dagger}\eta)+\lambda_{1}(H^{\dagger}H)^{2}+\lambda_{2}(\eta^{\dagger}\eta)^{2}+\lambda_{3}(H^{\dagger}H)(\eta^{\dagger}\eta)+\lambda_{4}(H^{\dagger}\eta)(\eta^{\dagger}H)$
(34)
$\displaystyle+\frac{\lambda_{5}}{2}\\{(H^{\dagger}\eta)(H^{\dagger}\eta)+h.c.\\}$

The doublets in the model can be parameterized as

$\displaystyle H=\begin{pmatrix}H^{+}\\\
v/\sqrt{2}+(h+i\zeta)/\sqrt{2}\end{pmatrix},\quad\eta=\begin{pmatrix}\eta^{+}\\\
(\eta_{R}+i\eta_{I})/\sqrt{2}\end{pmatrix}.$ (35)

The electroweak gauge symmetry is given by

$\displaystyle H=\begin{pmatrix}0\\\
v/\sqrt{2}\end{pmatrix},\quad\eta=\begin{pmatrix}0\\\ 0\end{pmatrix}.$ (36)

The above symmetry breaking pattern ensures that the $Z_{2}$ symmetry will
remain unbroken and results in two CP even scalars ($h,\eta_{R}$), one CP odd
neutral scalar $\eta_{I}$ in addition to a pair of charged scalars
($\eta^{\pm}$). Due to the dark $Z_{2}$ symmetry, there is no mixing between
$h$ and $\eta_{R}$, and $h$ plays the role of the SM Higgs boson. The $Z_{2}$
symmetry also ensures the stability of the lightest scalar ($\eta_{R}$ or
$\eta_{I}$) that can act as a dark matter candidate. The masses of all scalars
can be written in terms of the following parameters

$\displaystyle\\{\mu_{2},\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5}\\}.$
(37)

These parameters can be written in terms of physical masses of scalars as
Arhrib:2012ia

$\displaystyle\lambda_{1}=\frac{m_{h}^{2}}{2v^{2}},\quad\lambda_{3}=\frac{2}{v^{2}}(M_{\eta^{\pm}}^{2}-\mu_{2}^{2}),$
(38)
$\displaystyle\lambda_{4}=\frac{M_{\eta_{R}}^{2}+M_{\eta_{I}}^{2}-2M_{\eta^{\pm}}^{2}}{v^{2}},\quad\lambda_{5}=\frac{M_{\eta_{R}}^{2}-M_{\eta_{I}}^{2}}{v^{2}}.$
(39)

We can choose all the $\lambda$s as free parameters or, equivalently the four
physical scalar masses, $\lambda_{2}$ and $\mu_{2}$, namely

$\displaystyle\\{\mu_{2}^{2},m_{h},M_{\eta_{R}},M_{\eta_{I}},M_{\eta^{\pm}},\lambda_{2}\\}.$
(40)

The quartic couplings are constrained theoretically by perturbativity and
vacuum stability. We force the scalar potential to be perturbative which
requires all quartic couplings of the scalar potential to obey

$\displaystyle|\lambda_{i}|\leq 8\pi.$ (41)

To get the scalar potential to be bounded from below, the following conditions
can be obtained Lindner:2016kqk ; Branco:2011iw

$\displaystyle\lambda_{1,2}>0\quad\text{and}\quad\lambda_{3}+\lambda_{4}-|\lambda_{5}|+2\sqrt{\lambda_{1}\lambda_{2}}>0\quad\text{and}\quad\lambda_{3}+2\sqrt{\lambda_{1}\lambda_{2}}>0.$
(42)

Eq. (42) give constraints based on the bare couplings of the Lagrangian.
Another approach with running parameters of the model evaluated at the cut-off
scale $\Lambda$ of the theory is possible, see for instance Mandal:2021yph ;
Lindner:2016kqk ; Branco:2011iw ; Hernandez:2021tii ; CarcamoHernandez:2023wzf
. Apart from these theoretical constraints, $\lambda_{3}$, $\lambda_{4}$ and
$\lambda_{5}$ given in Eq. (39) can also be constrained from experimental and
phenomenological constraints. As we will discuss in the subsequent sections,
$\lambda_{5}$ is crucially relevant in determining the scotogenic Yukawa
coupling and hence is constrained from DM relic density, direct search
constraints as well as the neutrino phenomenology. Similarly, $\lambda_{3,4}$
can also be constrained from DM direct search as well as SM Higgs diphoton
signal strength.

The presence of the doublet scalar $\eta$ in our model can have important
consequences in the context of CDF-II $W$-boson mass anomaly CDF:2022hxs , for
instance see Babu:2022pdn ; Fan:2022dck ; Asadi:2022xiy ; Strumia:2022qkt ;
Arcadi:2022dmt ; Borah:2022zim ; Borah:2022obi ; Borah:2023hqw , as it can
affect the EW precision observables $S,T$, and $U$ ParticleDataGroup:2022pth .
Through the self-energy correction of the W-boson with the doublet scalar in
the loop, the $W$-boson mass can be increased from the SM prediction to the
value obtained by the CDF-II collaboration. Parameterizing the new physics
effects in terms of the $S,T,U$ parameters as Zhang:2006vt ;
CentellesChulia:2022vpz :

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{1}{12\pi}\ln\frac{M_{\eta^{0}}^{2}}{M_{\eta^{+}}^{2}},$
$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{4\sqrt{2}\pi^{2}\alpha_{em}}\Bigg{(}\frac{M_{\eta^{0}}^{2}+M_{\eta^{+}}^{2}}{2}-\frac{M_{\eta^{0}}^{2}M_{\eta^{+}}^{2}}{M_{\eta^{0}}^{2}-M_{\eta^{+}}^{2}}\ln\frac{M_{\eta^{+}}^{2}}{M_{\eta^{0}}^{2}}\Bigg{)},$
(43) $\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{1}{12\pi}\Bigg{[}\frac{(M_{\eta^{0}}^{2}+M_{\eta^{+}}^{2})(M_{\eta^{0}}^{4}-4M_{\eta^{0}}^{2}M_{\eta^{+}}^{2}+M_{\eta^{+}}^{4})\ln\big{(}\frac{M_{\eta^{+}}^{2}}{M_{\eta^{0}}^{2}}\big{)}}{(M_{\eta^{+}}^{2}-M_{\eta^{0}}^{2})^{3}}-\frac{5M_{\eta^{0}}^{4}-22M_{\eta^{0}}^{2}M_{\eta^{+}}^{2}+5M_{\eta^{+}}^{4}}{3(M_{\eta^{+}}^{2}-M_{\eta^{0}}^{2})}\Bigg{]},$

where $\eta_{0}=(\eta_{R}+i\eta_{I})/\sqrt{2}$, the W boson mass can be
written as Peskin:1991sw

$\displaystyle M_{W}^{2}=({M_{W}^{2}})_{\rm
SM}+\frac{\alpha_{em}\cos^{2}\theta_{W}}{\cos^{2}\theta_{W}-\sin^{2}\theta_{W}}M_{Z}^{2}\Bigg{[}-\frac{1}{2}S+\cos^{2}\theta_{W}T+\frac{(\cos^{2}\theta_{W}-\sin^{2}\theta_{W})}{4\sin^{2}\theta_{W}}U\Bigg{]}.$
(44)

where $\alpha_{em}$ is the fine structure constant, $\theta_{W}$ is the
Weinberg angle, and $({M_{W}^{2}})_{\rm SM}$ is the SM predicted value of W
boson mass. The dominant correction to $M_{W}$ comes from the $T$-parameter
which is very much sensitive to the mass difference between the charged scalar
and the neutral scalar components of the inert doublet. And the CDF-II $W$
mass can be obtained if the mass difference between $\eta^{+}$ and $\eta_{0}$
is around 80 to 100 GeV. However, we should stress that CDF-II data on $W$
mass are in contradiction with global electroweak $e^{+}e^{-}$ fits and recent
ATLAS LHC analysis, with systematic uncertainty improved by 15% ATLAS:2023fsi
and optimised reconstruction of the W-boson transverse momentum ATLAS:2023llf
.

