#  Neutrino Mass Model and Dark Matter with $Y=0$ Inert Triplet Scalar

Shilpa Jangid<EMAIL_ADDRESS>Asia Pacific Center for Theoretical
Physics (APCTP) - Headquarters San 31, Hyoja-dong, Nam-gu, Pohang 790-784,
Korea  Keiko I. Nagao<EMAIL_ADDRESS>Okayama University of Science, Faculty
of Science, Department of Physics, Ridaicho 1-1, Okayama, 700-0005, Japan
Hiroshi Okada<EMAIL_ADDRESS>Department of Physics, Kyushu
University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

###### Abstract

We study a one-loop induced neutrino mass model with an inert isospin triplet
scalar field of $Y=0$ and heavier Majorana right-handed fermions. We show
numerical analysis of neutrino oscillation and lepton flavor violations and
demonstrate our allowed regions in cases of normal and inverted hierarchies.
Then, we move on to the discussion of dark matter (DM) candidates to satisfy
the relic density where we have two candidates; fermionic DM (FDM) and bosonic
DM (BDM). And, we classify four cases NH+FDM, NH+BDM, IH+FDM, IH+BDM and
search for each of the allowed points in the model.

††preprint: KYUSHU-HET-273

##  I Introduction

The mechanism of the neutrino mass matrix must be one of the biggest issues on
how to be constructed or fixed, because the neutrinos have specific natures,
such as electrically neutral particles, and so tiny masses which would not
possess the other three fermion sectors in the standard model (SM). Moreover,
there exist some observables of CP Dirac phase, Majorana phases 111These
phases appear only when the neutrinos are assumed to be Majorana particles. In
this paper, however, we expect the neutrinos to be Majorana ones., and the
neutrino masses, which would not still exactly be measured by current
experiments yet. Thus, a neutrino model building DOES depend on what kind of
phenomenology one would like to explain or predict. For example, canonical
seesaw Yanagida (1979); Minkowski (1977); Mohapatra and Senjanovic (1980) can
explain the tiny neutrino masses by introducing heavier right-handed neutral
fermions; ${\it e.g.}$ ${\cal O}(10^{15})$ GeV, that leads to successful
explanation of the Baryon Asymmetry of the Universe via leptogenesis Fukugita
and Yanagida (1986). Then, alternative seesaw scenarios such as inverse seesaw
Mohapatra and Valle (1986); Wyler and Wolfenstein (1983) and linear seesaw
Wyler and Wolfenstein (1983); Akhmedov et al. (1996a, b) are developed to
reduce the energy scale in which one more neutral fermions with different
chirality are introduced. Another example is a scenario in which the neutrino
masses have to be induced at loop levels, and it is also a natural explanation
for the smallness of the neutrinos. 222Even though there are a lot of models,
we shall refer to a minimum model including DM candidate Ma (2006). This model
can often be realized at a lower energy scale, which can reach the current
experimental energy $\sim$TeV. It implies that the model testability could be
much better than the other scenarios. In addition, an intriguing DM candidate
can be involved in since it interacts with the lepton sector. Therefore, it
would be a flavorful (leptophillic) DM candidate, and it can be a clean signal
in collider experiments. In order to realize this model, we typically need to
impose an additional symmetry (such as $Z_{2}$ symmetry) to stabilize the DM
candidate and lepton flavor violations (LFVs) have to be taken into account,
which give stringent constraints for the corresponding Yukawa couplings and
masses. Thus, we can discuss various phenomenologies within reach of current
experimental energy and a vast alternative literature has been arisen along
this line of idea.

In this paper, we apply an isospin triplet boson with zero hypercharge
$\Delta_{0}$ to the neutrino mass matrix so that the neutrino masses are
radiatively generated via $\Delta_{0}$, imposing $Z_{2}$ symmetry to stabilize
the DM candidate. To completeness of the radiative neutrino mass, we also
introduce extra fermions and a singly-charged boson which have $Z_{2}$ odd. As
a result, we have two DM candidates; bosonic DM and fermionic one. Here,
$\Delta_{0}$ has intriguing features as a dark matter (DM) candidate Araki et
al. (2011); Jangid and Bandyopadhyay (2020). At first, the neutral component
of $\Delta_{0}$; $\delta_{0}$, is uniquely lighter than the singly-charged
component; $\delta^{\pm}$, even though the mass difference is induced at not
tree-level but radiative correction at one-loop level Cirelli and Strumia
(2009) as follows:

$\displaystyle m_{\delta^{\pm}}=m_{\delta_{0}}+166\ {\rm MeV},$ (I.1)

where $m_{\delta^{\pm}}$ and $m_{\delta_{0}}$ are respectively the mass of
singly charged and neutral boson of $\Delta_{0}$. Thus, if $\delta_{0}$ has a
mechanism to be stabilized, it is automatically a good DM candidate among the
component of $\Delta_{0}$. The second feature is that the constraint of the S
parameter is totally safe due to $Y=0$. Moreover, there is no bound of direct
detection of spin-independent scattering via the Z boson portal even though
there exists the bound via Higgs portal. It implies that this DM candidate
still has allowed region for the constraints of direct detection bounds since
contribution to the Z boson portal gives a much stronger bound than one to the
Higgs boson portal.

