††institutetext: Department of Applied Physics and Astronomy, University of
Sharjah, P.O. Box 27272 Sharjah, UAE.

# A Scotogenic Model with Two Inert Doublets

Amine Ahriche ${}^{\href https://orcid.org/0000-0003-0230-1774}$
<EMAIL_ADDRESS>
###### Abstract

In this work, we present a scotogenic model, where the neutrino mass is
generated at one-loop diagrams. The standard model (SM) is extended by three
singlet Majorana fermions and two inert scalar doublets instead of one doublet
as in the minimal scotogenic model. The model scalar sector includes two CP-
even, two CP-odd and two charged scalars in addition to the Higgs. The dark
matter (DM) candidate could be either the light Majorana fermion (Majorana
DM), or the lightest among the CP-even and the CP-odd scalars (scalar DM). We
show that the model accommodates both Majorana and scalar DM within a
significant viable parameter space, while considering all the relevant
theoretical and experimental constraints such as perturbativity, vacuum
stability, unitarity, the di-photon Higgs decay, electroweak precision tests
and lepton flavor violating constraints. In addition to the collider
signatures predicted by the minimal scotogenic model, our model predicts some
novel signatures that can be probed through some final states such as
$8~{}jets+\not{E}_{T}$, $1\ell+4~{}jets+\not{E}_{T}$ and $4b+\not{E}_{T}$.

##  1 Introduction

The standard model (SM) of particle physics was successful in describing the
properties of elementary particles and their interactions around the
electroweak (EW) scale. However, there are still unanswered questions such as
the neutrino oscillation data and the dark matter (DM) nature. Neutrino
oscillation data established that at least two of the three SM neutrinos have
mass and nonzero mixing. However, their properties, such as their nature and
the origin of the smallness of their mass, have no explanation within the SM,
which demands for new physics. One of the most popular mechanisms to explain
why neutrino mass is tiny; is the so-called seesaw mechanism [1], which
assumes the existence of right-handed (RH) neutrinos with masses that are
heavier by several orders of magnitude than the EW scale. Such heavy particles
decouple from the low-energy spectrum at the EW scale, and hence, cannot be
directly probed at high-energy physics experiments. Another approach that
requires lower new physics scale, where small neutrino mass is generated
naturally; is via quantum corrections, where loop suppression factor(s)
ensures the neutrino mass smallness. This can be realized at one loop [2], two
loops [3], three loops [4, 5], or four loops [6]. Interesting features for
this TeV scale new physics class of models can be tested at high energy
colliders [7, 8] (For a review, see [9]).

The simplest and most popular realization of a radiative neutrino mass
mechanism is the so-called scotogenic model [10], where the SM is extended by
an inert scalar doublet and three singlet Majorana fermions. In addition, this
model could accommodate two possible DM candidates, i.e., the lightest among
the Majorana fermions and the light neutral scalar in the inert dark doublet.
The scalar DM scenario is identical to the inert doublet Higgs model (IDM),
that is highly constrained by the DM direct detection experiments. In the
scenario of Majorana DM, the right amount of the DM relic density requires
relatively large values for the new Yukawa couplings that couple the inert
doublet with the Majorana fermions [11, 12]. This can be achieved only by
imposing a strong degeneracy between the masses of the CP-even and CP-odd
scalars, making the quartic coupling suppressed $\lambda_{5}\sim
10^{-10}-10^{-9}$. Such fine-tuning can be avoided by extending the minimal
scotogenic model (MSctM) by two real singlets and impose a global
$Z_{4}/Z_{2}$ symmetry [13]. Indeed, there have many models that have been
proposed and studied beyond the MSctM [14, 15]. For example, in [14] the
authors studied a generalized scotogenic model, or what they called ”general
scotogenic model”, by considering $n_{N}$ Majorana singlet fermions and
$n_{\eta}$ inert doublets, where many phenomenological aspects were discussed.

Based on the current/future results of both ATLAS and CMS, the IDM (and
therefore, the MSctM with scalar DM model) parameter space will become more
constrained as the precision measurements of the EW sector gets improved. In
addition, the recent results from the direct detection experiments such as
LUX-ZEPLIN experiment [16], would place more stringent bounds on the model.
This makes the extension of the IDM interesting, that might evade some of the
constraints and predict additional interesting signatures. On top of that,
among the important motivations to extend the IDM; is that it does not allow
CP violation in the scalar sector, where an additional $SU(2)_{L}$ Higgs
doublet is useful to do this task [17]. According to the previously mentioned
motivations, it is natural to consider the realization of a scotogenic model
where the SM is extended by three singlet Majorana fermions $N_{1,2,3}$ and
two inert doublets $\Phi_{1,2}$ that couple to the leptonic left-handed
doublets and the singlet Majorana’s. Here, the new additional fields transform
as odd fields under a global $Z_{2}$ symmetry
$\\{N_{1,2,3},\Phi_{1,2}\\}\rightarrow\\{-N_{1,2,3},-\Phi_{1,2}\\}$ to ensure
the DM stability.

This work is organized as follow. In section 2, we present the model and show
the neutrino mass generation at loop. In section 3, different theoretical and
experimental constraints are discussed. Sections 4 and 5 are devoted to show
the viable parameter space and the DM phenomenology, respectively, in both
scenarios of Majorana and scalar DM candidates. We conclude in section 6. In
addition, the paper includes three appendices: in Appendix A, we discuss how
to estimate the new Yukawa couplings where a generalized parameterization like
the Casas-Ibarra one is introduced. In Appendix B, we generalize our results
for the case of a scotogenic model with n inert scalar doublets instead of
two. In Appendix C, we give the amplitude matrices that are relevant to the
perturbative unitarity conditions.

##  2 The Model & Neutrino Mass

Here, we extend the SM by two inert doublets denoted by
$\Phi_{1,2}\sim(1,2,1)$ and three singlet Majorana fermions
$N_{i}\sim(1,1,0)$. The Lagrangian that involves the Majorana fermions can be
written as

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\overline{L}_{\alpha}\big{(}h_{\alpha,k}~{}\epsilon.\Phi_{1}+h_{\alpha,k+3}~{}\epsilon.\Phi_{2}\big{)}N_{k}+\frac{1}{2}\bar{N}_{k}^{C}M_{k}N_{k}+h.c.,$
(1)

where $L_{\alpha}$ are the left-handed lepton doublets, and
$\epsilon=i\sigma_{2}$ is an anti-symmetric tensor. Here, $h_{\alpha,k}$ is a
$3\times 6$ new Yukawa matrix, the mass matrix $M_{k}$ can be considered
diagonal.

The most general $Z_{2}$-symmetric, renormalizable, and gauge invariant
potential reads

$\displaystyle V\left(\mathcal{H},\Phi,S,\chi\right)$ $\displaystyle=$
$\displaystyle-\mu_{H}^{2}|\mathcal{H}|^{2}+\mu_{i}^{2}|\Phi_{i}|^{2}+\frac{\lambda_{H}}{6}|\mathcal{H}|^{4}+\frac{\lambda_{i}}{6}|\Phi_{i}|^{4}+\omega_{i}|\mathcal{H}|^{2}|\Phi_{i}|^{2}+\kappa_{i}|\mathcal{H}^{\dagger}\Phi_{i}|^{2}$
(2)
$\displaystyle+\varrho_{1}|\Phi_{1}|^{2}|\Phi_{2}|^{2}+\varrho_{2}|\Phi_{1}^{\dagger}\Phi_{2}|^{2}+\left\\{\mu_{3}^{2}\Phi_{1}^{\dagger}\Phi_{2}+\frac{1}{2}\xi_{i}(\mathcal{H}^{\dagger}\Phi_{i})^{2}+\xi_{3}(\mathcal{H}^{\dagger}\Phi_{1})(\mathcal{H}^{\dagger}\Phi_{2})\right.$
$\displaystyle\left.+\xi_{4}(\Phi_{1}^{\dagger}\mathcal{H})(\mathcal{H}^{\dagger}\Phi_{2})+h.c.\right\\},$

with $\mathcal{H}$, and $\Phi$ can be parameterized as follows

$\mathcal{H}=\left(\begin{array}[]{c}\chi^{+}\\\
\frac{1}{\sqrt{2}}(\upsilon+h+i\chi^{0})\end{array}\right),~{}\Phi_{i}=\left(\begin{array}[]{c}S_{i}^{+}\\\
\frac{1}{\sqrt{2}}(S_{i}^{0}+iQ_{i}^{0})\end{array}\right).$ (3)

The terms of the Lagrangian in (1) and (2) are invariant under a global
$Z_{2}$ symmetry according to the charges

$\Phi_{1,2}\rightarrow-\Phi_{1,2},\,N_{1,2,3}\rightarrow-N_{1,2,3},$ (4)

where all other fields are even. This $Z_{2}$ symmetry forbids such terms like
$(\mathcal{H}^{\dagger}\Phi_{i})(\Phi_{j}^{\dagger}\Phi_{k})$. In case of
complex values for $\mu_{3}^{2}$ and/or $\xi_{i}$, CP is explicitly broken,
where we obtain CP-violating interactions relevant to the four neutral inert
eigenstates in the basis {$S_{1}^{0},S_{2}^{0}$,$Q_{1}^{0},Q_{2}^{0}$}. This
case would be very interesting since the additional CP-violating sources are
required for matter antimatter asymmetry generation [18]. In what follows, we
consider real values for the parameters $\mu_{3}^{2}$ and $\xi_{i=1,2,3,4}$ to
avoid CP violation. The case of explicit CP violation (complex values for
$\mu_{3}^{2}$ and/or $\xi_{i}$) requires an independent investigation since it
may have important cosmological consequences .

After the electroweak symmetry breaking (EWSB), we are left with three
$CP$-even scalars $(h,H_{1,2}^{0})$, two $CP$-odd scalar $A_{1,2}^{0}$ and two
pair of charged scalars $H_{1,2}^{\pm}$. The Higgs mass is given
$m_{h}^{2}=2\mu_{H}^{2}=\frac{\lambda_{H}}{3}\upsilon^{2}$, and the inert
eigenstates are defined as

$\displaystyle\left(\begin{array}[]{c}H_{1}^{0}\\\
H_{2}^{0}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}c_{H}&s_{H}\\\
-s_{H}&c_{H}\end{array}\right)\left(\begin{array}[]{c}S_{1}^{0}\\\
S_{2}^{0}\end{array}\right),\left(\begin{array}[]{c}A_{1}^{0}\\\
A_{2}^{0}\end{array}\right)=\left(\begin{array}[]{cc}c_{A}&s_{A}\\\
-s_{A}&c_{A}\end{array}\right)\left(\begin{array}[]{c}Q_{1}^{0}\\\
Q_{2}^{0}\end{array}\right),$ (17)
$\displaystyle\left(\begin{array}[]{c}H_{1}^{\pm}\\\
H_{2}^{\pm}\end{array}\right)=\left(\begin{array}[]{cc}c_{C}&s_{C}\\\
-s_{C}&c_{C}\end{array}\right)\left(\begin{array}[]{c}S_{1}^{\pm}\\\
S_{2}^{\pm}\end{array}\right),$ (24)

with $c_{X}=\cos\theta_{X},\,s_{X}=\sin\theta_{X}$ and $\theta_{H}$ ,
$\theta_{A}$ and $\theta_{C}$ are the mixing angles that diagonalise the
$CP$-even, $CP$-odd and charged scalars mass matrices, respectively. The
charged, neutral $CP$-even and $CP$-odd scalar squared mass matrices in the
basis {$S_{1}^{\pm},S_{2}^{\pm}$}, {$S_{1}^{0},S_{2}^{0}$} and
{$Q_{1}^{0},Q_{2}^{0}$}, respectively, are given by

$\displaystyle M_{C}^{2}$
$\displaystyle=\left[\begin{array}[]{cc}\mu_{1}^{2}&\mu_{3}^{2}\\\
\mu_{3}^{2}&\mu_{2}^{2}\end{array}\right]+\frac{\upsilon^{2}}{2}\left[\begin{array}[]{cc}\omega_{1}&0\\\
0&\omega_{2}\end{array}\right],\,M_{H,A}^{2}=M_{C}^{2}+\frac{\upsilon^{2}}{2}\left[\begin{array}[]{cc}\kappa_{1}&\xi_{4}\\\
\xi_{4}&\kappa_{2}\end{array}\right]\pm\frac{\upsilon^{2}}{2}\left[\begin{array}[]{cc}\xi_{1}&\xi_{3}\\\
\xi_{3}&\xi_{2}\end{array}\right].$ (33)

The mass eigenvalues and the mixing are given by

$\displaystyle m_{X_{1,2}}^{2}$
$\displaystyle=\frac{1}{2}\bigg{(}[M_{X}^{2}]_{11}+[M_{X}^{2}]_{22}\mp\sqrt{([M_{X}^{2}]_{22}-[M_{X}^{2}]_{11})^{2}+4[M_{X}^{2}]_{12}^{2}}\bigg{)},$
$\displaystyle\tan(2\theta_{X})$
$\displaystyle=2[M_{X}^{2}]_{12}/\Big{(}[M_{X}^{2}]_{22}-[M_{X}^{2}]_{11}\Big{)},$
(34)

for
($[X_{1,2},\theta_{X}]=[H_{1,2}^{0},\theta_{H}],[A_{1,2}^{0},\theta_{A}],[H_{1,2}^{\pm},\theta_{C}]$),
respectively.