##  6 DM Phenomenology for the FSS1 model

In this FSS1 framework, both type-I seesaw and scotogenic mechanisms are
combined to obtain correct neutrino masses and mixing. The scotogenic
contribution contains two potential candidates for DM: the lightest neutral
scalar and the singlet fermion. Determining the DM relic density hinges on
these candidates’ production mechanisms during the early Universe. While the
literature extensively covers the scalar DM phenomenology444For scalar dark
matter phenomenology within the scoto-seesaw framework, see Ref.
Mandal:2021yph , which aligns with the inert doublet model (IDM) perspective,
our focus here is on the singlet fermion denoted as $f$, an odd $Z_{2}$
particle in the scoto-seesaw scenario. We explore various mechanisms that can
yield the correct relic density and delve into the associated parameter space.
Since $f$ is a gauge singlet, its production mechanism is intricately tied to
its Yukawa couplings, see Eq. (11) and Eq. (14), with SM leptons and the inert
doublet scalar $\eta$. The magnitude of these Yukawa couplings plays a pivotal
role in determining whether the correct relic density can be achieved through
thermal freeze-out or freeze-in mechanism.

Figure 5: Correlation between $\lambda_{5}$ defined in Eqs. (34) and (39), and
$\kappa$ defined in Eq. (14), consistent with the constrain from neutrino
oscillation data, i.e. $|b|\in[0.0048,0.0056]$ eV (see Fig. 1).

Relic Density of DM:
As outlined in the preceding section, in our FSS1 model, the scotogenic
contribution to neutrino mass is parameterized in terms of the parameter $b$
given in Eq.(14), which also plays a crucial role in explaining the observed
neutrino oscillation data, is constrained in a range $|b|\in[0.0048,0.0056]$
eV. Thus to obtain the magnitude of Yukawa couplings that can satisfy this
constraint with the masses of the loop particles with masses of the order
$\mathcal{O}(1-10^{3})$ GeV, we perform a numerical scan, the result of which
is shown in the plane of $\kappa$ versus $\lambda_{5}$ in Fig. 5. We have used
Eq. (14) and Eq. (39) to obtain the estimations of $\kappa$ and $\lambda_{5}$.
It is worth noticing from the neutrino mass expression that in the limit
$\lambda_{5}\to 0$, the scotogenic contribution to neutrino mass vanishes.
This is due to the fact that in this limit, the CP-even and CP-odd scalars
$\eta_{R}$ and $\eta_{I}$ become degenerate and thus
$\mathcal{F}(M_{\eta_{R}},M_{\eta_{I}},M_{f})\to 0$. Thus to satisfy the
constraints on $b$, from neutrino oscillation data, if $\lambda_{5}$ is made
small, then $\kappa$ can be enhanced and vice versa. We see that with this
constraint on $b$, it is not possible to obtain Yukawa couplings smaller than
$\mathcal{O}(10^{-6})$ even if $\lambda_{5}$ is $\mathcal{O}(1)$.
Consequently, the singlet fermion also remains in thermal equilibrium with the
SM bath. This equilibrium is guaranteed by the doublet scalar $\eta$, which,
due to its gauge interactions, consistently maintains equilibrium with the SM
bath during the early stages of the Universe. Hence the DM relic density is
governed through the WIMP mechanism. Several pertinent processes contribute to
the relic density of DM. Specifically, the essential parameters influencing
the relic density are the Yukawa couplings and the mass difference between the
singlet fermion $f$ and other particles in the dark sector, namely
$\eta_{R,I},\eta^{\pm}$.

For WIMP type DM which is produced thermally in the early universe, its
thermal relic abundance can be obtained by solving the Boltzmann equation for
the evolution of the DM number density

$\frac{dn}{dt}+3Hn=-\langle\sigma v\rangle_{\rm eff}(n^{2}-(n^{eq})^{2})$ (45)

where $n=\sum_{i}n_{i}$ represents the total number density of all the dark
sector particles and $n^{\rm eq}$ is the equilibrium number density.
$\langle\sigma v\rangle_{\rm eff}$ represents the effective annihilation
cross-section which takes into account all number changing processes for DM
freeze-out. It can be written as Griest:1990kh :

$\displaystyle\langle\sigma v\rangle_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{f}}{g^{2}_{\rm eff}}\langle\sigma
v\rangle_{ff}+\frac{g_{f}g_{\eta_{R}}}{g^{2}_{\rm eff}}\langle\sigma
v\rangle_{f\eta_{R}}(1+\Delta_{\eta_{R}})^{{3}/{2}}\exp(-x\Delta_{\eta_{R}})$
(46) $\displaystyle+$ $\displaystyle\frac{g_{f}g_{\eta_{I}}}{g^{2}_{\rm
eff}}\langle\sigma
v\rangle_{f\eta_{I}}(1+\Delta_{\eta_{I}})^{{3}/{2}}\exp(-x\Delta_{\eta_{I}})$
$\displaystyle+$ $\displaystyle\frac{g_{f}g_{\eta^{\pm}}}{g^{2}_{\rm
eff}}\langle\sigma
v\rangle_{f\eta^{\pm}}(1+\Delta_{\eta^{\pm}})^{{3}/{2}}\exp(-x\Delta_{\eta^{\pm}})+\frac{g^{2}_{\eta_{R}}}{g^{2}_{\rm
eff}}\langle\sigma
v\rangle_{\eta_{R}\eta_{R}}(1+\Delta_{\eta_{R}})^{3}\exp(-2x\Delta_{\eta_{R}})$
$\displaystyle+$ $\displaystyle\frac{g_{\eta_{R}}g_{\eta_{I}}}{g^{2}_{\rm
eff}}\langle\sigma
v\rangle_{\eta_{R}\eta_{I}}(1+\Delta_{\eta_{R}})^{3/2}(1+\Delta_{\eta_{I}})^{3/2}\exp\left(-x(\Delta_{\eta_{R}}+\Delta_{\eta_{I}})\right)$
$\displaystyle+$ $\displaystyle\frac{g_{\eta_{R}}g_{\eta^{\pm}}}{g^{2}_{\rm
eff}}\langle\sigma
v\rangle_{\eta_{R}\eta^{\pm}}(1+\Delta_{\eta_{R}})^{3/2}(1+\Delta_{\eta^{\pm}})^{3/2}\exp\left(-x(\Delta_{\eta_{R}}+\Delta_{\eta^{\pm}})\right)$
$\displaystyle+$ $\displaystyle\frac{g^{2}_{\eta_{I}}}{g^{2}_{\rm
eff}}\langle\sigma
v\rangle_{\eta_{I}\eta_{I}}(1+\Delta_{\eta_{I}})^{3}\exp(-2x\Delta_{\eta_{I}})+\frac{g^{2}_{\eta^{\pm}}}{g^{2}_{\rm
eff}}\langle\sigma
v\rangle_{\eta^{\pm}\eta^{\mp}}(1+\Delta_{\eta^{\pm}})^{3}\exp(-2x\Delta_{\eta^{\pm}})$
$\displaystyle+$ $\displaystyle\frac{g_{\eta_{I}}g_{\eta^{\pm}}}{g^{2}_{\rm
eff}}\langle\sigma
v\rangle_{\eta_{I}\eta^{\pm}}(1+\Delta_{\eta_{I}})^{3/2}(1+\Delta_{\eta^{\pm}})^{3/2}\exp(-x(\Delta_{\eta_{I}}+\Delta_{\eta^{\pm}})),$

where $g_{f},g_{\eta_{R}},g_{\eta_{I}}~{}{\rm and}~{}g_{\eta^{\pm}}$ represent
the internal degrees of $f,\eta_{R},\eta_{I}$ and $\eta^{\pm}$ respectively
and $\Delta_{i}$ stands for the ratio $(M_{i}-M_{f})/M_{f}$ with $M_{i}$
denoting masses of $\eta_{R},\eta_{I},\eta^{\pm}$. Here $g_{\rm eff}$ is the
effective degree of freedom which is given by