This paper is organized as follows. In Sec. II, we show our model, including
the Higgs sector, neutral fermion sector, LFVs, and muon anomalous magnetic
dipole moment. Then, we have chi square numerical analysis to satisfy the
neutrino oscillation data for the cases of NH and IH. In Sec. III, we analyze
fermionic and bosonic DM candidates to explain the relic density for each case
of NH and IH. We conclude and discuss in Sec. IV.

##  II  Model setup

| Lepton Fields | Scalar Fields
---|---|---
|  $L_{L}$ |  $e_{R}$ |  $N_{R}$ |  $L^{\prime}$ |  $H$ |  $\Delta_{0}$ |  $S^{+}$
$SU(2)_{L}$ | $\bm{2}$ | $\bm{1}$ | $\bm{1}$ | $\bm{2}$ | $\bm{2}$ | $\bm{3}$ | $\bm{1}$
$U(1)_{Y}$ | $-\frac{1}{2}$ | $-1$ | $0$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $0$ | $+1$
$Z_{2}$ | $+$ | $+$ | $-$ | $-$ | $+$ | $-$ | $-$
Table 1: Contents of fermion and scalar fields and their charge assignments
under $SU(2)_{L}\times U(1)_{Y}$, where $m\equiv\frac{1+N}{2}$ is the quantum
number of the electric charge.

In this section, we review our model setup. As for the fermion sector, we
introduce three heavy right-handed neutral fermions $N_{R}$ and three isospin
doublet vector-like leptons $L^{\prime}\equiv[n^{\prime},e^{\prime}]$. As for
the boson sector, we add an isospin inert triplet $\Delta_{0}$ with zero
hypercharge and an isospin singlet singly-charged boson $S^{+}$. Here, the SM
Higgs is denoted by $H$. All these new particles are imposed by $Z_{2}$ odd
number and their contents and charges are summarized in Tab. 1. Therefore, we
have a fermionic DM candidate or bosonic one, as discussed later. The
renormalizable Lagrangian in the lepton sector under these symmetries is found
as

$\displaystyle-\mathcal{L}_{Y}$
$\displaystyle=y_{\ell_{ii}}\bar{L}_{L_{i}}He_{R_{i}}+y_{S_{ij}}\bar{N}^{c}_{R_{i}}e_{R_{j}}S^{+}+y_{\Delta_{ij}}\bar{L}_{L_{i}}\Delta_{0}L^{\prime}_{R_{j}}+y_{N_{ij}}\bar{L}^{\prime}_{L_{i}}(i\tau_{2})H^{*}N_{R_{j}}$
$\displaystyle+M_{R_{ii}}\bar{N}^{c}_{R_{i}}N_{R_{i}}+M_{L_{ii}}\bar{L}^{\prime}_{L_{i}}L^{\prime}_{R_{i}}+{\rm
h.c.},$ (II.1)

where $i=1-3$, $j=1-3$, and $\tau_{2}$ being the second Pauli matrix. Without
loss of generality, we can work on the diagonal mass basis for
$y_{\ell},M_{R},M_{L}$.

###  II.1  Higgs potential

The renormalizable Higgs potential under these symmetries is given by

$\displaystyle{\cal V}_{tri.}$
$\displaystyle=\mu^{2}_{H}H^{\dagger}H+{\mu^{2}_{S}}S^{+}S^{-}+\mu^{2}_{\Delta}{\rm
Tr}[\Delta_{0}^{\dagger}\Delta_{0}]$
$\displaystyle+\lambda_{H}(H^{\dagger}H)^{2}+{\lambda_{S}}(S^{+}S^{-})^{2}+\lambda_{\Delta}{\rm
Tr}[(\Delta_{0}^{\dagger}\Delta_{0})^{2}]+\lambda_{HS}(H^{\dagger}H)(S^{+}S^{-})$
$\displaystyle+\lambda_{H\Delta}(H^{\dagger}H){\rm
Tr}[\Delta_{0}^{\dagger}\Delta_{0}]+\lambda^{\prime}_{H\Delta}\sum_{i=1}^{3}(H^{\dagger}\tau_{i}H){\rm
Tr}[\Delta_{0}^{\dagger}\tau^{i}\Delta_{0}],$ (II.2) $\displaystyle{\cal
V}_{non-tri.}$ $\displaystyle=m^{2}_{\Delta}{\rm
Tr}[\Delta_{0}^{2}]+\lambda_{0}(H^{T}\cdot\Delta_{0}H)S^{-}+{\rm h.c.},$
(II.3)

where $\tau_{i}(i=1,2,3)$ being Pauli matrices, and ${\cal V}_{tri.}({\cal
V}_{non-tri.})$ is trivial(nontrivial) potential. ${\cal V}_{non-tri.}$ is
more important in our scenario. The first term of ${\cal V}_{non-tri.}$ leads
us to the real DM particle since it gives the mass difference between the mass
of CP even and odd particle, and the mass difference is directly connected to
the nonzero neutrino mass matrix. The second one of ${\cal V}_{non-tri.}$
plays a role in generating the LFVs and muon $g-2$ connecting $S^{\pm}$ and
$\delta^{\pm}$ after the spontaneous symmetry breaking. Here, the scalar
fields can be parameterized as