Then, the independent free parameters are

$M_{1,2,3},~{}m_{H_{1,2}^{\pm}},~{}m_{H_{1,2}^{0}},~{}m_{A_{1,2}^{0}},~{}s_{H,A,C},~{}\omega_{1,2},~{}h_{\alpha
i},$ (35)

where

$\displaystyle\mu_{1}^{2}$
$\displaystyle=m_{H_{1}^{\pm}}^{2}c_{C}^{2}+m_{H_{2}^{\pm}}^{2}s_{C}^{2}-\frac{1}{2}\omega_{1}\upsilon^{2},\,\mu_{2}^{2}=m_{H_{1}^{\pm}}^{2}s_{C}^{2}+m_{H_{2}^{\pm}}^{2}c_{C}^{2}-\frac{1}{2}\omega_{2}\upsilon^{2},\,\mu_{3}^{2}=c_{C}s_{C}(m_{H_{2}^{\pm}}^{2}-m_{H_{1}^{\pm}}^{2}),$
$\displaystyle\kappa_{1}\upsilon^{2}$
$\displaystyle=m_{H_{1}^{0}}^{2}c_{H}^{2}+m_{H_{2}^{0}}^{2}s_{H}^{2}+m_{A_{1}^{0}}^{2}c_{A}^{2}+m_{A_{2}^{0}}^{2}s_{A}^{2}-2(m_{H_{1}^{\pm}}^{2}c_{C}^{2}+m_{H_{2}^{\pm}}^{2}s_{C}^{2}),$
$\displaystyle\kappa_{2}\upsilon^{2}$
$\displaystyle=m_{H_{2}^{0}}^{2}c_{H}^{2}+m_{H_{1}^{0}}^{2}s_{H}^{2}+m_{A_{2}^{0}}^{2}c_{A}^{2}+m_{A_{1}^{0}}^{2}s_{A}^{2}-2(m_{H_{2}^{\pm}}^{2}c_{C}^{2}+m_{H_{1}^{\pm}}^{2}s_{C}^{2}),$
$\displaystyle\xi_{1}\upsilon^{2}$
$\displaystyle=(m_{H_{1}^{0}}^{2}c_{H}^{2}+m_{H_{2}^{0}}^{2}s_{H}^{2})-(m_{A_{2}^{0}}^{2}s_{A}^{2}+m_{A_{1}^{0}}^{2}c_{A}^{2}),~{}\xi_{2}\upsilon^{2}=(m_{H_{1}^{0}}^{2}s_{H}^{2}+m_{H_{2}^{0}}^{2}c_{H}^{2})-(m_{A_{1}^{0}}^{2}s_{A}^{2}+m_{A_{2}^{0}}^{2}c_{A}^{2}),$
$\displaystyle\xi_{3}\upsilon^{2}$
$\displaystyle=s_{H}c_{H}(m_{H_{2}^{0}}^{2}-m_{H_{1}^{0}}^{2})-s_{A}c_{A}(m_{A_{2}^{0}}^{2}-m_{A_{1}^{0}}^{2}),$
$\displaystyle\xi_{4}\upsilon^{2}$
$\displaystyle=s_{H}c_{H}(m_{H_{2}^{0}}^{2}-m_{H_{1}^{0}}^{2})+s_{A}c_{A}(m_{A_{2}^{0}}^{2}-m_{A_{1}^{0}}^{2})-2c_{C}s_{C}(m_{H_{2}^{\pm}}^{2}-m_{H_{1}^{\pm}}^{2}).$
(36)

The first term in (1), can be written as

$\displaystyle\mathcal{L}$
$\displaystyle\supset\sum_{\alpha,j,k}\frac{1}{\sqrt{2}}\big{(}g_{\alpha
k}^{(j)}H_{j}^{0}+if_{\alpha k}^{(j)}A_{j}^{0}\big{)}\overline{\nu}_{\alpha
L}N_{k}-\zeta_{\alpha k}^{(j)}H_{j}^{\pm}\overline{\ell}_{\alpha L}N_{k}+h.c,$
(37)

with $\alpha=e,\mu,\tau$, $j=1,2$ and $k=1,3$; and

$\displaystyle g_{\alpha k}^{(1)}$
$\displaystyle=c_{H}h_{\alpha,k}+s_{H}h_{\alpha,k+3},\,\,g_{\alpha
k}^{(2)}=-s_{H}h_{\alpha,k}+c_{H}h_{\alpha,k+3},$ $\displaystyle f_{\alpha
k}^{(1)}$ $\displaystyle=c_{A}h_{\alpha,k}+s_{A}h_{\alpha,k+3},\,\,f_{\alpha
k}^{(2)}=-s_{A}h_{\alpha,k}+c_{A}h_{\alpha,k+3},$ $\displaystyle\zeta_{\alpha
k}^{(1)}$
$\displaystyle=c_{C}h_{\alpha,k}+s_{C}h_{\alpha,k+3},\,\,\zeta_{\alpha
k}^{(2)}=-s_{C}h_{\alpha,k}+c_{C}h_{\alpha,k+3}.$ (38)

The first two interaction terms in (37) give rise to the neutrino mass a la
scotogenic way; and the last term leads to the LFV processes
$\ell_{\beta}\rightarrow\ell_{\alpha}+\gamma$ and $\ell_{\beta}\rightarrow
3\ell_{\alpha}$. Then, the neutrino mass $[\alpha,\beta]$ matrix element comes
from the contribution 12 diagrams as shown in Fig. 1.

Figure 1: The 12 Feynman diagrams responsible for neutrino mass generation.

It can be written as

$\displaystyle m_{\alpha\beta}^{(\nu)}$
$\displaystyle=\sum_{k=1}^{3}\sum_{j=1}^{2}M_{k}\Big{\\{}g_{\alpha
k}^{(j)}\,g_{\beta
k}^{(j)}\,\mathcal{F}\Big{(}m_{H_{j}^{0}}/M_{k}\Big{)}-f_{\alpha
k}^{(j)}\,f_{\beta
k}^{(j)}\,\mathcal{F}\Big{(}m_{A_{j}^{0}}/M_{k}\Big{)}\Big{\\}},$ (39)

with $\mathcal{F}(x)=\frac{x^{2}\log(x)}{8\pi^{2}(x^{2}-1)}$. The divergent
parts of the diagrams in Fig. 1 that are mediated by $H_{j}^{0}$ and
$A_{j}^{0}$ cancels each other due to the identity111One has to mention that
this identity is equivalent to (18) in [14], where the scotogenic model is
generalized by considering many Majorana singlets and inert doublets.

$\sum_{j=1}^{2}\Big{\\{}g_{\alpha k}^{(j)}\,g_{\beta k}^{(j)}-f_{\alpha
k}^{(j)}\,f_{\beta k}^{(j)}\Big{\\}}=0.$ (40)

The neutrino mass matrix (39) can be parameterized as

$\displaystyle(m^{(\nu)})_{3\times 3}$ $\displaystyle=(h)_{3\times
6}.(\Lambda)_{6\times 6}.(h^{T})_{6\times 3}\,\,\,\,,$ (41)
$\displaystyle\Lambda$
$\displaystyle=\left(\begin{array}[]{cc}\varpi_{1}&\varpi_{3}\\\
\varpi_{3}&\varpi_{2}\end{array}\right),$ (44)

where $\varpi_{1,2,3}$ are $3\times 3$ diagonal matrices whose elements are
given by

$\displaystyle(\varpi_{1})_{ik}$
$\displaystyle=\delta_{ik}M_{k}\big{[}c_{H}^{2}\mathcal{F}\Big{(}m_{H_{1}^{0}}/M_{k}\Big{)}+s_{H}^{2}\,\,\mathcal{F}\Big{(}m_{H_{2}^{0}}/M_{k}\Big{)}-c_{A}^{2}\,\mathcal{F}\Big{(}m_{A_{1}^{0}}/M_{k}\Big{)}-s_{A}^{2}\,\,\mathcal{F}\Big{(}m_{A_{2}^{0}}/M_{k}\Big{)}\big{]},$
$\displaystyle(\varpi_{2})_{ik}$
$\displaystyle=\delta_{ik}M_{k}\big{[}s_{H}^{2}\mathcal{F}\Big{(}m_{H_{1}^{0}}/M_{k}\Big{)}+c_{H}^{2}\,\,\mathcal{F}\Big{(}m_{H_{2}^{0}}/M_{k}\Big{)}-s_{A}^{2}\,\mathcal{F}\Big{(}m_{A_{1}^{0}}/M_{k}\Big{)}-c_{A}^{2}\,\mathcal{F}\Big{(}m_{A_{2}^{0}}/M_{k}\Big{)}\big{]},$
$\displaystyle(\varpi_{3})_{ik}$
$\displaystyle=\delta_{ik}M_{k}\big{[}s_{H}c_{H}(\mathcal{F}\Big{(}m_{H_{1}^{0}}/M_{k}\Big{)}-\,\mathcal{F}\Big{(}m_{H_{2}^{0}}/M_{k}\Big{)})-s_{A}c_{A}(\,\mathcal{F}\Big{(}m_{A_{1}^{0}}/M_{k}\Big{)}-\,\mathcal{F}\Big{(}m_{A_{2}^{0}}/M_{k}\Big{)})\big{]},$
(45)

for $i,k=1,3$. Here, the matrix $\Lambda$ in (41) is not diagonal, and
therefore can cannot use the Casas-Ibarra parameterization [19] to estimate
the Yukawa couplings $h$. However, we can derive a similar formula in such
case ($\Lambda$ is not diagonal). We give a general solution for the Yukawa
coupling $3\times M$ matrix222Here, $M=6$ in our case. However, the solutions
(46) are valid for any value $M\geq 3$. as

$h_{3\times M}=(U_{\nu})_{3\times 3}.(D_{\sqrt{m_{\nu}}})_{3\times
3}.(T)_{3\times M}.(D_{(\Lambda^{\prime}_{i})^{-1/2}})_{M\times
M}.(Q)_{M\times M}\,\,\,\,\,\,,$ (46)

with $U_{\nu}$ is the Pontecorvo-Maki-Nakawaga-Sakata (PMNS) mixing matrix,
$D_{\sqrt{m_{\nu}}}=\textrm{diag}\\{m_{1}^{1/2}$ $,m_{2}^{1/2},m_{3}^{1/2}\\}$
with $m_{i}$ are the neutrino mass eigenvalues; and $T$ is an arbitrary
$3\times M$ matrix that fulfills the identity $T_{3\times M}.T_{M\times
3}^{T}=\boldsymbol{1}_{3\times 3}$. The matrix
$(D_{(\Lambda^{\prime}_{i})^{-1/2}})_{M\times
M}=\textrm{diag}\left\\{(\Lambda^{\prime}_{i})^{-1/2}\right\\}$, with
$\Lambda^{\prime}_{i}$ to be the eigenvalues of the matrix $\Lambda$ in (41),
that diagonalized using the matrix $Q$, i.e.,
$\textrm{diag}\left\\{\Lambda^{\prime}_{i}\right\\}=Q.\Lambda.Q^{T}$. In
Appendix A, we discuss the derivation of (46) and the possible representations
for this matrix. In addition, we discuss in Appendix B the generalization of
this model by considering n inert scalar doublets instead of two doublets.