$\displaystyle g_{\rm eff}$ $\displaystyle=$ $\displaystyle
g_{f}+g_{\eta_{R}}(1+\Delta_{\eta_{R}})^{3/2}\exp(-x\Delta_{\eta_{R}})+g_{\eta_{I}}(1+\Delta_{\eta_{I}})^{3/2}\exp(-x\Delta_{\eta_{I}})$
(47) $\displaystyle+$ $\displaystyle
g_{\eta^{\pm}}(1+\Delta_{\eta^{\pm}})^{3/2}\exp(-x\Delta_{\eta^{\pm}}),$

and $x$ is the dimensionless parameter $M_{f}/T$. The relic density of DM $f$
can then be evaluated as :

$\displaystyle\Omega_{f}h^{2}=\frac{1.09\times 10^{9}{\rm
GeV}^{-1}}{\sqrt{g_{*}}M_{Pl}}\left[\int_{x_{\rm
F.O.}}^{\infty}dx~{}\frac{\langle\sigma v\rangle_{\rm
eff}}{x^{2}}\right]^{-1}.$ (48)

Here $M_{Pl}$ is the Planck mass, $x_{\rm F.O.}=M_{f}/T_{\rm F.O.}$, and
$T_{\rm F.O.}$ denotes the freeze-out temperature of $f$. For this scenario we
have implemented the model in micrOMEGAs Belanger:2013oya  to calculate the
relic abundance of $f$. As evident from Eq.(46), the mass difference between
the dark sector particles, namely $f$ and $\eta$, along with the coupling
$\kappa$ is pivotal in determining the ultimate relic abundance of dark matter
in this configuration. Smaller mass splittings can induce effective co-
annihilations between $\eta$ and $f$, potentially reducing the relic abundance
to the observed ballpark. The dominant number changing processes relevant in
governing the relic density are as shown in Figs. 6, 7, and 8.

Figure 6: Annihilations of dark matter into SM leptons Figure 7:  Co-
annihilations among the dark matter and the other dark sector particles.
Figure 8: Annihilations of the scalar co-annihilation partners.

Clearly, the processes pivotal in establishing the relic abundance of dark
matter fall into three distinct categories: I) the annihilation of dark matter
particles into both charged and neutral SM leptons (Fig. 6), II) the co-
annihilation of dark matter particles with scalar particles from the dark
sector (Fig. 7), and III) the co-annihilation contribution arising from the
annihilation of dark-sector scalars (Fig. 8). As denoted by Eq. ( 46), the co-
annihilation contribution to the effective annihilation cross-section
$\langle\sigma v\rangle_{\rm eff}$ is predominantly shaped by the mass
difference between dark matter and the dark scalars.

Figure 9: DM relic density as a function of DM mass with the Yukawa couplings
(left panel) and mass difference $M_{\eta_{R}}-M_{f}$ (right panel) varied
randomly as mentioned in the inset of the figure. Horizontal cyan line
represents observed relic density Aghanim:2018eyx

.

To elucidate the influence of Yukawa couplings and mass splitting on the relic
abundance of dark matter, in Fig. 9, we illustrate the variation of relic
density with the dark matter mass. In the left panel of Fig. 9, the Yukawa
coupling $\kappa$ is varied within the range $\kappa\in[10^{-4},10^{-3}]$,
while the mass difference between the lightest neutral scalar $\eta_{R}$ and
$f$ (i.e., $M_{\eta_{R}}-M_{f}$) is varied in three different ranges, as
indicated in the figure’s inset. Evidently, an increase in the mass difference
leads to a corresponding increase in the relic density. This trend arises
because the co-annihilation contribution to $\langle\sigma v\rangle_{\rm eff}$
gradually diminishes with an increase in $(M_{\eta_{R}}-M_{f})$, thereby
boosting the relic abundance of $f$.

Expanding the analysis, in the right panel of Fig. 9, the mass difference
$(M_{\eta_{R}}-M_{f})$ is varied within a small range of $[50,60]$ GeV and the
variation of relic density with dark matter mass is then showcased for three
different ranges of Yukawa couplings, as outlined in the figure’s inset. It is
evident that an increase in Yukawa coupling leads to a decrease in relic
density, attributed to the increase in $\langle\sigma v\rangle_{\rm eff}$.
Additionally, an intriguing observation from this figure is that, when
$(M_{\eta_{R}}-M_{f})\in[50,60]$ GeV and Yukawa couplings are small (i.e.,
$\kappa\lesssim\mathcal{O}(10^{-3})$), the relic density does not change with
further reduction in Yukawa couplings, for dark matter masses exceeding 500
GeV, as indicated by the red and blue-colored points. This phenomenon can be
explained by the fact that, in scenarios with small Yukawa couplings, neither
the annihilation of DM nor the co-annihilation of DM with dark sector scalars
efficiently affects the relic density. Instead, it is primarily determined by
the co-annihilation contribution from the annihilation of dark scalar
partners. Conversely, in situations where Yukawa couplings are large and the
mass difference is substantial, the relic density is primarily influenced by
the annihilation of DM, as indicated by the purple-colored points.

Figure 10: Correct relic density satisfying points in the plane of DM mass
$M_{f}$ and $\kappa$. The color code shows the value of $M_{\eta_{R}}-M_{f}$.

Thus, in summary of the effects that affect the relic density, in scenarios
characterized by small Yukawa couplings and substantial mass differences
($\Delta_{i}$), the relic density is predominantly governed by the co-
annihilation contribution from the dark scalars. In such scenarios, DM
annihilation becomes subdominant, and co-annihilation among dark matter and
dark scalars is suppressed due to the large mass splitting. Conversely, when
the mass difference between dark matter and dark scalars is not considerably
large, co-annihilation among DM and dark scalars, as well as dark scalar
annihilations, play a crucial role in determining the relic density. Only in
cases where Yukawa couplings are significantly large, and the mass difference
is also substantial, dark matter annihilations become relevant for achieving
the correct relic density.

We present the parameter space satisfying correct relic density in the plane
of $M_{f}$ and $\kappa$ with the color code representing the corresponding
value of $M_{\eta_{R}}-M_{f}$ in Fig. 10. The grey-colored points are ruled
out by imposing a conservative limit on the doublet scalar mass given by the
LEP experiment of about $M_{\eta}\geq 100$ GeV. It is evident that, when DM
mass is small and $\kappa$ is small, the effective annihilation cross-section
is very small and thus it is not possible to achieve correct relic density
even with co-annihilation contributions. Thus we obtain a over abundant region
below $M_{f}$ around $30$ GeV and $\kappa\lesssim\mathcal{O}(10^{-2})$. In the
small DM mass range $M_{f}\lesssim 100$ GeV, correct relic density can be
obtained only when the Yukawa couplings are significant i.e.
$\kappa\sim\mathcal{O}(1)$ such that DM annihilation cross-section is
appropriate to match the thermal cross-section as in this region the co-
annihilation contributions are negligible. When DM mass is greater than $100$
GeV, and Yukawa coupling $\kappa\lesssim\mathcal{O}(10^{-2})$, we see that
with increase in DM mass, the $M_{\eta_{R}}-M_{f}$ shows a gradual decrease to
achieve the correct relic density. This is attributed to the fact that as the
DM mass increases, the effective cross-section gradually decreases thereby
increasing the relic density and thus it needs more effective co-annihilations
which is possible by decreasing the $M_{\eta_{R}}-M_{f}$, to bring the relic
density to correct ballpark. We also observe an under-abundant region when
$\kappa\gtrsim\mathcal{O}(10^{-1})$ even with very large $M_{\eta_{R}}-M_{f}$.
This is due to the fact that, with very large $\kappa$ the DM annihilation
cross-section is large. So even if the co-annihilation contribution is
suppressed because of large $M_{\eta_{R}}-M_{f}$, it is still not possible to
achieve correct density.