$\displaystyle H=\left[\begin{array}[]{c}w^{+}\\\
\frac{v+h+iz}{\sqrt{2}}\end{array}\right],\quad\Delta_{0}=\left[\begin{array}[]{cc}\frac{\delta_{R}+i\delta_{I}}{\sqrt{2}}&\sqrt{2}\delta^{+}\\\
\sqrt{2}\delta^{-}&-\frac{\delta_{R}+i\delta_{I}}{\sqrt{2}}\end{array}\right],$
(II.8)

where $v~{}\simeq 246$ GeV is vacuum expectation value (VEV) of the SM Higgs,
and $w^{\pm}$ and $z$ are respectively absorbed by the longitudinal component
of $W^{\pm}$ and $Z$ boson in the SM. Instead of describing the analysis of
just complicated Higgs potential, we define the mixing between singly-charged
bosons $S^{\pm}$ and $\delta^{\pm}$ and mass eigenvalues of $\delta_{R}$ and
$\delta_{I}$ as follows:

$\displaystyle\left[\begin{array}[]{c}S^{+}\\\
\delta^{+}\end{array}\right]=\left[\begin{array}[]{cc}c_{\alpha}&s_{\alpha}\\\
-s_{\alpha}&c_{\alpha}\end{array}\right]\left[\begin{array}[]{c}H_{1}^{+}\\\
H_{2}^{+}\end{array}\right],$ (II.15)

where $s_{\alpha}$ and $c_{\alpha}$ are respectively short-hand notations for
$\sin\alpha$ and $\cos\alpha$, and given as

$\displaystyle
s_{2\alpha}=\frac{2\lambda_{0}v^{2}}{m^{2}_{H_{1}}-m^{2}_{H_{2}}}.$ (II.16)

The mass eigenvalues are respectively given by

$\displaystyle m_{R}^{2}\equiv M^{2}+m^{2}_{\Delta},\quad m_{I}^{2}\equiv
M^{2}-m^{2}_{\Delta},$ (II.17)

where $M^{2}$ is written in terms of
$\mu^{2}_{\Delta},\lambda_{H\Delta}v^{2},\lambda^{\prime}_{H\Delta}v^{2}$ in
Higgs potential of Eq.(II.2). The mass square difference of $\delta_{R}$ and
$\delta_{I}$ plays an important role in a nonzero neutrino mass matrix as can
be seen later Ma (2006).

###  II.2  Neutral fermion mass matrix

Since we have three by three block heavier neutral fermion mass matrix based
on $\vec{N}=[N_{R},n^{\prime}_{R},n^{\prime C}_{L}]$ originated from $N_{R}$,
$L^{\prime}_{R}$, and $L^{\prime}_{L}$, we need to formulate their mass
eigenvalues and eigenvectors. After spontaneous symmetry breaking, the valid
mass terms are given by

$\displaystyle
m^{\prime}_{ij}\bar{n}^{\prime}_{L_{i}}N_{R_{j}}+M_{R_{ii}}\bar{N}^{c}_{R_{i}}N_{R_{i}}+M_{L_{ii}}\bar{L}^{\prime}_{L_{i}}L^{\prime}_{R_{i}}+{\rm
h.c.},$ (II.18)

where $m^{\prime}_{ij}(\equiv y_{N_{ij}}v/\sqrt{2})$ has to be within
perturbative limit which we set $m^{\prime}_{ij}\lesssim\sqrt{2\pi}v$. Then,
the mass matrix $M_{N}$ is given by

$\displaystyle M_{N}=\left[\begin{array}[]{ccc}M_{R}&0&m^{\prime T}\\\
0&0&M^{T}_{L}\\\ m^{\prime}&M_{L}&0\\\ \end{array}\right],$ (II.22)

and diagonalized by $V^{T}_{N}M_{N}V_{N}=D_{N}$ and $\vec{N}=V_{N}\vec{\psi}$.
It suggests that $N_{R}=\sum_{a=1}^{9}(V_{N})_{i,a}\psi_{R_{a}}$,
$n^{\prime}_{R}=\sum_{a=1}^{9}(V_{N})_{i+3,a}\psi_{R_{a}}$, and $n^{\prime
C}_{L}=\sum_{a=1}^{9}(V_{N})_{i+6,a}\psi_{R_{a}}$.