##  3 Theoretical & Experimental Constraints

Here, we discuss the theoretical and the experimental constraints relevant to
our model. These constraints are briefly discussed below:

  * •

Perturbativity: all the quartic vertices of the scalar fields should be
satisfy the perturbativity bounds, i.e.,

$\displaystyle\max\big{\\{}\lambda_{H},\lambda_{1,2},|\omega_{1,2}|,|\kappa_{1,2}+\omega_{1,2}|,|\omega_{1,2}+\kappa_{1,2}\pm\xi_{1,2}|,|\varrho_{1}|,|\varrho_{1}+\varrho_{2}|,$
$\displaystyle|\xi_{1,2}|,\frac{1}{2}|\kappa_{1,2}\pm\xi_{1,2}|,\frac{1}{2}|\varrho_{2}|\big{\\}}\leq
4\pi.$ (47)

  * •

Perturbative unitarity: the perturbative unitarity has to be preserved in all
processes involving scalars or gauge bosons. In the high-energy limit, the
gauge bosons should be replaced by their Goldstones; and the scattering
amplitude matrix of $\phi_{i}~{}\phi_{j}\rightarrow\phi_{k}\phi_{m}$ can be
easily calculated. It has been shown that the perturbative unitarity
conditions are achieved if the eigenvalues of the scattering amplitude matrix
to be smaller than $|\Lambda_{i}|<8\pi$ [20].

In our model, due to some exact symmetries such as the electric charge, CP and
the global $Z_{2}$ symmetry, the full scattering amplitude matrix can be
divided into six sub-matrices. These six sub-matrices are defined in the basis
of initial/final states that are: (1) CP-even, $Z_{2}$ even, & $Q_{em}=0$; (2)
CP-even, $Z_{2}$ odd, & $Q_{em}=0$, (3) CP-odd, $Z_{2}$ even, & $Q_{em}=0$,
(4) CP-odd, $Z_{2}$ odd, & $Q_{em}=0$, (5) $Z_{2}$ even, & $Q_{em}=\pm 1$; and
(6) $Z_{2}$ odd, & $Q_{em}=\pm 1$. These sub-matrices and their basis are
given in Appendix C in details.

  * •

Vacuum Stability: the scalar potential is required to be bounded from below in
all the directions of the field space. It is obvious that along the pure
directions ($\Phi_{1}=\Phi_{2}=0$), ($\mathcal{H}=\Phi_{2}=0$) and
($\mathcal{H}=\Phi_{1}=0$), the vacuum stability conditions, i.e., the
potential positivity at large field values, are
$\lambda_{H},\lambda_{1},\lambda_{2}>0$, respectively. However, along any
direction beside the pure ones, the potential positivity is guaranteed if all
quartic couplings are positive. In case of negative quartic coupling(s), the
vacuum stability conditions are [21]:

$\left|\begin{array}[]{ccc}\lambda_{H}&\overline{\omega_{1}+\kappa_{1}+\xi_{1}},&\overline{\omega_{2}+\kappa_{2}+\xi_{2}}\\\
\overline{\omega_{1}+\kappa_{1}+\xi_{1}}&\lambda_{1}&\overline{\varrho_{1}+\varrho_{2}}\\\
\overline{\omega_{2}+\kappa_{2}+\xi_{2}}&\overline{\varrho_{1}+\varrho_{2}}&\lambda_{2}\end{array}\right|>0,~{}~{}\left|\begin{array}[]{ccc}\frac{2}{3}\lambda_{H}&\overline{\omega_{1}+\kappa_{1}},&\overline{\omega_{2}+\kappa_{2}}\\\
\overline{\omega_{1}+\kappa_{1}}&\frac{2}{3}\lambda_{1}&\overline{\varrho_{1}+\varrho_{2}}\\\
\overline{\omega_{2}+\kappa_{2}}&\overline{\varrho_{1}+\varrho_{2}}&\frac{2}{3}\lambda_{2}\end{array}\right|>0,$
(48)

for the neutral and charged fields directions, respectively, with
$\overline{X}=\min\left(X,0\right)$.

In addition, we require that the vacuum $(<h>=\upsilon,<S_{i}^{0}>=0)$ would
be the deepest one. As a conservative choice in this work, we consider the
conditions that the potential (2) should not develop a vev in the inert
directions, i.e., we should have $<S_{i}^{0}>=0)$ either at $<h>=0$ or
$<h>=\upsilon$. These conditions can be translated into

$\mu_{1}^{2},\mu_{2}^{2},\mu_{1}^{2}+\mu_{2}^{2}-\sqrt{(\mu_{2}^{2}-\mu_{1}^{2})^{2}+4(\mu_{3}^{2})^{2}}>0,$
(49)

and obviously, $m_{H_{1}^{0}}^{2},m_{A_{1}^{0}}^{2},m_{H_{1}^{\pm}}^{2}>0$.

  * •

Gauge bosons decay widths: The decay widths of the $W/Z$-bosons were measured
with high precision at LEP. Therefore, we require that the decays of
$W/Z$-bosons to $Z_{2}$-odd scalars is closed. This is fulfilled if one
assumes that

$\min\big{(}m_{H_{1}^{0}}+m_{A_{1}^{0}},\,2m_{H_{1}^{\pm}}\big{)}>m_{Z},\,\min\big{(}m_{H_{1}^{\pm}}+m_{A_{1}^{0}},\,m_{H_{1}^{\pm}}+m_{H_{1}^{0}}\big{)}>m_{W}.$
(50)

  * •

Lepton flavor violating (LFV) decays: in this model, LFV decay processes arise
via one-loop diagrams mediated by the $H_{i=1,2}^{\pm}$ and $N_{k}$ particles.
The branching ratio of the decay
$\mathcal{B}(\ell_{\alpha}\rightarrow\ell_{\beta}+\gamma)$ due to the
contribution of the interactions (1) is [22]:

$\displaystyle\mathcal{B}(\ell_{\alpha}$
$\displaystyle\rightarrow\ell_{\beta}+\gamma)=\frac{3\alpha\upsilon^{4}}{32\pi}\left|\sum_{k=1}^{3}\sum_{j=1}^{2}\frac{\big{(}\zeta_{\alpha
k}^{(j)}\big{)}\big{(}\zeta_{\beta
k}^{(j)}\big{)}^{*}}{m_{H_{j}^{\pm}}^{2}}F\Big{(}\frac{M_{k}^{2}}{m_{H_{j}^{\pm}}^{2}}\Big{)}\right|^{2},$
(51)

where $\alpha=e^{2}/4\pi$ is the electromagnetic fine structure constant and
$F(x)=(1-6x+3x^{2}+2x^{3}-6x^{2}\log x)/6(1-x)^{4}.$ Since the bounds on
$\ell_{\alpha}\rightarrow\ell_{\beta}+\gamma$ are more severe, especially
$\mu\rightarrow e+\gamma$, we will consider only the LFV bounds on
$\ell_{\alpha}\rightarrow\ell_{\beta}+\gamma$ and not
$\ell_{\alpha}\rightarrow\ell_{\beta}\ell_{\beta}\ell_{\beta}$, since the
latter’s would implicitly fulfilled [23].

One has to notice that getting experimentally allowed values for the branching
ratios (51) can be achieved by considering very small Yukawa couplings
$h_{\alpha i}$ and/or very heavy charged scalars. Although, this choice is
preferred by the scalar DM case, but not interesting from Majorana DM relic
density and/or collider point of view. However, one can consider well chosen
values for the Yukawa couplings ($h_{\alpha i}$) and the Majorana and charged
scalar masses, where the terms in the summation in (51) cancel each other.
This allows relatively large Yukawa couplings as well relatively light charged
scalars in order to accommodate the DM relic density in the case of Majorana
DM [11, 13]; and provide interesting predictions at colliders [8, 23, 11, 12].
This can be clearly seen when comparing the Yukawa couplings and the Majorana
and scalar masses for the two benchmark points (BPs) in Table 1 that
corresponds to Majorana and scalar DM cases.

  * •

Direct searches of charginos and neutralinos at the LEP-II experiment: we use
the null results of neutralinos and charginos at LEP [24] to put lower bounds
on the masses of charged Higgs $H_{1}^{\pm}$ and the light neutral scalars of
the inert doublets ($H_{1}^{0},A_{1}^{0}$). The bound obtained from a re-
interpretation of neutralino searches [25] cannot apply to our model since the
decays $A_{1}^{0}\to H_{1}^{\pm}Z(\to\ell^{+}\ell^{-})$ are kinematically
forbidden. On the other hand, in most regions of the parameter space, the
charged Higgs decays exclusively into a Majorana fermion and a charged lepton.
For Yukawa couplings of order one $h_{ei}\simeq\mathcal{O}(1)$, the following
bounds are derived $m_{H^{\pm}}>100~{}\textrm{GeV}$ [12].

  * •

The electroweak precision tests: in this model, the oblique parameters acquire
contributions from the existence of inert scalars. We take $\Delta U=0$ in our
analysis, the oblique parameters are given by [26]

$\displaystyle\varDelta T$ $\displaystyle=\frac{1}{16\pi
s_{\mathrm{w}}^{2}M_{W}^{2}}\left\\{c_{H}^{2}c_{C}^{2}\,F(m_{H_{1}^{0}}^{2},m_{H_{1}^{\pm}}^{2})+c_{H}^{2}s_{C}^{2}\,F(m_{H_{1}^{0}}^{2},m_{H_{2}^{\pm}}^{2})+s_{H}^{2}c_{C}^{2}\,F(m_{H_{2}^{0}}^{2},m_{H_{1}^{\pm}}^{2})\right.$
$\displaystyle+s_{H}^{2}s_{C}^{2}\,F(m_{H_{2}^{0}}^{2},m_{H_{2}^{\pm}}^{2})+c_{A}^{2}c_{C}^{2}F(m_{A_{1}^{0}}^{2},m_{H_{1}^{\pm}}^{2})+c_{A}^{2}s_{C}^{2}F(m_{A_{1}^{0}}^{2},m_{H_{2}^{\pm}}^{2})+s_{A}^{2}c_{C}^{2}F(m_{A_{1}^{0}}^{2},m_{H_{1}^{\pm}}^{2})$
$\displaystyle+s_{A}^{2}s_{C}^{2}F(m_{A_{1}^{0}}^{2},m_{H_{2}^{\pm}}^{2})-c_{H}^{2}c_{A}^{2}F(m_{H_{1}^{0}}^{2},m_{A_{1}^{0}}^{2})-c_{H}^{2}s_{A}^{2}F(m_{H_{1}^{0}}^{2},m_{A_{2}^{0}}^{2})-s_{H}^{2}c_{A}^{2}F(m_{H_{2}^{0}}^{2},m_{A^{0}}^{2})$
$\displaystyle\left.-s_{H}^{2}s_{A}^{2}F(m_{H_{2}^{0}}^{2},m_{A_{2}^{0}}^{2})\right\\},$
$\displaystyle\varDelta S$
$\displaystyle=\frac{1}{24\pi}\left\\{(2s_{\mathrm{w}}^{2}-1)^{2}\left[c_{C}^{4}G(m_{H_{1}^{\pm}}^{2},m_{H_{1}^{\pm}}^{2},m_{Z}^{2})+s_{C}^{4}G(m_{H_{2}^{\pm}}^{2},m_{H_{2}^{\pm}}^{2},m_{Z}^{2})+2c_{C}^{2}s_{C}^{2}G(m_{H_{1}^{\pm}}^{2},m_{H_{2}^{\pm}}^{2},m_{Z}^{2})\right]\right.$
$\displaystyle+c_{H}^{2}c_{A}^{2}\,G(m_{H_{1}^{0}}^{2},m_{A_{1}^{0}}^{2},m_{Z}^{2})+c_{H}^{2}s_{A}^{2}\,G(m_{H_{1}^{0}}^{2},m_{A_{2}^{0}}^{2},m_{Z}^{2})+s_{H}^{2}c_{A}^{2}\,G(m_{H_{2}^{0}}^{2},m_{A_{1}^{0}}^{2},m_{Z}^{2})$
$\displaystyle+s_{H}^{2}s_{A}^{2}\,G(m_{H_{2}^{0}}^{2},m_{A_{2}^{0}}^{2},m_{Z}^{2})+c_{H}^{2}c_{C}^{2}\,\log\Big{(}\tfrac{m_{H_{1}^{0}}^{2}}{m_{H_{1}^{\pm}}^{2}}\Big{)}+c_{H}^{2}s_{C}^{2}\,\log\Big{(}\tfrac{m_{H_{1}^{0}}^{2}}{m_{H_{2}^{\pm}}^{2}}\Big{)}+s_{H}^{2}c_{C}^{2}\,\log\Big{(}\tfrac{m_{H_{2}^{0}}^{2}}{m_{H_{1}^{\pm}}^{2}}\Big{)}$
$\displaystyle+s_{H}^{2}s_{C}^{2}\,\log\Big{(}\tfrac{m_{H_{2}^{0}}^{2}}{m_{H_{2}^{\pm}}^{2}}\Big{)}+c_{H}^{2}c_{A}^{2}\log\Big{(}\tfrac{m_{A_{1}^{0}}^{2}}{m_{H_{1}^{\pm}}^{2}}\Big{)}+c_{H}^{2}s_{A}^{2}\log\Big{(}\tfrac{m_{A_{2}^{0}}^{2}}{m_{H_{1}^{\pm}}^{2}}\Big{)}+s_{H}^{2}c_{A}^{2}\log\Big{(}\tfrac{m_{A_{1}^{0}}^{2}}{m_{H_{2}^{\pm}}^{2}}\Big{)}$
$\displaystyle\left.+s_{H}^{2}s_{A}^{2}\log\Big{(}\tfrac{m_{A_{2}^{0}}^{2}}{m_{H_{2}^{\pm}}^{2}}\Big{)}\right\\},$
(52)

where $s_{\mathrm{W}}\equiv\sin~{}\theta_{W}$, with $\theta_{W}$ is the
Weinberg mixing angle, and $F(x,y)$ and $G(x,y,z)$ are one-loop functions that
can be found in [26].

