In this chapter, we consider a generic U(1)X model for pseudo-scalar DM and
discuss its phenomenological implications in this study. The interesting
aspect of the $U(1)_{X}$ models is that the three generations of right-handed
neutrinos (RHNs) are required to eliminate the gauge and mixed gauge-gravity
anomalies [147, 144, 145, 146]. The RHNs mix with active neutrinos of SM via
type I seesaw mechanism [71, 72, 73, 74] to generate the required light
neutrino masses and flavor mixing. A model that explains both the neutrino
mass problem and the nature of dark matter would be a major step forward in
high-energy physics [148, 149, 150, 151, 152, 153].

The pseudo-scalar is not protected by any symmetry; therefore, for it to be a
DM candidate, it must have a lifetime much greater than the age of the
universe. We study the feasible parameter space allowed by the lifetime
constraint on our pseudo-scalar particle, then we scan for the allowed
parameter space by relic and direct detection bounds while respecting several
other theoretical and experimental constraints.

The chapter is organised as follows: In sec. 4.2, we introduce the model and
discuss the details of the new fields and their interactions. In sec. 4.3, we
discuss some of the relevant theoretical and experimental constraints. In sec.
4.4, we discuss relic density, direct detection, and other phenomenologically
relevant studies. Finally, in sec. 4.5, we summarize the chapter.

|  SU(3)c |  SU(2)L |  U(1)Y |  U(1)X
---|---|---|---|---
$q_{L}^{i}$ | 3 | 2 | $\frac{1}{6}$ | $\frac{1}{6}x_{H}+\frac{1}{3}x_{\Phi}$
$u_{R}^{i}$ | 3 | 1 | $\frac{2}{3}$ | $\frac{2}{3}x_{H}+\frac{1}{3}x_{\Phi}$
$d_{R}^{i}$ | 3 | 1 | $-\frac{1}{3}$ | $-\frac{1}{3}x_{H}+\frac{1}{3}x_{\Phi}$
$\ell_{L}^{i}$ | 1 | 2 | $-\frac{1}{2}$ | $-\frac{1}{2}x_{H}-x_{\Phi}$
$e_{R}^{i}$ | 1 | 1 | $-1$ | $-x_{H}-x_{\Phi}$
$N_{R}$ | 1 | 1 | $0$ | $-x_{\Phi}$
$H$ | 1 | 2 | $\frac{1}{2}$ | $\frac{x_{H}}{2}$
$\Phi$ | 1 | 1 | $0$ | $2x_{\Phi}$
$\chi$ | 1 | 1 | $0$ | $-x_{\Phi}$
Table 4.1:  Particle content of the minimal U$(1)_{X}$ model where $i(=1,2,3)$
represents the family index.

### 4.2 Model

We consider a BSM framework based on the $U(1)_{X}$ gauge group, which has
been studied well in the literature [144, 145, 146, 30]. The model has three
RHNs ($N_{R}^{i}$) and two new scalars $\Phi,\chi$ additionally other than SM
particles. The particle content and their respective charges are given in
Table 4.1, where the family index $i$ runs from 1 to 3. The $U(1)_{X}$ charges
of the particles can be written in terms of only two charges, $x_{H}$ and
$x_{\Phi}$, as shown in Table 4.1.

#### 4.2.1 Scalar Sector

The scalar part of the Lagrangian is given by,

$\displaystyle\mathcal{L}_{s}=(D^{\mu}H)^{\dagger}(D_{\mu}H)+(D^{\mu}\Phi)^{\dagger}(D_{\mu}\Phi)+(D^{\mu}\chi)^{\dagger}(D_{\mu}\chi)-V(H,\Phi,\chi),$
(4.1)

here the covariant derivative can be defined as

$\displaystyle
D_{\mu}=\partial_{\mu}-ig_{s}T^{a}G^{a}_{\mu}-igT^{a}W^{a}_{\mu}-ig_{1}YB^{1}_{\mu}-ig_{X}Y_{X}B^{2}_{\mu},$
(4.2)

The scalar potential $V(H,\Phi,\chi)$ is given by,

$\displaystyle V(H,\Phi,\chi)=$
$\displaystyle\mu_{H}^{2}H^{\dagger}H+\mu_{\Phi}^{2}\Phi^{\dagger}\Phi+m_{\chi}^{2}\chi^{\dagger}\chi-\lambda_{H}(H^{\dagger}H)^{2}-\lambda_{\Phi}(\Phi^{\dagger}\Phi)^{2}-\lambda_{\chi}(\chi^{\dagger}\chi)^{2}$
$\displaystyle-\lambda_{H\Phi}(H^{\dagger}H)(\Phi^{\dagger}\Phi)-\lambda_{\Phi\chi}(\Phi^{\dagger}\Phi)(\chi^{\dagger}\chi)-\lambda_{H\chi}(H^{\dagger}H)(\chi^{\dagger}\chi)+(\kappa_{T}\Phi^{\dagger}\chi^{2}+\text{h.c.})$

We parameterize the scalar fields as,

$\displaystyle H=\frac{1}{\sqrt{2}}\begin{pmatrix}-i(\phi_{1}-i\phi_{2})\\\
v_{H}+R_{1}+i\phi_{3}\\\ \end{pmatrix},\hskip
5.69046pt\chi=\frac{1}{\sqrt{2}}(v_{\chi}+R_{2}+i\eta_{\chi}),\hskip
5.69046pt\Phi=\frac{1}{\sqrt{2}}(v_{\Phi}+R_{3}+i\eta_{\Phi})$ (4.3)

$v_{H},v_{\Phi}$ and $v_{\chi}$ are the VEVs of Higgs doublet and SM singlets
$\Phi$ and $\chi$ respectively. $w^{\pm}=\phi_{1}\mp i\phi_{2}$ would be the
Goldstone boson of $W^{\pm}$, while $\phi_{3},\eta_{\Phi}$ and $\eta_{\chi}$
will mix to give $G^{0},G^{\prime}$ and $A$, that would be the Goldstone
bosons of the $Z$ and $Z^{{}^{\prime}}$ bosons, and the physical pseudo-Nambu
Goldstone boson respectively. The real scalars $R_{1},R_{2}$ and $R_{3}$ are
not the mass eigenstates due to mixing which implies following mass matrix

$\displaystyle
M^{2}_{R}=\begin{pmatrix}2\lambda_{H}v^{2}_{H}&\lambda_{H\chi}v_{H}v_{\chi}&\lambda_{H\Phi}v_{H}v_{\phi}\\\
\lambda_{H\chi}v_{H}v_{\chi}&2\lambda_{\chi}v^{2}_{\chi}&(-\sqrt{2}\kappa_{T}+\lambda_{\Phi\chi}v_{\phi})v_{\chi}\\\
\lambda_{H\Phi}v_{H}v_{\phi}&(-\sqrt{2}\kappa_{T}+\lambda_{\Phi\chi}v_{\phi})v_{\chi}&2\lambda_{\Phi}v^{2}_{\phi}+\frac{\kappa_{T}v^{2}_{\chi}}{\sqrt{2}v_{\phi}}\\\
\end{pmatrix}$ (4.4)

The matrix $M^{2}_{R}$ can be diagonalized by an orthogonal matrix as follows
[193, 194],

$\displaystyle\mathcal{O}_{R}M_{R}^{2}\mathcal{O}_{R}^{T}=\text{diag}(m_{h_{1}}^{2},m_{h_{2}}^{2},m_{h_{3}}^{2}),$
(4.5)

where

$\displaystyle\left(\begin{array}[]{c}h_{1}\\\ h_{2}\\\ h_{3}\\\
\end{array}\right)=\mathcal{O}_{R}\left(\begin{array}[]{c}R_{1}\\\ R_{2}\\\
R_{3}\\\ \end{array}\right).$ (4.12)

We assume the mass eigenstates to be ordered by their masses $m_{h_{1}}\leq
m_{h_{2}}\leq m_{h_{3}}$. We will use the standard parameterization
$\mathcal{O}_{R}=R_{23}R_{13}R_{12}$ where

$R_{12}=\left(\begin{array}[]{ccc}c_{12}&s_{12}&0\\\ -s_{12}&c_{12}&0\\\
0&0&1\end{array}\right),\quad
R_{13}=\left(\begin{array}[]{ccc}c_{13}&0&s_{13}\\\ 0&1&0\\\
-s_{13}&0&c_{13}\end{array}\right),\quad
R_{23}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{array}\right)$ (4.13)

$c_{ij}=\cos\theta_{ij},s_{ij}=\sin\theta_{ij}$, where the angles
$\theta_{ij}$ can be chosen to lie in the range
$-\frac{\pi}{2}\leq\theta_{ij}\leq\frac{\pi}{2}$. The rotation matrix
$\mathcal{O}_{R}$ is re-expressed in terms of the mixing angles in the
following way:

$\displaystyle\mathcal{O}_{R}=\left(\begin{array}[]{ccc}c_{12}c_{13}&c_{13}s_{12}&s_{13}\\\
-c_{23}s_{12}-c_{12}s_{13}s_{23}&c_{23}c_{12}-s_{12}s_{13}s_{23}&c_{13}s_{23}\\\
-c_{12}c_{23}s_{13}+s_{23}s_{12}&-c_{23}s_{12}s_{13}-c_{12}s_{23}&c_{13}c_{23}\end{array}\right).$
(4.17)

It is possible to express the ten parameters of the scalar potential through
the three physical Higgs masses, the three mixing angles, the three VEVs and
$\kappa_{T}$ [194]. These relations are given by

$\displaystyle\lambda_{H}$
$\displaystyle=\frac{1}{2v_{H}^{2}}m_{h_{i}}^{2}(\mathcal{O}_{R}^{i1})^{2}\,,$
$\displaystyle\lambda_{\chi}$
$\displaystyle=\frac{1}{2v_{\chi}^{2}}m_{h_{i}}^{2}(\mathcal{O}_{R}^{i2})^{2}\,,$
$\displaystyle\lambda_{\Phi}$
$\displaystyle=\frac{1}{2v_{\Phi}^{2}}m_{h_{i}}^{2}(\mathcal{O}_{R}^{i3})^{2}-\kappa_{T}\frac{v_{\chi}^{2}}{2\sqrt{2}v_{\Phi}^{3}}\,,$
$\displaystyle\lambda_{H\chi}$
$\displaystyle=\frac{1}{v_{H}v_{\chi}}m_{h_{i}}^{2}\mathcal{O}_{R}^{i1}\mathcal{O}_{R}^{i2}\,,$
$\displaystyle\lambda_{H\Phi}$
$\displaystyle=\frac{1}{v_{H}v_{\Phi}}m_{h_{i}}^{2}\mathcal{O}_{R}^{i1}\mathcal{O}_{R}^{i3}\,,$
$\displaystyle\lambda_{\Phi\chi}$
$\displaystyle=\frac{1}{v_{\Phi}v_{\chi}}m_{h_{i}}^{2}\mathcal{O}_{R}^{i2}\mathcal{O}_{R}^{i3}+\sqrt{2}\frac{\kappa_{T}}{v_{\Phi}}\,,$
(4.18)

The squared mass matrix for the CP-odd scalars in the weak basis
($\eta_{\chi}$, $\eta_{\Phi}$) is given as

