# FIMP DM in the Extended Hyperchargeless Higgs Triplet Model

Pritam Das<EMAIL_ADDRESS>Mrinal Kumar Das<EMAIL_ADDRESS>Department of Physics, Tezpur University, Assam-784028, India.  Najimuddin
Khan<EMAIL_ADDRESS>School of Physical Sciences, Indian Association for
the Cultivation of Science 2A $\&$ 2B, Raja S.C. Mullick Road, Kolkata 700032,
India.  School of physics, Institute for Research in Fundamental Sciences
(IPM), P.O.Box 19395-5531, Tehran, Iran.

###### Abstract

We perform an exclusive study on the Feebly Interacting Massive Particle
(FIMP) dark matter candidate in an extended hyperchargeless ($Y=0$) Higgs
triplet model. The additional $Z_{2}$ odd neutral fermion singlet plays the
role of dark matter with support from two other vector-like fermion doublets.
The mixing between the neutral component of a doublet and singlet fermions
controls the current relic density through the Freeze-in mechanism, whereas
the additional doublet fermion helps to get the neutrino mass and mixing
angles. We obtain a broad region of the parameter spaces satisfying the
current relic density and neutrino mass and mixing angles.

##  I Introduction

Through meticulous measurements of cosmic microwave background (CMB)
anisotropies, cosmology based experiments such as PLANCK Aghanim  _et al._
(2018), WMAP Bennett  _et al._ (2013) have suggested the existence dark matter
(DM), giving rise to around 26% of the present universe’s energy density.
Astrophysical pieces of evidence like galaxy cluster observations by F. Zwicky
Zwicky (1937), observations of galaxy rotation curvesFreese (2009) to the more
recent observation of the bullet cluster Clowe  _et al._ (2006) also agrees
with the cosmological observations. Now, one can explain the dark matter relic
density as, $\Omega h^{2}=0.1198\pm 0.0026$ Aghanim  _et al._ (2018) using
various theories Kolb and Turner (1990); Hall  _et al._ (2010). Interestingly,
all those evidence are based on gravitational interactions only, and until
date, not a single evidence has reported in support of the particle
interaction of DM. The cosmological and astrophysical data can only explain
how much dark matter is there in the universe, but to understand what it is,
one must look from the particle physics perspective. The explanation from the
particle physics perspective is quite obvious, and no SM particle has the
quality to fulfill the criteria to be a DM candidate. There is still no
general agreement on what it is composed of. The non-baryonic dark matter
component can be put into three categories based on their velocities, e.g.,
hot dark matter (HDM), warm dark matter (WDM), and cold dark matter (CDM). The
neutrino has high velocities as an HDM is ruled out Murayama (2007). The WDM
and CDM have moderate velocities that have explained the dark matter densities
and other experimental constraints.

One of the most intriguing experimental findings in particle physics is the
phenomenon of neutrino oscillation. The atmospheric, solar, reactor and
accelerator neutrino oscillation experiments Abe  _et al._ (2016); An  _et
al._ (2012); Abe  _et al._ (2012) have predicted that the three flavor of the
neutrinos mix among themselves and they have a tiny mass, which is the only
solid experimental evidence available in favour of the beyond the standard
model (BSM) framework. The oscillation experiments are only sensitive to the
mixing angles and mass square differences ($\Delta
m_{ij}^{2}=m_{i}^{2}-m_{j}^{2}$). From various observations, we get a
constraints on the sum of the all neutrino mass eigenvalues
($\sum_{m_{i}}<0.117$ eV Roy Choudhury and Choubey (2018), with $i=1,2,3$).

From the SM point of view, there is no sufficient candidate left to propose as
a dark matter candidate or explain the neutrino mass. Therefore, one must head
towards BSM frameworks to explain the current dark matter density and neutrino
low energy variables i.e. mass and mixing angles. The recent LHC Higgs signal
strength data Sirunyan  _et al._ (2018, 2019a) also allows us to include
additional fields of the new physics beyond the SM. Among the various BSM
scenarios that are proposed in the literature to explain the tiny neutrino
masses,the seesaw mechanisms are the most popular one Mohapatra and Senjanovic
(1980); Bernabeu  _et al._ (1987). It is known that in order to get a neutrino
mass of the sub-eV scale, one has to take the new particles to be extremely
heavy (right-handed neutrino) or else take the new couplings to be extremely
small or need a vast fine-tuning in the new Yukawa sector.

A model that can address both light neutrino mass and dark matter with a
minimum particle content is much appealing and well-motivated in the model
building business. A working model is considered to be completed when it can
simultaneously explain light neutrino observables and dark matter, and both
sectors are well connected. There are many popular frameworks in the
literature Babu (2010); Borah  _et al._ (2018); Khan  _et al._ (2014); Das
_et al._ (2020); Bhattacharya  _et al._ (2017); Ghosh  _et al._ (2018); Das
_et al._ (2021a); Das (2021); Das and Dey (2014); Barry  _et al._ (2011);
Burgess  _et al._ (2001); Cohen  _et al._ (2012); Grimus  _et al._ (2009);
Bhupal Dev  _et al._ (2014); Toma and Vicente (2014). The framework proposed
by E.Ma Ma (2006), that gained severe attention in accommodating both dark
matter and neutrino mass at loop level is known as the $scotogenic$ model,
where the dimension-5 operator is realized at the one-loop level Weinberg
(1979). The notable feature of this framework is the way it connects neutrino
and DM. Due to the additional $Z_{2}$ discrete symmetry, new fields
contributing to the loop to produce sizable neutrino mass acquire opposite
parity to the SM fields. Hence, The new field becomes stable due to odd
$Z_{2}$ charge imposed on it, and can be addressed as a viable dark matter
candidate. With these impressive features in addressing neutrino mass and dark
matter, the scotogenic model has gained popularity in recent days Gonzalez
Felipe  _et al._ (2004); Fraser  _et al._ (2014); Merle and Platscher (2015);
Law and McDonald (2013); Mahanta and Borah (2019); Klein  _et al._ (2019);
Sarma  _et al._ (2021).

Many of the DM genesis theories are based upon the thermal Freeze-out
mechanism. It is to be noted that the assumption behind the weakly interacting
massive particle (WIMP) dark matter is that the coupling between the dark
matter and SM particles cannot be too small; otherwise, it will never reach
thermal equilibrium. Similarly, we cannot have large coupling for a small dark
matter mass, and it will violate the perturbative-unitarity and the direct
detection limits Aprile  _et al._ (2018). So far, we do not have any signature
in those direct detection experiments Aprile  _et al._ (2018). Moreover, most
of the single component WIMP dark matter model Burgess  _et al._ (2001);
Deshpande and Ma (1978); Khan (2018); Chaudhuri  _et al._ (2015) are tightly
constrained by the recent direct detection limits Aprile  _et al._ (2018).
People have also introduced the multi-component of the dark matter models
Dienes and Thomas (2012) to avoid these limits. Recently the authors of the
Ref. Hall  _et al._ (2010), have suggested the idea of a Freeze-in mechanism
where the dark matter candidate interacts with the other particle feebly,
called Feebly Interacting Massive Particles (FIMPs). It is assumed that,
initially the dark matter density remains tiny in the early universe and get
populated through the decay and/or annihilation of the other heavy particles.
We found manly two types of Freeze-in mechanisms in the literature. The infra-
red (IR) Freeze-in Borah  _et al._ (2018); Hall  _et al._ (2010); Biswas and
Gupta (2016), which is based on renormalizable theory while the ultra-violet
(UV) Freeze-in Elahi  _et al._ (2015) model consists of higher dimension
operators, and the relic density in the UV Freeze-in mechanism depends on the
reheat temperature.

The lepton flavour violating (LFV) processes ($\mu\rightarrow e\gamma$) along
with the electron and muon anomalous magnetic moment are also a striking
indication of the BSM frameworks. Recently, the muon g-2 collaboration at
Fermilab has reported an staggering measurement of the anomalous magnetic
dipole moment of $\mu^{\pm}$ with a 3.3$\sigma$ deviation from the SM
prediction achieving a combined experimental average of Abi  _et al._ (2021);
Bennett  _et al._ (2006a); Parker  _et al._ (2018): $\delta
a_{\mu}=a_{\mu}^{\text{exp}}-a_{\mu}^{\text{SM}}=(2.51\pm 0.59)\times
10^{-9}$. This gives new hope in the BSM physics for a long-standing tension
between SM and experimental data that was previously reported by the E821
experiment at Brookhaven National Laboratory Bennett  _et al._ (2006b), and it
gets serious attention within a short period Borah  _et al._ (2021); Dasgupta
_et al._ (2021); Cen  _et al._ (2021); Yang  _et al._ (2021); Das  _et al._
(2021b); Escribano  _et al._ (2021); Athron  _et al._ (2021); Kowalska and
Sessolo (2017); Calibbi  _et al._ (2018); Kowalska and Sessolo (2020);
Kawamura  _et al._ (2020); Baker  _et al._ (2021). To explain these data, one
can also alternatively use the lattice QCD, the hadronic vacuum polarization
contribution to the muon anomalous magnetic moment based on gauge ensembles
with $N_{f}=2+1$ flavours of $O(a)$ improved Wilson quarks Gérardin  _et al._
(2019); Gérardin (2021); Chao  _et al._ (2021); Davies  _et al._ (2020).