  * •

The di-photon Higgs decay: the Higgs couplings to the two charged scalar can
change drastically the value of the Higgs boson loop-induced decay into two
photons. These loop interactions depends on the signs and the values of
$\omega_{1,2}$, as well on the mixing and $\theta_{C}$. The ratio
$R_{\gamma\gamma}=\mathcal{B}(h\to\gamma\gamma)/\mathcal{B}(h\to\gamma\gamma)^{\mathrm{SM}}=1.02_{-0.12}^{+0.09}$
[27], that characterizes the di-photon Higgs decay modification can be written
as

$\displaystyle R_{\gamma\gamma}$
$\displaystyle=\big{(}1-\mathcal{B}_{BSM}\big{)}\left|1+\frac{\upsilon^{2}}{2}\frac{\sum_{j=1}^{2}\frac{r_{j}}{m_{H_{j}^{+}}^{2}}A_{0}^{\gamma\gamma}\big{(}\frac{m_{h}^{2}}{4m_{H_{j}^{+}}^{2}}\big{)}}{A_{1}^{\gamma\gamma}\big{(}\frac{m_{h}^{2}}{4m_{W}^{2}}\big{)}+\frac{4}{3}A_{1/2}^{\gamma\gamma}\big{(}\frac{m_{h}^{2}}{4m_{t}^{2}}\big{)}}\right|^{2},$
(53)

with
$r_{1}=c_{C}^{2}\omega_{1}+s_{C}^{2}\omega_{2},\,r_{2}=s_{C}^{2}\omega_{1}+c_{C}^{2}\omega_{2}$;
and $\mathcal{B}_{BSM}$ represents any non-SM decay for the Higgs like
$h\rightarrow H_{i}^{0}H_{k}^{0},\,A_{i}^{0}A_{k}^{0}$. The loop functions
$A_{0,1/2,1}^{\gamma\gamma}$ are given in the literature [28]. The
$R_{\gamma\gamma}$ ratio estimation in (53) is based on the assumption of the
SM production rates for the SM Higgs boson [29]. Similarly, the ratio
$R_{\gamma Z}=\mathcal{B}(h\to\gamma Z)/\mathcal{B}(h\to\gamma
Z)^{\mathrm{SM}}$ can be written as

$\displaystyle R_{\gamma Z}$
$\displaystyle=\big{(}1-\mathcal{B}_{BSM}\big{)}\left|1-\frac{\upsilon^{2}}{2}\frac{1-2s_{W}^{2}}{c_{W}}\frac{\sum_{j=1}^{2}\frac{r_{j}}{m_{H_{j}^{+}}^{2}}A_{0}^{\gamma
Z}\big{(}\frac{m_{h}^{2}}{4m_{H_{j}^{+}}^{2}},\frac{m_{Z}^{2}}{4m_{H_{j}^{+}}^{2}}\big{)}}{c_{W}~{}A_{1}^{\gamma
Z}\big{(}\frac{m_{h}^{2}}{4m_{W}^{2}},\frac{m_{Z}^{2}}{4m_{W}^{2}}\big{)}+\frac{6-16s_{W}^{2}}{3c_{W}}A_{1/2}^{\gamma
Z}\big{(}\frac{m_{h}^{2}}{4m_{t}^{2}},\frac{m_{Z}^{2}}{4m_{t}^{2}}\big{)}}\right|^{2},$
(54)

with $c_{W}=\cos\theta_{W},~{}s_{W}=\sin\theta_{W}$ and $\theta_{W}$ is the
Weinberg mixing angle; and the loop functions $A_{0,1/2,1}^{\gamma Z}$ are
given in the literature [28].

In Appendix B, we discuss the generalization of these results and constraints
in the case of a $n$ inert scalar doublets model.

##  4 Viable Parameter Space

In our analysis, we consider both cases where the DM candidate could be a
Majorana fermion $N_{1}$ and the light inert scalar $H_{1}^{0}$. By taking
into account all the above mentioned theoretical and experimental constraints,
we perform a full numerical scan over the model free parameters (35). In our
scan, we consider the following ranges for the free parameters

$\displaystyle|s_{H,A,C}|\leq
1,~{}|\omega_{1,2}|<4\pi,~{}m_{H_{1}^{0},A_{1}^{0}}<3~{}\textrm{TeV},~{}78~{}\textrm{GeV}<m_{H_{1}^{\pm}}<3~{}\textrm{TeV},$
(55)

while focusing on the parameter space regions that makes this model different
than a two inert doublets model SM extension, i.e., the case with non
negligible new Yukawa couplings $\min|h_{\alpha i}|\geq 10^{-3}$.

In Fig. 2, using 4k BPs for each (scalar and Majorana) DM case, we show the
mixing angles sine (left-up), the strength of the new Yukawa couplings
$h_{\alpha i}$ (right-up) the masses; and the inert masses (bottom).

Figure 2: Left-top: the CP-even mixing angle sine vs the corresponding CP-odd
one, where the palette shows the mixing of the charged scalar sector. Right-
top: the strength of the new Yukawa couplings $h_{\alpha i}$ where the palette
shows the ratio
$\delta^{0}=(m_{A_{1}^{0}}^{2}-m_{H_{1}^{0}}^{2})/m_{H_{1}^{0}}^{2}$. Left-
bottom: the mass ratio
$\delta^{0}=(m_{A_{1}^{0}}^{2}-m_{H_{1}^{0}}^{2})/m_{H_{1}^{0}}^{2}$ versus
the light CP-even scalar mass $m_{H_{1}^{0}}^{2}$, where the palette shows the
light charged scalar mass $m_{H_{1}^{\pm}}^{2}$. Right-bottom: the ratios
$\delta_{even}=(m_{H_{2}^{0}}^{2}-m_{H_{1}^{0}}^{2})/m_{H_{1}^{0}}^{2}$,
$\delta_{even}=(m_{A_{2}^{0}}^{2}-m_{A_{1}^{0}}^{2})/m_{A_{1}^{0}}^{2}$ and
$\delta_{charged}=(m_{H_{2}^{\pm}}^{2}-m_{H_{1}^{\pm}}^{2})/m_{H_{1}^{\pm}}^{2}$
that represent the mass relative difference in each sector.

Clearly from Fig. 2, the mixing angles of the CP-even, CP-odd and charged
sectors can take most of the possible ranges, however, there some preferred
values around $s_{H}=0$ and $s_{A}=s_{H}$. All the above mentioned constraints
together with the neutrino oscillation data that are taken into account via
(46); lead to the new Yukawa couplings $h_{\alpha i}$ strength as shown in
Fig. 2-top-right. It is clear that all the couplings can suppressed which
makes the LFV constraints easily fulfilled and the model indistinguishable
from the SM extended by two inert singlets. As mentioned previously, here we
are interested in the opposite case where $|h_{\alpha i}|\geq 10^{-3}$, then,
the couplings can be large as $|h_{\alpha i}|<\sqrt{4\pi}$ and a significant
hierarchy between the largest and smallest couplings strength does exist as
$\max(|h_{\alpha i}|)>1.8~{}\min(|h_{\alpha i}|)$. According to Fig. 2-bottom,
the masses of the inert eigenstates lie within large values intervals.

In Fig. 3, we present the Higgs decay modifiers of
$h\rightarrow\gamma\gamma,\gamma\,Z$ (left) and the oblique parameters
(right).

Figure 3: Left: The Higgs decay modifiers of
$h\rightarrow\gamma\gamma,\gamma\,Z$, where the colored band represents the
allowed experimental values. Right: the oblique parameters $\Delta~{}T$ vs
$\Delta~{}S$, where the palette shows the light charged scalar mass.

One notices that for allowed values for the Higgs branching ratio
$h\rightarrow\gamma\gamma$, the branching ratio $h\rightarrow\gamma\,Z$ can be
modified with respect to the SM within the range $[-5.2\%,\,5,1\%]$.
Concerning the electroweak precision tests, as it is expected the $T$
parameter is more sensitive to the quantum corrections due to the interactions
of the gauge bosons to the two extra doublets, while the $S$ parameter is
sensitive to the corrections only in case of light charged scalars.

In Fig. 4, we show the branching ratios for the LFV processes
$\ell_{\alpha}\rightarrow\ell_{\beta}+\gamma$ compared by their experimental
bounds.

Figure 4: The branching ratio of the LFV processes $\mu\rightarrow e+\gamma$,
$\tau\rightarrow e+\gamma$ and $\tau\rightarrow\mu+\gamma$ versus the light
Majorana singlet fermion, where the palette shows the light charged scalar
mass. The dashed horizontal line in each panel represents the branching ratio
experimental bounds.

Like in many models, the constraint form the branching ratio $\mu\rightarrow
e\gamma$ is the most severe one and the bounds from the other LFV decays
($\tau\rightarrow e\gamma$ and $\tau\rightarrow\mu\gamma$) are at least three
orders of magnitude smaller.

At colliders, the model may predict many interesting signatures. For instance,
it predicts all the signatures relevant to the MSctM [11]. At the LHC, charged
scalars can be pair produced as $pp\rightarrow H_{1,2}^{+}H_{1,2}^{-}$ or
associated with neutral inert scalars $pp\rightarrow
H_{1,2}^{\pm}H_{1,2}^{0},H_{1,2}^{\pm}A_{1,2}^{0}$. These channels can be
probed via the final states mono-lepton $\ell+\not{E}_{T}$ and di-lepton
$2\ell+\not{E}_{T}$ in case if the channels $H_{1}^{\pm}\rightarrow
W^{\pm}H_{1,2}^{0},W^{\pm}A_{1,2}^{0}$ are closed. If these channels are open,
then the final states $4~{}jets+\not{E}_{T}$ and $1\ell+2~{}jets+\not{E}_{T}$
could be useful to probe this model as it is useful to probe the MSctM [11].
In order to look for beyond MSctM signatures, one has to consider the heavy
inert scalar production at the LHC: $pp\rightarrow
H_{2}^{\pm}H_{1,2}^{\mp},H_{2}^{\pm}H_{1,2}^{0},H_{2}^{\pm}A_{1,2}^{0},$
$H_{2}^{0}H_{1,2}^{\pm},A_{2}^{0}H_{1,2}^{\pm},H_{2}^{0}H_{1,2}^{0},A_{2}^{0}H_{1,2}^{0},A_{2}^{0}A_{1,2}^{0}$,
where the heavy scalars should decay as $H_{2}^{+}\rightarrow
W^{+}H_{1}^{0},~{}W^{+}A_{1}^{0},~{}hH_{1}^{+}$, $H_{2}^{0}\rightarrow
ZA_{1}^{0},~{}hH_{1}^{0}$ and $A_{2}^{0}\rightarrow ZH_{1}^{0},~{}hA_{1}^{0}$.
These novel signatures can be probed through the final states
$8~{}jets+\not{E}_{T}$, $1\ell+4~{}jets+\not{E}_{T}$,
$1\ell+2~{}jets+\not{E}_{T}$, $4b+\not{E}_{T}$, $2~{}jets+2b+\not{E}_{T}$,
$2b+\not{E}_{T}$, that may not be seen in the MSctM or in other neutrino mass
and DM motivated SM extensions. The investigation of these novel signatures
requires a full and precise numerical scan to define the relevant regions of
the parameter space, that should be confronted with the existing analyses.