$\displaystyle
M^{2}_{\eta}=\frac{\kappa_{T}}{\sqrt{2}}\begin{pmatrix}4v_{\Phi}&-2v_{\chi}\\\
-2v_{\chi}&\frac{v^{2}_{\chi}}{v_{\Phi}}\\\ \end{pmatrix}$ (4.19)

This mass matrix can be diagonalized as
$\mathcal{O}_{\eta}M^{2}_{\eta}\mathcal{O}_{\eta}^{T}=\text{diag}(0,m_{A}^{2})$,
where

$\displaystyle\mathcal{O}_{\eta}=\begin{pmatrix}\cos\theta_{\eta}&\sin\theta_{\eta}\\\
-\sin\theta_{\eta}&\cos\theta_{\eta}\\\
\end{pmatrix},\,\,m^{2}_{A}=\frac{\kappa_{T}(4v^{2}_{\Phi}+v^{2}_{\chi})}{\sqrt{2}v_{\Phi}}$
(4.20)

and the null mass corresponds to the would-be Goldstone boson $G^{\prime}$,
which will be eaten by $Z^{\prime}$. The gauge eigenstates
$(\eta_{\chi}\,\,\eta_{\Phi})$ are related with the mass eigenstates
$(G^{\prime}\,\,A)$ as

$\displaystyle\begin{pmatrix}\eta_{\chi}\\\ \eta_{\Phi}\\\
\end{pmatrix}=\begin{pmatrix}\cos\theta_{\eta}&-\sin\theta_{\eta}\\\
\sin\theta_{\eta}&\cos\theta_{\eta}\\\
\end{pmatrix}\begin{pmatrix}G^{\prime}\\\ A\\\ \end{pmatrix},\text{ where
}\tan\theta_{\eta}=\frac{2v_{\Phi}}{v_{\chi}}$ (4.21)

Note that fixing one of the Higgs masses to the mass of the observed Higgs
boson, $m_{h_{1}}=125$ GeV, and fixing the Higgs doublet vev to its SM value,
$v_{H}=246$ GeV, leaves eight free input parameters from the scalar sector:

$\displaystyle
m_{h_{2}},\,\,m_{h_{3}},\,\,m_{A},\,\,\theta_{12},\,\,\theta_{13},\,\,\theta_{23},\,\,v_{\chi}\text{
and }v_{\Phi}$ (4.22)

#### 4.2.2 Gauge Sector

We note that a kinetic mixing can occur provided there are two or more field
strength tensors $B^{1}_{\mu\nu}$ and $B^{2}_{\mu\nu}$ which are neutral under
some gauge symmetry. This only arises for the abelian $U(1)$ gauge group. Thus
in our case with the gauge group $SU(3)_{c}\otimes SU(2)_{L}\otimes
U(1)_{Y}\otimes U(1)_{X}$, there can be kinetic mixing between two abelian
gauge group $U(1)_{Y}$ and $U(1)_{X}$. The kinetic terms can be expressed as
follows

$\displaystyle\mathcal{L}_{\rm
Kinetic}=-\frac{1}{4}B^{1\mu\nu}B^{1}_{\mu\nu}-\frac{1}{4}B^{2\mu\nu}B^{2}_{\mu\nu}-\frac{\kappa}{2}B^{1\mu\nu}B^{2}_{\mu\nu},$
(4.23)

where $B^{1}_{\mu\nu}$ and $B^{2}_{\mu\nu}$ are the filed strength tensors of
the gauge groups $U(1)_{Y}$ and $U(1)_{X}$, respectively. The requirement of
positive kinetic energy implies that kinetic coefficient $|\kappa|<1$. One can
diagonalize the kinetic mixing term as follows

$\displaystyle\begin{pmatrix}\tilde{B}_{1}\\\ \tilde{B}_{2}\\\
\end{pmatrix}=\begin{pmatrix}1&\kappa\\\ 0&\sqrt{1-\kappa^{2}}\\\
\end{pmatrix}\begin{pmatrix}B_{1}\\\ B_{2}\\\ \end{pmatrix}$ (4.24)

Let’s first set kinetic mixing $\kappa=0$ to fix the notation. The diagonal
component of the $SU(2)_{L}$ gauge filed $W_{3\mu}$ will mix with the
$U(1)_{Y}$ and $U(1)_{X}$ gauge fields $B^{1}_{\mu}$ and $B^{2}_{\mu}$. To
determine the gauge boson mass spectrum, we have to expand the scalar kinetic
terms

$\displaystyle\mathcal{L}_{s}=(D^{\mu}H)^{\dagger}(D_{\mu}H)+(D^{\mu}\Phi)^{\dagger}(D_{\mu}\Phi)+(D^{\mu}\chi)^{\dagger}(D_{\mu}\chi)$
(4.25)

and have to replace the fields $H,\Phi$ and $\chi$ by the following
expressions such as

$\displaystyle H=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
v_{H}+R_{1}\end{pmatrix},\,\,\braket{\Phi}=\frac{1}{\sqrt{2}}(v_{\Phi}+R_{2})\text{
and }\braket{\chi}=\frac{1}{\sqrt{2}}(v_{\chi}+R_{3}).$ (4.26)

With this above replacement, we can expand the scalar kinetic terms
$(D^{\mu}H)^{\dagger}(D_{\mu}H)$, $(D^{\mu}\Phi)^{\dagger}(D_{\mu}\Phi)$ and
$(D^{\mu}\chi)^{\dagger}(D_{\mu}\chi)$ as follows

$\displaystyle(D^{\mu}H)^{\dagger}(D_{\mu}H)\equiv\frac{1}{2}\partial^{\mu}R_{1}\partial_{\mu}R_{1}+\frac{1}{8}(R_{1}+v_{H})^{2}\Big{(}g^{2}|W_{1}^{\mu}-iW_{2}^{\mu}|^{2}+(gW_{3}^{\mu}-g_{1}B^{1\mu}-\tilde{g}B^{2\mu})^{2}\Big{)}$
(4.27)
$\displaystyle(D^{\mu}\Phi)^{\dagger}(D_{\mu}\Phi)\equiv\frac{1}{2}\partial^{\mu}R_{2}\partial_{\mu}R_{2}+\frac{1}{2}(R_{2}+v_{\Phi})^{2}(2g_{1}^{{}^{\prime}}B^{2\mu})^{2}$
(4.28)
$\displaystyle(D^{\mu}\chi)^{\dagger}(D_{\mu}\chi)\equiv\frac{1}{2}\partial^{\mu}R_{3}\partial_{\mu}R_{3}+\frac{1}{2}(R_{3}+v_{\chi})^{2}(g_{1}^{{}^{\prime}}B^{2\mu})^{2}.$
(4.29)

where we have defined $\tilde{g}=g_{X}x_{H}$ and
$g_{1}^{{}^{\prime}}=g_{X}x_{\Phi}$. SM charged gauge boson $W^{\pm}$ can be
easily recognized with mass $M_{W}=\frac{gv_{H}}{2}$. On the other hand, the
mass matrix of the neutral gauge bosons is given by

$\displaystyle\mathcal{L}_{M}=\frac{1}{2}V_{0}^{T}M_{G}^{2}V_{0}$ (4.30)

where

$\displaystyle
V_{0}^{T}=\begin{pmatrix}B^{1}_{\mu}&W_{3\mu}&B^{2}_{\mu}\end{pmatrix}\text{
and
}M_{G}^{2}=\begin{pmatrix}\frac{1}{4}g^{2}_{1}v_{H}^{2}&-\frac{1}{4}gg_{1}v_{H}^{2}&\frac{1}{4}g_{1}\tilde{g}v_{H}^{2}\\\
-\frac{1}{4}gg_{1}v_{H}^{2}&\frac{1}{4}g^{2}v_{H}^{2}&-\frac{1}{4}g\tilde{g}v_{H}^{2}\\\
\frac{1}{4}g_{1}\tilde{g}v_{H}^{2}&-\frac{1}{4}g\tilde{g}v_{H}^{2}&\frac{1}{4}\tilde{g}^{2}v_{H}^{2}+4g_{1}^{{}^{\prime}2}v_{T}^{2}\end{pmatrix}$
(4.31)

Following linear combination of $B_{1}^{\mu}$, $W_{3}^{\mu}$ and $B_{2}^{\mu}$
gives definite mass eigenstates $A_{0}^{\mu}$, $Z_{0}^{\mu}$ and
$Z_{0}^{{}^{\prime}\mu}$ (when $\kappa=0$),

$\displaystyle\begin{pmatrix}B_{1}^{\mu}\\\ W_{3}^{\mu}\\\
B_{2}^{\mu}\end{pmatrix}=\begin{pmatrix}\cos\theta_{w}&-\sin\theta_{w}~{}\cos\theta_{0}&\sin\theta_{w}~{}\sin\theta_{0}\\\
\sin\theta_{w}&\cos\theta_{w}\cos\theta_{0}&-\cos\theta_{w}~{}\sin\theta_{0}\\\
0&\sin\theta_{0}&\cos\theta_{0}\\\ \end{pmatrix}\begin{pmatrix}A_{0}^{\mu}\\\
Z_{0}^{\mu}\\\ Z_{0}^{{}^{\prime}\mu}\\\ \end{pmatrix}$ (4.32)

where $\theta_{w}$ is the Weinberg mixing angle and,

$\displaystyle\text{tan}2\theta_{0}=\frac{2\tilde{g}\sqrt{g^{2}+g_{1}^{2}}}{\tilde{g}^{2}+16\left(\frac{v_{T}}{v_{H}}\right)^{2}g_{1}^{{}^{\prime}2}-g^{2}-g_{1}^{2}}\text{
with }v_{T}=\frac{\sqrt{v_{\chi}^{2}+4v_{\Phi}^{2}}}{2}.$ (4.33)

When the kinetic mixing $\kappa$ is non-zero, the fields $A_{0},Z_{0}$ and
$Z^{\prime}_{0}$ are not orthogonal. In the kinetic term diagonalized basis
$\tilde{V}_{0}^{T}=(\tilde{B}_{1}\,\,W_{3}\,\,\tilde{B}_{2})$, the mass matrix
of the neutral gauge boson can be written as

$\displaystyle\mathcal{L}_{M}=\frac{1}{2}\tilde{V}_{0}^{T}S^{T}M_{G}^{2}S\tilde{V}_{0}=\frac{1}{2}\tilde{V}_{0}^{T}\tilde{M}_{G}^{2}\tilde{V}_{0}$
(4.34)

where

$\displaystyle S=\begin{pmatrix}1&0&-\frac{\kappa}{\sqrt{1-\kappa^{2}}}\\\
0&1&0\\\
0&0&\frac{1}{\sqrt{1-\kappa^{2}}}\end{pmatrix},\tilde{M}_{G}^{2}=S^{T}M_{G}^{2}S=\begin{pmatrix}\frac{1}{4}g_{1}v_{H}^{2}&-\frac{1}{4}gg_{1}v_{H}^{2}&\frac{1}{4}g_{1}\tilde{g}_{t}v_{H}^{2}\\\
-\frac{1}{4}gg_{1}v_{H}^{2}&\frac{1}{4}g^{2}v_{H}^{2}&-\frac{1}{4}g\tilde{g}_{t}v_{H}^{2}\\\
\frac{1}{4}g_{1}\tilde{g}_{t}v_{H}^{2}&-\frac{1}{4}g\tilde{g}_{t}v_{H}^{2}&\frac{1}{4}\tilde{g}_{t}^{2}v_{H}^{2}+4g_{1}^{{}^{\prime\prime}2}v_{T}^{2}\end{pmatrix}$
(4.35)

with $\tilde{g}_{t}=\frac{\tilde{g}-g_{1}\kappa}{\sqrt{1-\kappa^{2}}}$. Hence,
comparing $M_{G}^{2}$ and $\tilde{M}_{G}^{2}$ we see that the overall effect
of kinetic mixing introduction is just the modification of $\tilde{g}$ to
$\tilde{g}_{t}$ and $g_{1}^{\prime}$ to $g_{1}^{{}^{\prime\prime}}$. Hence,
they can be related to orthogonal fields $A,Z$ and $Z^{\prime}$ by the same
transformation as in Eq. 4.32 but now $\theta_{0}$ is replaced by
$\theta_{12}$