In this work we add a $Z_{2}$-odd real scalar Higgs triplet with hypercharge
$Y=0$ to the SM. This model is popularly termed as the hyperchargeless Higgs
triplet model in the literature Blank and Hollik (1998). Along with this Higgs
triplet, two additional vector-like $SU(2)_{L}$ doublets111In this model, we
need two generations of fermion doublet to address all the light neutrino
parameters Das  _et al._ (2021a) and neutral singlet fermions are also added
in this model (to cancel the effect of the anomaly), which can help to explain
the dark matter as well as all the light neutrino parameters. In a parallel
footing, we also probe the couplings and other models parameters to the latest
results on muon anomalous magnetic moment and check whether this model can
accommodate those results or not. The Higgs triplet Khan (2018) alone can
produce the relic density for dark matter mass $M_{DM}>2.3$ TeV onward but, it
fails to provide the exact relic density since the co-annihilation channels
dominate at the low mass region. Similarly, the extension of $Z_{2}$-odd
vector-like neutral fermion could explain the DM, however, most of the
parameter space ruled out by the present direct detection experiments. It is
worthy mentioning that the region $M_{H}+M_{H^{\pm}}<M_{W}$ is ruled out from
the decay of $W$-boson. Though, in the region $2M_{H},2M_{H^{\pm}}<M_{h}$,
there still have some allowed parameters even from the 14 TeV LHC. In the only
HTM (i.e., without vector-like fermion) model, the Higgs-125 GeV invisible
decay width put only strong bounds in the region $2M_{H},2M_{H^{\pm}}<M_{h}$.
Interestingly, the $H^{\pm}$ could be long-lived in the HTM model as the mass
is almost degenerate, the decay time we found $\sim 10^{7}$ sec. However, in
the presence of vector-like fermion $H^{\pm}$ could decay quickly. We find
that Higgs triplet and vector-like fermions in this model can explain a large
region of the dark matter parameter spaces including the neutrino low energy
variables, allowed by all theoretical and experimental constants. To the best
of our knowledge, this model has not discussed in the literature, which
motivates us to go for a detailed analysis.

The paper is organized as follows. Section II starts with a detailed
description of the model. We revisit the detailed constraints for the extended
HTM model in Sec. III. The new constraints from the LFV decay and
contributions to the muon anomalous magnetic moment are discussed in the Sec.
IV. We discuss the neutrino mass and mixing angle in Sec. V. The FIMP dark
matter parameter spaces are shown in detail in Sec. VI. Finally, we conclude
our work in Sec. VIII.

##  II Model

We extended the SM particle content by a triplet $T$, a vector-like fermion
(VLF) singlet $N_{S}$ and two VLF doublet
$F_{D,i}=(N_{D}~{}~{}E_{D}^{-})^{T}_{i=1,2}$. The choice of a vector-like
particle in this model will keep the model anomaly free Pal (2014). We impose
a discrete $Z_{2}$ symmetry on this model such that the scalar triplet and
vector-like fermions are $odd$ ($\Psi_{BSM}\rightarrow-\Psi_{BSM}$), whereas
the SM fields are $even$ ($\Psi_{SM}\rightarrow\Psi_{SM}$) under this
transformation. The extra scalar triplet consists of a pair of singly-charged
fields $H^{\pm}$ and a $CP$-even neutral scalar field $H$. The doublet and
triplet scalar are conventionally written as Khan (2018); Chen  _et al._
(2008); Fiaschi  _et al._ (2019)

$\displaystyle\Phi=\left(\begin{array}[]{c}{G_{1}^{+}}\\\
\frac{1}{\sqrt{2}}(v_{1}+{h^{0}}+i{G^{0}})\end{array}\right)\,,\qquad\qquad
T=\left(\begin{array}[]{c}{H^{+}}\\\ {H}\\\
-{H^{-}}\end{array}\right)\,\equiv\left(\begin{array}[]{cc}\frac{H}{\sqrt{2}}&H^{+}\\\
H^{-}&-\frac{H}{\sqrt{2}}\end{array}\right).$ (8)

The kinetic part of the Lagrangian is given by,

${\cal L}_{k}=\mid D_{\mu}\Phi\mid^{2}+\frac{1}{2}\mid D_{\mu}T\mid^{2}\,,$
(9)

where, the covariant derivatives are defined as,

$D_{\mu}\Phi=\biggl{(}\partial_{\mu}+i\frac{g_{2}}{2}\sigma^{a}W^{a}+i\frac{g_{1}}{2}YB_{\mu}\biggr{)}\Phi~{}~{}~{}\text{and}~{}~{}~{}D_{\mu}T=\biggl{(}\partial_{\mu}+ig_{2}t_{a}W^{a}\biggr{)}T\,,$
(10)

here, $W^{a}_{\mu}$ ($a$=1,2,3) are the ${SU(2)_{L}}$ gauge bosons,
corresponding to three generators of ${SU(2)_{L}}$ group and $B_{\mu}$ is the
${U(1)}_{Y}$ gauge boson. $\sigma^{a}~{}(a=1,2,3)$ are the Pauli matrices, and
$t_{a}$ can be written as follows

$t_{1}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&1\\\
0&1&0\end{array}\right),\quad
t_{2}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&-i&0\\\ i&0&-i\\\
0&i&0\end{array}\right),\quad t_{3}=\left(\begin{array}[]{ccc}1&0&0\\\
0&0&0\\\ 0&0&-1\end{array}\right)\,.$ (11)

The tree-level scalar potential with the Higgs doublet and the real scalar
triplet is invariant under $SU(2)_{L}\times U(1)_{Y}$ transformation. This is
given by,

$\displaystyle V(\Phi,T)$ $\displaystyle=$
$\displaystyle\mu_{1}^{2}\mid\Phi\mid^{2}+\frac{\mu_{2}^{2}}{2}\mid
T\mid^{2}+{\lambda_{1}}\mid\Phi\mid^{4}+\frac{\lambda_{2}}{4}\mid
T\mid^{4}+\frac{\lambda_{3}}{2}\mid\Phi\mid^{2}\mid T\mid^{2}.$ (12)

The scalar fields of the triplet do not mix with the scalar fields of SM
doublet. After the EWSB, the scalar potential in Eq. 12 is then given by

$\displaystyle V(h,H,H^{\pm})$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[2\mu_{1}^{2}(h+v)^{2}+\lambda_{1}(h+v)^{4}+2\mu_{2}^{2}(H^{2}+2H^{+}H^{-})\right.$
(13)
$\displaystyle\left.+\lambda_{2}(H^{2}+2H^{+}H^{-})^{2}+\lambda_{3}(h+v)^{2}(H^{2}+2H^{+}H^{-})\right].$

Here, $v\equiv v_{SM}$ and the mass of these scalar fields $h$, $H$ and
$H^{\pm}$ are given by

$\displaystyle M_{h}^{2}$ $\displaystyle=$ $\displaystyle 2\lambda_{1}v^{2},$
$\displaystyle M_{H}^{2}$ $\displaystyle=$
$\displaystyle\mu_{2}^{2}+\frac{\lambda_{3}}{2}v^{2},$ (14) $\displaystyle
M_{H^{\pm}}^{2}$ $\displaystyle=$
$\displaystyle\mu_{2}^{2}+\frac{\lambda_{3}}{2}v^{2}\,.$

At tree-level, the mass of the neutral scalar $H$ and the charged particles
$H^{\pm}$ are degenerate. If we include one-loop radiative correction, the
charged particles become slightly heavier Cirelli and Strumia (2009); Cirelli
_et al._ (2006) than the neutral ones. The mass difference between them is
given by,

$\Delta M=(M_{H^{\pm}}-M_{H})_{1\text{-}loop}=\frac{\alpha
M_{H}}{4\pi}\Big{[}f\Big{(}\frac{M_{W}}{M_{H}}\Big{)}-c_{W}^{2}f\Big{(}\frac{M_{Z}}{M_{H}}\Big{)}\Big{]},$
(15)

with, $f(x)=-\frac{x}{4}\Big{\\{}2x^{3}~{}{\rm
log}(x)+(x^{2}-4)^{\frac{3}{2}}~{}{\rm
log}\left(\frac{x^{2}-2-x\sqrt{x^{2}-4}}{2}\right)\Big{\\}}$ and $c_{W}$ being
the $cosine$ of the Weinberg angle. It has been shown in Refs Cirelli and
Strumia (2009); Cirelli  _et al._ (2006) that the mass splitting between
charged and neutral scalars remains $\sim 150$ MeV for $M_{H}=0.1-5$ TeV.

The Lagrangian for the new fermionic interaction are given by,

$\displaystyle\mathcal{L_{F}}$ $\displaystyle=$
$\displaystyle\overline{F}_{D,i}\gamma^{\mu}D_{\mu}F_{D,i}+\overline{N}_{S}\gamma^{\mu}D_{\mu}{N}_{S}-M_{ND}^{ik}\overline{F}_{D,i}F_{D,k}-M_{NS}\overline{N}_{S}{N}_{S},$
$\displaystyle\mathcal{L}_{int}$ $\displaystyle=$ $\displaystyle-
Y_{N,i}\overline{F}_{D,i}\phi{N}_{S}-Y_{fli}\overline{L}_{l}F_{D,i}\,T+h.c.~{},$
(16)

here, $D_{\mu}$ stands for the corresponding covariant derivative of the
doublet and singlet fermions. The indices expand as $l=e,\mu,\tau$ and
$i=1,2$. The mass matrix for these neutral fermion fields is given by,

$\displaystyle\mathcal{M}=\begin{pmatrix}M_{NS}&M_{X}^{1}&M_{X}^{2}\\\
M_{X}^{1,\dagger}&M_{ND}^{11}&M_{ND}^{12}\\\
M_{X}^{2,\dagger}&M_{ND}^{21}&M_{ND}^{22}\\\ \end{pmatrix},$ (17)

where, $M_{X}^{1}=\frac{v\,Y_{N,1}}{\sqrt{2}}$ and
$M_{X}^{2}=\frac{v\,Y_{N,2}\,}{\sqrt{2}}$. The neutral component of the
fermion doublets ($N_{D,1}$ and $N_{D,2}$) and the singlet charged fermion
(${N}_{S}$) mix at tree level. For simplicity, Let us assume the second VLF
doublet is decoupled, i.e., $Y_{N,2}=0$ and $M_{ND}^{21}=M_{ND}^{12}=0$ 222We
discuss the diagonalization of these fermions in Appendix A for the non-zero
of-diagonal components in Eq. 17. The mass eigenstates are obtained by
diagonalizing the mass matrix with a rotation of the $({N}_{S}$ $N_{D,1}$
$N_{D,2})$ basis as,