##  5 Dark Matter: Majorana or Scalar?

In this model, DM candidate could be either the light Majorana fermion
($N_{1}$), the light $CP$-even ($CP$-odd) scalar $H_{1}^{0}$ ($A_{1}^{0}$), or
a mixture of all these components if they are degenerate in mass. In the case
of scalar DM, the possible annihilation channels are $W^{\pm}W^{\mp}$, $ZZ$,
$q_{i}\bar{q}_{i}$, $hh$, $\bar{\ell}_{\alpha}\ell_{\beta}$ and
$\bar{\nu}_{\alpha}\nu_{\beta}$. In this case, very small couplings $h_{\alpha
i}$ could be favored by the LFV constraints, and therefore the contributions
of the channels $\nu_{\alpha}\bar{\nu}_{\beta},\ell_{\alpha}\ell_{\beta}$ to
the annihilation across section would be negligible. Then, this case is almost
identical to the SM extended by two inert doublets with scalar DM. The co-
annihilation effect along the channels
$H_{1}^{0}A_{1}^{0}(A_{1}^{0}A_{1}^{0})\rightarrow X_{SM}X^{\prime}_{SM}$
could be important if the mass difference
$(m_{A_{1}^{0}}^{2}-m_{H_{1}^{0}}^{2})/(m_{A_{1}^{0}}^{2}+m_{H_{1}^{0}}^{2})$
is small enough. This makes the couplings $h_{\alpha i}$ non-suppressed and
therefore the annihilation into
$\nu_{\alpha}\bar{\nu}_{\beta},\ell_{\alpha}\ell_{\beta}$ is more important.
Such effect could make this setup (spin-$0$ DM) better than the minimal IDM.
In case of Majorana DM scenario, the DM (co-)annihilation could occur into
charged leptons $\ell_{\alpha}^{-}\ell_{\beta}^{+}$ and light neutrinos
$\nu_{\alpha}\bar{\nu}_{\beta}$; via $t$-channel diagrams mediated by the
charged scalar $H_{1,2}^{\pm}$; and the neutral scalars
$H_{1,2}^{0},A_{1,2}^{0}$, respectively. Here, the annihilation cross section
is fully triggered by the non-suppressed values of the couplings $h_{\alpha
i}$.

In case of scalar DM, negative searches from DM direct detection (DD) at
underground detectors impose more constraints on the parameters space. This is
translated into upper bounds on the DM-nucleon cross section. In our model,
the DM interaction with nucleons occurs via a single t-channel diagram
mediated by the Higgs, and the scattering cross-section is given by333In the
scalar DM scenario, the vertex $ZH_{1}^{0}H_{1}^{0}$ is vanishing since
violates CP symmetry. However, the vertex $ZH_{1}^{0}A_{1}^{0}$ does not
vanish, but the process $H_{1}^{0}\mathcal{N}\rightarrow A_{1}^{0}\mathcal{N}$
is kinematically forbidden. In case of Majorana DM, the vertex $hN_{1}N_{1}$
($ZN_{1}N_{1}$) is vanishing at tree-level due to the fact that the fermion
$N_{1}$ is a singlet (a Majorana one). At loop level, it has been shown that
DD cross section is few orders of magnitude suppressed [37, 11].

$\sigma_{det}=\frac{(\lambda_{hH_{1}^{0}H_{1}^{0}})^{2}}{4\pi\upsilon^{2}m_{h}^{4}}\frac{m_{\mathcal{N}}^{2}(m_{\mathcal{N}}-\frac{7}{9}m_{\mathcal{B}})^{2}}{(m_{\mathcal{N}}+m_{H_{1}^{0}})^{2}},$
(56)

where $m_{\mathcal{N}}$ and $m_{\mathcal{B}}$ are, respectively, the nucleon
and baryon masses in the chiral limit [30]; and
$\lambda_{hH_{1}^{0}H_{1}^{0}}=\frac{\upsilon}{2}[c_{H}^{2}(\kappa_{1}+\omega_{1}+\xi_{1})+s_{H}^{2}(\kappa_{2}+\omega_{2}+\xi_{2})+2s_{H}c_{H}(\xi_{3}+\xi_{4})]$
is the DM Higgs triple coupling. In Fig. 5, we show the DD cross section (56)
versus the DM mass for the BPs used in Fig. 2 that correspond to scalar DM
cases, compared with recent experimental bounds from the PandaX-4T experiment
[31] and LUX-ZEPLIN [16].

Figure 5: The DD cross section vs the DM mass for the scalar DM BPs used in
Fig. 2. The violet, blue and orange lines correspond to the bounds from LUX-
ZEPLIN [16], Xenon-1T bound [32] and PandaX-4T [31], respectively, while the
yellow one represents the neutrino floor [33].

In order to show that both scalar and Majorana DM scenarios are viable in this
model, we consider the two BPs among the BPs used in Fig. 2, where their free
parameters values are given Table 1, where BP1 and BP2 correspond to Majorana
and scalar DM cases, respectively.

BP1 |  $s_{H,A,C}=\\{-0.053734,~{}0.72187,~{}0.82912\\},~{}\omega_{1,2}=\\{-0.15715,~{}-0.92515\\}$,
---|---
|  $m_{H_{i}^{0}}=\\{2755.6,2820.6\\},\,m_{A_{i}^{0}}=\\{2766.7,2823\\}$,
|
$m_{H_{i}^{\pm}}=\\{2318.7,~{}2327.8\\},\,M_{i}=\\{2317.1,~{}2318.7,~{}2327.8\\}$,
| $h_{\alpha
i}=\left(\begin{array}[]{cccccc}-1.3664-i0.066927,&-0.0063156-i1.1746,&-0.020184+i0.28501,&-0.0063503-i1.1811,&-1.3739-i0.067294,&-0.020295+i0.28657\\\
1.8625+i0.046833,&-0.09083-i2.2339,&-0.0063982-i0.33045,&-0.091329-i2.2461,&1.8727+i0.04709,&-0.0064333-i0.33226\\\
-2.4404-i0.065694,&-0.667-i1.8413,&0.037294-i0.36281,&-0.67066-i1.8514,&-2.4538-i0.066055,&0.037499-i0.36481\end{array}\right)$
BP2 |  $s_{H,A,C}=\\{-0.38546,\,-0.27092,\,-0.019603\\},~{}\omega_{1,2}=\\{-1.0362,\,5.1668\\}$,
|  $m_{H_{i}^{0}}=\\{1228.9,~{}1246\\},\,m_{A_{i}^{0}}=\\{1229.3,~{}1266\\}$,
|
$m_{H_{i}^{\pm}}=\\{1230.4,~{}1270.1\\},\,M_{i}=\\{1633.4,~{}1850.7,~{}1853.5\\}$,
| $h_{\alpha
i}=10^{-3}\times\left(\begin{array}[]{cccccc}2.5475-i51.865,&15.027+i1.4322,&19.457+i0.69002,&0.44292-i9.0175,&2.6139+i0.24914,&3.3845+i0.12003\\\
-1.7657+i70.885,&-31.844+i0.82879,&4.9703-i0.46673,&-0.307+i12.325,&-5.5394+i0.14417,&0.86461-i0.08119\\\
2.5076-i92.163,&-33.608+i0.041734,&1.0967-i6.4609,&0.43599-i16.024,&-5.8463+i0.0072598,&0.19077-i1.1239\end{array}\right)$

Table 1: The values of the free parameters for BP1 and BP2. Here, all the
masses are given in GeV.

Then, we use the package MadDM [34] to estimate the relic density and the
direct detection cross section for the scalar DM case by varying both the DM
mass and the Yukawa couplings $h_{\alpha i}$ strength as shown in Fig. 6. In
order to do so, we used the package FeynRules [35] to implement the model and
produce the UFO files [36].

Figure 6: The DM relic density versus the DM mass (left) and versus the
couplings combination $\sum_{i,\alpha}|h_{\alpha,i}|^{2}$ (right). The yellow
band represents the observed the DM relic density value. One has to mention
that in the left panel, we make all masses changing simultaneously, i.e., all
the mass fulfill the condition $m_{i}/m_{DM}=constant$ with all the DM mass
interval.

According to Fig. 6, it is clear that the right amount of the DM relic density
can be achieved in this model whether the DM is a Majorana or a scalar
particle. From Fig. 6-left, the dependence of the relic density on the DM mass
in case of Majorana DM seems to be quadratic $\Omega_{DM}h^{2}\sim m_{DM}^{2}$
(or $\sigma_{DM}\sim 1/m_{DM}^{2}$). This can be understood due to the fact
that the Majorana DM (co-)annihilation occurs via simple channels, i.e.,
$N_{1}N_{1}(N_{2,3})\rightarrow\nu_{\alpha}\bar{\nu}_{\beta},\ell_{\alpha}\ell_{\beta}$.
However, in the scalar DM case, its dependence on the DM mass is not trivial
due to the existence of many annihilation channels
($N_{1}N_{1}(N_{2,3})\rightarrow W^{\pm}W^{\mp}$, $ZZ$, $q_{i}\bar{q}_{i}$,
$hh$, $\bar{\ell}_{\alpha}\ell_{\beta}$ and $\bar{\nu}_{\alpha}\nu_{\beta}$),
where each channel could be dominant within a specific DM mass range. From
Fig. 6-right, one learns that in the scalar DM case the DM annihilation into
$\nu_{\alpha}\bar{\nu}_{\beta},\ell_{\alpha}\ell_{\beta}$ could be efficient
only if $\sum_{i,\alpha}|h_{\alpha,i}|^{2}\sim 0.1$. However, since the DM is
annihilated only via the channels
$\nu_{\alpha}\bar{\nu}_{\beta},\ell_{\alpha}\ell_{\beta}$ for the Majorana DM
case, the Yukawa coupling magnitude is very important, and it is solely
dictated by the relic density right amount. Therefore, Yukawa couplings of
order ${\cal O}(0.1\sim 1)$ could be problematic for the LFV constraints. In
order to fulfill the requirements of DM relic density, the LFV constraints and
the neutrino oscillation data together, one has to impose that the six terms
in the summation in (51) should cancel together. Although, this is not
surprising since it has been shown in the MSctM with Majorana DM [11] that
these requirements are fulfilled by the imposing the cancellation of three
terms in a formula similar to (51); for a large DM mass range. Indeed, in our
model, the Feynman diagrams relevant to the neutrino mass, LFV and DM
annihilation are more numerous; and involve more free parameters with respect
the MSctM, and hence make the DM relic density, the LFV constraints and the
neutrino oscillation data easily relaxed with respect to the case in [11].

##  6 Conclusion

In this paper, we proposed a SM extension by three singlet Majorana fermions
and two scalar inert doublets. After diagonalizing the scalar squared mass
sub-matrices, we found that the model includes two CP-even ($H_{1,2}^{0}$),
two CP-odd ($A_{1,2}^{0}$) and two charged scalar ($H_{1,2}^{\pm}$)
eigenstates, with different mixing angles. Then, the values of many physical
observables got modified with respect to the MSctM. For instance, the neutrino
mass is generated at one-loop level via 12 diagrams instead of 6 in the MSctM.
Similar remarks hold for the observables relevant to the electroweak precision
tests, LFV and the di-photon Higgs decay, where many masses and mixing angles
are included. This made the neutrino oscillation data easily explained without
being in conflict with the different theoretical and experimental constraints
for large parameter space regions.

The dark matter (DM) in this model could be either fermionic ($N_{1}$) or
scalar (the lightest among $H_{1}^{0}$ and $A_{1}^{0}$). We have shown that
the right amount of the relic density can be easily achieved in both scenarios
for different DM mass ranges. In addition, for the scalar DM scenario the
model can accommodate the constraints for the direct detection cross section
of the DM-nucleon scattering.

The model predicts all the signatures predicted by the MSctM, however, novel
signature relevant to the production and decay of the heavy scalars
($H_{2}^{\pm},H_{2}^{0},A_{2}^{0}$) can be probed through some final states
like: $8~{}jets+\not{E}_{T}$, $1\ell+4~{}jets+\not{E}_{T}$,
$1\ell+2~{}jets+\not{E}_{T}$, $4b+\not{E}_{T}$, $2~{}jets+2b+\not{E}_{T}$ and
$2b+\not{E}_{T}$. This point requires an independent investigation.

###### Acknowledgements.

This work is funded by the University of Sharjah under the research projects
No 21021430100 “Extended Higgs Sectors at Colliders: Constraints &
Predictions” and No 21021430107 “Hunting for New Physics at Colliders”.