Direct Detection of DM:
As the sole interaction connecting $f$ with SM particles is the Yukawa term in
Eq. (8), direct interactions between quarks and dark matter are absent at the
tree-level. However, at the one-loop level, $f$ can have effective couplings
with various SM particles, such as the photon, $Z$ boson, and Higgs boson.
Specifically, the exchange of the $Z$-boson results in the emergence of an
effective axial-vector interaction which gives rise to a Spin-dependent DM-
nucleon scattering and is dominant only when the couplings between Higgs and
$\eta$ are very small. The constraints of spin dependent DM nucleon scattering
is also relatively less constrained as compared to the spin-independent
scattering cross-sections. Thus we focus here on the spin-independent DM-
nucleon scattering rate as the direct-search experiments very stringently
constrain it.

Figure 11: Spin-independent elastic DM-nucleon scattering arising at one loop.

The detection rate of dark matter particles within a detector can experience
an amplification if the quartic couplings $\lambda_{3}$ and $\lambda_{4}$ are
significant. When this condition is met, the exchange of Higgs bosons, as
depicted in Fig. 11, leads to the emergence of an effective scalar interaction
term between the quark $q$ and the dark matter particle $f$. This interaction
is effectively described by $S_{q}~{}\bar{q}q~{}\bar{f}f$, where

$\displaystyle S_{q}$
$\displaystyle=-\frac{\kappa^{2}}{16\pi^{2}M_{h}^{2}M_{f}}\left[\lambda_{3}\mathcal{G}\left(\frac{M_{f}^{2}}{M_{\eta^{\pm}}^{2}}\right)+\frac{(\lambda_{3}+\lambda_{4})}{2}\mathcal{G}\left(\frac{M_{f}^{2}}{M_{0}^{2}}\right)\right],$
(49)

with the loop function $\mathcal{G}(x)$ defined as:

$\displaystyle\mathcal{G}_{1}(x)$ $\displaystyle=\frac{x+(1-x)\ln(1-x)}{x},$
(50)

and its value spans between 0 and 1 for $0\leq x\leq 1$.

Figure 12: Spin-independent DM-nucleon scattering cross-section as a function
of DM mass for the points satisfying correct relic density. The color code
represents the value of $\kappa$. The black solid line represents the most
recent constraint from the Lux-Zeplin experiment. The green and red dot-dashed
line represents the projected sensitivities of XENON-nT and DARWIN
experiments, respectively.

This interaction then results in the computation of the spin-independent cross
section $\sigma_{SI}$ for the interaction of $f$ with a proton and expression
for $\sigma_{\rm SI}$ is given by:

$\displaystyle\sigma_{\rm SI}$
$\displaystyle=\frac{4}{\pi}\frac{M_{f}^{2}m_{p}^{2}}{(M_{f}+m_{p})^{2}}m_{p}^{2}S^{2}_{q}f_{p}^{2},$
(51)

where $f_{p}$ represents the scalar form factor. We show the DM-nucleon
scattering cross-section as a function of DM mass for the points satisfying
correct relic density in Fig. 12. Because of the loop suppression, we observe
that even when the Yukawas and scalar quartic couplings are large, none of the
points are ruled out, and the parameter space remains safe from the DM direct
search constraints. However, interestingly, future experiments like XENON-nT
XENON:2020kmp  and DARWIN DARWIN:2016hyl  with enhanced sensitivity can probe
the Yukawa coupling $\kappa$ down to $\mathcal{O}(0.1)$.

##  7 Higgs boson in the diphoton decay channel

The SM Higgs boson has a mass of $m_{h}\simeq 125$ GeV CMS:2021kom ;
ATLAS:2022tnm , and one of the main decay channels is the diphoton, where the
SM rate for $h\to\gamma\gamma$ is dominated by the W-boson loop contribution.

Figure 13:  $R_{\gamma\gamma}$ is plotted against $m_{\eta^{+}}$ in the plane
of $\lambda_{3}$. The white-shaded region is the allowed region determined by
the ATLAS experiment ATLAS:2022tnm

.

The signal strength of $h\to\gamma\gamma$ is the ratio between the observed
cross section $pp\to h\to\gamma\gamma$ and the same quantity computed in the
SM. The observed cross section $pp\to h\to\gamma\gamma$ should match the FSS1
model prediction. Since the dominant process of the Higgs boson production is
the gluon fusion, in the first approximation, the production cross-section of
the Higgs boson in the FSS1 model is the same as in the SM. As a result,
following Arhrib:2012ia , after using narrow width approximation, the signal
strength of $h\to\gamma\gamma$ in the model can be written as

$\displaystyle R_{\gamma\gamma}=\frac{\big{[}\sigma(gg\to h)\times{\rm
Br}(h\to\gamma\gamma)\big{]}_{\rm FSS_{1}}}{\big{[}\sigma(gg\to h)\times{\rm
Br}(h\to\gamma\gamma)\big{]}_{\rm SM}}=\frac{\Gamma_{\rm SM}^{h}}{\Gamma_{\rm
FSS_{1}}^{h}}\frac{\Gamma(h\to\gamma\gamma)_{\rm
FSS_{1}}}{\Gamma(h\to\gamma\gamma)_{\rm SM}}.$ (52)

Here, the quantities with the FSS1 and SM suffixes are computed in the
flavored scoto-seesaw and the Standard Model, respectively. $\Gamma^{h}_{{{\rm
FSS}_{1}},{\rm SM}}$ is the total decay width for these models. The
$h\to\gamma\gamma$ decay is experimentally well established, and in the LHC,
the signal strength of $h\to\gamma\gamma$ is $R=1.04_{-0.09}^{+0.10}$
ATLAS:2022tnm . While computing $R_{\gamma\gamma}$, we take for the total
decay width of the Higgs boson $\Gamma_{\rm SM}^{h}=4.07\times 10^{-3}$ GeV
with a relative uncertainty of ${}^{+4.0\%}_{-3.9\%}$
ParticleDataGroup:2022pth . For a theoretical error estimate, see also
Freitas:2019bre . For detailed studies on $h\to\gamma\gamma$ decays within
miscellaneous beyond SM scenarios, see Cacciapaglia:2009ky ; Posch:2010hx ;
Arhrib:2012ia ; Hundi:2013lca ; Hundi:2023tdq . In the framework of the FSS1
model, this decay can be enhanced with the charged scalars ($\eta^{\pm}$) in
the loop, over the SM contribution with charged fermions and $W$ bosons in the
loop. Using Eq. (52), the expression for the partial decay width of
$h\to\gamma\gamma$ in the FSS1 model induced by the $\eta^{\pm}$ loop can be
written as Shifman:1979eb