###  II.3  Neutrino mass matrix

Now that we have all the definitions for constructing the neutrino mass
matrix, we discuss the neutrino sector. At first, we write down the relevant
Lagrangian as follows:

$\displaystyle\frac{1}{\sqrt{2}}(Y_{\Delta})_{ia}\bar{\nu}_{L_{i}}\psi_{R_{a}}(\delta_{R}+i\delta_{I})+{\rm
h.c.},$ (II.23)

where we define
$\sum_{j=1}^{3}(Y_{\Delta})_{ia}\equiv(y_{\Delta})_{ij}(V_{N})_{j+3,a}$ which
is three by nine dimensionless matrix. The resultant neutrino mass matrix is
then given by

$\displaystyle(m_{\nu})_{ij}=\sum_{a=1}^{9}\frac{(Y_{\Delta})_{ia}D_{N_{a}}(Y^{T}_{\Delta})_{aj}}{2(4\pi)^{2}}\left(\frac{m^{2}_{R}}{m^{2}_{R}-D^{2}_{N_{a}}}\ln\left[\frac{m^{2}_{R}}{D^{2}_{N_{a}}}\right]-\frac{m^{2}_{I}}{m^{2}_{I}-D^{2}_{N_{a}}}\ln\left[\frac{m^{2}_{I}}{D^{2}_{N_{a}}}\right]\right).$
(II.24)

In order to apply the Casas-Ibarra parametrization Casas and Ibarra (2001), we
redefine the above formula as follows:

$\displaystyle(m_{\nu})_{ij}$
$\displaystyle\equiv\frac{1}{2(4\pi)^{2}}(y_{\Delta})_{i\alpha}\left[(V_{N})_{\alpha+3,a}F_{a}(V_{N}^{T})_{a,\beta+3}\right](y^{T}_{\Delta})_{\beta,j}\equiv\frac{1}{2(4\pi)^{2}}(y_{\Delta})_{i\alpha}(\mu_{M})_{\alpha\beta}(y^{T}_{\Delta})_{\beta,j},$
(II.25) $\displaystyle F_{a}$
$\displaystyle\equiv\left(\frac{m^{2}_{R}}{m^{2}_{R}-D^{2}_{N_{a}}}\ln\left[\frac{m^{2}_{R}}{D^{2}_{N_{a}}}\right]-\frac{m^{2}_{I}}{m^{2}_{I}-D^{2}_{N_{a}}}\ln\left[\frac{m^{2}_{I}}{D^{2}_{N_{a}}}\right]\right).$
(II.26)

Then, the symmetric parameter $\mu_{M}$ is uniquely decomposed by a lower unit
triangular $R_{N}$ as $\mu_{M}=R_{N}R_{N}^{T}$ Nomura et al. (2017). Since
$m_{\nu}$ is diagonalized by Pontecorvo-Maki-Nakagawa-Sakata mixing matrix
$V_{\rm MNS}$ Maki et al. (1962) as $D_{\nu}\equiv V_{\rm
MNS}^{T}m_{\nu}V_{\rm MNS}$ due to the diagonal charged-lepton sector, we
obtain the following Casas-Ibarra parametrization

$\displaystyle y_{\Delta}=\sqrt{2}(4\pi)V_{\rm
MNS}^{T}D_{\nu}^{1/2}O_{mix}R_{N}^{-1}\lesssim\sqrt{4\pi},$ (II.27)

where $O_{mix}$ is three by three orthogonal complex matrix;
$O_{mix}O_{mix}^{T}=O_{mix}^{T}O_{mix}=1$ and perturbative limit be imposed
for $y_{\Delta}$. We impose $V_{\rm MNS}$ and $D_{\nu}$ be within the current
experimental results (except Dirac CP phase and two Majorana ones) where we
refer to Nufit 5.2 Esteban et al. (2020) without SK. Also, we impose the
cosmological constant telling us $\sum D_{\nu}\lesssim 120$ meV Vagnozzi et
al. (2017); Aghanim et al. (2020) where $\sum D_{\nu}\equiv
D_{\nu_{1}}+D_{\nu_{2}}+D_{\nu_{3}}$. The effective mass for the neutrinoless
double beta decay is defined by

$\displaystyle\langle m_{ee}\rangle=\left|\sum_{i=1}^{3}D_{\nu_{i}}(V_{\rm
MNS})_{ei}^{2}\right|,$ (II.28)

where it is constrained by the current KamLAND-Zen data Gando et al. (2016)
and could be measured in its future experiment Abe et al. (2023). The upper
bound is given by $\langle m_{ee}\rangle<(61-165)$ meV at 90 % confidential
level (CL).