##  Appendix A Generalized Casas-Ibarra parameterization

It is well known that a (complex) $3\times 3$ fermionic mass matrices can be
diagonalized via two unitary matrices $U$ and $V$ à la singular decomposition
value problem [38], as

$m^{(\nu)}=U.D_{m}.V^{\dagger},$ (57)

with $D_{m}=\mathrm{diag\big{\\{}m_{1},m_{2},m_{3}\big{\\}}}$. Here,
$U_{3\times 3}$ and $V_{3\times 3}$ can be defined as the unitary matrices
that diagonalize the matrices products $m^{(\nu)}.(m^{(\nu)})^{\dagger}$ and
$(m^{(\nu)})^{\dagger}.m^{(\nu)}$, respectively. In other words
$U^{\dagger}.[m^{(\nu)}.(m^{(\nu)})^{\dagger}].U=D_{m}^{\dagger}.D_{m}=D_{m^{2}}=D_{m}.D_{m}^{\dagger}=V^{\dagger}.[(m^{(\nu)})^{\dagger}.m^{(\nu)}].V$.
In the case of neutrino Majorana mass, the matrix (57) is complex symmetric,
which implies $U=V^{*}$ and therefore

$m^{(\nu)}=U.D_{m}.U^{T}\,\,.$ (58)

This allows to write a general form for the diagonal matrix $D_{m}$ as

$\displaystyle D_{m}=\mathcal{P}_{1}.U^{T}.m^{(\nu)}.U.\mathcal{P}_{2}\,\,\,$
, (59)

where the matrices
$\mathcal{P}_{1,2}=\mathrm{diag}\big{\\{}1,e^{i\zeta_{1,2}},e^{i\vartheta_{1,2}}\big{\\}}$
can be absorbed in a redefinition of the Majorana phase matrix in the PNMS
mixing matrix; and similarly, $D_{m}$ in (59) can be converted to a diagonal
matrix with real positive elements. This parameterization (59) is in agreement
with the so-called Takagi diagonalization [39].

In what follows, we consider the new Yukawa couplings are given by the matrix
$h_{3,M}$, with $M=3n$ and is the inert doublets number that are considered in
the scotogenic model generalization. In this case, the matrix $\Lambda$ in
(41) should be a $M\times M$ matrix rather than $6\times 6$ one. In our case,
we have $n=2$ and then the Yukawa couplings are presented by $3\times 6$
complex matrix. Following the Casas-Ibarra approach [19], the Yukawa couplings
are assumed to have the form of a product of many matrices as

$h=S_{1}.S_{2}.S_{3}.S_{4}.S_{5}.$ (60)

Then by replacing (60) in (41); and match it with (58), one immediately finds
that $S_{1}=U_{3\times 3}$ is the $3\times 3$ PNMS mixing matrix. In order to
identify the remaining matrices, we simplify the product
$U^{\dagger}.m^{(\nu)}.(m^{(\nu)})^{\dagger}.U$ and match it with $D_{m^{2}}$.
Then, in case where the matrix $S_{5}=Q$ is chosen to be the orthogonal
$M\times M$ matrix that diagonalizes the symmetric real matrix $\Lambda$ as
$Q_{M\times M}.(\Lambda)_{M\times M}.Q_{M\times
M}^{T}=D_{\Lambda^{\prime}}=\mathrm{diag}\big{\\{}\Lambda_{1}^{\prime},\Lambda_{2}^{\prime},...,\Lambda_{M}^{\prime}\big{\\}}$;
and the matrix $S_{4}$ to be the diagonal $M\times M$ matrix
$S_{4}=D_{(\Lambda^{\prime})^{-1/2}}=\mathrm{diag}\big{\\{}(\Lambda_{1}^{\prime})^{-1/2},(\Lambda_{2}^{\prime})^{-1/2},...,(\Lambda_{M}^{\prime})^{-1/2}\big{\\}}$,
then the product $U^{\dagger}.m^{(\nu)}.(m^{(\nu)})^{\dagger}.U$ can be
written as

$\displaystyle U^{\dagger}.m^{(\nu)}.(m^{(\nu)})^{\dagger}.U$
$\displaystyle=(S_{2}.S_{3})_{3\times M}.(S_{3}^{T}.S_{2}^{T})_{M\times
3}.(S_{2}^{*}.S_{3}^{*})_{3\times
M}.(S_{3}^{\dagger}.S_{2}^{\dagger})_{M\times 3}.$ (61)

Here, $\Lambda^{\prime}_{i}$ are the eigenvalues of the matrix $\Lambda$. In
order to match (61) with$D_{m^{2}}$, it is enough to have
$S_{2}=D_{m^{1/2}}=\mathrm{diag\big{\\{}m_{1}^{1/2},m_{2}^{1/2},m_{3}^{1/2}\big{\\}}}$
and $S_{3}=T$ with $(T)_{3\times M}.(T^{T})_{M\times
3}=\boldsymbol{1}_{3\times 3}$. Obviously, for $M=3$ the matrix $T$ is just a
$3\times 3$ orthogonal matrices $T.T^{T}=T^{T}.T=\boldsymbol{1}_{3\times 3}$.
However, for $M=3\times k$, we claim that the matrix $T$ has the most general
form

$\displaystyle T_{3\times(3k)}$ $\displaystyle=$
$\displaystyle\big{(}\begin{array}[]{ccccc}\alpha^{(1)}R_{3\times
3}^{(1)},&\alpha^{(2)}R_{3\times 3}^{(2)},&.......&.&\alpha^{(k)}R_{3\times
3}^{(k)}\end{array}\big{)},$ (63)

with $R^{(i)}$ are $3\times 3$ orthogonal matrices
$R^{(i)}.(R^{(i)})^{T}=(R^{(i)})^{T}.R^{(i)}=\boldsymbol{1}_{3\times 3}$, and
$\alpha^{(i)}$ are real numbers that fulfill the identity
$\sum_{i}(\alpha^{(i)})^{2}=1$.

In order to get large Yukawa couplings as favored by collider searches or
required by Majorana DM relic density, it is enough to choose a
diagonalization matrix $(Q)_{M\times M}$ that leads to the order
$|\Lambda^{\prime}_{1}|<|\Lambda^{\prime}_{2}|<..<<|\Lambda^{\prime}_{M}|$;
and $\alpha^{(1)}=1,\,\alpha^{(i>1)}=0$.

##  Appendix B Scotogenic Model with $n$ inert Doublets

In case of a scotogenic model with N inert doublets ($\Phi_{i=1,n}$) with the
global $Z_{2}$ symmetry $N_{i}\rightarrow-
N_{i},~{}\Phi_{i}\rightarrow-\Phi_{i}$, the Lagrangian (1) can be generalized
as

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\sum_{i=1}^{n}\sum_{k=1}^{3}\overline{L}_{\alpha}\big{(}(h_{\alpha,k+3(i-1)})\epsilon\Phi_{i}\big{)}N_{k}+\frac{1}{2}\bar{N}_{k}^{C}M_{k}N_{k}+h.c.,$
(64)

where the new Yukawa couplings $h_{\alpha,i}$ here are a $3\times(3n)$ matrix.
The scalar potential (2) can be generalized as

$\displaystyle V\left(\mathcal{H},\Phi_{i},S,\chi\right)$ $\displaystyle=$
$\displaystyle-\mu_{H}^{2}|\mathcal{H}|^{2}+\mu_{i}^{2}|\Phi_{i}|^{2}+\frac{\lambda_{H}}{6}|\mathcal{H}|^{4}+\frac{\lambda_{i}}{6}|\Phi_{i}|^{4}+\omega_{i}|\mathcal{H}|^{2}|\Phi_{i}|^{2}+\kappa_{i}|\mathcal{H}^{\dagger}\Phi_{i}|^{2}$
(65)
$\displaystyle+\varrho_{i,j}^{(1)}|\Phi_{i}|^{2}|\Phi_{j}|^{2}+\varrho_{i,j}^{(2)}|\Phi_{i}^{\dagger}\Phi_{j}|^{2}+\left\\{\mu_{3,ij}^{2}\Phi_{i}^{\dagger}\Phi_{j}+\frac{1}{2}\xi_{i}^{(1)}(\mathcal{H}^{\dagger}\Phi_{i})^{2}\right.$
$\displaystyle\left.+\xi_{i,j}^{(2)}(\mathcal{H}^{\dagger}\Phi_{i})(\mathcal{H}^{\dagger}\Phi_{j})+\xi_{i,j}^{(3)}(\Phi_{i}^{\dagger}\mathcal{H})(\mathcal{H}^{\dagger}\Phi_{j})+h.c.\right\\}.$

In case of CP-conservation (real values of all potential parameters), there
are $n$ charged eigenstates ($H_{j=1,n}^{\pm}$), $n$ neutral CP-even
eigenstates ($H_{j=1,n}^{0}$) and $n$ neutral CP-odd eigenstates
($A_{j=1,n}^{0}$) with similar couplings to (38). In (38), the mixing
($c_{H,A,C}$ and $s_{H,A,C}$ should be replaced by the elements of the mixing
matrices ${\cal U}^{(H,A,C)}$ that diagonalize the CP-even, CP-odd and charged
scalars squared $n\times n$ mass matrices, respectively, where the summation
should be performed over $j=1,n$. Consequently, the neutrino mass is generated
via $4\times n$ diagrams mediated by $N_{1,2,3}$ and
$H_{j=1,n}^{0}/A_{j=1,n}^{0}$. Therefore, the formulas of the mass matrix
elements (39) and the values of the new Yukawa couplings in (46) hold for
$j=1,n$ and $M=3\times n$. Similar conclusions can be achieved for the LFV
branching ratios (51) and the di-photon Higgs decay ratio (53). For the
oblique parameters (52), the summation should be performed over $j=1,n$; and
the mixing ($c_{H,A,C}$ and $s_{H,A,C}$ should be replaced by the elements of
the mixing matrix ${\cal U}^{(H,A,C)}$. By considering $n_{N}$ Majorana
singlet fermions instead of three, the description of this generalization
corresponds what called the general scotogenic model [14].

##  Appendix C The Unitarity Amplitude Matrices

Here, we give the relevant amplitude matrices for the unitarity. We have the
following basis that characterize the initial/final states:

CP-even, $Z_{2}$ even, & $Q_{em}=0$: the basis is {$hh$, $\chi^{0}\chi^{0}$,
$S_{1}^{0}S_{1}^{0}$, $S_{2}^{0}S_{2}^{0}$, $Q_{1}^{0}Q_{1}^{0}$,
$Q_{2}^{0}Q_{2}^{0}$, $\chi^{+}\chi^{-}$, $S_{1}^{+}S_{1}^{-}$,
$S_{2}^{+}S_{2}^{-}$, $S_{1}^{+}S_{2}^{-}$}; and the matrix is

$\left[\begin{array}[]{cccccccccc}\lambda_{H},&\frac{1}{3}\lambda_{H},&\omega_{1}+\kappa_{1}+\xi_{1},&\omega_{2}+\kappa_{2}+\xi_{2},&\omega_{1}+\kappa_{1}-\xi_{1},&\omega_{2}+\kappa_{2}-\xi_{2},&\frac{1}{3}\lambda_{H},&\omega_{1},&\omega_{2},&0\\\
\frac{1}{3}\lambda_{H},&\lambda_{H},&\omega_{1}+\kappa_{1}-\xi_{1},&\omega_{2}+\kappa_{2}-\xi_{2},&\omega_{1}+\kappa_{1}+\xi_{1},&\omega_{2}+\kappa_{2}+\xi_{2},&\frac{1}{3}\lambda_{H},&\omega_{1},&\omega_{2},&0\\\
\omega_{1}+\kappa_{1}+\xi_{1},&\omega_{1}+\kappa_{1}-\xi_{1},&\lambda_{1},&\varrho_{1}+\varrho_{2},&\frac{1}{3}\lambda_{1},&\varrho_{1}+\varrho_{2},&\omega_{1},&\frac{1}{3}\lambda_{1},&\varrho_{1},&0\\\
\omega_{2}+\kappa_{2}+\xi_{2},&\omega_{2}+\kappa_{2}-\xi_{2},&\varrho_{1}+\varrho_{2},&\lambda_{2},&\varrho_{1}+\varrho_{2},&\frac{1}{3}\lambda_{2},&\omega_{2},&\varrho_{1},&\frac{1}{3}\lambda_{2},&0\\\
\omega_{1}+\kappa_{1}-\xi_{1},&\omega_{1}+\kappa_{1}+\xi_{1},&\frac{1}{3}\lambda_{1},&\varrho_{1}+\varrho_{2},&\lambda_{1},&\varrho_{1}+\varrho_{2},&\omega_{1},&\frac{1}{3}\lambda_{1},&\varrho_{1},&0\\\
\omega_{2}+\kappa_{2}-\xi_{2},&\omega_{2}+\kappa_{2}+\xi_{2},&\varrho_{1}+\varrho_{2},&\frac{1}{3}\lambda_{2},&\varrho_{1}+\varrho_{2},&\lambda_{2},&\omega_{2},&\varrho_{1},&\frac{1}{3}\lambda_{2},&0\\\
\frac{1}{3}\lambda_{H},&\frac{1}{3}\lambda_{H},&\omega_{1},&\omega_{2},&\omega_{1},&\omega_{2},&\frac{2}{3}\lambda_{H},&\kappa_{1}+\omega_{1},&\kappa_{2}+\omega_{2},&\xi_{4}\\\
\omega_{1},&\omega_{1},&\frac{1}{3}\lambda_{1},&\varrho_{1},&\frac{1}{3}\lambda_{1},&\varrho_{1},&\kappa_{1}+\omega_{1},&\frac{2}{3}\lambda_{1},&\varrho_{1}+\varrho_{2},&0\\\
\omega_{2},&\omega_{2},&\varrho_{1},&\frac{1}{3}\lambda_{2},&\varrho_{1},&\frac{1}{3}\lambda_{2},&\kappa_{2}+\omega_{2},&\varrho_{1}+\varrho_{2},&\frac{2}{3}\lambda_{2},&0\\\
0,&0,&0,&0,&0,&0,&\xi_{4},&0,&0,&\varrho_{1}+\varrho_{2}\end{array}\right]$