$\displaystyle\Gamma(h\rightarrow\gamma\gamma)=\frac{G_{F}\alpha_{em}^{3}m_{h}^{3}}{128\sqrt{2}\pi^{3}}\Bigg{|}\sum_{f}N_{f}Q_{f}^{2}F_{1/2}(\beta_{f})+F_{1}(\beta_{W})+\frac{\lambda_{3}v^{2}}{2M_{\eta^{+}}^{2}}F_{0}(\beta_{\eta^{\pm}})\Bigg{|}^{2},$
(53)

where $\beta_{i}=4M_{i}^{2}/m_{h}^{2}$, $i=f,W,\eta^{+}$. $N_{f}$ is the color
factor, and $Q_{f}$ is the charge of quarks. $\alpha_{em}$ and $G_{F}$ are the
fine structure constant and Fermi constant. The $F$-functions in Eq. (53) are
the form factors of spin $1/2,1,0$ fields for the $h\to\gamma\gamma$ decay

$\displaystyle F_{1/2}(\beta_{f})$ $\displaystyle=$
$\displaystyle-2\beta[1+(1-\beta)f(\beta)],$ (54) $\displaystyle
F_{1}(\beta_{W})$ $\displaystyle=$
$\displaystyle[2+3\beta+3\beta(2-\beta)f(\beta)],$ (55) $\displaystyle
F_{0}(\beta_{\eta^{\pm}})$ $\displaystyle=$ $\displaystyle\beta[1-\beta
f(\beta)],$ (56)

where

$\displaystyle f(\beta)$ $\displaystyle=$
$\displaystyle\big{(}\sin^{-1}\frac{1}{\sqrt{\beta}}\big{)}^{2},\quad\beta\geq
1$ (57) $\displaystyle=$ $\displaystyle-\frac{1}{4}\Big{[}{\rm
ln}\frac{1+\sqrt{1-\beta}}{1-\sqrt{1-\beta}}-i\pi\Big{]}^{2},\quad\beta<1.$
(58)

In the FSS1 model, therefore, the total decay width of the Higgs boson can be
written as

$\displaystyle\Gamma^{\rm h}_{{\rm FSS}_{1}}=\Gamma^{h}_{\rm
SM}+\Gamma(h\to\eta_{R}\eta_{R})+\Gamma(h\to\eta_{I}\eta_{I})+\Gamma(h\to\eta^{+}\eta^{-}).$
(59)

In the above equation, the decay width of the Higgs boson to different scalar
particles is calculated using tree-level couplings

$\displaystyle\lambda_{h\eta_{R}\eta_{R}}=\frac{2}{v}(M_{\eta_{R}}^{2}-\mu_{2}^{2}),$
(60)
$\displaystyle\lambda_{h\eta_{I}\eta_{I}}=\frac{2}{v}(M_{\eta_{I}}^{2}-\mu_{2}^{2}).$
(61)

Doing numerical analysis for $\Gamma(h\to\gamma\gamma)$, we scanned the
parameters of the FSS1 model in the range

$\displaystyle 100\hskip
2.84526pt\text{GeV}<M_{\eta_{R}},M_{\eta_{I}},M_{\eta^{+}}<2000\hskip
2.84526pt\text{GeV},\quad|\lambda_{3,4,5}|\leq 4\pi.$ (62)

In Eq. (59), the total decay width of SM Higgs $h$ in the FSS1 model has three
extra contributions over the SM. In the FSS1 framework, the scalars $\eta_{R}$
and $\eta_{I}$ are not the lightest neutral $Z_{2}$ odd particles of the
theory, $f$ being the DM candidate. Thus with a judicious choice of the DM
mass $M_{f}$ (satisfying relic density and direct search constraints), the
Higgs boson decays to $\eta_{R}\eta_{R}$ and $\eta_{I}\eta_{I}$ can be made
kinematically forbidden. The result of numerical analysis is shown in Fig. 13
where the signal strength of $R_{\gamma\gamma}$ in the wide $\lambda_{3}$
range is given as a function of the charged scalar mass $M_{\eta^{+}}$. The
horizontal white region ($R_{\gamma\gamma}=1.04^{+0.10}_{-0.09}$) represents
the currently allowed region measured by the ATLAS experiment using 139 fb-1
of $pp$ collision data at $\sqrt{s}=13$ TeV ATLAS:2022tnm . This shows that
the $M_{\eta^{+}}$ masses heavier than 1000 GeV are completely safe from LHC
constraints. As follows from Eq. (53), if $\lambda_{3}<0$, the partial decay
width of $h$ is smaller than in the SM while positive $\lambda_{3}$ will give
the enhancement beyond the SM value. So, depending on the positive or negative
values of $\lambda_{3}$, we get $R_{\gamma\gamma}>1$ or $R_{\gamma\gamma}<1$,
respectively. This behaviour can be seen in Fig. 13.

##  8 Lepton Flavor Violation

The constraints on lepton flavor violating processes is an important aspect of
the FSS1 model under consideration. The model offers specific predictions,
given that the flavor structure of the Yukawa couplings is entirely dictated
by the $A_{4}$ discrete flavor symmetry and the alignment of flavon vacuums.
Along with neutrino masses, mixing and DM phenomenology, LFV decays also give
valuable insight on the FSS1 model parameters. As a consequence of the
considered flavor symmetry, the Yukawa couplings in the charged lepton sector
are diagonal, see Eq. (7). However, Yukawa couplings $y_{N}$ and $y_{s}$ in
Eq. (8) associated with the type-I seesaw and scotogenic mechanisms,
respectively, contribute to the LFV decays. These Yukawa couplings can
generate lepton flavor violating processes like $l_{\alpha}\to
l_{\beta}\gamma$ and $l_{\alpha}\to 3l_{\beta}$ ($\alpha,\beta=e,\mu,\tau$)555
To study lepton flavor violation in the pure scotogenic model, see Refs.
Toma:2013zsa ; Vicente:2014wga ; Hagedorn:2018spx .. Studies on these LFV
decays completely depend on the FSS1 model construction as described below.

In our framework, the branching ratios of the $l_{\alpha}\rightarrow
l_{\beta}\gamma$ decays for the scotogenic contribution can be written as
Rojas:2018wym ; Toma:2013zsa

$\displaystyle{\rm Br}(l_{\alpha}\rightarrow
l_{\beta}\gamma)\approx\frac{3\pi{\alpha_{em}}}{64G_{F}^{2}}|Y_{F}^{\beta*}Y_{F}^{\alpha}|^{2}\frac{1}{M_{\eta^{+}}^{4}}\Bigg{(}F_{1}\Bigg{(}\frac{M_{f}^{2}}{M_{\eta^{+}}^{2}}\Bigg{)}\Bigg{)}^{2}{\rm
Br}(l_{\alpha}\rightarrow l_{\beta}\nu_{\alpha}\bar{\nu}_{\beta}).$ (63)

Here $G_{F}$ is the Fermi constant, $Y_{F}$ is the Yukawa coupling matrix from
the scotogenic contribution given in Eq. (11). The expression for the function
$F_{1}$ is given by

$\displaystyle F_{1}(x)=\frac{1-6x+3x^{2}+2x^{3}-6x^{2}{\rm
log}x}{6(1-x)^{4}}.$ (64)

As mentioned earlier, the Yukawa couplings are determined by the considered
discrete symmetries of the model. Due to the specific VEV structure of the
$A_{4}$ triplet flavon, $Y_{F}^{\tau}=0$ as given in Eq. (11). Therefore, the
scotogenic contribution alone yields a vanishing contribution for
$\tau\rightarrow e\gamma$ and $\tau\rightarrow\mu\gamma$ lepton flavor
violating decays. So, the only non-vanishing contribution arising in the
decays of the form $\l_{\alpha}\rightarrow\l_{\beta}\gamma$ comes from the
$\mu\rightarrow e\gamma$ decay, the expression of branching fraction of which
is given by Rojas:2018wym ; Toma:2013zsa