$\displaystyle\begin{pmatrix}\tilde{B}_{1}^{\mu}\\\ W_{3}^{\mu}\\\
\tilde{B}_{2}^{\mu}\end{pmatrix}=\begin{pmatrix}\cos\theta_{w}&-\sin\theta_{w}~{}\cos\theta&\sin\theta_{w}~{}\sin\theta\\\
\sin\theta_{w}&\cos\theta_{w}\cos\theta&-\cos\theta_{w}~{}\sin\theta\\\
0&\sin\theta&\cos\theta\\\ \end{pmatrix}\begin{pmatrix}A^{\mu}\\\ Z^{\mu}\\\
Z^{{}^{\prime}\mu}\\\ \end{pmatrix}$ (4.36)

where

$\displaystyle\text{tan}2\theta=\frac{2\tilde{g}_{t}\sqrt{g^{2}+g_{1}^{2}}}{\tilde{g}_{t}^{2}+16\left(\frac{v_{T}}{v_{H}}\right)^{2}g_{1}^{{}^{\prime\prime}2}-g^{2}-g_{1}^{2}}\text{
with }\tilde{g}_{t}=\frac{\tilde{g}-g_{1}\kappa}{\sqrt{1-\kappa^{2}}}\text{
and }g_{1}^{{}^{\prime\prime}}=\frac{g_{1}^{\prime}}{\sqrt{1-\kappa^{2}}}.$
(4.37)

Masses of physical gauge bosons $A$, $Z$ and $Z^{{}^{\prime}}$ are given by,

$\displaystyle
M_{A}=0,\,\,M_{Z,Z^{{}^{\prime}}}^{2}=\frac{1}{8}\left(Cv_{H}^{2}\mp\sqrt{-D+v_{H}^{4}C^{2}}\right),$
(4.38)

where,

$\displaystyle
C=g^{2}+g_{1}^{2}+\tilde{g}_{t}^{2}+16\left(\frac{v_{T}}{v_{H}}\right)^{2}g_{1}^{{}^{\prime\prime}2},\hskip
14.22636ptD=64v_{H}^{2}v_{T}^{2}(g^{2}+g_{1}^{2})g_{1}^{{}^{\prime\prime}2}.$
(4.39)

The covariant derivative also can be expressed in terms of the orthogonal
fields $\tilde{B}_{1}$ and $\tilde{B}_{2}$ as

$\displaystyle
D_{\mu}=\partial_{\mu}-ig_{s}T^{a}G^{a}_{\mu}-igT^{a}W^{a}_{\mu}-ig_{1}Y\tilde{B}^{1}_{\mu}-i\left(g_{X}Y_{X}\frac{1}{\sqrt{1-\kappa^{2}}}-g_{1}Y\frac{\kappa}{\sqrt{1-\kappa^{2}}}\right)\tilde{B}^{2}_{\mu}$
(4.40)

#### 4.2.3 Yukawa Sector

The general form of the Yukawa interactions is given by,

$\displaystyle\mathcal{L}_{y}$
$\displaystyle=-y_{u}^{ij}\overline{q_{L}^{i}}\tilde{H}u^{j}_{R}-y_{d}^{ij}\overline{q_{L}^{i}}Hd^{j}_{R}-y_{e}^{ij}\overline{\ell_{L}^{i}}He^{j}_{R}-y_{\nu}^{ij}\overline{\ell_{L}^{i}}\tilde{H}\nu^{j}_{R}-\frac{1}{2}y_{M}^{ij}\Phi\overline{N_{R}^{ic}}N^{j}_{R}+\text{H.c.}$
(4.41)

The last two terms are responsible for Dirac and Majorana masses of neutrinos.

#### 4.2.4 Neutrino Mass

Relevant light neutrino masses will come from the fourth and fifth terms of
Eq. 4.41. After the electroweak symmetry breaking, we can write the mass terms
as,

$\displaystyle-\mathcal{L}_{M}=\sum_{i=1}^{3}\sum_{j=1}^{3}\overline{\nu_{jL}}m_{ij}^{D}N_{jR}+\frac{1}{2}\sum_{i,j=1}^{3}\overline{(N_{R})_{i}^{c}}M_{ij}^{R}N_{jR}+\text{H.c.},$
(4.42)

where $m_{ij}^{D}=\frac{y_{\nu}^{ij}v_{H}}{\sqrt{2}}$ and
$M_{ij}^{R}=\frac{y_{M}^{ij}}{\sqrt{2}}v_{\Phi}$. Now we can write the
$\mathcal{L}_{M}$ in the following matrix form,

$\displaystyle-\mathcal{L}_{M}=\frac{1}{2}\begin{pmatrix}\overline{\nu_{L}}&\overline{(N_{R})^{c}}\\\
\end{pmatrix}\begin{pmatrix}{\bf 0}&m_{D}\\\ m_{D}^{T}&M_{R}\\\
\end{pmatrix}\begin{pmatrix}(\nu_{L})^{c}\\\ N_{R}\\\
\end{pmatrix}+\text{H.c.}$ (4.43)

In the seesaw approximation, this leads to the usual light neutrino mass
matrix $m_{\nu}\approx-m_{D}M_{R}^{-1}m_{D}^{T}$. The $6\times 6$ matrix in
Eq. 4.43 is symmetric and can be diagonalized by the unitary matrix (up to
$\mathcal{O}(M_{R}^{-2})$) [74]

$\displaystyle\begin{pmatrix}U&S\\\
-S^{\dagger}&V\end{pmatrix}^{\dagger}\begin{pmatrix}{\bf 0}&m_{D}\\\
m_{D}^{T}&M_{R}\\\ \end{pmatrix}\begin{pmatrix}U&S\\\
-S^{\dagger}&V\end{pmatrix}^{*}=\begin{pmatrix}\widehat{M_{\nu}}&{\bf 0}\\\
{\bf 0}&\widehat{M_{N}}\end{pmatrix}$ (4.44)

where, $\widehat{M_{\nu}}=\text{diag}\\{m_{1},m_{2},m_{3}\\}$,
$\widehat{M_{N}}=\text{diag}\\{M_{N_{1}},M_{N_{2}},M_{N_{3}}\\}$, $U\approx
U_{\text{PMNS}}$, $V\approx{\bf I}$ and $S\approx m_{D}M_{R}^{-1}$.

### 4.3 Theoretical and experimental constraints

We discuss the relevant theoretical and experimental constraints in this
section.

#### 4.3.1 Vacuum Stability condition

The scalar potential $V(H,\Phi,\chi)$ must be bounded from below [195] and it
can be determined from the following symmetric matrix which comes from the
quadratic part of the potential,

$\displaystyle
V^{4}_{s}=\begin{pmatrix}\lambda_{H}&\frac{\lambda_{H\Phi}}{2}&\frac{\lambda_{H\chi}}{2}\\\
\frac{\lambda_{H\Phi}}{2}&\lambda_{\Phi}&\frac{\lambda_{\Phi\chi}}{2}\\\
\frac{\lambda_{H\chi}}{2}&\frac{\lambda_{\Phi\chi}}{2}&\lambda_{\chi}\\\
\end{pmatrix}$ (4.45)

Above matrix will be positive-definite if following conditions are satisfied,

$\displaystyle\lambda_{H}>0,\hskip
28.45274pt4\lambda_{H}\lambda_{\Phi}-\lambda_{H\Phi}^{2}>0,$
$\displaystyle(-\lambda_{H}\lambda_{\Phi\chi}^{2}+\lambda_{H\Phi}\lambda_{\Phi\chi}\lambda_{H\chi}-\lambda_{\Phi}\lambda_{H\chi}^{2}+4\lambda_{H}\lambda_{\Phi}\lambda_{\chi}-\lambda_{H\Phi}^{2}\lambda_{\chi})>0.$
(4.46)

A stable vacuum can be achieved if conditions given in Eq. (4.46) are
satisfied. The perturbativity condition on quartic coupling $\lambda_{i}\leq
4\pi$ and gauge coupling $g_{X}\leq\sqrt{4\pi}$ is being ensured in the model.

#### 4.3.2 Invisible Higgs width constraint

In our model, $h_{1}$ scalar is SM Higgs by choice, which is the mixed state
of three real scalar fields from eq. 4.12.

$\displaystyle h_{\text{SM}}=\sum_{i}\mathcal{O}_{R_{1i}}R_{i}$ (4.47)

If $m_{h_{1}}>m_{A}/2$, then the partial decay width to $AA$ is given by:

$\displaystyle\Gamma(h_{1}\to AA)$
$\displaystyle=\frac{\lambda^{2}_{AAh_{1}}}{32\pi
m_{h_{1}}}\sqrt{1-\frac{4m_{A}^{2}}{m_{h_{1}}^{2}}}$ (4.48)

here $\lambda_{AAh_{1}}$ is trilinear coupling for vertex $A-A-h_{1}$.

$\displaystyle\lambda_{AAh_{1}}=2\sqrt{2}\kappa_{T}\mathcal{O}_{\eta_{21}}^{\dagger}\mathcal{O}_{\eta_{22}}^{\dagger}\mathcal{O}_{R_{12}}-$
$\displaystyle\left(\mathcal{O}_{\eta_{22}}^{\dagger}\right)^{2}(\lambda_{H\Phi}v_{H}\mathcal{O}_{R_{11}}+\lambda_{\Phi\chi}v_{\chi}\mathcal{O}_{R_{12}}+2\lambda_{\Phi}v_{\phi}\mathcal{O}_{R_{13}})\
-$
$\displaystyle\left(\mathcal{O}_{\eta_{21}}^{\dagger}\right)^{2}(\lambda_{\Phi\chi}v_{H}\mathcal{O}_{R_{11}}+2\lambda_{\chi}v_{\chi}\mathcal{O}_{R_{12}}+(\sqrt{2}\kappa_{T}+\lambda_{\Phi\chi}v_{\phi})\mathcal{O}_{R_{13}})$

The invisible decay width of SM Higgs boson $h_{1}$ is given by
$\Gamma^{\text{inv}}(h_{1})=\Gamma(h_{1}\to AA)$. Then, one can calculate the
invisible Higgs branching ratio as follow [196, 194]

$\displaystyle\text{BR}^{\text{inv}}(h_{1})=\frac{\Gamma^{\text{inv}}(h_{1})}{\mathcal{O}^{2}_{R_{1i}}\Gamma^{\text{SM}}(h_{1})+\Gamma^{\text{inv}}(h_{1})},$
(4.49)

here $\Gamma^{\text{SM}}(h_{1})=4.1$ MeV. The current upper limit on the
invisible Higgs branching ratio is from ATLAS experiment [197, 168, 167],

$\displaystyle\text{BR}^{\text{inv}}(h_{1})\leq 0.11.$ (4.50)

#### 4.3.3 Relic density constraint

The relic density bound is from Planck satellite data [49].