(66)

CP-even, $Z_{2}$ even, & $Q_{em}=0$: the basis is {$hS_{1}^{0}$, $hS_{2}^{0}$,
$\chi^{0}Q_{1}^{0}$, $\chi^{0}Q_{2}^{0}$, $\chi^{+}S_{1}^{-}$,
$\chi^{+}S_{2}^{-}$}; and the matrix is

$\left[\begin{array}[]{cccccc}\omega_{1}+\kappa_{1}+\xi_{1},&\xi_{3}+\xi_{4},&\xi_{1},&\xi_{3},&\frac{1}{2}(\kappa_{1}+\xi_{1}),&\frac{1}{2}(\xi_{3}+\xi_{4})\\\
\xi_{3}+\xi_{4},&\omega_{2}+\kappa_{2}+\xi_{2},&\xi_{3},&\xi_{2},&\frac{1}{2}(\xi_{3}+\xi_{4}),&\frac{1}{2}(\kappa_{2}+\xi_{2})\\\
\xi_{1},&\xi_{3},&\omega_{1}+\kappa_{1}+\xi_{1},&\xi_{3}+\xi_{4},&\frac{1}{2}(\kappa_{1}+\xi_{1}),&\frac{1}{2}(\xi_{3}+\xi_{4})\\\
\xi_{3},&\xi_{2},&\xi_{3}+\xi_{4},&\omega_{2}+\kappa_{2}+\xi_{2},&\frac{1}{2}(\xi_{3}+\xi_{4}),&\frac{1}{2}(\kappa_{2}+\xi_{2})\\\
\frac{1}{2}(\kappa_{1}+\xi_{1}),&\frac{1}{2}(\xi_{3}+\xi_{4}),&\frac{1}{2}(\kappa_{1}+\xi_{1}),&\frac{1}{2}(\xi_{3}+\xi_{4}),&\kappa_{1}+\omega_{1},&\xi_{4}\\\
\frac{1}{2}(\xi_{3}+\xi_{4}),&\frac{1}{2}(\kappa_{2}+\xi_{2}),&\frac{1}{2}(\xi_{3}+\xi_{4}),&\frac{1}{2}(\kappa_{2}+\xi_{2}),&\xi_{4},&\kappa_{2}+\omega_{2}\end{array}\right]$

(67)

CP-even, $Z_{2}$ even, & $Q_{em}=0$: the basis is {$h\chi^{0}$,
$S_{1}^{0}Q_{1}^{0}$, $S_{1}^{0}Q_{2}^{0}$, $S_{2}^{0}Q_{1}^{0}$,
$S_{2}^{0}Q_{2}^{0}$, $\chi^{+}\chi^{-}$, $S_{1}^{+}S_{1}^{-}$,
$S_{2}^{+}S_{2}^{-}$, $S_{1}^{+}S_{2}^{-}$}; and the matrix is

$\left[\begin{array}[]{ccccccccc}\frac{1}{3}\lambda_{H},&\xi_{1},&\xi_{3},&\xi_{3},&\xi_{2},&0,&0,&0,&0\\\
\xi_{1},&\frac{1}{3}\lambda_{1},&0,&0,&0,&0,&0,&0,&0\\\
\xi_{3},&0,&\varrho_{1}+\varrho_{2},&0,&0,&0,&0,&0,&-\frac{i}{2}\varrho_{2}\\\
\xi_{3},&0,&0,&\varrho_{1}+\varrho_{2},&0,&0,&0,&0,&\frac{i}{2}\varrho_{2}\\\
\xi_{2},&0,&0,&0,&\frac{1}{3}\lambda_{2},&0,&0,&0,&0\\\
0,&0,&0,&0,&0,&\frac{2}{3}\lambda_{H},&\kappa_{1}+\omega_{1},&\kappa_{2}+\omega_{2},&\xi_{4}\\\
0,&0,&0,&0,&0,&\kappa_{1}+\omega_{1},&\frac{2}{3}\lambda_{1},&\varrho_{1}+\varrho_{2},&0\\\
0,&0,&0,&0,&0,&\kappa_{2}+\omega_{2},&\varrho_{1}+\varrho_{2},&\frac{2}{3}\lambda_{2},&0\\\
0,&0,&\frac{i}{2}\varrho_{2},&-\frac{i}{2}\varrho_{2},&0,&\xi_{4},&0,&0,&\varrho_{1}+\varrho_{2}\end{array}\right]$
(68)

CP-even, $Z_{2}$ even, & $Q_{em}=0$: the basis is {$hQ_{1}^{0}$, $hQ_{2}^{0}$,
$S_{1}^{0}\chi^{0}$, $S_{2}^{0}\chi^{0}$, $\chi^{+}S_{1}^{-}$,
$\chi^{+}S_{2}^{-}$}; and the matrix is

$\left[\begin{array}[]{cccccc}\omega_{1}+\kappa_{1}-\xi_{1},&-\xi_{3}+\xi_{4},&\xi_{1},&\xi_{3},&-\frac{i}{2}(\kappa_{1}-\xi_{1}),&\frac{i}{2}(\xi_{3}-\xi_{4})\\\
-\xi_{3}+\xi_{4},&\omega_{2}+\kappa_{2}-\xi_{2},&\xi_{3},&\xi_{2},&\frac{i}{2}(\xi_{3}-\xi_{4}),&-\frac{i}{2}(\kappa_{2}-\xi_{2})\\\
\xi_{1},&\xi_{3},&\omega_{1}+\kappa_{1}-\xi_{1},&-\xi_{3}+\xi_{4},&\frac{i}{2}(\kappa_{1}-\xi_{1}),&-\frac{i}{2}(\xi_{3}-\xi_{4})\\\
\xi_{3},&\xi_{2},&-\xi_{3}+\xi_{4},&\omega_{2}+\kappa_{2}-\xi_{2},&-\frac{i}{2}(\xi_{3}-\xi_{4}),&\frac{i}{2}(\kappa_{2}-\xi_{2})\\\
\frac{i}{2}(\kappa_{1}-\xi_{1}),&-\frac{i}{2}(\xi_{3}-\xi_{4}),&-\frac{i}{2}(\kappa_{1}-\xi_{1}),&\frac{i}{2}(\xi_{3}-\xi_{4}),&\kappa_{1}+\omega_{1},&\xi_{4}\\\
-\frac{i}{2}(\xi_{3}-\xi_{4}),&\frac{i}{2}(\kappa_{2}-\xi_{2}),&\frac{i}{2}(\xi_{3}-\xi_{4}),&-\frac{i}{2}(\kappa_{2}-\xi_{2}),&\xi_{4},&\kappa_{2}+\omega_{2}\end{array}\right]$

(69)

CP-even, $Z_{2}$ even, & $Q_{em}=0$: the basis is {$h\chi^{+}$,
$\chi^{0}\chi^{+}$, $S_{1}^{0}S_{1}^{+}$, $S_{1}^{0}S_{2}^{+}$,
$S_{2}^{0}S_{1}^{+}$, $S_{2}^{0}S_{2}^{+}$, $Q_{1}^{0}S_{1}^{+}$,
$Q_{1}^{0}S_{2}^{+}$, $Q_{2}^{0}S_{1}^{+}$, $Q_{2}^{0}S_{2}^{+}$}; and the
matrix is

$\left[\begin{array}[]{cccccccccc}\frac{1}{3}\lambda_{H},&0,&\frac{1}{2}(\kappa_{1}+\xi_{1}),&\frac{1}{2}(\xi_{3}+\xi_{4}),&\frac{1}{2}(\xi_{3}+\xi_{4}),&\frac{1}{2}(\kappa_{2}+\xi_{2}),&\frac{i}{2}(\kappa_{1}-\xi_{1}),&-\frac{i}{2}(\xi_{3}-\xi_{4}),&-\frac{i}{2}(\xi_{3}-\xi_{4}),&\frac{i}{2}(\kappa_{2}-\xi_{2})\\\
0,&\frac{1}{3}\lambda_{H},&-\frac{i}{2}(\kappa_{1}-\xi_{1}),&\frac{i}{2}(\xi_{3}-\xi_{4}),&\frac{i}{2}(\xi_{3}-\xi_{4}),&-\frac{i}{2}(\kappa_{2}-\xi_{2}),&\frac{1}{2}(\kappa_{1}+\xi_{1}),&\frac{1}{2}(\xi_{3}+\xi_{4}),&\frac{1}{2}(\xi_{3}+\xi_{4}),&\frac{1}{2}(\kappa_{2}+\xi_{2})\\\
\frac{1}{2}(\kappa_{1}+\xi_{1}),&\frac{i}{2}(\kappa_{1}-\xi_{1}),&\frac{1}{3}\lambda_{1},&0,&0,&\frac{1}{2}\varrho_{2},&0,&0,&0,&\frac{i}{2}\varrho_{2}\\\
\frac{1}{2}(\xi_{3}+\xi_{4}),&-\frac{i}{2}(\xi_{3}-\xi_{4}),&0,&\varrho_{1},&\frac{1}{2}\varrho_{2},&0,&0,&0,&-\frac{i}{2}\varrho_{2},&0\\\
\frac{1}{2}(\xi_{3}+\xi_{4}),&-\frac{i}{2}(\xi_{3}-\xi_{4}),&0,&\frac{1}{2}\varrho_{2},&\varrho_{1},&0,&0,&-\frac{i}{2}\varrho_{2},&0,&0\\\
\frac{1}{2}(\kappa_{2}+\xi_{2}),&\frac{i}{2}(\kappa_{2}-\xi_{2}),&\frac{1}{2}\varrho_{2},&0,&0,&\frac{1}{3}\lambda_{2},&\frac{i}{2}\varrho_{2},&0,&0,&0\\\
-\frac{i}{2}(\kappa_{1}-\xi_{1}),&\frac{1}{2}(\kappa_{1}+\xi_{1}),&0,&0,&0,&-\frac{i}{2}\varrho_{2},&\frac{1}{3}\lambda_{1},&0,&0,&\frac{1}{2}\varrho_{2}\\\
\frac{i}{2}(\xi_{3}-\xi_{4}),&\frac{1}{2}(\xi_{3}+\xi_{4}),&0,&0,&\frac{i}{2}\varrho_{2},&0,&0,&\varrho_{1},&\frac{1}{2}\varrho_{2},&0\\\
\frac{i}{2}(\xi_{3}-\xi_{4}),&\frac{1}{2}(\xi_{3}+\xi_{4}),&0,&\frac{i}{2}\varrho_{2},&0,&0,&0,&\frac{1}{2}\varrho_{2},&\varrho_{1},&0\\\
-\frac{i}{2}(\kappa_{2}-\xi_{2}),&\frac{1}{2}(\kappa_{2}+\xi_{2}),&-\frac{i}{2}\varrho_{2},&0,&0,&0,&\frac{1}{2}\varrho_{2},&0,&0,&\frac{1}{3}\lambda_{2}\end{array}\right]$

(70)