$\displaystyle{\rm Br}(\mu\rightarrow e\gamma)$ $\displaystyle\approx$
$\displaystyle\frac{3\pi{\alpha_{em}}}{64G_{F}^{2}}|2y_{s}y_{s}^{*}\epsilon^{4}|^{2}\frac{1}{M_{\eta^{+}}^{4}}\Bigg{(}F_{1}\Bigg{(}\frac{M_{f}^{2}}{M_{\eta^{+}}^{2}}\Bigg{)}\Bigg{)}^{2}{\rm
Br}(\mu\rightarrow e\nu_{\mu}\bar{\nu}_{e})$ (65) $\displaystyle=$
$\displaystyle\frac{3\pi{\alpha_{em}}}{16G_{F}^{2}M_{f}^{2}}\Bigg{(}\frac{|b|}{\mathcal{F}(M_{\eta_{R}},M_{\eta_{I}},M_{f})}\Bigg{)}^{2}\frac{1}{M_{\eta^{+}}^{4}}\Bigg{(}F_{1}\Bigg{(}\frac{M_{f}^{2}}{M_{\eta^{+}}^{2}}\Bigg{)}\Bigg{)}^{2}{\rm
Br}(\mu\rightarrow e\nu_{\mu}\bar{\nu}_{e}),$ (66)

where we have substituted Eq. (11) in Eq. (65) to obtain Eq. (66). In the
above, $\epsilon=v_{f}/\Lambda$, where for simplicity we have assumed all
flavon VEVs to be the same, $i.e.$ $v_{\xi}=v_{s,a}=v_{f}$. Clearly, ${\rm
Br}(\mu\to e\gamma)$ depends on the parameter $|b|$, which is constrained from
neutrino oscillation data ranging from 0.0048 to 0.0056 eV as given in Fig. 1.

Figure 14: Branching ratio of $\mu\to e\gamma$ ($\mu\to 3e$) is plotted
against the dark matter mass $M_{f}$ in the left (right) panel. In both
panels, the red line corresponds to the present upper bound and the pink line
corresponds to the future sensitivity (see text for details).  Figure 15:
Branching ratio for $\mu\to e\gamma$ is plotted against $|b|$ for the points
satisfying correct relic density of DM. The variation of color legends is for
the variation of $\kappa$. Cyan shaded regions rules-out values of $|b|$ other
than that consistent with the neutrino oscillation data $(0.0048-0.0056)$ eV.
Solid red and dot-dashed brown lines represent the recent updated constraint
from MEG-II and its projected sensitivity, respectively.

Another type of the LFV decays appearing in our FSS1 framework are the
$l_{\alpha}\rightarrow 3l_{\beta}$ ($\l_{\alpha}\rightarrow
l_{\beta}\bar{l}_{\beta}l_{\beta}$) processes. The corresponding branching
ratios are given by Toma:2013zsa

$\displaystyle{\rm Br}(l_{\alpha}\rightarrow
3l_{\beta})\approx\frac{3{\alpha_{em}}^{2}}{512G_{F}^{2}}|Y_{F}^{\beta^{*}}Y_{F}^{\alpha}|^{2}\frac{1}{M_{\eta^{+}}^{4}}\mathcal{G}\Big{(}\frac{m_{\alpha}}{m_{\beta}}\Big{)}\Bigg{(}F_{2}\Bigg{(}\frac{M_{f}^{2}}{M_{\eta^{+}}^{2}}\Bigg{)}\Bigg{)}^{2}{\rm
Br}(l_{\alpha}\rightarrow l_{\beta}\nu_{\alpha}\bar{\nu}_{\beta}),$ (67)

where

$\displaystyle F_{2}(x)$ $\displaystyle=$
$\displaystyle\frac{2-9x+18x^{2}-11x^{3}+6x^{3}{\rm log}x}{6(1-x)^{4}},$ (68)
$\displaystyle\mathcal{G}\Big{(}\frac{m_{\alpha}}{m_{\beta}}\Big{)}$
$\displaystyle=$ $\displaystyle\Big{(}\frac{16}{3}{\rm
log}\Big{(}\frac{m_{\alpha}}{m_{\beta}}\Big{)}-\frac{22}{3}\Big{)}.$ (69)

Again, following Eq. (11), we find that $Y_{F}^{\tau}=0$, hence the branching
fractions for $\tau\rightarrow 3e$ and $\tau\rightarrow 3\mu$ decays vanish.
The only non-vanishing contribution originates from the $\mu\rightarrow 3e$
decay, and the branching fraction can be written as Toma:2013zsa

$\displaystyle{\rm Br}(\mu\rightarrow 3e)$ $\displaystyle\approx$
$\displaystyle\frac{3{\alpha_{em}}^{2}}{512G_{F}^{2}}|2y_{s}y_{s}^{*}\epsilon^{4}|^{2}\frac{1}{M_{\eta^{+}}^{4}}\mathcal{G}\Big{(}\frac{m_{\mu}}{m_{e}}\Big{)}\Bigg{(}F_{2}\Bigg{(}\frac{M_{f}^{2}}{M_{\eta^{+}}^{2}}\Bigg{)}\Bigg{)}^{2},$
(70) $\displaystyle=$
$\displaystyle\frac{3\alpha_{em}^{2}}{128M_{f}^{2}G_{F}^{2}}\Bigg{(}\frac{|b|}{\mathcal{F}(M_{\eta_{R}},M_{\eta_{I}},M_{f})}\Bigg{)}^{2}\frac{1}{M_{\eta^{+}}^{4}}\mathcal{G}\Big{(}\frac{m_{\mu}}{m_{e}}\Big{)}\Bigg{(}F_{2}\Bigg{(}\frac{M_{f}^{2}}{M_{\eta^{+}}^{2}}\Bigg{)}\Bigg{)}^{2},$
(71)

where we have substituted Eq. (11) in Eq. (70) to obtain Eq. (71) with
$\epsilon=v_{f}/\Lambda$. Similar to Eq. (66), here we also find that ${\rm
Br}(\mu\rightarrow 3e)$ depends on the scotogenic mass parameters
$M_{f,{\eta^{+}},{\eta_{R}},{\eta_{I}}}$ as well as $|b|$, the parameter
involved in explaining correct neutrino oscillation parameter and DM relic
density. The variation of the corresponding coupling $\lambda_{5}$ is given in
the inset. In Fig. 14, we have shown plots for $\mu\to e\gamma$ (left panel)
and $\mu\to 3e$ (right panel) branching ratios against the dark matter mass
$M_{f}$, satisfying bound on $|b|$ obtained from Fig. 1. The current
constraints (denoted by magenta lines) for the branching ratio of the $\mu\to
e\gamma$ decay is given by the MEG-II experiment as ${\rm Br}(\mu\to
e\gamma)\leq 3.1\times 10^{-13}$ MEGII:2023ltw  whereas for $\mu\rightarrow
3e$ decay the constraint by SINDRUM experiment is given as ${\rm
Br}(\mu\rightarrow 3e)\leq 1\times 10^{-12}$ SINDRUM:1987nra . In both plots,
a current upper bound on both these decays, constrains the dark matter mass
$M_{f}$ specifically in the low mass region. $M_{f}$ will be further
constrained by the future MEG-II (Proj.) MEGII:2018kmf  and Mu3e Phase-I
Perrevoort:2018ttp  experiments. To illustrate the dependence of the LFV
branching ratio on the neutrino oscillation parameters and its consistency
with the DM phenomenology, in Fig. 15, we have plotted ${\rm
Br}(\mu\rightarrow e\gamma)$ against $|b|$. Here, the white-shaded region is
consistent with correct neutrino masses and mixing given in the right panel of
Fig. 1. Hence, the cyan-shaded regions are ruled out by neutrino oscillation
data. This plot also depicts the dependence of the branching ratio on the
scotogenic Yukawa coupling shown by the variation of $\kappa$. The upper
shaded region is already ruled out by the recent updated constraint from MEG-
II MEGII:2023ltw , the projected sensitivity of MEG-II can probe $\kappa$ of
the order $\mathcal{O}(10^{-2})$.