$\displaystyle\Omega_{\text{DM}}h^{2}=0.12\pm 0.001$ (4.51)

Any DM candidate must satisfy the relic bound given in equation 4.51.

### 4.4 Dark Matter analysis

The general charge assignment for $U(1)_{X}$ model can be described in terms
of only two free charges $x_{H},~{}x_{\phi}$. The charges $x_{H}$, and
$x_{\Phi}$ are the real parameters. For simplicity, one can fix $x_{\Phi}=1$
and vary $x_{H}$ only to characterize the $U(1)$ models. We can obtain the
$B-L$ case by choosing $x_{H}=0$. When $x_{H}=-2$, the left-handed fermions
have no interactions with the $Z^{\prime}$ which leads to an $U(1)_{R}$ model.
The interactions of $e_{R}$ and $d_{R}$ with $Z^{\prime}$ are absent when
$x_{H}=-1$ and $1$ respectively. Thus, one can study any $U(1)$ models by
changing only one parameter $x_{H}$ in our generic model. We have utilized
this generality of our model in doing dark matter analysis.

Benchmark : We have fixed the following independent parameters throughout the
paper for the study of our pseudo scalar particle candidate to DM.

$\displaystyle\textbf{BP :}\ \ \ \ x_{\phi}=1,\ \ \ \theta_{13,23}\approx 0,\
\ \ m_{h_{3}}=10^{13}~{}\text{GeV},\ \ \ g_{X}=0.1,\ \ \
M_{Z^{\prime}}=10^{14}~{}\text{GeV}$

We have also set $m_{A}<<M_{N_{i}}$, therefore RHNs are not relevant for our
DM analysis. Remaining free parameters such as $x_{H}$, $\theta_{12}$,
$\kappa$, $m_{h_{2}}$, $v_{\phi}$, $v_{\chi}$ and $m_{A}$ are relevant for DM
phenomenology.

Figure 4.1: The relevant two and three-body decay modes for pseudo scalar DM
are shown here.

#### 4.4.1 DM lifetime study

A good candidate for DM must be stable or have a lifetime greater than the age
of the universe. A conservative limit on the lifetime of dark matter is
$\tau_{DM}\geq 10^{27}$ sec, or in terms of decay width, $\Gamma_{DM}\leq
6.6\times 10^{-52}$ GeV, from the gamma rays observation of dwarf spheroidal
galaxies [28]. Our pseudo-scalar should also follow this criterion to be a
good DM candidate.

There are two possible body decay modes, namely
$A\rightarrow\nu\nu,~{}h_{i}Z$. Particles $Z^{\prime}$ and $N_{i}$ are chosen
to be very heavy, such that $h_{i}Z^{\prime}$ and $NN$ decay modes are not
relevant to our study of DM phenomenology. The $A-\nu-\nu$ coupling arises due
to scalar and active-sterile mixing. The $\nu\nu$ mode is proportional to
neutrino mass and therefore strongly suppressed. $\Phi,\chi$ are SM singlets;
therefore, the resultant pseudo-scalars do not couple to SM fermions and gauge
bosons. The only two body decay channels are $h_{i}Z$; however, $h_{3}$ is
also a massive particle. Hence, only $h_{1}Z,~{}h_{2}Z$ modes are
kinematically allowed in our interested mass range of DM.

Apart from the two body decays, there could be two possible; three body decays
such as $A\rightarrow Zf\bar{f},\ h_{i}f\bar{f}$ are also relevant and could
be dominant or subdominant depending on the kinematics. The former process
contributes to non-zero-gauge kinetic mixing. The latter process is mediated
by $Z,~{}Z^{\prime}$, and a dominant three-body decay when kinematically
allowed. We have shown the Feynman diagrams for these decay modes in figure D
and calculated the decay width expressions for all these modes as given in the
following equations

$\displaystyle\Gamma(A\to
h_{i}X)=\frac{|\lambda_{AXh_{i}}|^{2}m_{A}^{3}}{16\pi
M_{X}^{2}}\lambda^{\frac{3}{2}}\Big{(}1,\frac{M_{X}^{2}}{m_{A}^{2}},\frac{m_{h_{i}}^{2}}{m_{A}^{2}}\Big{)},$
(4.52) $\displaystyle\Gamma(A\to N_{i}N_{i})=\frac{\cos^{2}\theta_{\eta}}{8\pi
v_{\Phi}^{2}}m_{A}M_{N_{i}}^{2}\sqrt{1-4\frac{M_{N_{i}}^{2}}{m_{A}^{2}}},$
(4.53) $\displaystyle\Gamma(A\to
Xf\bar{f})=\frac{|H_{X}(z)|^{2}m^{9}x_{f}^{2}}{128\pi^{3}M_{X}^{2}v_{H}^{2}}\big{(}z-4x_{f}^{2}\big{)}\lambda^{\frac{3}{2}}\big{(}1,\frac{M_{X}^{2}}{m_{A}^{2}},z\big{)}\frac{dz}{z}\lambda^{\frac{1}{2}}\big{(}1,x_{f}^{2},x_{f}^{2}\big{)},$
(4.54) $\displaystyle d\Gamma(A\to
h_{i}f\bar{f})=\frac{m_{A}^{5}}{384\pi^{3}}\Big{[}\big{(}|A_{L}^{if}(z)|^{2}+|A_{R}^{if}(z)|^{2}\big{)}\big{\\{}\lambda\big{(}1,x_{h_{i}}^{2},z\big{)}+\frac{x_{f}^{2}}{z}\big{(}2(1-x_{h_{i}}^{2})^{2}+2(1+x_{h_{i}}^{2})z-z^{2}\big{)}\big{\\}}$
$\displaystyle-6\text{Re}\big{(}A_{L}^{if*}(z)A_{R}^{if}(z)\big{)}x_{f}^{2}(2(1+x_{h_{i}}^{2})-z)\Big{]}\frac{dz}{z}\lambda^{\frac{1}{2}}\big{(}1,x_{h_{i}}^{2},z\big{)}\lambda^{\frac{1}{2}}\big{(}1,x_{f}^{2},x_{f}^{2}\big{)};\,\,\,4x_{f}^{2}\leq
z\leq(1-x_{h_{i}})^{2},$ (4.55)

where $X=Z,~{}Z^{\prime}$ and $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2bc-2ca$
and couplings,

$\displaystyle\lambda_{AZh_{i}}=\frac{g_{X}x_{\Phi}\sin\theta}{\sqrt{1-\kappa^{2}}}(2\cos\theta_{\eta}O_{R_{i3}}-\sin\theta_{\eta}O_{R_{i2}}),$
$\displaystyle\lambda_{AZ^{\prime}h_{i}}=\frac{g_{X}x_{\Phi}\cos\theta}{\sqrt{1-\kappa^{2}}}(2\cos\theta_{\eta}O_{R_{i3}}-\sin\theta_{\eta}O_{R_{i2}}),$
$\displaystyle
A_{L(R)}^{if}(z)=\frac{\lambda_{AZh_{i}}g_{L(R)}^{Zf}}{zm_{A}^{2}-M_{Z}^{2}+i\Gamma_{Z}M_{Z}}+\frac{\lambda_{AZ^{\prime}h_{i}}g_{L(R)}^{Z^{\prime}f}}{zm_{A}^{2}-M_{Z}^{{}^{\prime}2}+i\Gamma_{Z^{\prime}}M_{Z^{\prime}}},$
$\displaystyle
H_{Z}(z)=\sum_{i=1}^{3}\frac{\lambda_{AZh_{i}}\mathcal{O}_{R_{i1}}}{zm_{A}^{2}-m_{h_{i}}^{2}+i\Gamma_{h_{i}}m_{h_{i}}},\,\,H_{Z^{\prime}}(z)=\sum_{i=1}^{3}\frac{\lambda_{AZ^{\prime}h_{i}}\mathcal{O}_{R_{i1}}}{zm_{A}^{2}-m_{h_{i}}^{2}+i\Gamma_{h_{i}}m_{h_{i}}},$
$\displaystyle
g^{Z\ell}_{R}=-\frac{1}{\sqrt{1-\kappa^{2}}}\Big{(}g_{X}\sin\theta\big{(}x_{H}+x_{\Phi}\big{)}-g_{Z}\big{(}\sin^{2}\theta_{W}\cos\theta\sqrt{1-\kappa^{2}}+\kappa\sin\theta\sin\theta_{W}\big{)}\Big{)},$
$\displaystyle
g^{Z\ell}_{L}=-\frac{1}{2\sqrt{1-\kappa^{2}}}\Big{(}g_{X}\sin\theta\big{(}x_{H}+2x_{\Phi}\big{)}-g_{Z}\big{(}\kappa\sin\theta\sin\theta_{W}-(1-2\sin^{2}\theta_{W})\cos\theta\sqrt{1-\kappa^{2}}\big{)}\Big{)},$
$\displaystyle
g^{Zu}_{R}=\frac{1}{3\sqrt{1-\kappa^{2}}}\Big{(}g_{X}\sin\theta\big{(}2x_{H}+x_{\Phi}\big{)}-2g_{Z}\big{(}\cos\theta\sin^{2}\theta_{W}\sqrt{1-\kappa^{2}}+\kappa\sin\theta\sin\theta_{W}\big{)}\Big{)},$
$\displaystyle
g^{Zu}_{L}=\frac{1}{6\sqrt{1-\kappa^{2}}}\Big{(}g_{X}\sin\theta\big{(}x_{H}+2x_{\Phi}\big{)}-g_{Z}\big{(}\cos\theta\sqrt{1-\kappa^{2}}(4\sin^{2}\theta_{W}-3)+\kappa\sin\theta\sin\theta_{W}\big{)}\Big{)},$
$\displaystyle
g^{Zd}_{R}=\frac{1}{3\sqrt{1-\kappa^{2}}}\Big{(}g_{X}\sin\theta\big{(}x_{\Phi}-x_{H}\big{)}+g_{Z}\big{(}\cos\theta\sin^{2}\theta_{W}\sqrt{1-\kappa^{2}}+\kappa\sin\theta\sin\theta_{W}\big{)}\Big{)},$
$\displaystyle
g^{Zd}_{L}=\frac{1}{6\sqrt{1-\kappa^{2}}}\Big{(}g_{X}\sin\theta\big{(}x_{H}+2x_{\Phi}\big{)}-g_{Z}\big{(}\cos\theta\sqrt{1-\kappa^{2}}(3-2\sin^{2}\theta_{W})+\kappa\sin\theta\sin\theta_{W}\big{)}\Big{)},$

where as
$g^{Z^{\prime}\ell/u/d}_{L/R}=g^{Z\ell/u/d}_{L/R}\Big{(}\sin\theta\to\cos\theta;\cos\theta\to-\sin\theta\Big{)}$.