$\displaystyle\begin{pmatrix}\chi_{1}\\\ \chi_{2}\\\ \chi_{3}\\\
\end{pmatrix}=\begin{pmatrix}\cos\beta&\sin\beta&0\\\
-\sin\beta&\cos\beta&0\\\ 0&0&1\\\ \end{pmatrix}\begin{pmatrix}N_{S}\\\
N_{D,1}\\\ N_{D,2}\\\ \end{pmatrix},{~{}\rm with~{}}\tan
2\beta=\frac{2M_{X}^{1}}{M_{ND}^{11}-M_{NS}}$ (18)

Diagonalization of Eq. 17 gives the following eigenvalues for the new leptons
($M_{ND}^{11}-M_{NS}\gg M_{X}^{1}$) as,

$\displaystyle
M_{\chi_{1}}=M_{NS}-\frac{2(M_{X}^{1})^{2}}{M_{ND}^{11}-M_{NS}},\,M_{\chi_{2}}=M_{ND}^{11}+\frac{2(M_{X}^{1})^{2}}{M_{ND}^{11}-M_{NS}},\,M_{E_{1}^{\pm}}=M_{ND}^{11},\,M_{E_{2}^{\pm}}=M_{\chi_{3}}=M_{ND}^{22},$

$E_{1}^{\pm}\equiv E_{D,1}^{\pm}$ and $E_{2}^{\pm}\equiv E_{D,2}^{\pm}$ are
the charged component fermion fields. The fermion state $\chi_{1}$ can serve
as a dark matter candidate for the choice of mass
$M_{\chi_{1}}<M_{\chi_{2,3},E_{1,2}^{\pm},H,H^{\pm}}$. With this choice of
parameters $Y_{N,2}=0$ and $M_{ND}^{21}=M_{ND}^{12}=0$, the second doublet
becomes decoupled from the first doublet and singlet vector-like fermion.
Hence, it does not affect the dark matter density calculations; however, it is
important to get all the neutrino low energy variables. In the absence of the
second doublet, we can not explain one of the neutrino parameters: mass
differences or mixing angles. We will provide a detailed discussion on the new
region of the allowed parameter spaces and the effect of the presence of
additional $Z_{2}$-odd fermion in the sections of dark matter VI and neutrino
mass V.

The parameter space of this model is constrained by various bounds arising
from theoretical considerations like absolute vacuum stability and unitarity
of the scattering matrix, observation phenomenons like dark matter relic
density. The direct search limits at LEP and electroweak precision
measurements also put severe restrictions on the model. The recent
measurements of the Higgs invisible decay width and signal strength at the LHC
put additional constraints. The dark matter (DM) requirement saturates the DM
relic density all alone restricts the allowed parameter space considerably.

##  III Constraints on the model

###  III.1 Vacuum stability bounds

The condition for the absolute stability of the vacuum comes from requiring
that the scalar potential is bounded from below, i.e., it should not approach
negative infinity along any direction of the field space for large field
values Khan (2018). One can use the copositivity criteria Kannike (2012) and
calculate the required conditions for the absolute stability/bounded from
below the scalar potential. The tree-level scalar potential is absolutely
stable if,

$\lambda_{1}(\Lambda)\geq 0,\quad\lambda_{2}(\Lambda)\geq
0,\quad\lambda_{3}(\Lambda)\geq-2~{}\sqrt{\lambda_{1}(\Lambda)\lambda_{2}(\Lambda)}.$
(19)

The coupling constants are evaluated at a scale $\Lambda$ using RGEs.

###  III.2 Perturbativity and Unitarity bounds

To ensure that the radiatively improved scalar potential remains perturbative
at any energy scale ($\Lambda$), one must put the following conditions,

$\mid\lambda_{1,2,3}\mid\lesssim 4\pi~{}~{}~{}{\rm
and}~{}~{}~{}\Big{|}\frac{\lambda_{4}}{\Lambda}\Big{|}\lesssim 4\pi.$ (20)

The most stringent unitary bound comes from the sub-matrix ${\cal M}_{3}$ (see
Ref. Khan (2018)) corresponds to scattering fields
$(G_{1}^{+}~{}G_{1}^{-},~{}\eta^{+}~{}\eta^{-},$
$~{}\frac{G^{0}~{}G^{0}}{\sqrt{2}},~{}\frac{h^{0}~{}h^{0}}{\sqrt{2}},$ and
$\frac{\eta^{0}~{}\eta^{0}}{\sqrt{2}})$ and it is given by,

${\cal
M}_{3}=\begin{pmatrix}4\lambda_{1}&\lambda_{3}&\sqrt{2}\lambda_{1}&\sqrt{2}\lambda_{1}&\frac{\lambda_{3}}{\sqrt{2}}\\\
\lambda_{3}&4\lambda_{2}&\frac{\lambda_{3}}{\sqrt{2}}&\frac{\lambda_{3}}{\sqrt{2}}&\sqrt{2}\lambda_{2}\\\
\sqrt{2}{\lambda_{1}}&\frac{\lambda_{3}}{\sqrt{2}}&3\lambda_{1}&\lambda_{1}&\frac{\lambda_{3}}{2}\\\
\sqrt{2}\lambda_{1}&\frac{\lambda_{3}}{\sqrt{2}}&\lambda_{1}&3\lambda_{1}&\frac{\lambda_{3}}{2}\\\
\frac{\lambda_{3}}{\sqrt{2}}&\sqrt{2}\lambda_{2}&\frac{\lambda_{3}}{2}&\frac{\lambda_{3}}{2}&3\lambda_{2}\end{pmatrix}.$
(21)

Unitary constraints of the scattering processes demand that the eigenvalues
$2\lambda_{1},~{}2\lambda_{1},~{}2\lambda_{2},$ and
$\frac{1}{2}\left(6\lambda_{1}+5\lambda_{2}\pm\sqrt{(6\lambda_{1}-5\lambda_{2})^{2}+12\lambda_{3}^{2}}\right)$
of the matrix should be less than $8\pi$ Khan (2018).

###  III.3 Bounds from electroweak precision experiments

Electroweak precision data has imposed severe bounds on new physics models via
Peskin-Takeuchi $S,~{}T,~{}U$ parameters Peskin and Takeuchi (1992). The
additional contributions from the scalar triplet of this model are given by
Forshaw  _et al._ (2003, 2001),

$\displaystyle S$ $\displaystyle\simeq$ $\displaystyle 0,$ (22) $\displaystyle
T$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\,\frac{1}{\sin^{2}\theta_{W}\cos^{2}\theta_{W}}\left[\frac{M^{2}_{H}+M^{2}_{H^{\pm}}}{M^{2}_{Z}}\;-\;\frac{2M^{2}_{H^{\pm}}M^{2}_{H}}{M^{2}_{Z}(M^{2}_{H}-M^{2}_{H^{\pm}})}\log\left(\frac{M^{2}_{H}}{M^{2}_{H^{\pm}}}\right)\right]$
(23) $\displaystyle\simeq$
$\displaystyle\frac{1}{6\pi}\,\frac{1}{\sin^{2}\theta_{W}\cos^{2}\theta_{W}}\;\frac{(\Delta
M)^{2}}{M^{2}_{Z}},$ $\displaystyle U$ $\displaystyle=$
$\displaystyle-\frac{1}{3\pi}\left(M^{4}_{H}\log\left(\frac{M^{2}_{H}}{M^{2}_{H^{\pm}}}\right)\frac{(3M^{2}_{H^{\pm}}-M^{2}_{H})}{(M^{2}_{H}-M^{2}_{H^{\pm}})^{3}}+\frac{5(M^{4}_{H}+M^{4}_{H^{\pm}})-22M^{2}_{H^{\pm}}M^{2}_{H}}{6(M^{2}_{H}-M^{2}_{H^{\pm}})^{2}}\right)$
(24) $\displaystyle\simeq$ $\displaystyle\frac{\Delta M}{3\pi M_{H^{\pm}}},$

where, $\Delta M=M_{H^{\pm}}-M_{H}\approx 160$ MeV Khan (2018). $M_{H^{\pm}}$
and $M_{H}$ are almost degenerate for $M_{H^{\pm},H}>M_{h}$. The contributions
to the $T~{}{\rm and}~{}U$ parameters from this model are also neglected Khan
(2018); Baak  _et al._ (2014). Similarly the contribution from the $SU(2)$
singlet neutral fermion is zero whereas in the absence of mixing and
$M_{N_{D,i}},M_{E_{D,i}^{\pm}}>>\Delta
M_{D,i}=M_{E_{D,i}^{\pm}}-M_{N_{D,i}}(i=1,2)$, the contribution from the
doublet fermion doublet are given by Peskin and Takeuchi (1992); Hieu  _et
al._ (2020),

$\displaystyle S$ $\displaystyle\simeq$
$\displaystyle\frac{2}{3\pi}\,\sum_{i=1,2}\frac{\Delta M_{D,i}}{M_{N_{D,i}}},$
(25) $\displaystyle T$ $\displaystyle\simeq$
$\displaystyle\frac{1}{6\pi}\frac{1}{\sin^{2}\theta_{W}}\,\sum_{i=1,2}\frac{(\Delta
M_{D,i})^{2}}{M^{2}_{W}}$ (26) $\displaystyle U$ $\displaystyle\simeq$
$\displaystyle\frac{11}{30\pi}\,\sum_{i=1,2}\frac{(\Delta
M_{D,i})^{2}}{M^{2}_{W}},$ (27)

In the tree-level, $\Delta M_{D,i}=0$ and the loop-level separation is also
small, hence the contributions to the $S,T$ and $U$ parameters from these
additional fermion are almost negligible.