###  II.4  Lepton Flavor Violations and muon $g-2$

In our model, there exist $\ell_{b}\to\ell_{a}\gamma$ processes and muon $g-2$
without chiral suppression arise from the following terms

$\displaystyle(Y_{S})_{aj}\bar{\psi}^{C}_{R_{a}}e_{R_{j}}(c_{\alpha}H_{1}^{+}+s_{\alpha}H_{2}^{+})+\sqrt{2}(Y_{\Delta})_{ia}\bar{\ell}_{L_{i}}\psi_{R_{a}}(-s_{\alpha}H_{1}^{+}+c_{\alpha}H_{2}^{+})+{\rm
h.c.},$ (II.29)

where $(Y_{S})_{aj}\equiv\sum_{i=1}^{3}(V^{T}_{N})_{ai}(y_{S})_{ij}$. Then the
branching ratio of ${\rm BR}(\ell_{b}\to\ell_{a}\gamma)$ is defined by

$\displaystyle{\rm
BR}(\ell_{i}\to\ell_{j}\gamma)=\frac{48\pi^{3}C_{ij}\alpha_{\rm em}}{{\rm
G_{F}^{2}}m_{i}^{2}}(|a_{R_{ij}}|^{2}+|a_{R_{ij}}|^{2}),$ (II.30)

where $\alpha_{\rm em}\approx 1/137$ is the fine-structure constant,
$C_{ij}\approx(1,0.1784,0.1736)$ for ($(ij)=(21),(31),(32)$), ${\rm
G_{F}}\approx 1.17\times 10^{-5}$ GeV-2 is the Fermi constant, $a_{L}$ and
$a_{R}$ are respectively computed as

$\displaystyle a_{R_{ij}}$
$\displaystyle=\sqrt{2}\sum_{a=1}^{9}(Y_{\Delta})_{ja}D_{N_{a}}(Y_{S})_{ai}s_{\alpha}c_{\alpha}\left[-G(D_{N_{a}},m_{H_{1}})+G(D_{N_{a}},m_{H_{2}})\right],$
(II.31) $\displaystyle a_{L_{ij}}$
$\displaystyle=\sqrt{2}\sum_{a=1}^{9}(Y^{\dagger}_{S})_{ja}D_{N_{a}}(Y_{\Delta}^{\dagger})_{ai}s_{\alpha}c_{\alpha}\left[-G(D_{N_{a}},m_{H_{1}})+G(D_{N_{a}},m_{H_{2}})\right],$
(II.32) $\displaystyle G(m_{1},m_{2})$ $\displaystyle\equiv\int
dxdydz\delta(x+y+z-1)\frac{y}{xm_{1}^{2}+(y+z)m_{2}^{2}}.$ (II.33)

These branching ratios have to be compared with experimental values in Table
2. Notice here that three body decay processes
$\ell_{i}\to\ell_{j}\ell_{k}\bar{\ell}_{\ell}$ at the one-loop box type of
diagrams are almost negligible comparing to the ${\rm
BR}(\ell_{i}\to\ell_{j}\gamma)$ types of LFVs Lindner et al. (2018). Even
though the upper bound of the branching ratio of $\mu\to e\gamma$ is recently
updated by $3.1\times 10^{-13}$ in ref. Afanaciev et al. (2023), we will apply
the previous upper bound in our numerical analysis.

Process | $(ij)$ | Experimental bounds ($90\%$ CL) | References
---|---|---|---
$\mu^{+}\to e^{+}\gamma$ | $(21)$ | $\text{Br}(\mu\to e\gamma)<4.2(3.1)\times 10^{-13}$ |  Baldini et al. (2016)(Afanaciev et al. (2023))
$\tau^{\pm}\to e^{\pm}\gamma$ | $(31)$ | $\text{Br}(\tau\to e\gamma)<3.3\times 10^{-8}$ | Aubert et al. (2010)
$\tau^{\pm}\to\mu^{\pm}\gamma$ | $(32)$ | $\text{Br}(\tau\to\mu\gamma)<4.4\times 10^{-8}$ | Aubert et al. (2010)
Table 2: Summary of $\ell_{i}\to\ell_{j}\gamma$ process and the upper bound of
experimental data, where upper bound of branching ratio of $\mu\to e\gamma$ is
recently updated by $3.1\times 10^{-13}$ in ref. Afanaciev et al. (2023).

###  II.5 Muon anomalous magnetic moment

In using the same formula as LFVs, we also describe a formula for muon $g-2$;
$\Delta a_{\mu}$, as follows:

$\displaystyle\Delta
a_{\mu}\approx-\frac{m_{\mu}}{2(4\pi)^{2}}(a_{R_{22}}+a_{L_{22}}).$ (II.34)

The above formula should be within the current experimental value whose new
results on the muon $g-2$ were recently published by the E989 collaboration at
Fermilab Abi et al. (2021):

$\displaystyle a^{\rm FNAL}_{\mu}=116592040(54)\times 10^{-11}.$ (II.35)

Combined with the previous BNL result, the result indicates that the muon
$g-2$ deviates from the SM prediction by 4.2$\sigma$ level and its deviation
from the SM prediction at 1$\sigma$ is given by

$\displaystyle\Delta a^{\rm exp}_{\mu}=a^{\rm FNAL}_{\mu}-a_{\mu}^{\rm
SM}=(25.1\pm 5.9)\times 10^{-10},$ (II.36)

where $\Delta a^{\rm exp}_{\mu}$ is space of explanation by the new physics.