The relevant parameters for the study of DM lifetime constraints on our
pseudo-scalar DM are U(1)X charge $x_{H}$, scalar mixing $\theta_{12}$ between
Higgs h1 and h2, vev $v_{\phi}$ and $v_{\chi}$, gauge kinetic mixing $\kappa$
and masses $m_{A}$ and $m_{h_{2}}$. Other free parameters are the same as in
the benchmark.

Figure 4.2: Plots are displayed for $x_{H}=-2,-1,1,2$. In each plot,
$\theta_{12}=0.1$ is fixed. The allowed parameter space for a pseudo-scalar is
shown in the $(m_{A},v_{\phi})$ plane. The coloured region is disfavored. The
purple, pink, orange, and grey regions are not allowed by the Planck mass
limit on vev, the DM lifetime constraint ($\tau>10^{27}$ sec) [28], and the
perturbative unitarity bound on gauge coupling $g_{X}$. The orange (pink)
region corresponds to kinetic mixing 0 ($1/\sqrt{2}$). The dark (light) orange
and pink regions correspond to $m_{h_{2}}=250~{}(500)$ GeV.

Figure 4.3: Plots are displayed for $x_{H}=-2,-1,1,2$. The allowed parameter
space for a pseudo-scalar is shown in the $(m_{A},v_{\phi})$ plane. The
disfavored region is coloured. The purple, orange, and grey regions are
disfavored by the Planck mass limit on vev, the lifetime bound [28], and the
perturbative unitarity limit on quartic coupling $\lambda_{\chi}$. The orange
region (from darker to lighter) corresponds to mixing
$\theta_{12}=10^{-1,-2,-3}$.

In fig 4.2, we show the allowed parameter space for DM in the
$(m_{A},v_{\phi})$ plane. The four plots correspond to $x_{H}=-2,-1,1,2$ from
top left to bottom right. We have a fixed mixing angle $\theta_{12}=0.1$ in
all the plots. Vev $v_{\chi}$ is a free parameter that is chosen from the
range ($\sim 10v_{H}-10^{4}v_{H}$), whereas $\kappa_{T}$ is determined using
eq 4.20, which is of the order of
$\kappa_{T}\sim\mathcal{O}(\frac{m^{2}_{A}}{v_{\Phi}})$. The purple region on
top of each plot is not allowed as vev $v_{\phi}$ gets larger than the Planck
scale $M_{P}=1.2\times 10^{19}$ GeV. The grey region on the bottom is
disfavored by the perturbative unitarity bound of coupling
$g_{X}>\sqrt{4\pi}$, which allows vev $v_{\phi}$ to be very high and small
$\kappa_{T}$. The pink and orange regions in the middle of each plot
correspond to kinetic mixing $\kappa=0,~{}1/\sqrt{2}$, respectively, and are
disfavored by the DM lifetime constraint. The light (dark) orange and red
region is due to $m_{h_{2}}=250~{}(500)$ GeV. When allowed kinematically, the
decay modes $h_{i}f\bar{f},~{}h_{i}Z$ are active in both pink and orange
regions. However, in the pink region, where we turn on the kinetic mixing,
which opens up an additional decay mode $Zf\bar{f}$, the lesser parameter
space available by lifetime bound on the $(m_{A},v_{\phi})$ plane,. One can
also see that vev, $v_{\phi}>10^{13}$ GeV, is favored for DM mass around
$\sim$ 100 GeV.

In fig 4.3, we have done the analysis similar to above; however, we have fixed
$m_{h_{2}}=250$ GeV, gauge kinetic mixing $\kappa=0$, and varied the scalar
mixing $\theta_{12}$. The U(1)X charge $x_{H}$ is labelled in each plot. The
orange region (from darker to lighter) corresponds to mixing
$\theta_{12}=10^{-1,-2,-3}$. Here, channels $h_{i}f\bar{f}$ and $h_{i}Z$ are
the only relevant decay modes. The disallowed region increases as one
increases the scalar mixing due to decay modes $h_{1}Z,~{}h_{2}Z$ contributing
significantly. The difference is significant only in a tiny region of
parameter space, simply because of kinematics.

#### 4.4.2 Relic density analysis

Figure 4.4: Plots are displayed for $x_{H}=-2,-1,1,2$ when $\kappa=0$. Allowed
parameter spaces are shown in the $(m_{A},v/v_{\chi})$ plane. The disfavored
region is coloured. The grey, orange, and purple regions are disfavored by the
perturbative unitarity bound on quartic coupling $\lambda_{\chi}$, the
invisible Higgs width constraint 4.50, and the conservative lifetime bound
($\tau>10^{27}$ sec) [28] for $v_{\phi}=10^{14,15}$ GeV. Red and orange curves
represent the correct thermal relic abundance satisfied by Planck data 4.51.
The two dips in the red (orange) curve at 62.5, 125 (250) GeV are the
resonances due to $h_{1}$ and $h_{2}$.

Figure 4.5: Plots are displayed for $x_{H}=-2,-1,1,2$ when
$\kappa=1/\sqrt{2}$.

We can now study the relic density constraint on our pseudo-scalar candidate
to DM, which we did by micrOMEGAs package [58]. In fig 4.4 and 4.5, we show
the relic density analysis for our pseudo-scalar DM in the
$(m_{A},~{}v/v_{\chi})$ plane. The plots in Fig. 4.4 (4.5) correspond to gauge
kinetic mixing $\kappa=0~{}(1/\sqrt{2})$. The scalar mixing $\theta_{12}=0.1$
is fixed in each plot. The U(1)X charge $x_{H}$ is also labelled in each plot.
The grey region on top of each plot is disfavored by the perturbative
unitarity of quartic coupling $\lambda_{\chi}<8\pi/3$. The light and dark
purple regions are disfavored by the DM lifetime bound, and they correspond to
$v_{\phi}=10^{14,15}$ GeV, respectively. The light orange region in the middle
is not allowed by the invisible Higgs width constraint 4.50. The red and
orange curves correspond to $m_{h_{2}}=250,~{}500$ GeV, respectively, and they
represent the correct relic abundance following the Planck data 4.51. The main
annihilation channels that contribute to relic abundances are:
$W^{+}W^{-},~{}ZZ,~{}h_{1}h_{1},~{}h_{2}h_{2}$. The two dips in each plot are
the two resonances due to two Higgs poles at $m_{h_{1}}/2$ and $m_{h_{2}}/2$.
One can see the allowed parameter space in the $(m_{A},~{}v/v_{\chi})$ plane
can be increased by increasing vev $v_{\phi}$ and decreasing gauge kinetic
mixing $\kappa$.

#### 4.4.3 Direct detection

The main advantage of having a pseudo-scalar DM is that the DM-nucleon
scattering cross section vanishes in the non-relativistic limit, and we need
to check that this is the case in our model too. The relevant interaction
vertices for scattering matrix is A$-$A$-h_{i}$, which simplify in the limit
$v_{\phi}>>v_{H},~{}v_{\chi}$ and given as follows:

$\displaystyle g_{AAh_{1}}\approx
v_{H}\lambda_{H\chi}\cos\theta_{12}+2.v_{\chi}\lambda_{\chi}\sin\theta_{12},\hskip
14.22636ptg_{AAh_{2}}\approx-
v_{H}\lambda_{H\chi}\sin\theta_{12}+2.v_{\chi}\lambda_{\chi}\cos\theta_{12},\
\hskip 8.5359ptg_{AAh_{3}}\approx 0$

Hence, the scattering matrix is some of only two Feynman diagrams mediated by
the two lighter Higgs $h_{1},~{}h_{2}$.

$\displaystyle\mathcal{M}\sim\frac{g_{AAh_{1}}g_{h_{1}f\bar{f}}}{q-m^{2}_{h_{1}}}+\frac{g_{AAh_{2}}g_{h_{2}f\bar{f}}}{q-m^{2}_{h_{2}}}$

Here $g_{h_{i}f\bar{f}}$ is the coupling for Higgs-SM fermions interaction and
q is the momentum transfer. One can see by using equation 4.18 that the tree
label amplitude vanishes in the non-relativistic limit, as it is shown for the
simpler case in [185]. However, one loop contribution could be finite, and it
has been studied in ref [198]. There will be three types of Feynman diagrams,
namely: 1. self-energy; 2. vertex corrections; and 3. box and triangle
diagrams, as shown in ref [198], which contribute to one loop. The generic
expression for the scattering cross-section, which is the sum of all these
Feynman diagrams, is given by

$\displaystyle\sigma^{N}_{\text{SI}}=\frac{1}{\pi}\left(\frac{m_{N}}{m_{A}+m_{N}}\right)^{2}|f^{N}_{\text{scalar}}+f^{N}_{\text{twist2}}|^{2}$
(4.56)

Here $m_{N}$ is the mass of the nucleon, and the definition of the functions
$f^{N}_{\text{scalar}},~{}f^{N}_{\text{twist2}}$ is given in ref [198].

Figure 4.6: We have fixed $m_{h_{2}}=250$ GeV and $v_{\chi}=0.5~{}v_{H}$ in
each plot. The red and pink curves in each plot are the DM-nucleon spin-
independent cross section ($\sigma_{SI}$) for scalar mixing
$\theta_{12}=0.1,0.01$, respectively. The black and dark-green curves are the
direct detection limits from the XENON1T [18] and LZ [21] experiments. The
orange region is neutrino floor data [29] for various targets such as Helium
(He), Argon (Ar), Xenon (Xe), Germanium (Ge), NaI, and CaWO4 as labeled in
plots.

Figure 4.7: We have fixed $m_{h_{2}}=1$ TeV and $v_{\chi}=2~{}v_{H}$ here in
all the plots.

In our model, the third Higgs $h_{3}$ and Z′ are too heavy to play any
significant role in the scattering matrix calculation; therefore, it is
straightforward to follow the formalism developed in the reference to evaluate
the scattering cross section at one loop level using equation 4.56 in our
model.

In fig 4.6, 4.7, we check for the direct detection constraint on the
$(\sigma_{\text{SI}},m_{A})$ plane. We have fixed $m_{h_{2}}=250$ GeV and
$v_{\chi}=0.5~{}v_{H}$ in all plots of fig 4.6, similarly $m_{h_{2}}=1$ TeV
and $v_{\chi}=2~{}v_{H}$ are fixed in fig 4.7. The red and blue curves in each
plot are the predictions from the model for the $h_{1}-h_{2}$ mixing angle
$\theta_{12}=0.1,~{}0.01$, respectively. The black and dark-green curve is the
limit from XENON1T [18] and LZ [21]. The light orange region is the neutrino
background in each plot due to six different targets: Helium (He), Argon (Ar),
Xenon (Xe), Germanium (Ge), NaI, and CaWO4. The neutrino floor data is taken
from [29]. One can see the theoretical prediction, e.g., $\theta_{12}=0.1$, is
above the neutrino floor around $m_{A}=200,20$ GeV when
$v_{\chi}=0.5v_{H},~{}2v_{H}$, respectively, and the theoretical value is
comparable to the LZ and XENON1T data for most of the $m_{A}$ range. The
neutrino floor limit from six different target materials suggests an improved
experiment setup to search for direct detection signals. The direct detection
signals can be searched in the allowed parameter space of our model, as shown
in fig. 4.6, 4.7.