CP-even, $Z_{2}$ even, & $Q_{em}=0$: the basis is {$hS_{1}^{+}$, $hS_{2}^{+}$,
$\chi^{0}S_{1}^{+}$, $\chi^{0}S_{2}^{+}$, $S_{1}^{0}\chi^{+}$,
$S_{2}^{0}\chi^{+}$, $Q_{1}^{0}\chi^{+}$, $Q_{2}^{0}\chi^{+}$}; and the matrix
is

$\left[\begin{array}[]{cccccccc}\omega_{1},&0,&0,&0,&\frac{1}{2}(\kappa_{1}+\xi_{1}),&\frac{1}{2}(\xi_{3}+\xi_{4}),&-\frac{i}{2}(\kappa_{1}-\xi_{1}),&\frac{i}{2}(\xi_{3}-\xi_{4})\\\
0,&\omega_{2},&0,&0,&\frac{1}{2}(\xi_{3}+\xi_{4}),&\frac{1}{2}(\kappa_{2}+\xi_{2}),&\frac{i}{2}(\xi_{3}-\xi_{4}),&-\frac{i}{2}(\kappa_{2}-\xi_{2})\\\
0,&0,&\omega_{1},&0,&\frac{i}{2}(\kappa_{1}-\xi_{1}),&-\frac{i}{2}(\xi_{3}-\xi_{4}),&\frac{1}{2}(\kappa_{1}+\xi_{1}),&\frac{1}{2}(\xi_{3}+\xi_{4})\\\
0,&0,&0,&\omega_{2},&-\frac{i}{2}(\xi_{3}-\xi_{4}),&\frac{i}{2}(\kappa_{2}-\xi_{2}),&\frac{1}{2}(\xi_{3}+\xi_{4}),&\frac{1}{2}(\kappa_{2}+\xi_{2})\\\
\frac{1}{2}(\kappa_{1}+\xi_{1}),&\frac{1}{2}(\xi_{3}+\xi_{4}),&-\frac{i}{2}(\kappa_{1}-\xi_{1}),&\frac{i}{2}(\xi_{3}-\xi_{4}),&\omega_{1},&0,&0,&0\\\
\frac{1}{2}(\xi_{3}+\xi_{4}),&\frac{1}{2}(\kappa_{2}+\xi_{2}),&\frac{i}{2}(\xi_{3}-\xi_{4}),&-\frac{i}{2}(\kappa_{2}-\xi_{2}),&0,&\omega_{2},&0,&0\\\
\frac{i}{2}(\kappa_{1}-\xi_{1}),&-\frac{i}{2}(\xi_{3}-\xi_{4}),&\frac{1}{2}(\kappa_{1}+\xi_{1}),&\frac{1}{2}(\xi_{3}+\xi_{4}),&0,&0,&\omega_{1},&0\\\
-\frac{i}{2}(\xi_{3}-\xi_{4}),&\frac{i}{2}(\kappa_{2}-\xi_{2}),&\frac{1}{2}(\xi_{3}+\xi_{4}),&\frac{1}{2}(\kappa_{2}+\xi_{2}),&0,&0,&0,&\omega_{2}\end{array}\right]$

(71)

## References

  * [1] R. N. Mohapatra, Phys. Rev. Lett. 56 (1986), 561-563. R. N. Mohapatra and J. W. F. Valle, Phys. Rev. D 34 (1986), 1642
  * [2] A. Zee, Phys. Lett. 161B, 141 (1985). E. Ma, Phys. Rev. Lett. 81, 1171 (1998) [hep-ph/9805219]. T. P. Cheng and L. F. Li, Phys. Rev. D 22 (1980), 2860
  * [3] A. Zee, Nucl. Phys. B 264, 99 (1986); K. S. Babu, Phys. Lett. B 203, 132 (1988); M. Aoki, S. Kanemura, T. Shindou, and K. Yagyu, J. High Energy Phys. 10 1007 (2010) 084; J. High Energy Phys. 11 (2010) 049; G. Guo, X. G. He, and G. N. Li, J. High Energy Phys. 10 (2012) 044; Y. Kajiyama, H. Okada, and K. Yagyu, Nucl. Phys. B874, 198 (2013).
  * [4] M. Aoki, S. Kanemura, and O. Seto, Phys. Rev. Lett. 102, 051805 (2009); M. Aoki, S. Kanemura, and O. Seto, Phys. Rev. D 80, 033007 (2009).
  * [5] L. M. Krauss, S. Nasri, and M. Trodden, Phys. Rev. D 67, 085002 (2003). A. Ahriche and S. Nasri, JCAP 07 (2013), 035 [arXiv:1304.2055 [hep-ph]]. A. Ahriche, K. L. McDonald and S. Nasri, Phys. Rev. D 92 (2015) no.9, 095020 [arXiv:1508.05881 [hep-ph]].
  * [6] T. Nomura and H. Okada, Phys. Lett. B 755, 306 (2016) [arXiv:1601.00386 [hep-ph]].
  * [7] K. Cheung and O. Seto, Phys. Rev. D 69 (2004), 113009 [arXiv:hep-ph/0403003 [hep-ph]].
  * [8] A. Ahriche, S. Nasri and R. Soualah, Phys. Rev. D 89 (2014) no.9, 095010 [arXiv:1403.5694 [hep-ph]]. C. Guella, D. Cherigui, A. Ahriche, S. Nasri and R. Soualah, Phys. Rev. D 93, no. 9, 095022 (2016) [arXiv:1601.04342 [hep-ph]], D. Cherigui, C. Guella, A. Ahriche and S. Nasri, Phys. Lett. B 762, 225 (2016) [arXiv:1605.03640 [hep-ph]],
  * [9] Y. Cai, J. Herrero-Garcia, M. A. Schmidt, A. Vicente and R. R. Volkas, Front. in Phys. 5, 63 (2017) [arXiv:1706.08524 [hep-ph]]. S. M. Boucenna, S. Morisi and J. W. F. Valle, Adv. High Energy Phys. 2014 (2014), 831598 [arXiv:1404.3751 [hep-ph]].
  * [10] E. Ma, Phys. Rev. D 73 (2006), 077301 [arXiv:hep-ph/0601225 [hep-ph]].
  * [11] A. Ahriche, A. Jueid and S. Nasri, Phys. Rev. D 97 (2018) no.9, 095012 [arXiv:1710.03824 [hep-ph]].
  * [12] A. Ahriche, A. Arhrib, A. Jueid, S. Nasri and A. de La Puente, Phys. Rev. D 101 (2020) no.3, 035038 [arXiv:1811.00490 [hep-ph]].
  * [13] A. Ahriche, A. Jueid and S. Nasri, Phys. Lett. B 814 (2021), 136077 [arXiv:2007.05845 [hep-ph]].
  * [14] P. Escribano, M. Reig and A. Vicente, JHEP 07 (2020), 097 [arXiv:2004.05172 [hep-ph]].
  * [15] A. Ahriche, K. L. McDonald and S. Nasri, JHEP 06 (2016), 182 [arXiv:1604.05569 [hep-ph]]. A. Beniwal, J. Herrero-García, N. Leerdam, M. White and A. G. Williams, JHEP 21 (2020), 136 [arXiv:2010.05937 [hep-ph]]. D. M. Barreiros, F. R. Joaquim, R. Srivastava and J. W. F. Valle, JHEP 04 (2021), 249 [arXiv:2012.05189 [hep-ph]]. Z. L. Han, R. Ding, S. J. Lin and B. Zhu, Eur. Phys. J. C 79 (2019) no.12, 1007 [arXiv:1908.07192 [hep-ph]]. C. H. Chen and T. Nomura, JHEP 10 (2019), 005 [arXiv:1906.10516 [hep-ph]]. Z. L. Han and W. Wang, Eur. Phys. J. C 79 (2019) no.6, 522 [arXiv:1901.07798 [hep-ph]]. W. Wang, R. Wang, Z. L. Han and J. Z. Han, Eur. Phys. J. C 77 (2017) no.12, 889 [arXiv:1705.00414 [hep-ph]]. D. M. Barreiros, H. B. Camara and F. R. Joaquim, [arXiv:2204.13605 [hep-ph]]. D. Hehn and A. Ibarra, Phys. Lett. B 718 (2013), 988-991 [arXiv:1208.3162 [hep-ph]]. G. Cacciapaglia and M. Rosenlyst, JHEP 09 (2021), 167 [arXiv:2010.01437 [hep-ph]].
  * [16] J. Aalbers et al. [LUX-ZEPLIN], [arXiv:2207.03764 [hep-ex]].
  * [17] B. Grzadkowski, O. M. Ogreid and P. Osland, Phys. Rev. D 80 (2009), 055013 [arXiv:0904.2173 [hep-ph]].
  * [18] L. Sarma, P. Das and M. K. Das, Nucl. Phys. B 963 (2021), 115300 [arXiv:2004.13762 [hep-ph]]. L. Sarma, B. B. Boruah and M. K. Das, Eur. Phys. J. C 82 (2022) no.5, 488 [arXiv:2106.04124 [hep-ph]].
  * [19] J. A. Casas and A. Ibarra, Nucl. Phys. B 618 (2001), 171-204 [arXiv:hep-ph/0103065 [hep-ph]].
  * [20] A. G. Akeroyd, A. Arhrib and E. M. Naimi, Phys. Lett. B 490 (2000), 119-124 [arXiv:hep-ph/0006035 [hep-ph]].
  * [21] K. Kannike, Eur. Phys. J. C 72 (2012), 2093 [arXiv:1205.3781 [hep-ph]]. K. Kannike, Eur. Phys. J. C 76 (2016) no.6, 324 [erratum: Eur. Phys. J. C 78 (2018) no.5, 355] [arXiv:1603.02680 [hep-ph]]. K. Kannike, Eur. Phys. J. C 81 (2021) no.10, 940 [arXiv:2109.01671 [hep-ph]]. A. Ahriche, G. Faisel, S. Y. Ho, S. Nasri and J. Tandean, Phys. Rev. D 92 (2015) no.3, 035020 [arXiv:1501.06605 [hep-ph]].
  * [22] T. Toma and A. Vicente, JHEP 01 (2014), 160 [arXiv:1312.2840 [hep-ph]].
  * [23] M. Chekkal, A. Ahriche, A. B. Hammou and S. Nasri, Phys. Rev. D 95, no. 9, 095025 (2017). [arXiv:1702.04399 [hep-ph]].
  * [24] J. Abdallah et al. [DELPHI], Eur. Phys. J. C 31 (2003), 421-479 [arXiv:hep-ex/0311019 [hep-ex]].
  * [25] E. Lundstrom, M. Gustafsson and J. Edsjo, Phys. Rev. D 79 (2009), 035013 [arXiv:0810.3924 [hep-ph]].
  * [26] W. Grimus, L. Lavoura, O. M. Ogreid and P. Osland, Nucl. Phys. B 801 (2008), 81-96 [arXiv:0802.4353 [hep-ph]].
  * [27] [ATLAS], ATLAS-CONF-2017-047.
  * [28] A. Djouadi, Phys. Rept. 457 (2008), 1-216 [arXiv:hep-ph/0503172 [hep-ph]].
  * [29] [ATLAS], ATLAS-CONF-2018-031.
  * [30] X. G. He, T. Li, X. Q. Li, J. Tandean and H. C. Tsai, Phys. Rev. D 79 (2009), 023521 [arXiv:0811.0658 [hep-ph]].
  * [31] Y. Meng et al. [PandaX-4T], Phys. Rev. Lett. 127 (2021) no.26, 261802 [arXiv:2107.13438 [hep-ex]].
  * [32] E. Aprile et al. [XENON], JCAP 04 (2016), 027 [arXiv:1512.07501 [physics.ins-det]].
  * [33] J. Billard, L. Strigari and E. Figueroa-Feliciano, Phys. Rev. D 89 (2014) no.2, 023524 [arXiv:1307.5458 [hep-ph]].
  * [34] F. Ambrogi, C. Arina, M. Backovic, J. Heisig, F. Maltoni, L. Mantani, O. Mattelaer and G. Mohlabeng, Phys. Dark Univ. 24 (2019), 100249 [arXiv:1804.00044 [hep-ph]].
  * [35] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, Comput. Phys. Commun. 185 (2014), 2250-2300 [arXiv:1310.1921 [hep-ph]].
  * [36] https://feynrules.irmp.ucl.ac.be/wiki/2IDM3N
  * [37] A. Ahriche, S. M. Boucenna and S. Nasri, Phys. Rev. D 93 (2016) no.7, 075036 [arXiv:1601.04336 [hep-ph]].
  * [38] R.A. Horn and C.R. Johnson, ”Topics in Matrix Analysis”, Cambridge University Press 1994.
  * [39] T. Takagi, Japan J. Math 1 (1925) 83.