###  III.4 LHC diphoton signal strength constraints

Gluon fusion at LHC gives the dominant production cross-section of $h$, and
the Higgs to diphoton signal strength $\mu_{\gamma\gamma}$ in this extended
HTM model can be written as,

$\mu_{\gamma\gamma}=\frac{\sigma(gg\rightarrow
h\rightarrow\gamma\gamma)_{EHTM}}{\sigma(gg\rightarrow
h\rightarrow\gamma\gamma)_{\rm SM}}=\frac{\sigma(gg\rightarrow
h)_{EHTM}}{\sigma(gg\rightarrow
h)_{SM}}\frac{Br(h\rightarrow{\gamma\gamma})_{\rm
EHTM}}{Br(h\rightarrow{\gamma\gamma})_{\rm SM}}.$ (28)

We use the narrow width approximation as
$\Gamma_{h}^{total}/{M_{h}}\rightarrow 0$. The $\mu_{\gamma\gamma}$ can be
simplified as,

$\mu_{\gamma\gamma}=~{}\frac{\Gamma^{total}_{h,\rm SM}}{\Gamma^{total}_{h,\rm
EHTM}}~{}\frac{\Gamma(h\rightarrow\gamma\gamma)_{\rm
EHTM}}{\Gamma(h\rightarrow\gamma\gamma)_{\rm SM}}.$ (29)

Since, the couplings $h\rightarrow f\bar{f}/VV$, ($V$ stands for vector
bosons) are same as in SM.

The new charged fermions $E_{1,2}^{\pm}$ and charged Higgs $H^{\pm}$ in this
model will alter the decay width of $h\rightarrow\gamma\gamma$, $Z\gamma$
through one-loop. At the same time, if the mass of the extra scalar particles
($\chi_{1,2,3},E_{1,2}^{\pm},H,H^{\pm}$) happen to be lighter than the half of
the Higgs mass $M_{h}/2$, then they might also contribute to the invisible
decay of the Higgs boson. We are using the global fit analysis Belanger  _et
al._ (2013), that such an invisible branching ratio is less than $\sim 20\%$.
In Eq. 29, the first ratio provides a suppression of $\sim 0.8-1$. For
$M_{\chi_{1,2,3},E_{1,2}^{\pm},H,H^{\pm}}>M_{h}/2$, the ratio becomes
$\frac{\Gamma^{total}_{h,\rm SM}}{\Gamma^{total}_{h,\rm EHTM}}\approx 1$.
Therefore, the modified Higgs to diphoton signal strength,
$\mu_{\gamma\gamma}$ can be written as,

$\mu_{\gamma\gamma}\approx\frac{\Gamma(h\rightarrow\gamma\gamma)_{\rm
EHTM}}{\Gamma(h\rightarrow\gamma\gamma)_{\rm SM}}\,.$ (30)

In this model, the additional contribution arising due to $E_{12}^{\pm}$ and
$H^{\pm}$ are incorporated in the total decay width as given by Djouadi
(2008),

$\displaystyle\Gamma(h\rightarrow\gamma\gamma)_{\rm EHTM}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}M_{h}^{3}}{256\pi^{3}v^{2}}\Big{|}\sum_{f}N^{c}_{f}Q_{f}^{2}y_{f}F_{1/2}(\tau_{f})+y_{W}F_{1}(\tau_{W})+Q_{E_{1}^{\pm}}^{2}y_{hE_{1}^{+}E_{1}^{-}}F_{1/2}(\tau_{E_{1}^{\pm}})$
(31)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+Q_{E_{2}^{\pm}}^{2}y_{hE_{2}^{+}E_{2}^{-}}F_{1/2}(\tau_{E_{2}^{\pm}})+Q_{H^{\pm}}^{2}\frac{v\mu_{{hH^{+}H^{-}}}}{2M_{H^{\pm}}^{2}}F_{0}(\tau_{H^{\pm}})\Big{|}^{2},$

where $\tau_{i}=M_{h}^{2}/4M_{i}^{2}$. $Q_{X}$
($X=f,{E_{1}^{\pm}},{E_{2}^{\pm}},H^{\pm}$) denotes electric charges of
corresponding particles. $N_{f}^{c}$ is the color factor. $y_{f}$ and $y_{W}$
denote the Higgs couplings to $f\bar{f}$ and $W^{+}W^{-}$.
$\mu_{hH^{+}H^{-}}=\lambda_{3}v_{SM}$ stands for the coupling constant of
$hH^{+}H^{-}$ vertex and $y_{hE_{1,2}^{+}E_{1,2}^{-}}=0$ as $E_{1,2}^{\pm}$
are $Z_{2}$ odd. The loop functions $F_{(0,\,1/2,\,1)}$ can be found in Ref
Djouadi (2008).

Recently, the diphoton rates $\mu_{\gamma\gamma}$ of the observed Higgs to the
SM prediction are measured by the ATLAS Aad  _et al._ (2014); Aaboud  _et al._
(2019) and CMS Khachatryan  _et al._ (2014); Sirunyan  _et al._ (2019b)
collaborations. At present the combined result from these experiments of
$\mu_{\gamma\gamma}$ is $1.14^{+0.19}_{-0.18}$ Aad  _et al._ (2016).

In $\Gamma(h\rightarrow\gamma\gamma)_{\rm EHTM}$ (see Eq. 31), a positive
$\lambda_{3}$ leads to a destructive interference between Higgs triplet and
the SM contributions and vice versa. One can see from the Eq. 31, the
contribution to the Higgs diphoton channel is proportional to
$\frac{\lambda_{3}}{M_{H^{\pm}}^{2}}$. If the charged scalar mass is greater
than $300$ GeV, then the contributions of $H^{\pm}$ to the diphoton signal is
negligibly small.

##  IV Lepton flavour violating decay and anomalous magnetic moments

Among the different lepton number violating processes, the radiative (through
one-loop) muon decay $\Gamma(\mu\rightarrow e\gamma)$ is one of the popular
and tightly constraints. In this present model, the diagram is mediated by
charged particles $E_{1}^{\pm},E_{2}^{\pm},H^{\pm}$ and
$\chi_{1},\chi_{2},\chi_{3},H$ present in the internal lines of the one-loop
diagrams 1.

Figure 1: The diagrams contribute to LFV and both electron and muon anomalous
magnetic moments.

The mixing angle between the neutral component of singlet fermion with other
neutral component of doublet fermions are very small, hence the contributions
through $\chi_{1}$ is neglected. The corresponding expression for the
branching ratio is given by Akeroyd  _et al._ (2009),

$\displaystyle{\rm BR}(\mu\rightarrow
e\gamma)\approx\sum_{i=1}^{2}\frac{3\alpha_{em}}{32\pi
G_{F}^{2}}\Big{|}Y_{f\mu
i}Y_{fei}\frac{F(M_{E_{i}^{\pm}}^{2}/M_{H}^{2})}{M_{H}^{2}}\Big{|}^{2},$ (32)

where, $F(x)=\frac{x^{3}-6x^{2}+3x+2++xln(x)}{6\,(x-1)^{4}}$. The most recent
experimental bounds for LFV could be found in Ref. Baldini  _et al._ (2018) is
given by ${\rm BR}(\mu\rightarrow e\gamma)<4.2\times 10^{-13}$ at $90\%$ CL.
We need small Yukawa coupling $Y_{f\mu i}$ and/or $Y_{fei}$ to get the allowed
parameter spaces in a desirable range. It is noted that these Yukawa couplings
can also alter the neutrino mass and mixing angle (will be discussed in the
next section) and the muon as well as electron anomalous magnetic moments. The
numerical expression for additional contribution to the electron ($l=e$) and
muon ($l=\mu$) anomalous magnetic moments in this model is expressed as Chao
(2008); Chen and Nomura (2019),

$\displaystyle\Delta\alpha_{i}$ $\displaystyle=$
$\displaystyle\frac{Re[Y_{fli}^{2}]m_{l}^{2}}{8\pi^{2}}\int_{0}^{1}\,\frac{x(1-x)^{2}}{(x-x^{2})m_{l}^{2}+(x-1)M_{E_{i}^{\pm}}^{2}-xM_{H}^{2}}$
(33)
$\displaystyle+\,\frac{Re[Y_{fli}^{2}]m_{l}^{2}}{16\pi^{2}}\int_{0}^{1}\,\frac{x(1-x)^{2}}{(x^{2}-x)m_{l}^{2}+xM_{E_{i}^{\pm}}^{2}-(x-1)M_{H^{\pm}}^{2}}.$

This model shows that the muon decay $\Gamma(\mu\rightarrow e\gamma)$ and
neutrino mass and mixing angles put tighter constraints on the model parameter
spaces. We find that the contributions to both anomalous magnetic moments are
almost negligible in this model. Interestingly, one can keep a large $Y_{f\mu
i}(i=1,2)$ to achieve the discrepancy between the SM predictions $\delta
a_{\mu}=a_{\mu}^{\text{exp}}-a_{\mu}^{\text{SM}}=(2.51\pm 0.59)\times 10^{-9}$
Abi  _et al._ (2021). In this case, we choose to keep a very small
$Y_{fei}(i=1,2)$ to satisfy ${\rm BR}(\mu\rightarrow e\gamma)<4.2\times
10^{-13}$. With these choices of parameters, one cannot get all the neutrino
low energy variables as measured by the experiments. In the next section we
will discuss the neutrino oscillation parameters for this anomaly.

##  V Neutrino Mass and mixing angles

In this study, we will provide a brief overview of the neutrino mass
generation at the one-loop level. The second term of the interaction
Lagrangian from Eq. (16) is responsible for the neutrino mass generation via
1-loop level, shown in Fig. 2. The Yukawa interaction associated with the
interaction term, $Y_{fli}$ does violates lepton number conservation in this
model. The neutral component of the fermion doublets ($N_{D,i}$) and singlet
fermion ($N_{S}$) mixed at tree level via the mixing angle $\beta$. Hence, the
Yukawa interaction also modified accordingly. One can find the modified Yukawa
interaction $A_{fli}=\cos\beta~{}Y_{fli}$ involved in the neutrino mass
generation process. The neutral $Z_{2}$-odd vector-like fermions and scalar
involved in the radiative neutrino mass generation after the EWSB are shown in
Fig. 2.