#### 4.4.4 Constraining $\sin{\theta_{12}}-m_{h_{2}}$ plane

In this section, we constraint the ($\sin\theta_{12}$, $m_{h_{2}}$) plane
using the direct detection study done in fig 4.6, 4.7. To do this, we
calculate the DM-nucleon cross section using equation 4.56 for
$(m_{A},v_{\chi})=P1\sim(20~{}\text{GeV},~{}2v_{H}),~{}P2\sim(200~{}\text{GeV},~{}0.5v_{H})$,
which is allowed points for $\theta_{12}=0.1$ by bounds as shown in direct
detection plots. We then constrained the scattering cross section using limits
from LZ and neutrino floor data for the corresponding points while taking
$\theta_{12}$ and $m_{h_{2}}$ as free parameters. Since neutrino floor data is
from five different targets such as Ar, Xe, Ge, NaI, and CaWO4, we have five
distinct plots for scalar mixing as shown in fig 4.8. We do not consider
neutrino floor data from the Helium target since it gives a much larger cross-
section, as seen in figure 4.6(4.7); thus, ‘He’ is not a good choice for
direct searches, and hence, we do not consider it for further analysis.

Figure 4.8: The plot here is adapted from [30]. Several bounds from colliders
such as LHC [15], LEP [16], prospective colliders such as ILC [17], and CLIC
[15] on the scalar mixing $\sin^{2}\theta_{12}$ as a function of $m_{h_{2}}$
are shown here. Shaded regions are ruled out by corresponding experiments.
Limits on the scalar mixing from Eq. 4.18 are also shown for different
$\lambda_{H\chi}=$0.1, 0.05, respectively. Constraints from direct detection
for $(m_{A},v_{\chi})=$ P1$\sim(20~{}\text{GeV},~{}2v_{H})$,
P2$\sim(200~{}\text{GeV},~{}0.5v_{H})$ are shown when the neutrino floor is
due to Ar, Xe, Ge, NaI, and CaWO4.

There are several colliders bounds such as LHC [15], LEP [16], prospective
colliders e.g., ILC [17], and CLIC [15] that are adapted here from the
reference [30]. The LEP bounds shown by the blue solid line consider both
$Z\to\mu^{+}\mu^{-}$ and $e^{+}e^{-}$ modes. Bounds from LHC and High
Luminosity LHC (HL-LHC) at 8 TeV are shown by the brown dot-dashed and dashed
lines, respectively [15]. LHC bounds at 13 TeV are shown by the black dashed
line from the ATLAS results [170]. The prospective bounds were estimated from
$h_{2}\to ZZ$ mode at the LHC at 300 (3000) fb-1 luminosity using an orange
dot-dashed (dotted) line from [172]. The prospective bounds from the combined
CLIC analysis at 380 GeV, 1.5 TeV, and 3 TeV are shown by the black dot-dashed
line.

We also superimpose the limits on the scalar mixing $\theta_{12}$ using Eq.
4.18 by considering two different choices of $\lambda_{H\chi}$ as $0.1$,
$0.05$ for each point P1 and P2. The scalar mixing $\sin^{2}\theta_{12}<0.001$
is respected by each constraint, as seen in fig 4.8.

### 4.5 Summary

We have considered a generic $U(1)_{X}$ extension of SM that accommodates a
pseudo-scalar DM candidate and neutrino mass generation mechanism. In the
model, we have three RHNs and two complex scalars $\Phi$ and $\chi$
additionally, and all are charged under the $U(1)_{X}$ gauge group. $U(1)_{X}$
gauge is broken by the vev of $\Phi$. $\chi$ field also gets vev $v_{\chi}$,
which results in a massive pseudo-scalar particle. The pseudo-scalar has
coupling with Higgs and neutral gauge bosons, and its phenomenology is
controlled by a few parameters such as $x_{H}$, $\kappa$, $m_{h_{i}}$,
$g^{\prime}_{1}$, $M_{Z^{\prime}}$, and $m_{A}$. We have found that pseudo-
scalars have sufficient parameter space, which is satisfied by the DM lifetime
constraint while remaining consistent. We found the allowed parameter space
using the relic density constraints, the invisible Higgs width constraint, and
other bounds. A direct detection check is also done in our model. The tree-
level DM-nucleon scattering amplitude vanishes in a non-relativistic limit,
but one loop contribution is finite. We have shown the one-loop contribution
to scattering cross-section against the experimental limit from XENON1T, LZ,
and neutrino floors from five different targets. In our model, the pseudo-
scalar DM has sufficient parameter space, which is consistent with several
theoretical and experimental constraints and can be tested at future colliders
and direct detection experiments.

##  5 Summary

This thesis examines several models of Weakly Interacting Massive Particle
(WIMP) dark matter and their implications from a phenomenological standpoint.
These models are particularly intriguing due to the stringent constraints of
direct detection experiments. In chapters 2 and 4, we explore models where the
pseudo-scalar properties serve as both a mediator and a dark matter candidate,
allowing them to evade the direct detection bounds. Additionally, in chapter
3, we delve into a two-component dark matter model and its relevant
phenomenological features. Here, we provide a concise overview of each of
these chapters.

Chapter 2 focuses on an ALP portal fermion dark matter model. This model is
minimal, consisting of three right-handed neutrinos (RHNs) and one pseudo-
scalar mediator known as ALP. The two RHNs mix with the three active
neutrinos, generating neutrino masses through Type I seesaw mechanism
(detailed in Appendix-C). The stability of the third RHN is ensured by a Z2
symmetry, making it a viable dark matter candidate. We then proceed to analyze
the constraints imposed by neutrino mixing and ALP couplings to Standard Model
(SM) particles. By incorporating these experimental bounds, we investigate the
parameter space that allows for fermion dark matter, taking into account relic
density, direct detection, and indirect detection constraints. Remarkably, the
model exhibits a significant parameter space that satisfies the relic density
constraint, and it also possesses detectable signatures for indirect searches
such as Fermi-LAT or HESS.

In chapter 3, we explore a model of dark matter consisting of a scalar and
fermion as components. This model is based on a generic U(1)X extension of the
Standard Model (SM). To ensure the consistency of the model, we introduce
three right-handed neutrinos, which necessitate the presence of the U(1)X
gauge symmetry. Additionally, we incorporate two new complex scalars into the
model. These scalars are singlets under the SM but carry charges under the
U(1)X gauge symmetry. One of the scalars breaks the U(1)X gauge symmetry,
while the other scalar is responsible for the scalar dark matter (DM).

In this model, we also include two Z2 symmetries. One of these symmetries
applies to the right-handed neutrinos, while the other symmetry applies to the
new scalar. As a result, both the scalars and fermions can potentially act as
DM candidates. Furthermore, the two remaining right-handed neutrinos mix with
the three active neutrinos, leading to the generation of neutrino masses
through a Type I seesaw mechanism.

We proceed to analyze various constraints that exist for this model. These
constraints include the Higgs mixing angle ($\theta_{12}$) and the mass of the
second Higgs boson ($m_{h_{2}}$), as well as the gauge coupling
($g^{\prime}_{1}$) and the mass of the $Z^{\prime}$ boson ($M_{Z}^{\prime}$).
By considering these experimental bounds, we investigate the parameter space
that is consistent with relic density bounds for both scalar and fermion DM,
taking into account scenarios where the two types of DM either interact or do
not interact with each other. Additionally, we perform a scan to identify
feasible parameter space that satisfies the direct detection bounds.

Finally, we explore a few more U(1) models that allow for parameter space
accommodating the two-component DM. This model offers a rich phenomenology and
provides opportunities for testing and collaboration with colleagues.

In chapter 4, we explore a dark matter model with a pseudo-scalar in a generic
U(1)X extension of the SM. This model shares similar particle and symmetry
content with the two-component model discussed in chapter 3, but it does not
possess Z2 symmetries. The U(1)X gauge symmetry is broken by one complex
scalar, while the second scalar is responsible for the pseudo-scalar dark
matter. The mixing of three right-handed neutrinos (RHNs) with three active
neutrinos generates neutrino masses through a Type I seesaw mechanism. We then
analyze various theoretical and experimental constraints, including the Higgs
invisible width and the Planck bound, in the corresponding section. By
imposing these constraints, we investigate the parameter space that satisfies
the lifetime bound for the pseudo-scalar in different U(1) models.
Additionally, we examine the relic constraint on the model by considering the
lifetime bound in various U(1) models. Our findings reveal that the model
possesses a sufficient parameter space that satisfies both the lifetime and
relic bounds. Subsequently, we assess the feasible parameter space by
considering the direct detection bounds when the neutrino floor is caused by
multiple target materials such as helium (He), silver (Ag), xenon (Xe),
germanium (Ge), sodium iodide (NaI), and calcium tungstate (CaWO4). Our
analysis demonstrates that the model exhibits a significant allowed space
according to the direct bounds. Finally, we constrain the higgs mixing angle
$(\theta_{12})$-higgs mass ($m_{h_{2}}$) plane based on our direct detection
study. The model successfully passes all the bounds, both those related to
dark matter and non-dark matter, and it can be experimentally tested at
colliders or through direct searches.

We have previously discussed the WIMP DM models within the framework of the
ALP portal and U(1)X extension. These models have been thoroughly analyzed,
taking into account various theoretical and experimental constraints. The
model we have discussed addresses certain issues associated with WIMP models,
particularly the lack of direct detection of WIMP DM. Additionally, our models
incorporate neutrino physics, including oscillation and mass mechanisms. These
bottom-up models offer a fascinating range of phenomenology that can be
explored through collider-based experiments. However, it is worth noting that
the standard DM mass in these WIMP DM models is limited to the range of a few
GeV to TeV, which restricts the scope of our explored DM scenarios. There are
other intriguing mass ranges for DM, such as ultralight dark matter, sub-GeV
DM, and primordial black holes, among others. I intend to investigate these
scenarios in future research. Furthermore, apart from considering different DM
candidates, exploring the production mechanisms of DM is also an intriguing
avenue for further exploration and study.

##  Appendix A Standard model

The SM [199, 200] is a renormalizable, Lorentz-invariant quantum field theory
based on the gauge group $SU(3)_{c}\otimes SU(2)_{L}\otimes U(1)_{Y}$, which
contains the fundamental set of particles, the leptons, quarks, gauge bosons,
and the Higgs.

|  SU(3)C |  SU(2)L |  U(1)Y
---|---|---|---
$q^{i}_{L}$ | 3 | 2 | $\frac{1}{6}$
$u^{i}_{R}$ | 3 | 1 | $\frac{2}{3}$
$d^{i}_{R}$ | 3 | 1 | $-\frac{1}{3}$
$\ell^{i}_{L}$ | 1 | 2 | $-\frac{1}{2}$
$e^{i}_{R}$ | 1 | 1 | $-1$
$H$ | 1 | 2 | $\frac{1}{2}$

Gauge bosons | Force
---|---
$G_{\mu}^{a}$ | Strong
$W_{\mu}^{\pm},\,Z_{\mu}^{0}$ | Weak
$A_{\mu}$ | Electromagnetic

Table A.1: The SM particle contents are shown in the table and $i=1,2,3$ is
the generation index.