Decay Modes | Scotogenic | Seesaw | Remarks
---|---|---|---
$\mu\to e\gamma$ | ✓ | ✗ | $Y_{N}^{e}=0$
$\tau\to e\gamma$ | ✗ | ✗ | $Y_{F}^{\tau}=0,Y_{N}^{e}=0$
$\tau\to\mu\gamma$ | ✗ | ✓ | $Y_{F}^{\tau}=0$
$\mu\to 3e$ | ✓ | ✗ | $Y_{N}^{e}=0$
$\tau\to 3e$ | ✗ | ✗ | $Y_{F}^{\tau}=0,Y_{N}^{e}=0$
$\tau\to 3\mu$ | ✗ | ✓ | $Y_{F}^{\tau}=0$
Table 3: Possible lepton flavor violating decay modes and their origin in the
FSS1 framework. The ✓and ✗ symbols stand for allowed and disallowed regimes.

For the type-I seesaw contributions to LFV decays, the branching fractions for
$l_{\alpha}\to l_{\beta}\gamma$ decays can be cast in the following form

$\displaystyle{\rm Br}(l_{\alpha}\rightarrow l_{\beta}\gamma)$
$\displaystyle\approx$ $\displaystyle\frac{3{\alpha_{em}}v^{4}}{8\pi
M_{N}^{4}}\Bigg{|}Y_{N}^{\beta}Y_{N}^{\alpha^{*}}f\Bigg{(}\frac{M^{2}_{N}}{M_{W}^{2}}\Bigg{)}\Bigg{|}^{2},$
(72)

where $Y_{N}$ is given in Eq. (10). The loop function $f(x)$ in Eq. (72) is

$\displaystyle f(x)=\frac{x(2x^{3}+3x^{2}-6x-6x^{2}{\rm
log}(x)+1)}{2(1-x)^{4}}.$ (73)

Similar to the scotogenic contribution, the $A_{4}$ discrete symmetry and the
VEV alignment of the flavon $\phi_{s}$ plays a crucial role in estimating the
branching ratio for $l_{\alpha}\rightarrow l_{\beta}\gamma$. The VEV alignment
of the flavon $\phi_{s}$ is such that it gives $Y_{N}^{e}=0$ as a result of
which the branching fraction for $\mu\rightarrow e\gamma$ and $\tau\rightarrow
e\gamma$ decays vanish. The only non-vanishing contribution arising in the
type of $l_{\alpha}\rightarrow l_{\beta}\gamma$ decay is
$\tau\rightarrow\mu\gamma$ and the branching fraction is given by

$\displaystyle{\rm Br}(\tau\rightarrow\mu\gamma)$ $\displaystyle=$
$\displaystyle\frac{3{\alpha_{em}}v^{4}}{8\pi
M_{N}^{4}}\Bigg{|}(y_{N}y_{N}^{*}\epsilon^{2}f\Bigg{(}\frac{M^{2}_{N}}{M_{W}^{2}}\Bigg{)}\Bigg{|}^{2}=\frac{3{\alpha_{em}}}{8\pi
M_{N}^{2}}|a|^{2}\Bigg{|}f\bigg{(}\frac{M_{N}^{2}}{M_{W}^{2}}\bigg{)}\Bigg{|}^{2}.$
(74)

For $M_{N}\sim 10^{4}$ GeV and $|a|=0.0250$ eV, the branching fraction in Eq.
(74) gives $5.4\times 10^{-33}$ which is very small compared to the
experimental limit ( $4.4\times 10^{-8}$) BaBar:2009hkt . For higher $M_{N}$
values, the branching ratio will be even more suppressed. Similarly, the
branching ratio for the $\tau\rightarrow 3\mu$ conversion is found to be very
small compared to the experimental bound Hayasaka:2010np .

In Tab. 3, we have summarized the allowed LFV decays in the FSS1 model. The
considered discrete flavor symmetry and corresponding vacuum alignment of the
flavons completely disallow decay channels such as $\tau\rightarrow e\gamma$
and $\tau\rightarrow 3e$. Such a decisive prediction can be made since we have
vanishing values for the Yukawa couplings $Y_{F}^{\tau}$ and $Y_{N}^{e}$, see
Eq. (10) and Eq. (11), associated with the scotogenic and type-I seesaw
contributions, respectively. Present experiments already exclude branching
ratios larger than about $\mathcal{O}(10^{-8})$. Any positive signal by the
future experiments will essentially test the validity of the FSS1 framework.

##  9 Summary of Phenomenological Analysis

Both type-I seesaw and the scotogenic contribution within our FSS1 framework
are crucial in explaining the hierarchy associated with the neutrino mass-
squared differences. The scotogenic contribution is characterized by the
parameter $b$ and its magnitude is restricted within a narrow range
$0.0048-0.0056$ eV.

Figure 16: Final parameter space for the two important parameters of the FSS1
model, namely the Yukawa coupling $\kappa$ and the DM fermion mass $M_{f}$,
after imposing the constraints from DM relic density, direct search, neutrino
oscillation data, and LFV decays.

The estimation of DM relic density depends on the scotogenic contribution
Yukawa coupling $\kappa$ associated with $|b|$ as given in Eq. (11). This
dependence is shown in Fig. 10 in the DM mass $M_{f}-\kappa$ plane. The
allowed parameter space gets further constrained to satisfy correct neutrino
oscillation data and experimental limits on LFV decays discussed in Section 4
and 8, respectively. Although the allowed range of $|b|$ is tightly
constrained from neutrino oscillation data, interplay of DM $f$ and other dark
sector particles $\eta_{I,R,\pm}$ can satisfy correct DM relic density with
contributions from various annihilation and co-annihilation contributions
mentioned in Fig. 6 \- 8. Hence, updating Fig. 10, in Fig. 16, we have plotted
the final parameter space, which includes constraints from DM relic density,
neutrino oscillation data, and LFV decays. The points with the color code
represent the parameter space consistent with the DM relic density and direct
search constraints. Once we impose the constraints for $|b|$ from the neutrino
oscillation data obtained from Fig. 1, we get the magenta-colored points.
Finally, we obtain the red star points when we impose the constraint from LFV
decays along with the constraints mentioned above from DM phenomenology and
neutrino oscillation. LFV constraints restrict the maximum allowed Yukawa
coupling to be less than $\mathcal{O}(10^{-2})$ and DM masses between
$(100-1000)$ GeV are found to be simultaneously consistent with neutrino
oscillation data, DM relic density, direct search, and LFV decay constraints.