Figure 2:  One loop diagram for neutrino mass generation.

Summing over all the two-point function contributions, we arrive at the
neutrino mass matrix component as Fileviez Perez and Wise (2009),

$\displaystyle(M_{\nu})_{ll^{\prime}}=\sum_{i,l,l^{\prime}}\frac{A_{fli}^{\dagger}A_{fl^{\prime}i}}{32\,\pi^{2}}\,(\lambda_{3}v^{2})\,\mathcal{I}(M_{\chi_{i}},M_{H}),$
(34)

where, $\lambda_{3}$ is the Higgs portal quartic coupling, and $v=264.221$ GeV
is the SM vacuum expectation value. The indices $i=1,2,3$ stand for the three
neutral fermion. $M_{\chi_{i=1,2,3}}$ are the masses for the neutral heavy
fermions. The loop function $I(M_{\chi_{i}},M_{DM})$ is defined as Fileviez
Perez and Wise (2009),

$\displaystyle\mathcal{I}(M_{\chi_{i}},M_{H})=4M_{\chi_{i}}\frac{M_{H}^{2}-M_{\chi_{i}}^{2}+M_{\chi_{i}}\,log(\frac{M_{\chi_{i}}^{2}}{M_{H}^{2}})}{(M_{H}^{2}-M_{\chi_{i}}^{2})^{2}}.$
(35)

Neutrino mass eigenvalues are obtained by diagonalizing the above mass matrix
$(M_{\nu})_{ij}$ using the well established PMNS matrix, consists of three
mixing angles ($\theta_{12},\theta_{13},\theta_{23}$) and three phases (one
Dirac and two Majorana). The diagonal mass matrix can be written as
$m_{Diag}=U_{\rm PMNS}^{\dagger}M_{\nu}U_{\rm PMNS}$. Moreover, it is
essential to ensure that the choice of Higgs portal quartic coupling
$\lambda_{3}$, Yukawa couplings, as well as other parameters involved in the
light neutrino mass matrix component.. From the above expressions (eqns. 34
and 35), light neutrino masses and mixing angles can exactly be explained by
adjusting the Higgs portal quartic coupling $\lambda_{3}$, Yukawa couplings
and mass parameters present in equation (34). For a few hundred GeV, heavy
scalar and neutral fermions, one can choose small
$\lambda_{3}\sim\mathcal{O}(10^{-8})$ to get the small neutrino masses. It is
important to highlight that the neutrino mass and mixing angles are almost
independent of the dark matter mass $M_{\chi_{1}}$ and tiny mixing angle
$\beta$.

A tiny $\beta$ is essential to get the exact relic density through the freeze-
in mechanism. We also find that the second generation of the fermion doublet
is necessary to generate all the neutrino mass differences and mixing angles
according to the experimental data. In the absence of this doublet, it is
unable to produce one or more light neutrino variables. For example, we can
generate mixing angles $\theta_{12}=32.7^{\circ}$, $\theta_{13}=8.4^{\circ}$,
$\theta_{23}=44.71^{\circ}$ and mass differences $\Delta m_{21}^{2}=7.44\times
10^{-5}$ eV2 and $\Delta m_{31}^{2}=4.9\times 10^{-4}$ eV2 with phases
$\alpha=\delta=45^{\circ}$. Although $\Delta m_{31}^{2}$ is within the present
3$\sigma$ bound yet, $\Delta m_{21}^{2}$ deviates from the actual range for
this choice of the parameters. For the other option of the input parameters,
we can get $\Delta m_{21}^{2}$ is within the present 3$\sigma$ bound yet,
$\Delta m_{31}^{2}$ is deviating from the actual range for this choice of the
parameters. Therefore, we added another doublet without affecting dark matter
results and with the choice of tiny couplings we can explain all the light
neutrino parameters within the latest bound.

Figure 3: The plot stand for the neutrino low energy variable bounds in $\sum
Tr|Y_{fli}^{\dagger}Y_{fli}|$ against the quartic coupling $\lambda_{3}$. We
consider the fermion masses $M_{\chi_{2}}=1000$ GeV, $M_{\chi_{3}}=2500$ GeV
and $M_{H}=1000$ GeV. The red region will violate one and/or more neutrino
experimental parameters.

Parameters | Benchmark Points
---|---
|  BM-I (NH) |  BM-II (NH) |  BM-III (IH)
$M_{\chi_{2}}$ GeV | $800$ | $1000$ | $880$
$M_{\chi_{3}}$ GeV | $2500$ | $2500$ | $2000$
$M_{H}$ GeV | $500$ | $850$ | $995$
$|Y_{fe1}|$ | $0.00360362$ | $0.0141421$ | $0.0232594$
$|Y_{fe2}|$ | $0.0462481$ | $0.0269258$ | $0.0282843$
$|Y_{f\mu 1}|$ | $0.000128141$ | $0.206155$ | $0.200679$
$|Y_{f\mu 2}|$ | $0.00420078$ | $0.400009$ | $0.632533$
$|Y_{f\tau 1}|$ | $0.00415482$ | $0.0613105$ | $0.0250098$
$|Y_{f\tau 2}|$ | $0.0492443$ | $0.0141421$ | $0.0150333$
$|\lambda_{3}|$ | $6.6468\times 10^{-8}$ | $1.35\times 10^{-9}$ | $3.657\times 10^{-9}$
Outputs | Corresponding Low-energy variables
$\Delta m^{2}_{21}/10^{-5}~{}{\rm eV^{2}}$ | 7.552 | 7.735 | 2.5425 $\times 10^{2}$ (IH)
$\Delta m^{2}_{31}/10^{-3}~{}{\rm eV^{2}}$ | 2.504 | 2.46 | 7.46257 $\times 10^{-2}$ (IH)
$\theta_{12}$ rad | 0.5719 | 0.5708 | 0.5720
$\theta_{23}$ rad | 0.7803 | 0.7803 | 0.7803
$\theta_{13}$ rad | 0.1469 | 0.1434 | 0.1469
$\delta_{PMNS}$ rad | $0.75$ | $0.7$ | $0.15$
$\alpha$ rad | $0.5$ | $0.6$ | $0.6$
$m_{i}$ eV | $0.057634,0.0298795,0.0285878$ | $0.0553511,0.0260174,0.0245639$ | $0.0238301,0.0550976,0.0222092$
$Br(\mu\rightarrow e~{}\gamma)$ | $7.38\times 10^{-19}$ | $9.51\times 10^{-15}$ | $3.06\times 10^{-15}$

Table 1: Three sets of benchmark points used in our analysis. These BPs are
used to carry out our numerical analysis satisfying all the low energy
constraints.

Depending on the parameter space, here we can explain both the normal
hierarchy (NH: $m_{1}<m_{2}<m_{3}$) and inverted hierarchy (IH:
$m_{3}<m_{1}<m_{2}$). Three sets of such benchmark points and corresponding
neutrino mass and mixing angles (within $3\sigma$) are displayed in table 1.
All three BPs are also allowed by the lepton flavor violation data. One can
see that the masses of the heavy fermions and scalar particle are
$\sim\mathcal{O}(1)$ TeV and $Y_{f\mu 2}\sim\mathcal{O}(10^{-2})$. On the
contrary, to achieve the precise value of the muon anomalous moment $\delta
a_{\mu}=a_{\mu}^{\text{exp}}-a_{\mu}^{\text{SM}}=(2.51\pm 0.59)\times
10^{-9}$, one needs fine-tuning the Yukawa coupling $Y_{f\mu 2}$ and this
fine-tuning demands $Y_{f\mu 2}>\mathcal{O}(1)$. Therefore, the region allowed
by the muon $g-2$ data may also violate the unitary-perturbativity limits. One
can see from Fig. 3, the current bounds in the light neutrino parameters are
possible only with tiny Higgs portal coupling $\lambda_{3}$ and/or the Yukawa
couplings $Y_{fli}$ for fermion masses $\mathcal{O}(1)$ TeV. The red region is
the forbidden region for this model, as all the light neutrino parameters can
not be explained at the same time in that region. Therefore, after discussing
the model phenomenology and bounds achieved in the context of neutrino mass
and mixing, we are now well equipped to carry out the dark matter and collider
analysis. In the following section, we have studied FIMP dark matter in the
context of a simplified version of this model with tiny coupling constants.

##  VI Dark Matter

We already discussed that one needs additional symmetry such that the dark
matter could not decay and/or the decay lifetime is much larger than the
lifetime of the universe. We have such $Z_{2}$-symmetry, which prohibits the
lightest neutral $Z_{2}$-odd particle (dark matter) decay. In this model, we
have total four such neutral $Z_{2}$-odd ($\chi_{1,2,3}$ and $H$) particles.
The eigenstate $\chi_{1}$ is the lightest neutral $Z_{2}$-odd fermion in this
model and can be considered as a viable dark matter candidate for
$M_{\chi_{1}}<M_{H}$. The other $Z_{2}$-odd neutral scalar field can also
behave as a viable dark matter candidate for $M_{H}<M_{\chi_{1}}$. The co-
annihilation channels ($\sigma(H^{\pm}H\rightarrow f_{SM}f_{SM})$) dominate in
the relic density calculation for the $Z_{2}$-odd neutral scalar field through
the Freeze-out mechanism. One can get the relic density for the scalar dark
matter mass $M_{H}>2$ TeV, though the low mass region is completely ruled out.
On the other hand, for the fermionic case, we checked that various parameter
space could give the exact relic density through the Freeze-out mechanism,
however the large region of the parameter space is ruled out from the present
direct detection experiment Aprile  _et al._ (2018). Again, the low mass
regions (except $\frac{M_{h}}{2}\pm 5$ GeV region) are forbidden from either
the relic density and/or direct detection constraints. This study will mainly
focus on the FIMP dark matter, i.e., the dark matter relic density achieved
through the Freeze-in mechanism. Perhaps, the $Z_{2}$-odd neutral scalar field
can also produce sufficient relic density, however, we do not discuss it here
as this choice may not provide both normal as well as inverted hierarchy
neutrino mass pattern in this model. Also, the region
$M_{H}<\frac{M_{W,Z}}{2}$ are strictly prohibited from the invisible decay
width constraints of the gauge bosons ($W,Z$). Therefore, we only focus on the
fermionic dark matter candidate for the mass region $keV-TeV$ in this study.