The SM lagrangian can be written as

$\displaystyle\mathcal{L}_{\text{SM}}=\mathcal{L}_{\text{kinetic}}+\mathcal{L}_{\text{gauge}}+\mathcal{L}_{\text{Higgs}}+\mathcal{L}_{\text{Yukawa}},$
(A.1)

where,

$\displaystyle\mathcal{L}_{\text{kinetic}}=\mathcal{L}_{\text{kinetic}}^{\text{fermion}}+\mathcal{L}_{\text{kinetic}}^{\text{gauge}},$
(A.2)

The kinetic term for the fermions (leptons and quarks) is given by

$\displaystyle\mathcal{L}_{\text{kinetic}}^{\text{fermion}}$ $\displaystyle=\
\bar{\ell_{L}}\gamma^{\mu}D^{\mu}\ell_{L}+\overline{e_{R}}\gamma^{\mu}D_{\mu}e_{R}+\overline{q_{L}}\gamma^{\mu}D_{\mu}q_{L}$
$\displaystyle+\overline{u_{R}}\gamma^{\mu}D_{\mu}u_{R}+\overline{d_{R}}\gamma^{\mu}D_{\mu}d_{R}$
(A.3)

where $D_{\mu}$ is the covariant derivative. The kinetic term for the gauge
particles can be written as

$\displaystyle\mathcal{L}^{gauge}_{kinetic}=-\frac{1}{4}W_{\mu\nu}^{a}W^{a^{\mu\nu}}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}G^{a}_{\mu\nu}G^{a\mu\nu},$
(A.4)

In SM, the masses of all the fermions and $SU(2)_{L}$ gauge bosons are
generated through the spontaneous symmetry breaking (SSB). The Lagrangian for
the Higgs sector is given by

$\displaystyle\mathcal{L}_{\text{Higgs}}=\left(D_{\mu}H\right)\left(D^{\mu}H\right)-\left(-\mu^{2}H^{\dagger}H+\lambda\left(H^{\dagger}H\right)^{2}\right)$
(A.5)

The Yukawa interactions for the leptons and quarks are given by

$\displaystyle\mathcal{L}_{\text{Yukawa}}$
$\displaystyle=Y_{e}\overline{\ell_{L}}He_{R}+Y_{u}\overline{q_{L}}\tilde{H}u_{R}+Y_{d}\overline{q_{L}}Hd_{R}+H.c.$
(A.6)

##  Appendix B Friedmann equations

Our universe at a large scale can be described well if we assume isotropy and
homogeneity of space. The Friedmann-Lemaitre-Robertson-Walker (FLRW) metric
hold these assumptions, which is given by,

$\displaystyle ds^{2}=dt^{2}-a^{2}(t)\left(\frac{dr^{2}}{1-\kappa
r^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})\right)$ (B.1)

here r,$\theta$,$\phi$, t are comoving coordinates, a(t) is the scale factor.
$\kappa$ is the curvature scalar and can take values +1,0,-1 correspond to
positive, flat and negatively curved Universe respectively. Friedmann
equations can be derived using Einstein’s equation,

$\displaystyle\mathcal{R}_{\mu\nu}-\frac{1}{2}\mathcal{R}g_{\mu\nu}=8\pi
GT_{\mu\nu}+\Lambda g_{\mu\nu}$ (B.2)

here $\mathcal{R}$, $\mathcal{R}_{\mu\nu}$, $T_{\mu\nu}$ and $\Lambda$ are
Ricci scalar, Ricci tensor, energy-momentum tensor and cosmological constant
respectively. Assuming the Universe is composed of fluid, the energy-momentum
tensor can be written as

$\displaystyle T_{\mu\nu}=-pg_{\mu\nu}+(p+\rho)u_{\mu}u_{\nu}$ (B.3)

here p and $\rho$ are the pressure and energy density of the fluid
respectively whereas $u_{\mu}$ is the velocity vector in comoving coordinates.
One can now derive the Friedmann equations as

$\displaystyle\frac{\dot{a}^{2}}{a^{2}}=\frac{8\pi
G\rho}{3}-\frac{\kappa}{a^{2}}+\frac{\Lambda}{3}$ (B.4)
$\displaystyle\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3p)+\frac{\Lambda}{3}$
(B.5)

Using the above two equations, one can derive the density evolution equation,

$\displaystyle\dot{\rho}=-3H(\rho+p)$ (B.6)

where H is the Hubble parameter.

##  Appendix C Type I seasaw mechanism

In the type I seesaw mechanism, we have right-handed neutrinos $\nu_{R}$,
which can have both Dirac and Majorana mass terms as follow [71, 72, 73, 74],

$\displaystyle\mathcal{L}^{\nu}_{mass}=m_{D}\bar{\nu}_{R}\nu_{L}+\frac{1}{2}m_{L}\bar{\nu}^{c}_{L}\nu_{L}+\frac{1}{2}M_{R}\bar{\nu}^{c}_{R}\nu_{R}+h.c.$
(C.1)

here subscript c stands for charge conjugation. Now one can write the above
expression as follows

$\displaystyle\mathcal{L}^{\nu}_{mass}=\begin{pmatrix}\overline{\nu^{c}_{L}}&\overline{\nu_{R}}\\\
\end{pmatrix}\begin{pmatrix}m_{L}&m_{D}\\\ m_{D}^{T}&M_{R}\\\
\end{pmatrix}\begin{pmatrix}\nu_{L}\\\ \nu_{R}\\\ \end{pmatrix}$ (C.2)

One can diagonalize the above mass matrix and can get mass eigenstates
corresponding to

$\displaystyle
m_{1,2}=\frac{1}{2}|(m_{L}+m_{R})\pm\sqrt{(m_{L}-m_{R})^{2}+4m^{2}_{D}}|$
(C.3)

In the limit $m_{D}<<M_{R}$, mass eigen values simplifies to

$\displaystyle m_{1}\approx\frac{m^{2}_{D}}{M_{R}}$ (C.4) $\displaystyle
m_{2}\approx M_{R}$ (C.5)

This naturally explains the smallness of neutrino masses.

##  Appendix D Feynman diagrams in two-component DM model

The relevant Feynman diagram for relic density analysis of scalar and fermion
DM are shown in Figs. D.1 and D.2, respectively.

Figure D.1: Annihilation and coannihilation tree-level Feynman diagrams
contributing to the relic abundance of $\chi_{R}$. Figure D.2: Annihilation
and coannihilation tree-level Feynman diagrams contributing to the relic
abundance of $N_{R}^{3}$.