##  10 Conclusions and Outlook

We propose the flavor-scoto-seesaw (FSS) model, which explains the observed
hierarchy between the solar and atmospheric neutrino mass scales,
experimentally allows the trimaximal mixing scheme, and naturally accommodates
viable dark matter candidates. In this framework, type-I seesaw and 1-loop
scotogenic mechanisms contribute to the effective light neutrino mass. With
only one right-handed neutrino, the type-I seesaw contribution dominantly
contributes to generating atmospheric neutrino mass scale, and the scotogenic
contribution (with the involvement of the dark fermion $f$ and scalar $\eta$)
is mainly responsible for the solar neutrino mass scale. The whole framework
is embedded within $A_{4}\times Z_{4}\times Z_{3}\times Z_{2}$ discrete flavor
symmetry predicting the lightest neutrino to massless and one non-vanishing
Majorana phase. The model also contains a few flavon fields to realize
appropriate flavor structure to explain observed neutrino mixing. The
inclusion of auxiliary $Z_{N}$ ($N=$ 4, 3, 2) symmetries is a generic feature
of discrete flavor symmetric models to forbid several unwanted terms, and the
charged lepton mass matrix is found to be a diagonal one. These $Z_{N}$
symmetries are broken down to a dark $Z_{2}$ symmetry, ensuring the stability
of dark matter under which only $f$ and $\eta$ are odd. With a judicious
choice of the flavon vacuum alignments, the TM1 mixing scheme can be realized,
and hence we call our flavor symmetric scoto-seesaw model an FSS1 model.

Considered flavor symmetry completely dictates the flavor structure of the
model and makes it highly predictive. The FSS1 model provides rich
phenomenology for neutrino masses, mixing, LFV decays, and collider studies
and accommodates potential dark matter candidates with DM $f$ fermion and
$\eta$ scalar. With both type-I and scotogenic contributions, a rank-2 light
neutrino mass matrix is obtained, predicting normal ordering of light neutrino
mass. The presence of flavor symmetry in FSS1 implies a preference for the
higher octant of the atmospheric mixing angle $\theta_{23}$ where the allowed
ranges given by $0.531\leq\sin^{2}\theta_{23}\leq 0.544$ and
$0.580\leq\sin^{2}\theta_{23}\leq 0.595$. The model also tightly constrains
the TM1 prediction for the Dirac CP phase $\delta_{\rm CP}$ (within the range
$\pm(1.44-1.12)$ radian) and the Jarlskog CP invariant. Moreover, correlations
among neutrino mixing parameters within the FSS1 model (see Figs. 3 and 4)
give a strict determination of the Majorana CP phase, thus giving an accurate
prediction for $m_{\beta\beta}$ (see Tab. 2) within the range $1.61-3.85$ meV.
Here, the dark fermion $f$ is considered as the DM candidate whose production
mechanism is connected with its Yukawa coupling with SM leptons and the inert
doublet scalar $\eta$. The magnitude of these Yukawa couplings plays a
critical role in determining correct neutrino mixing and DM relic density
through the thermal freeze-out mechanism. With the flavor structure of the
FSS1 framework, only the scotogenic part contributes to the lepton flavor
violating decays such as $\mu\rightarrow e\gamma,\mu\rightarrow 3e$. On the
other hand, though they are very small, the seesaw part of FSS1 only
contributes to decays such as $\tau\rightarrow\mu\gamma,\tau\rightarrow 3\mu$.
Interestingly, owing to the flavor symmetry and vacuum alignment of the
flavons, LFV decays such as $\tau\rightarrow e\gamma$ and $\tau\rightarrow 3e$
are completely disallowed, and any positive signal LFV for these two decays
will test the viability of this model. Within the FSS1 framework, the WIMP DM
masses between $(100-1000)$ GeV are simultaneously consistent with the
constraints from neutrino oscillation data, DM relic density, direct search,
and LFV decays.

The FSS1 model can also be tested at the colliders via a wide range of
phenomenological studies. For example, FSS1 can contribute to the Higgs boson
diphoton decay channel $h\to\gamma\gamma$. Fig. 13 shows that $M_{\eta^{+}}$
masses up to 1 TeV can have implications at the diphoton Higgs decay channel
using the present LHC experimental results. With the increasing data
collection at LHC and HL-LHC, the precision of $R_{\gamma\gamma}$ will
improve, giving prospects for better determination of allowed regions for
specific flavor model parameters. Thus, phenomenology-based $R_{\gamma\gamma}$
constraints can be used for further studies and predictions for producing
exotic discrete flavor model signals at present and future colliders. The same
statement is valid for further phenomenological studies of the model based on
DM and LFV constraints. Thus, in alignment with all pertinent constraints, the
model retains its predictiveness across LFV experiments, direct detection of
DM, as well as collider experiments.

##

##  Appendix A Appendix: $A_{4}$ symmetry

$A_{4}$ is a discrete group of even permutations of four objects666For a
detailed discussion on $A_{4}$ see Refs. 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 relations

$\displaystyle S^{2}=T^{3}=(ST)^{3}=1$ (75)

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
Ishimori:2010au

$\displaystyle 1$ $\displaystyle\otimes$ $\displaystyle
1=1;~{}1^{\prime}\otimes 1^{\prime\prime}=1,$ (76) $\displaystyle 1^{\prime}$
$\displaystyle\otimes$ $\displaystyle
1^{\prime}=1^{\prime\prime};~{}1^{\prime\prime}\otimes
1^{\prime\prime}=1^{\prime},$ (77) $\displaystyle 3$ $\displaystyle\otimes$
$\displaystyle 3=1\oplus 1^{\prime}\oplus 1^{\prime\prime}\oplus 3_{s}\oplus
3_{a},$ (78)

where the subscripts $``s"$ and $``a"$ denote symmetric and antisymmetric
parts, respectively. Writing two $A_{4}$ triplets as
$X=(x_{1},x_{2},x_{3})^{T}$ and $Y=(y_{1},y_{2},y_{3})^{T}$ respectively,
their product can be written as Ishimori:2010au

$\displaystyle X\otimes Y$ $\displaystyle=$ $\displaystyle(X\otimes
Y)_{1}\oplus(X\otimes Y)_{1^{\prime}}\oplus(X\otimes
Y)_{1^{\prime\prime}}\oplus(X\otimes Y)_{3s}\oplus(X\otimes Y)_{3a}$ (79)

where

$\displaystyle(X\otimes Y)_{1}$ $\displaystyle\sim$ $\displaystyle
x_{1}y_{1}+x_{2}y_{3}+x_{3}y_{2},$ (80) $\displaystyle(X\otimes
Y)_{1^{\prime}}$ $\displaystyle\sim$ $\displaystyle
x_{3}y_{3}+x_{1}y_{2}+x_{2}y_{1},$ (81) $\displaystyle(X\otimes
Y)_{1^{\prime\prime}}$ $\displaystyle\sim$ $\displaystyle
x_{2}y_{2}+x_{1}y_{3}+x_{3}y_{1},$ (82)

$\displaystyle(X\otimes Y)_{3s}$ $\displaystyle\sim$
$\displaystyle\begin{pmatrix}2x_{1}y_{1}-x_{2}y_{3}-x_{3}y_{2}\\\
2x_{3}y_{3}-x_{1}y_{2}-x_{2}y_{1}\\\
2x_{2}y_{2}-x_{1}y_{3}-x_{3}y_{1}\end{pmatrix},$ (83) $\displaystyle(X\otimes
Y)_{3a}$ $\displaystyle\sim$
$\displaystyle\begin{pmatrix}x_{2}y_{3}-x_{3}y_{2}\\\ x_{1}y_{2}-x_{2}y_{1}\\\
x_{3}y_{1}-x_{1}y_{3}\end{pmatrix}.$ (84)

These relations are used in the construction of the mass matrices given in Eq.
(7) and Eq. (12).

###### Acknowledgements.

This work has been supported in part by the Polish National Science Center
(NCN) under grant 2020/37/B/ST2/02371, the Freedom of Research (Swoboda Badań)
and the Research Excellence Initiative of the University of Silesia in
Katowice. BK would like to thank José W. F. Valle and Claudia Hagedorn for
useful discussions. BK also acknowledges hospitality at the Korea Institute of
Advanced Study, Seoul, where part of this work has been completed. SM
acknowledges the financial support from the National Research Foundation of
Korea grant 2022R1A2C1005050.

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