The central assumption is that the dark matter particle gets populated through
the decay or annihilation of the heavy particles (SM and/or BSM) until the
number density of the corresponding heavy particles species becomes Boltzmann-
suppressed. In the existing literature Borah  _et al._ (2018); Hall  _et al._
(2010); Biswas and Gupta (2016), it has been proved that if the same couplings
are involved in both decay as well as scattering processes, then the former
has the dominant contribution to DM relic density over the latter one. In this
mechanism, it has been proved that the dark matter relic density depends only
on the partial decay width (DM production channels only) of the mother
particles Hall  _et al._ (2010). At the same time, the other decay channel
(like, $\psi_{heavy}^{Mother}\rightarrow\psi_{light}\psi_{light}$) may reduce
the mother particles density for the universe temperature temperature
$T<M_{DM}$ rapidly. However, for $T>M_{DM}$ the reverse processes (
$\psi_{light}\psi_{light}\rightarrow\psi_{heavy}^{Mother}$) can compensate the
mother particles density from the other bath particles. Hence, the mother
particle was thermally equilibrium with the other particles in the early
universe Hall  _et al._ (2010). One can easily get ${\Gamma\over H(T)}\geq 1$
for all the mother particles which produce dark matter through decay and/or
annihilation, where $\Gamma$ is the relevant decay width or annihilation rate
for the production of mother particles from the annihilation of other
particles and $H(T)$ is the Hubble parameter
$H(T)=\left(g^{*}\,\frac{\pi^{2}}{90}\,\frac{T^{4}}{M_{\rm
Pl}^{2}}\right)^{1/2}$. $M_{\rm Pl}=2.435\times 10^{18}$ GeV is the reduced
Planck mass and $T$ is the temperature Bauer and Plehn (2019); Hall  _et al._
(2010). For the dark matter production channels, the forward as well backward
processes are prolonged due to the smaller coupling strengths, hence the
forward dark matter production processes are always dominant, and the reverse
rate is almost zero during the expansion of the universe. Therefore, the other
decay and or annihilation channels of the mother particles will not affect the
relic density calculations in the Freeze-in mechanism.

Thus, we have to solve only one Boltzmann equation for the evolution of the DM
from various mother particles. In this case, we have to consider the sum of
all DM production through the decay and annihilation channels of different
mother particles. It is noted that if the mother particles are not in thermal
equilibrium in the early universe, then one must need to solve the evaluation
of the mother particles and at the same time has to solve the evaluation for
the DM particle Biswas and Gupta (2016). For our case, the relic density can
be calculated as Bauer and Plehn (2019); Hall  _et al._ (2010); Biswas and
Gupta (2016)

$\displaystyle\Omega h^{2}\approx 1.09\times
10^{27}\,M_{\chi_{1}}\,\sum_{i}\frac{g_{\psi_{Heavy,i}}\,\Gamma_{\psi_{Heavy,i}}}{M_{\psi_{Heavy,i}}^{2}},$
(36)

where, $\Gamma_{\psi_{Heavy,i}}$ is the partial decay width of various dark
matter production channels. In this model, we find that the annihilation
contributions are tiny as compared to the decay channels. The neutral
component of the first vector fermion doublet and singlet fermion can mix, and
this mixing angle ($\beta$) help us to produce the dark matter density at the
right ballpark. We noticed that it is impossible to have low mass (below
$\mathcal{O}(1)$ GeV) range dark matter without these mixings, either via
freeze-in or freeze-out mechanism. Most of the high-mass (above
$\mathcal{O}(1)$ GeV) range dark matter cannot produce the right relic density
and is ruled out from the direct detection experiments. The mixing angle helps
us to get the allowed dark matter mass within $keV-TeV$ region. In this model,
we have various channels which can produce dark matter through the decay of
the heavy model particles. All the possible decay channels and their partial
decay width are given by

$\displaystyle\Gamma(h\rightarrow\chi_{1}\chi_{1})$ $\displaystyle=$
$\displaystyle\frac{M_{h}}{8\pi}\,|g_{{\chi_{1}}\,{\chi_{1}}\,h}|^{2}\left(1-\frac{4M_{\chi_{1}}^{2}}{M_{h}^{2}}\right)^{\frac{3}{2}},$
$\displaystyle\Gamma(H^{\pm}\rightarrow\chi_{1}l)$ $\displaystyle=$
$\displaystyle\frac{|g_{\overset{-}{l}\,{\chi_{1}}\,H^{+}}|^{2}}{16\pi
M_{H^{\pm}}^{3}}\,\left(M_{H^{\pm}}^{2}-M_{\chi_{1}}^{2}-m_{l}^{2}\right)\,\sqrt{\left(M_{H^{\pm}}^{2}-M_{\chi_{1}}^{2}-m_{l}^{2}\right)^{2}-4\,M_{\chi_{1}}^{2}\,m_{l}^{2}},$
$\displaystyle\Gamma(H\rightarrow\chi_{1}\nu_{l})$ $\displaystyle=$
$\displaystyle\frac{|g_{\overset{-}{{\nu_{l}}}\,{\chi_{1}}\,{H}}|^{2}}{16\pi
M_{H}^{3}}\,\left(M_{H}^{2}-M_{\chi_{1}}^{2}\right)^{2},$ (37)
$\displaystyle\Gamma(\chi_{2}\rightarrow\chi_{1}h)$ $\displaystyle=$
$\displaystyle\frac{|g_{{\chi_{1}}\,{\chi_{2}}\,h}|^{2}}{16\pi
M_{\chi_{2}}^{3}}\,\left(M_{\chi_{2}}^{2}-M_{\chi_{1}}^{2}-M_{h}^{2}\right)\,\sqrt{\left(M_{\chi_{2}}^{2}-M_{\chi_{1}}^{2}-M_{h}^{2}\right)^{2}-4\,M_{\chi_{1}}^{2}\,M_{h}^{2}},$
$\displaystyle\Gamma(\chi_{2}\rightarrow\chi_{1}Z)$ $\displaystyle=$
$\displaystyle\frac{|{{cb}\,\,{sb}\,\sqrt{g_{1}^{2}+g_{2}^{2}}}/{2}|^{2}}{16\pi
M_{\chi_{2}}^{3}}\,\sqrt{\left(M_{\chi_{2}}^{2}-M_{\chi_{1}}^{2}-M_{Z}^{2}\right)^{2}-4\,M_{\chi_{1}}^{2}\,M_{Z}^{2}},$
$\displaystyle\times\,2\left(M_{\chi_{2}}^{2}+M_{\chi_{1}}^{2}+2M_{\chi_{2}}M_{\chi_{1}}-M_{Z}^{2}\right),$
$\displaystyle\Gamma(E_{1}^{\pm}\rightarrow\chi_{1}W)$ $\displaystyle=$
$\displaystyle\frac{|{g_{2}\,{sb}}/{\sqrt{2}}|^{2}}{16\pi
M_{\chi_{2}}^{3}}\,\,\sqrt{\left(M_{\chi_{2}}^{2}-M_{\chi_{1}}^{2}-M_{W}^{2}\right)^{2}-4\,M_{\chi_{1}}^{2}\,M_{W}^{2}},$
$\displaystyle\times\,2\left(M_{\chi_{2}}^{2}+M_{\chi_{1}}^{2}+2M_{\chi_{2}}M_{\chi_{1}}-M_{W}^{2}\right),$
$\displaystyle\Gamma(Z\rightarrow\chi_{1}\chi_{1})$ $\displaystyle=$
$\displaystyle\frac{|{{sb}\,^{2}\sqrt{g_{1}^{2}+g_{2}^{2}}}/{2}|^{2}\,M_{Z}}{48\pi}.\,$

The coupling strength are defined as:

$\displaystyle
g_{{\chi_{1}}\,{\chi_{1}}\,h}={2i\,{cb}\,^{2}\,{sb}\,^{2}\,\Delta
M_{\chi}}/{{v_{SM}}},\,g_{{\chi_{1}}\,{\chi_{2}}\,h}={i\,{cb}\,\,{sb}\,\left(\,{cb}\,^{2}-\,{sb}\,^{2}\right)\Delta
M_{\chi}}/{{v_{SM}}},$ $\displaystyle
g_{\overset{-}{l}\,{\chi_{1}}\,H^{+}}=-i\,{sb}\,Y_{fl1},\,g_{\overset{-}{{\nu_{l}}}\,{\chi_{1}}\,{H}}=-{i\,{sb}\,Y_{fl1}}/{\sqrt{2}},\,g_{{\chi_{1}}\,{E1}\,W}={ig_{2}\,{sb}\,\gamma^{\mu}}/{\sqrt{2}},$
(38) $\displaystyle
g_{{\chi_{1}}\,{\chi_{2}}\,Z}=-{i\,{cb}\,\,{sb}\,\sqrt{g_{1}^{2}+g_{2}^{2}}\,\gamma^{\mu}\gamma^{5}}/{2},\,g_{{\chi_{1}}\,{\chi_{1}}\,Z}=-{i\,{sb}\,^{2}\sqrt{g_{1}^{2}+g_{2}^{2}}\,\gamma^{\mu}\gamma^{5}}/{2},$

where, $\Delta M_{\chi}=M_{\chi_{2}}-M_{\chi_{1}}$, $cb\equiv\cos\beta$ and
$sb\equiv\sin\beta$.