###  II.6 Numerical analysis

Here we show our allowed space to satisfy the neutrino oscillation data in the
case of NH and IH, and LFVs and discuss whether our result would reach the
sizable value of muon $g-2$. We randomly scan our input absolute values in
range $[0.01,\sqrt{4\pi}]$ for nine complex values of $y_{S}$, three complex
values $a,b,c$ of $O_{mix}$, $[-\pi/10,\pi/10]$ for mixing of singly-charged
bosons $\alpha$, $[-\pi,\pi]$ for two Majorana phases,
$|m^{\prime}|\lesssim\sqrt{2\pi}v$, $0\leq|M_{R}|\lesssim 10^{5}$ GeV, and
$100{\rm GeV}\leq|M_{L^{\prime}}|\lesssim 10^{5}$GeV. We assume that
$m_{I}^{2}=m_{R}^{2}+\delta m_{1}^{2}$, and
$m_{H_{2}}^{2}=m_{H_{1}}^{2}+\delta m_{2}^{2}$ with $0\lesssim\delta
m^{2}_{1,2}\leq(0.1v)^{2}$ and $10^{2}{\rm GeV}\leq(m_{R},m_{H_{1}})\leq
10^{4}$GeV that would be an appropriate hypothesis to satisfy the oblique
parameters and LEP bounds.

Furthermore, we classify FDM candidate or BDM one imposing the following
conditions. In case of FDM, we have to impose
$D_{N_{1}}<m_{R},m_{I},m_{H_{1}},m_{H_{2}}$ and $0.9\leq|(V_{N})_{11}|$ where
the second condition implies DM is the lightest $N_{R}$ dominant. Notice here
that the DM is ruled out when DM is $n^{\prime}_{L/R}$ dominant because of the
Z boson portal in direct detection searches. In case of BDM, assuming real
inert boson $\delta_{R}$, we have to impose
$m_{R}<D_{N_{1}},m_{I},m_{H_{1}},m_{H_{2}}$. Note that the relic density of DM
is not taken into account at this stage.

####  II.6.1 NH in case of FDM

Figure 1: Numerical results for the case of FDM candidate with NH: The left
side figure represents the scattering plot in the DM mass $D_{N_{1}}$ and the
singly-charged boson mass $m_{H_{1}}$. The middle one represents the
scattering plot of the neutrinoless double beta decay (green color) and the
sum of neutrino masses (blue color) in terms of $D_{N_{1}}$. The right one
shows the scattering plot of muon $g-2$ in terms of LFVs where ${\rm
BR}(\mu\to e\gamma)$ is blue, ${\rm BR}(\tau\to e\gamma)$ green and ${\rm
BR}(\tau\to\mu\gamma)$ red.

In Fig. 1, we show numerical results for the case of FDM candidate with NH
where we have obtained 4312 allowed plots to satisfy the FDM conditions and
cosmological constant in 22,791 allowed points that satisfy the neutrino
oscillation data and LFVs. The left side figure represents the scattering plot
in the DM mass $D_{N_{1}}$ and the singly-charged boson mass $m_{H_{1}}$,
which suggests that a wide mass range is allowed when $D_{N_{1}}\leq 1$ TeV.
The middle one represents the scattering plot of the neutrinoless double beta
decay (green color) and the sum of neutrino masses (blue color) in terms of
$D_{N_{1}}$. The neutrinoless double beta decay is within the range of [5
meV-30 meV], while the sum of neutrino masses is larger than 58 meV. Here, we
impose cosmological constant $\sum D_{\nu}\leq$120 meV. The right one shows
the scattering plot of muon $g-2$ in terms of LFVs where ${\rm BR}(\mu\to
e\gamma)$ is blue, ${\rm BR}(\tau\to e\gamma)$ green and ${\rm
BR}(\tau\to\mu\gamma)$ red. This figure implies that many plots of ${\rm
BR}(\mu\to e\gamma)$ are located in places near the experimental limit of this
LFV that would be well-tested by future experiments, but the other two
processes are difficult to test soon.

####  II.6.2 NH in case of BDM

Figure 2: Numerical results for the case of BDM candidate with NH where all
the figure-legends are the same as the NH one.

In Fig. 2, we show numerical results for the case of BDM candidate with NH
where we have obtained 325 allowed plots to satisfy the FDM conditions and
cosmological constant in 22,791 allowed points that satisfy the neutrino
oscillation data and LFVs. Clearly, FDM candidate would be favorable than the
BDM one in this model. The left side figure represents the scattering plot in
the DM mass $m_{R}$ and the singly-charged boson mass $m_{H_{1}}$, which
suggests that masses below 1 TeV are favored. The middle one represents the
scattering plot of the neutrinoless double beta decay (green color) and the
sum of neutrino masses (blue color) in terms of $m_{R}$. The neutrinoless
double beta decay is within the range of [5 meV-30 meV], while the sum of
neutrino masses is larger than 58 meV. Even though the allowed points are less
than the one of FDM, the allowed region is the same as the case of NH. The
right one shows the scattering plot of muon $g-2$ in terms of LFVs where ${\rm
BR}(\mu\to e\gamma)$ is blue, ${\rm BR}(\tau\to e\gamma)$ green and ${\rm
BR}(\tau\to\mu\gamma)$ red. This figure implies that many plots of ${\rm
BR}(\mu\to e\gamma)$ are located in places near the experimental limit of this
LFV that would be well-tested by future experiments, but the other two
processes are difficult to test soon.