Channel |  $~{}~{}M_{\chi_{1}}$ (GeV) |  $~{}~{}M_{\chi_{2}}$ (GeV) |  $~{}~{}~{}sin\beta~{}~{}~{}$ | $\Omega_{DM}h^{2}$ |  Percentage
---|---|---|---|---|---
|  |  |  |  |  $\Gamma(\chi_{2}\rightarrow\chi_{1}h)\quad~{}93.56\%$
BP-1 |  $10^{-5}$ |  1000 |  $~{}~{}6.006\times 10^{-9}$ | 0.1194 |  $\Gamma(\chi_{2}\rightarrow\chi_{1}Z)\quad 1.58\%$
|  |  |  |  |  $\Gamma({E_{1}^{\pm}}\rightarrow\chi_{1}W)\quad~{}4.86\%$
|  |  |  |  |  $\Gamma(\chi_{2}\rightarrow\chi_{1}h)\quad~{}93.58\%$
BP-2 |  $10^{-3}$ |  1000 |  $~{}~{}6.015\times 10^{-10}$ | 0.1198 |  $\Gamma(\chi_{2}\rightarrow\chi_{1}Z)\quad 1.60\%$
|  |  |  |  |  $\Gamma({E_{1}^{\pm}}\rightarrow\chi_{1}W)\quad~{}4.82\%$
|  |  |  |  |  $\Gamma(\chi_{2}\rightarrow\chi_{1}h)\quad~{}93.58\%$
BP-4 |  $5$ |  1000 |  $~{}~{}8.543\times 10^{-12}$ | 0.1194 |  $\Gamma(\chi_{2}\rightarrow\chi_{1}Z)\quad 1.60\%$
|  |  |  |  |  $\Gamma({E_{1}^{\pm}}\rightarrow\chi_{1}W)\quad~{}4.82\%$
|  |  |  |  |  $\Gamma(\chi_{2}\rightarrow\chi_{1}h)\quad~{}86.07\%$
BP-5 |  $200$ |  1000 |  $~{}~{}1.626\times 10^{-12}$ | 0.112 |  $\Gamma(\chi_{2}\rightarrow\chi_{1}Z)\quad 3.42\%$
|  |  |  |  |  $\Gamma({E_{1}^{\pm}}\rightarrow\chi_{1}W)\quad~{}10.51\%$
|  |  |  |  |  $\Gamma(\chi_{2}\rightarrow\chi_{1}h)\quad~{}35.94\%$
BP-6 |  $600$ |  1000 |  $~{}~{}1.967\times 10^{-12}$ | 0.127 |  $\Gamma(\chi_{2}\rightarrow\chi_{1}Z)\quad 15.69\%$
|  |  |  |  |  $\Gamma({E_{1}^{\pm}}\rightarrow\chi_{1}W)\quad~{}48.35\%$
|  |  |  |  |  $\Gamma(\chi_{2}\rightarrow\chi_{1}h)\quad~{}93.94\%$
BP-7 |  $600$ |  2000 |  $~{}~{}8.549\times 10^{-13}$ | 0.127 |  $\Gamma(\chi_{2}\rightarrow\chi_{1}Z)\quad 1.49\%$
|  |  |  |  |  $\Gamma({E_{1}^{\pm}}\rightarrow\chi_{1}W)\quad~{}4.567\%$
|  |  |  |  |  $\Gamma(\chi_{2}\rightarrow\chi_{1}h)\quad~{}33.96\%$
BP-8 |  $1500$ |  2000 |  $~{}~{}1.916\times 10^{-12}$ | 0.119 |  $\Gamma(\chi_{2}\rightarrow\chi_{1}Z)\quad 16.02\%$
|  |  |  |  |  $\Gamma({E_{1}^{\pm}}\rightarrow\chi_{1}W)\quad~{}48.92\%$
|  |  |  |  |  $\Gamma(\chi_{2}\rightarrow\chi_{1}h)\quad~{}2.54\%$
BP-9 |  $1800$ |  2000 |  $~{}~{}1.916\times 10^{-12}$ | 0.112 |  $\Gamma(\chi_{2}\rightarrow\chi_{1}Z)\quad 23.48\%$
|  |  |  |  |  $\Gamma({E_{1}^{\pm}}\rightarrow\chi_{1}W)\quad~{}73.98\%$
Table 2: The benchmark points allowed by all the theoretical and experimental
constraints. We use $M_{H}=500$ GeV and $Y_{fl,i}~{}(i=1,2,3)<0.01$,
$\lambda_{3}\sim 10^{-9}$. These choices of parameters are consistent with
neutrino mass, mixing angles and other experimental data.

We show the contour plot 4 in the mixing angle ($\sin\beta$) vs. dark matter
mass plane for two different neutral fermion mass $M_{\chi_{2}}=1000$ GeV and
$2000$ GeV respectively. The left plot 4 corresponds to the dark matter mass
from $keV-MeV$ range while the $MeV-TeV$ mass range is shown in the right plot
4.

Figure 4:  The blue lines satisfy the exact relic denisty. The upper line
stands for the $M_{\chi_{2}}=1000$ GeV and lower one for $M_{\chi_{2}}=2000$
GeV. The slope of these blue lines depend on the mass gap
$M_{\chi_{2}}-M_{\chi_{1}}$.The triplet Higgs mass $M_{H,H^{\pm}}$ and Yukawa
coupling $Y_{fl1}$ are assumed in such a way that it provide the neutrino mass
and mixing angles and allowed by all the theoretical and experimental
constraints (see the table 1).

We fixed the triplet Higgs mass $M_{H}$ and Yukawa coupling $Y_{fl1}$ such
that it can provide the neutrino low energy variables and allowed by all the
theoretical and experimental constraints (e.g., see the table 1). We find that
the $\Gamma(\chi_{2}\rightarrow\chi_{1}h)$ is always dominant than the other
contributions. $\Gamma(h\rightarrow\chi_{1}\chi_{1})$ and
$\Gamma(Z\rightarrow\chi_{1}\chi_{1})$ have very tiny contribution. The other
channels also have smaller effect as compared to the
$\Gamma(\chi_{2}\rightarrow\chi_{1}h)$. The mass difference of the initial
heavy and final particles would change the contributions to the relic density.
We show a few related benchmark points, including the percentage of the
contributions of various channels in table 2 for the neutral fermion mass
$M_{\chi_{2}}=1000$ and $2000$ GeV respectively. We use the heavy Higgs mass
at $M_{H}=500$ GeV and the Yukawa couplings $Y_{fl,i}~{}(i=1,2,3)<0.01$ and
Higgs portal quartic coupling $\lambda_{3}\sim 10^{-9}$. Parameters are chosen
in such way that consistent with neutrino mass, mixing angles and other
experimental data.

It is to be highlighted that the second generation of vector fermion doublets
is needed to explain the neutrino mass and mixing angle accurately.
Interestingly, these new doublet fermions do not have any mixing; hence the
fermions $\chi_{3}$ and $E_{2}^{\pm}$ do not have two-body decay channels with
at least one dark matter component in the final state. However, they can decay
into dark matter through $>3$-body decay channels, e.g.,
$E_{2}^{\pm}\rightarrow H^{\pm}\nu_{l},H^{\pm}\rightarrow\chi_{1}l$ and
$\chi_{3}\rightarrow Hl,H\rightarrow\chi_{1}\nu_{l}$, etc. We also checked
that, these new fermion doublets can annihilate to the dark matter via $Z$ or
$h$ bosons, e.g., $\chi_{3}\chi_{3}\rightarrow
Z(h)\rightarrow\chi_{1}\chi_{1}$ and $E_{2}^{\pm}E_{2}^{\mp}\rightarrow
Z(h)\rightarrow\chi_{1}\chi_{1}$, etc. However, the effect of these production
from the second doublets are negligibly small as compared to the other
production channels.

##  VII Collider signature as charged track

The tiny fermionic mixing, i.e. $\sin\beta\sim 10^{-11}$ provides the exact
relic density for the dark matter candidate in this model. The direct
detection cross-section for the dark matter candidate is much below the
XENON-1T bound Aprile  _et al._ (2018). However, the interactions of the new
vector-like fermions with standard model fermions make them more feasible to
probe at the high energetic collider. The existing collider searches No  _et
al._ (2020) includes possible ways to explore dark matter particles. The
authors of the Ref. No  _et al._ (2020) have primarily investigated the FIMP
dark matter candidate in the context of 14 TeV LHC experiments with a future
high luminosity at the MATHUSLA surface detector. In this model, there have a
possibility to form a charged track due to the heavy vector like-charged
particle $E_{1}^{\pm}$ decay into two SM fermions and a dark matter. The decay
width of the vector like-charged fermion
($\Gamma(E_{1}^{\pm}\rightarrow\chi_{1}W^{*})$) if proportional to (see eqn.
VI) $\sin^{2}\beta$, we find the decay width $\sim 9\times 10^{-21}$ GeV and
corresponding decay length for these charged fermions to be $\mathcal{O}(20)$
Kilometer for this choice of $\sin\beta\sim 10^{-11}$. Now, it is interesting
to see if we can get a sufficient number of collider events from these charged
tracks for the detection; it mainly depends on the production cross-section of
the mother particle $E_{1}^{\pm}$ and luminosity. In this model, the dominant
production of the vector fermions come through the Drell-Yan processes. We
find the production cross-section of the vector-like charged fermions at $14$
TeV LHC energy for mass $M_{E_{1}^{\pm}}=1000$ GeV is $\sim 2$ fb. The total
number of events at the LHC-MATHUSLA can be found in Ref. No  _et al._ (2020)
and is given by $N_{events}=\sigma^{\rm LHC}_{\sqrt{s}}\,\mathcal{L}\,\int
P^{\rm MATH}_{\rm Decay},~{}{\rm with}~{}P^{\rm MATH}_{\rm
Decay}=0.05(e^{-\frac{L_{a}}{\beta\,c\,\tau_{E_{1}^{\pm}}}}-e^{-\frac{L_{b}}{\beta\,c\,\tau_{E_{1}^{\pm}}}}).$
Hence it might need larger luminosity and/or energy to get significant events
at the present MATHUSLA surface detector for our analysis. For this choice of
$\sin\beta\sim 10^{-11}$, we find $N_{events}>3$ at 14 TeV LHC with an
integrated luminosity $\mathcal{L}=5.0\times 10^{4}~{}{\rm fb}^{-1}$.

##  VIII Discussion and Conclusion

We study the possibility of FIMP dark matter in an extended hyperchargeless
Higgs triplet model in this work. This model contains additional two vector-
like doublet fermions and a neutral singlet fermion. We also analyze the
neutrino sector keeping an eye on the latest muon anomalous magnetic moment
data along with dark matter study. A detailed discussion on this model has not
yet explored with these recent results, which motivates us to carry out our
study.

We impose a discrete $Z_{2}$ symmetry such that the scalar triplet and
additional fermions are $odd$ under this $Z_{2}$ transformation while SM
fields do retain even. The extra scalar triplet consists of a pair of singly-
charged fields $H^{\pm}$ and a $CP$-even neutral scalar field $H$. The neutral
component of the first vector-like fermion doublet can mix with the neutral
singlet fermion, and the lightest eigenstate (mainly composed of the neutral
singlet fermion) behaves as a viable dark matter candidate, and this mixing
angle will help us to get the dark matter density through Freeze-in mechanism.
We assume that the second vector-like fermion does not mix with the other two
new fermions; however, it contributes to neutrino mass and mixing angles. The
other fermion mass eigenstates and the neutral component of the triplet scalar
$H$ can also serve as dark matter candidate depending on the mass. In such a
case, one can get the relic density through the freeze-out mechanism only.
However, for the fermionic dark matter candidate (mostly composed of neutral
component of first vector doublet) case,
$M_{\chi_{2}}<M_{\chi_{1,3},E^{\pm}_{1,2},H}$, and most of the regions are
ruled out by the present direct detection experiments. It has additional
$Z$-portal diagram for the direct searches. One can get the scalar dark matter
for $M_{H}<M_{\chi_{1,2,3},E^{\pm}_{1,2}}$. In an earlier study Khan (2018),
it was found that the exact relic density can be found at $M_{H}\sim 2$ TeV.
The low mass region gives under abundance due to the large co-annihilation
channels $HH^{\pm}\rightarrow W\rightarrow SM~{}particles$. We also see that
the additional enhancement come from the diagram $HH^{\pm}\,(HH)\rightarrow
SM~{}leptons$ through $t$-channels which having $\chi_{1,2,3},E^{\pm}_{1,2}$
propagators. Incidentally, one can get relic density $M_{H}\sim 2.8$ TeV
through freeze-out mechanism only. In such a scenario, the low mass region for
the scalar dark matter gives under abundance; conversely, it is ruled out from
the relic density and/or direct detection for doublet-type neutral fermion.
These components also ruled out as a FIMP dark matter candidate, give the
overabundance of the dark matter relic density through the Freeze-in
mechanism. We also check that $\chi_{1}$ (mostly compose of singlet neutral
fermion) as a WIMP dark matter is ruled out as it gives overabundance through
Freeze-out mechanism. Hence, $\chi_{1}$ is the most suitable dark matter
candidate in this model via Freeze-in mechanism, which gives relic density for
all mass range $keV-TeV$. We show two different plots in the mixing angle
$vs.$ dark matter mass plane. It is noted that the allowed dark matter region
can also provide the exact neutrino mass and mixing angles by adjusting the
other parameters in the model, mainly the Higgs portal coupling $\lambda_{3}$,
Yukawa couplings $Y_{fli}$ for a fixed $M_{\chi_{2}}$. The effect of these
coupling adjustments are negligibly small in the relic density.

This model also discusses the numerical insights to neutrino phenomenology,
and neutrino masses achieved via one-loop correction scotogenic model. A tiny
Higgs portal coupling, $\lambda_{3}$ and/or the Yukawa couplings, $Y_{fli}$
with fermion masses $\mathcal{O}(1)$ TeV can give exact mass order $\sum
m_{\nu}<0.1$ eV. We first check with only one generation of the vector-like
fermion doublet with every possible way to generate all the neutrino low
energy variables. Eventually, we find that minimum of two doublets are
essential to address all the light neutrino parameters. Therefore, we add
another doublet without affecting dark matter results. This second fermion
doublet extension helps us generate all the neutrino variables in both the
normal and inverted hierarchy mass pattern.

In the LFV study, one can have additional corrections (positive or negative
depending on the parameters) to the muon and electron anomalous magnetic
moments at one-loop level. The model also has lepton flavor violation decay
(especially, $\Gamma(\mu\rightarrow e\gamma)$) through one-loop correction.
This decay process provides tight constraints on the model parameters that
involve the neutrino mass, dark matter density, and anomalous magnetic
moments. We find that the allowed parameter by the LFV process cannot explain
the recent muon and electron anomalous magnetic moments data. We need very
large new Yukawa couplings (non-perturbative) to explain muon anomalous
magnetic moments data, which violates LFV and relic density bounds. Therefore,
as a conclusive remark on this model, one can explain the neutrino low energy
variables, extensive range of dark matter masses using the parameters allowed
by the LFV data with tiny couplings and mixing angles.

Acknowledgements:
PD and MKD would like to acknowledge the Department of Science and Technology,
Government of India under project number EMR/2017/001436 for financial aids.
NK would like to thank Prof. Dilip Kumar Ghosh for the support at IACS-
KOLKATA. NK also would like to thank the organizers of the Invisibles21
virtual Workshop of the European Unions Horizon research and innovation
program. This project has received funding /support from the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska
-Curie grant agreement No 860881-HIDDeN”

##  Appendix A Mixing of three new neutral neutral and charged fermions

If we consider, the effect of the second generation of fermion, then we mast
need to diagonalize the following matrix.

$\displaystyle\mathcal{M}=\begin{pmatrix}M_{NS}&M_{X}^{1}&M_{X}^{2}\\\
M_{X}^{1,\dagger}&M_{ND}^{11}&M_{ND}^{12}\\\
M_{X}^{2,\dagger}&M_{ND}^{21}&M_{ND}^{22}\\\ \end{pmatrix},$ (39)

where, $M_{X}^{1}=\frac{v\,Y_{N,1}}{\sqrt{2}}$ and
$M_{X}^{2}=\frac{v\,Y_{N,2}\,}{\sqrt{2}}$. The neutral component of the
fermion doublets ($N_{D,1}$ and $N_{D,2}$) and the singlet charged fermion
(${N}_{S}$) mix at tree level. Let us assume the second VLF doublet is not
decoupled, i.e., $Y_{N,2}\neq 0$ and $M_{ND}^{21}=M_{ND}^{12}\neq 0$. Then,
the mass eigenstates are obtained by diagonalizing the mass matrix (similar to
the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix with $zero$ phases) with a
rotation of the $({N}_{S}$ $N_{D,1}$ $N_{D,2})$ basis as,

$\displaystyle\begin{pmatrix}\chi_{1}\\\ \chi_{2}\\\ \chi_{3}\\\
\end{pmatrix}=U_{PMNS}\begin{pmatrix}N_{S}\\\ N_{D,1}\\\ N_{D,2}\\\
\end{pmatrix},$ (40)

where,

$\displaystyle U_{PMNS}=\begin{pmatrix}\cos\beta_{12}&\sin\beta_{12}&0\\\
-\sin\beta_{12}&\cos\beta_{12}&0\\\ 0&0&1\\\
\end{pmatrix}\begin{pmatrix}\cos\beta_{13}&0&\sin\beta_{13}\\\ 0&1&0\\\
-\sin\beta_{13}&0&\cos\beta_{13}\\\ \end{pmatrix}\begin{pmatrix}1&0&0\\\
0&\cos\beta_{23}&\sin\beta_{23}\\\ 0&-\sin\beta_{23}&\cos\beta_{23}\\\
\end{pmatrix}.$ (41)

The new charged fermion also do get mixed together and the mass matrix can be
written as:

$\displaystyle\mathcal{M}_{c}=\begin{pmatrix}M_{ND}^{11}&M_{ND}^{12}\\\
M_{ND}^{21}&M_{ND}^{22}\\\ \end{pmatrix},$ (42)

With the condition $M_{ND}^{12}=M_{ND}^{21}$, the mass eigenstates are
obtained by diagonalizing the mass matrix with a rotation of the (
$E_{D,1}^{\pm}$ $E_{D,2}^{\pm}$ ) basis as:

$\displaystyle\begin{pmatrix}E_{1}^{\pm}\\\ E_{2}^{\pm}\\\
\end{pmatrix}=\begin{pmatrix}\cos\beta_{23}&\sin\beta_{23}\\\
-\sin\beta_{23}&\cos\beta_{23}\\\ \end{pmatrix}\begin{pmatrix}E_{D,1}^{\pm}\\\
E_{D,2}^{\pm}\\\ \end{pmatrix},{~{}\rm with~{}}\tan
2\beta_{23}=\frac{2M_{ND}^{12}}{M_{ND}^{22}-M_{ND}^{11}}.$ (43)

The masses for $M_{ND}^{22}-M_{ND}^{11}>>M_{ND}^{12}$ can be written as:

$\displaystyle M_{E_{1}^{\pm}}$ $\displaystyle=$ $\displaystyle
M_{ND}^{11}-\frac{2(M_{ND}^{12})^{2}}{M_{ND}^{22}-M_{ND}^{11}},~{}{\rm
and}~{}~{}M_{E_{2}^{\pm}}=M_{ND}^{22}+\frac{2(M_{ND}^{12})^{2}}{M_{ND}^{22}-M_{ND}^{11}}.$
(44)

All the non-zero of these off-diagonal term could change our results which we
left for our future analysis. Please note that the limits
$\beta_{13}=\beta_{23}=0$ produce our results.

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