In the NH case, the amount of muon $g-2$ is at most $8\times 10^{-14}$ that is
much lower than the sizable muon $g-2$ and the other source (or highly fine-
tuned parameters) would be expected.

####  II.6.3 IH in case of FDM

Figure 3: Numerical results for the case of FDM candidate with IH where all
the figure-legends are the same as the NH one.

In Fig. 3, we show numerical results for the case of FDM candidate with IH
where we have obtained 206 allowed plots to satisfy the FDM conditions and
cosmological constant in 21,004 allowed points that satisfy the neutrino
oscillation data and LFVs. The left side figure represents the scattering plot
in the DM mass $D_{N_{1}}$ and the singly-charged boson mass $m_{H_{1}}$,
which suggests that a wide mass range is allowed if $D_{N_{1}}\leq 1$ TeV. The
middle one represents the scattering plot of the neutrinoless double beta
decay (green color) and the sum of neutrino masses (blue color) in terms of
$D_{N_{1}}$. The neutrinoless double beta decay is within the range of [47
meV-52 meV], while the sum of neutrino masses is larger than 100 meV. The
right one shows the scattering plot of muon $g-2$ in terms of LFVs where ${\rm
BR}(\mu\to e\gamma)$ is blue, ${\rm BR}(\tau\to e\gamma)$ green and ${\rm
BR}(\tau\to\mu\gamma)$ red. This figure implies that the situation would
almost be the same as the case for NH.

####  II.6.4 IH in case of BDM

Figure 4: Numerical results for the case of BDM candidate with IH where all
the figure-legends are the same as the NH one.

In Fig. 4, we show numerical results for the case of BDM candidate with IH
where we have only obtained 20 allowed plots to satisfy the FDM conditions and
cosmological constant in 22,791 allowed points that satisfy the neutrino
oscillation data and LFVs. Clearly, the FDM candidate would also be favored in
this model. The left side figure represents the scattering plot in the DM mass
$m_{R}$ and the singly-charged boson mass $m_{H_{1}}$, which suggests DM mass
be within 0.45 TeV and $m_{H_{1}}\lesssim$4 TeV. The middle one represents the
scattering plot of the neutrinoless double beta decay (green color) and the
sum of neutrino masses (blue color) in terms of $m_{R}$. Each of the allowed
regions is almost the same as the case of FDM with IH. The right one shows the
scattering plot of muon $g-2$ in terms of LFVs where ${\rm BR}(\mu\to
e\gamma)$ is blue, ${\rm BR}(\tau\to e\gamma)$ green and ${\rm
BR}(\tau\to\mu\gamma)$ red. This figure demonstrates the situation is the same
as the other cases.

In the IH case, the amount of muon $g-2$ is at most $2\times 10^{-14}$ that is
much lower than the sizable muon $g-2$, and the other source (or highly fine-
tuned parameters) would be expected.

##  III Dark Matter Particles

In this section, we further propel our discussion on DM candidates FDM and its
mass $D_{N_{1}}$ or BDM and its mass $m_{R}$, where we have just estimated
their mass ordering among new particles in the neutrino and LFVs numerical
analysis. The points used in the analysis of this section are those allowed by
LFVs and derived in Subsection II.6.

The relic density of DM in the Universe is determined from observations to be

$\displaystyle\Omega_{DM}h^{2}=0.120\pm 0.001.$ (III.1)

In the thermal production scenario of DM, the relic abundance of DM is
obtained by solving the Boltzmann equation Jungman et al. (1996). It can be
approximated by

$\displaystyle\Omega_{DM}h^{2}\simeq\frac{s_{0}}{\rho_{c}/h^{2}}\left(\frac{45}{\pi^{2}g_{*}^{2}}\right)^{1/2}\frac{m_{DM}}{T_{f}M_{P}\langle\sigma
v\rangle},$ (III.2)

where $s_{0}$ is the present entropy density of the Universe, $\rho_{c}$ is
the critical density, $h$ is the scaled Hubble parameter, $g_{*}$ is the
number of relativistic degrees of freedom when DM freezes out, $T_{f}$ is the
freeze-out temperature, and $M_{P}$ is the Planck mass Baer et al. (2015).
Thermally averaged DM annihilation cross section times DM relative velocity is
denoted as $\langle\sigma v\rangle$. The numerical analysis of the relic
abundance is performed using MicrOmegas 5.2.7 Giomataris et al. (1996).

###  III.1  FDM ($D_{N_{1}}$)

Figure 5: The relic abundance of FDM for NH (left) and IH (right) cases. The
vertical axis represents the largest absolute value of $y_{S_{1j}}$ and
$y_{N_{1j}}$. All points are allowed by LFVs. Also, $\lambda_{H}$,
$\lambda_{S}$, $\lambda_{\Delta}$, $\lambda_{HS}$, $\lambda_{H\Delta}$,
$\lambda^{\prime}_{H\Delta}$ are randomly scanned in the range of 0.13 to 5.

We do not need to investigate constraints of direct detection bounds since FDM
has no interactions with the quark sector at tree level. 333At one-loop level,
there exist, but it would be easy to evade these bounds. See e.g. Abe et al.
(2019). Since the FDM interacts with SM particles only via Yukawa couplings,
we might have a relatively clear result for analysis of the relic density of
DM on whether we have solutions to satisfy the correct relic density.

Fig. 5 shows the relic abundance for the case of FDM. The left figure is for
the NH case, and the right figure is for the IH case. The large red points in
the figures are points where the relic density is within 5$\sigma$ of Eq.
(III.1), and the small gray points are points where the relic density is less
than that. The annihilation processes that mainly contribute to the dark
matter relic abundance calculation are the t and u channels of
$N_{R_{1}},\bar{N}_{R_{1}}\to H_{2}^{+}\to$ SM leptons such as $e,\bar{e}$.
Note that $m_{H_{1}}$ and its couplings $y_{S_{ij}}$, which determine the
cross sections, are derived from the LFVs. In many cases, the relic abundance
is too much larger than the observation, and only very few points are allowed,
especially in the case of IH.

###  III.2  BDM $m_{R}$

Figure 6: The relic abundance of BDM for NH (left) and IH (right) cases. The
vertical axis represents the largest absolute value of $y_{\Delta_{j1}}$,
$\lambda_{H}$, $\lambda_{S}$, $\lambda_{\Delta}$, $\lambda_{HS}$,
$\lambda_{H\Delta}$, and $\lambda^{\prime}_{H\Delta}$. As in Figure 5, all
points are allowed by LFVs.

In addition to the Yukawa interactions, BDM has potential interactions that
always provide solutions via the resonant region of the SM Higgs or additional
bosons. However, we need to consider the bounds on direct detection bounds
since BDM always couples to the quark sector via Higgs potential Kanemura et
al. (2010).

Fig. 6 shows the relic abundance for the case of BDM. The relic abundance
tends to be less than observed one, and even though LFVs are satisfied.
However, since $\Omega_{DM}h^{2}=10^{-8}-10^{-2}$ in many cases, few points
agree with the observation. Especially in the IH case, the $m_{R}$ allowed by
LFVs is small, roughly $m_{R}<$ 450 GeV. Since the relic abundance
$\Omega_{DM}h^{2}\propto m_{R}$, it cannot be enhanced with a relatively small
DM mass. In the parameter range, we scanned, no points consistent with the
observation are found. Here, the main annihilation processes for the
calculation of the relic abundance are gauge interactions, e.g.,
$H_{2}^{+},H_{2}^{-}\to W^{+},W^{-}$ and $\delta_{I},\delta_{I}\to
W^{+},W^{-}$. Note that since $m_{R}$ is degenerate with the masses of the
same scalar sector $m_{H_{2}^{+}},m_{H_{2}^{-}}$ and $m_{I}$, their
coannihilation processes contribute to the calculation. The scalar potential
interactions like $H_{2}^{+},H_{2}^{-}\to hh$ only work secondarily, so the
effects by $\lambda$ couplings are insignificant. Annihilation from
$\delta_{R,I}$ to SM leptons can also occur through $y_{\Delta_{ij}}$, but
this contribution is even smaller since $|\lambda|\gg|y_{\Delta_{ij}}|$ in
many cases.

##  IV  Conclusions and discussions

We have studied a one-loop induced radiative neutrino model with an inert
isospin triplet boson of zero hypercharge, in which we have discussed neutrino
oscillations, lepton flavor violations, muon anomalous magnetic dipole moment,
and shown the allowed region of our parameter space in our chi square
numerical analysis in both the cases for NH and IH. Then, we have discussed
FDM and BDM candidates to explain the relic density of DM in light of our
numerical analysis of the neutrino oscillation and LFVs. We have found that we
have had some allowed points satisfying all our phenomenologies except the
case of IH+BDM, but we need another source or highly fine-tuned parameters to
explain the muon $g-2$. As for the FDM candidate, the rather wide range within
1 TeV is satisfied, while BDM is allowed only when the mass is around 1 TeV or
greater than this scale. This would be a rather natural consequence when we
would consider the collider as well as oblique parameters.

## Acknowledgments

This research was supported by an appointment to the YST Program at the APCTP
through the Science and Technology Promotion Fund and Lottery Fund of the
Korean Government. This was also supported by the Korean Local Governments -
Gyeongsangbuk-do Province and Pohang city (S.J.). The work was also supported
by JSPS Grant-in-Aid for Scientific Research (C) 21K03562 and (C) 21K03583 (K.
I. N).

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