## Bibliography

  * [1] Y. Sofue, Y. Tutui, M. Honma, A. Tomita, T. Takamiya, J. Koda, and Y. Takeda, Central rotation curves of spiral galaxies, The Astrophysical Journal 523 no. 1, (Sep, 1999) 136–146. https://doi.org/10.1086/307731.
  * [2] D. Clowe, M. Bradač, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones, and D. Zaritsky, A direct empirical proof of the existence of dark matter, The Astrophysical Journal 648 no. 2, (Aug., 2006) L109–L113. http://dx.doi.org/10.1086/508162.
  * [3] K. Garrett and G. Duda, Dark matter: A primer, Advances in Astronomy 2011 (2011) 1–22. https://doi.org/10.1155%2F2011%2F968283.
  * [4] J. Cooley, Overview of non-liquid noble direct detection dark matter experiments, Physics of the Dark Universe 4 (Sep, 2014) 92–97. https://doi.org/10.1016%2Fj.dark.2014.10.005.
  * [5] Limits to dark matter annihilation cross-section from a combined analysis of MAGIC and fermi-LAT observations of dwarf satellite galaxies, Feb, 2016\.  https://doi.org/10.1088%2F1475-7516%2F2016%2F02%2F039.
  * [6] G. Elor, N. L. Rodd, T. R. Slatyer, and W. Xue, Model-Independent Indirect Detection Constraints on Hidden Sector Dark Matter, JCAP 06 (2016) 024, arXiv:1511.08787 [hep-ph].
  * [7] HESS Collaboration, H. Abdallah et al., Search for $\gamma$-Ray Line Signals from Dark Matter Annihilations in the Inner Galactic Halo from 10 Years of Observations with H.E.S.S., Phys. Rev. Lett. 120 no. 20, (2018) 201101, arXiv:1805.05741 [astro-ph.HE].
  * [8] Fermi-LAT Collaboration, M. Ackermann et al., Search for Gamma-ray Spectral Lines with the Fermi Large Area Telescope and Dark Matter Implications, Phys. Rev. D 88 (2013) 082002, arXiv:1305.5597 [astro-ph.HE].
  * [9] ALEPH, DELPHI, L3, OPAL, LEP Electroweak Collaboration, S. Schael et al., Electroweak Measurements in Electron-Positron Collisions at W-Boson-Pair Energies at LEP, Phys. Rept. 532 (2013) 119–244, arXiv:1302.3415 [hep-ex].
  * [10] ATLAS Collaboration, G. Aad et al., Search for high-mass dilepton resonances using 139 fb-1 of $pp$ collision data collected at $\sqrt{s}=$13 TeV with the ATLAS detector, Phys. Lett. B 796 (2019) 68–87, arXiv:1903.06248 [hep-ex].
  * [11] CMS Collaboration, Search for a narrow resonance in high-mass dilepton final states in proton-proton collisions using 140$~{}\mathrm{fb}^{-1}$ of data at $\sqrt{s}=13~{}\mathrm{TeV}$, 2019.
  * [12] ATLAS Collaboration, Search for New Phenomena in Dijet Events using 139 fb-1 of $pp$ collisions at $\sqrt{s}$ = 13TeV collected with the ATLAS Detector, 3, 2019.
  * [13] CMS Collaboration, A. M. Sirunyan et al., Search for narrow and broad dijet resonances in proton-proton collisions at $\sqrt{s}=13$ TeV and constraints on dark matter mediators and other new particles, JHEP 08 (2018) 130, arXiv:1806.00843 [hep-ex].
  * [14] A. Das, P. S. B. Dev, Y. Hosotani, and S. Mandal, Probing the minimal $U(1)_{X}$ model at future electron-positron colliders via the fermion pair-production channel, arXiv:2104.10902 [hep-ph].
  * [15] J. de Blas et al., The CLIC Potential for New Physics, arXiv:1812.02093 [hep-ph].
  * [16] LEP Working Group for Higgs boson searches, ALEPH, DELPHI, L3, OPAL Collaboration, R. Barate et al., Search for the standard model Higgs boson at LEP, Phys. Lett. B 565 (2003) 61–75, arXiv:hep-ex/0306033.
  * [17] Y. Wang, M. Berggren, and J. List, ILD Benchmark: Search for Extra Scalars Produced in Association with a $Z$ boson at $\sqrt{s}=500$ GeV, arXiv:2005.06265 [hep-ex].
  * [18] XENON Collaboration, E. Aprile et al., Dark Matter Search Results from a One Ton-Year Exposure of XENON1T, Phys. Rev. Lett. 121 no. 11, (2018) 111302, arXiv:1805.12562 [astro-ph.CO].
  * [19] J. Billard, L. Strigari, and E. Figueroa-Feliciano, Implication of neutrino backgrounds on the reach of next generation dark matter direct detection experiments, Phys. Rev. D 89 no. 2, (2014) 023524, arXiv:1307.5458 [hep-ph].
  * [20] PandaX Collaboration, H. Zhang et al., Dark matter direct search sensitivity of the PandaX-4T experiment, Sci. China Phys. Mech. Astron. 62 no. 3, (2019) 31011, arXiv:1806.02229 [physics.ins-det].
  * [21] LUX-ZEPLIN Collaboration, D. S. Akerib et al., Projected WIMP sensitivity of the LUX-ZEPLIN dark matter experiment, Phys. Rev. D 101 no. 5, (2020) 052002, arXiv:1802.06039 [astro-ph.IM].
  * [22] LZ Collaboration, J. Aalbers et al., First Dark Matter Search Results from the LUX-ZEPLIN (LZ) Experiment, Phys. Rev. Lett. 131 no. 4, (2023) 041002, arXiv:2207.03764 [hep-ex].
  * [23] XENON Collaboration, E. Aprile et al., Projected WIMP sensitivity of the XENONnT dark matter experiment, JCAP 11 (2020) 031, arXiv:2007.08796 [physics.ins-det].
  * [24] E. Aprile and K. e. Abe, First dark matter search with nuclear recoils from the xenonnt experiment, Physical Review Letters 131 no. 4, (July, 2023) . http://dx.doi.org/10.1103/PhysRevLett.131.041003.
  * [25] GADMC Collaboration, C. Galbiati et al., Future Dark Matter Searches with Low-Radioactivity Argon, 2018.  https://indico.cern.ch/event/765096/contributions/3295671/attachments/1785196/2906164/DarkSide-Argo_ESPP_Dec_17_2017.pdf.
  * [26] DARWIN Collaboration, J. Aalbers et al., DARWIN: towards the ultimate dark matter detector, JCAP 11 (2016) 017, arXiv:1606.07001 [astro-ph.IM].
  * [27] J. Billard et al., Direct Detection of Dark Matter – APPEC Committee Report, arXiv:2104.07634 [hep-ex].
  * [28] M. G. Baring, T. Ghosh, F. S. Queiroz, and K. Sinha, New limits on the dark matter lifetime from dwarf spheroidal galaxies using fermi-LAT, Physical Review D 93 no. 10, (May, 2016) . https://doi.org/10.1103%2Fphysrevd.93.103009.
  * [29] C. A. J. O’Hare, New definition of the neutrino floor for direct dark matter searches, Physical Review Letters 127 no. 25, (Dec, 2021) . https://doi.org/10.1103%2Fphysrevlett.127.251802.
  * [30] A. Das, S. Gola, S. Mandal, and N. Sinha, Two-component scalar and fermionic dark matter candidates in a generic U(1)X model, Phys. Lett. B 829 (2022) 137117, arXiv:2202.01443 [hep-ph].
  * [31] S. Profumo, K. Sigurdson, and L. Ubaldi, Can we discover multi-component WIMP dark matter?, JCAP 12 (2009) 016, arXiv:0907.4374 [hep-ph].
  * [32] M. Lisanti, Lectures on Dark Matter Physics,  in Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, pp. 399–446.  2017\.  arXiv:1603.03797 [hep-ph].
  * [33] E. W. Kolb and M. S. Turner, The Early Universe, 1990.
  * [34] M. Bartelmann and P. Schneider, Weak gravitational lensing, Phys. Rept. 340 (2001) 291–472, arXiv:astro-ph/9912508.
  * [35] D. Clowe, A. Gonzalez, and M. Markevitch, Weak lensing mass reconstruction of the interacting cluster 1E0657-558: Direct evidence for the existence of dark matter, Astrophys. J. 604 (2004) 596–603, arXiv:astro-ph/0312273.
  * [36] D. Harvey, R. Massey, T. Kitching, A. Taylor, and E. Tittley, The non-gravitational interactions of dark matter in colliding galaxy clusters, Science 347 (2015) 1462–1465, arXiv:1503.07675 [astro-ph.CO].
  * [37] WMAP Collaboration, G. Hinshaw et al., Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results, Astrophys. J. Suppl. 208 (2013) 19, arXiv:1212.5226 [astro-ph.CO].
  * [38] Planck Collaboration, P. A. R. Ade et al., Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys. 594 (2016) A13, arXiv:1502.01589 [astro-ph.CO].
  * [39] Planck Collaboration, N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6, arXiv:1807.06209 [astro-ph.CO].
  * [40] W. T. Kelvin, The index, p. [695]-703, was issued separately with cover-title: Index to Lord Kelvin’s volume of Baltimore lectures. Cambridge, Printed at the University press, 1905, Cambridge, Printed at the University press (1904) .
  * [41] J. H. Oort, The force exerted by the stellar system in the direction perpendicular to the galactic plane and some related problems, bain 6 (Aug., 1932) 249.
  * [42] F. Zwicky, On the Masses of Nebulae and of Clusters of Nebulae, apj 86 (Oct., 1937) 217.
  * [43] V. C. Rubin, Dark matter in spiral galaxies, Scientific American 248 no. 6, (1983) 96–109. http://www.jstor.org/stable/24968923.
  * [44] D. Walsh, R. F. Carswell, and R. J. Weymann, 0957+561 A, B: twin quasistellar objects or gravitational lens?, nat 279 (May, 1979) 381–384.
  * [45] A. A. Penzias and R. W. Wilson, A Measurement of Excess Antenna Temperature at 4080 Mc/s., apj 142 (July, 1965) 419–421.
  * [46] G. Bertone and D. Hooper, History of dark matter, Oct, 2018.  https://doi.org/10.1103%2Frevmodphys.90.045002.
  * [47] C. A. Argüelles, K. J. Kelly, and V. M. Muñoz, Millicharged particles from the heavens: single- and multiple-scattering signatures, Journal of High Energy Physics 2021 no. 11, (Nov., 2021) . http://dx.doi.org/10.1007/JHEP11(2021)099.
  * [48] S. Profumo, L. Giani, and O. F. Piattella, An Introduction to Particle Dark Matter, Universe 5 no. 10, (2019) 213, arXiv:1910.05610 [hep-ph].
  * [49] Planck Collaboration, N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6, arXiv:1807.06209 [astro-ph.CO].
  * [50] G. Danby, J.-M. Gaillard, K. Goulianos, L. M. Lederman, N. Mistry, M. Schwartz, and J. Steinberger, Observation of high-energy neutrino reactions and the existence of two kinds of neutrinos, Phys. Rev. Lett. 9 (Jul, 1962) 36–44. https://link.aps.org/doi/10.1103/PhysRevLett.9.36.
  * [51] T. Kajita, Nobel Lecture: Discovery of atmospheric neutrino oscillations, Rev. Mod. Phys. 88 no. 3, (2016) 030501.
  * [52] A. de Gouvêa, Neutrino mass models, Annual Review of Nuclear and Particle Science 66 no. 1, (2016) 197–217, https://doi.org/10.1146/annurev-nucl-102115-044600. https://doi.org/10.1146/annurev-nucl-102115-044600.
  * [53] P. F. de Salas, D. V. Forero, S. Gariazzo, P. Martínez-Miravé, O. Mena, C. A. Ternes, M. Tórtola, and J. W. F. Valle, 2020 global reassessment of the neutrino oscillation picture, JHEP 02 (2021) 071, arXiv:2006.11237 [hep-ph].
  * [54] M. Lattanzi and M. Gerbino, Status of neutrino properties and future prospects - Cosmological and astrophysical constraints, Front. in Phys. 5 (2018) 70, arXiv:1712.07109 [astro-ph.CO].
  * [55] V. De Luca, A. Mitridate, M. Redi, J. Smirnov, and A. Strumia, Colored dark matter, Physical Review D 97 no. 11, (June, 2018) . http://dx.doi.org/10.1103/PhysRevD.97.115024.
  * [56] S.-M. Choi, J. Kim, P. Ko, and J. Li, A multi-component SIMP model with $U(1)_{X}\to Z_{2}\times Z_{3}$, JHEP 09 (2021) 028, arXiv:2103.05956 [hep-ph].
  * [57] S.-Y. Ho, P. Ko, and C.-T. Lu, Scalar and Fermion Two-component SIMP Dark Matter with an Accidental $\mathbb{Z}_{4}$ Symmetry, arXiv:2201.06856 [hep-ph].
  * [58] G. Bélanger, F. Boudjema, A. Goudelis, A. Pukhov, and B. Zaldivar, micrOMEGAs5.0 : Freeze-in, Comput. Phys. Commun. 231 (2018) 173–186, arXiv:1801.03509 [hep-ph].
  * [59] T. Bringmann, P. F. Depta, M. Hufnagel, J. T. Ruderman, and K. Schmidt-Hoberg, Dark matter from exponential growth, Physical Review Letters 127 no. 19, (Nov., 2021) . http://dx.doi.org/10.1103/PhysRevLett.127.191802.
  * [60] L. Di Luzio, M. Giannotti, E. Nardi, and L. Visinelli, The landscape of qcd axion models, Physics Reports 870 (July, 2020) 1–117. http://dx.doi.org/10.1016/j.physrep.2020.06.002.
  * [61] E. G. M. Ferreira, Ultra-light dark matter, The Astronomy and Astrophysics Review 29 no. 1, (Sept., 2021) . http://dx.doi.org/10.1007/s00159-021-00135-6.
  * [62] A. Boyarsky, M. Drewes, T. Lasserre, S. Mertens, and O. Ruchayskiy, Sterile neutrino dark matter, Progress in Particle and Nuclear Physics 104 (Jan., 2019) 1–45. http://dx.doi.org/10.1016/j.ppnp.2018.07.004.
  * [63] D. Hooper and S. Profumo, Dark matter and collider phenomenology of universal extra dimensions, Physics Reports 453 no. 2–4, (Dec., 2007) 29–115. http://dx.doi.org/10.1016/j.physrep.2007.09.003.
  * [64] C. e. Alcock, Eros and macho combined limits on planetary-mass dark matter in the galactic halo, The Astrophysical Journal 499 no. 1, (May, 1998) L9–L12. http://dx.doi.org/10.1086/311355.
  * [65] LUX Collaboration, D. S. Akerib et al., Results from a search for dark matter in the complete LUX exposure, Phys. Rev. Lett. 118 no. 2, (2017) 021303, arXiv:1608.07648 [astro-ph.CO].
  * [66] XENON Collaboration, E. Aprile et al., Dark Matter Search Results from a One Ton-Year Exposure of XENON1T, Phys. Rev. Lett. 121 no. 11, (2018) 111302, arXiv:1805.12562 [astro-ph.CO].
  * [67] IceCube Collaboration Collaboration, A. et al., Limits on a muon flux from neutralino annihilations in the sun with the icecube 22-string detector, Phys. Rev. Lett. 102 (May, 2009) 201302. https://link.aps.org/doi/10.1103/PhysRevLett.102.201302.
  * [68] M. A. et al., Antares: The first undersea neutrino telescope, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 656 no. 1, (2011) 11–38. https://www.sciencedirect.com/science/article/pii/S0168900211013994.
  * [69] PAMELA Collaboration, O. Adriani et al., An anomalous positron abundance in cosmic rays with energies 1.5-100 GeV, Nature 458 (2009) 607–609, arXiv:0810.4995 [astro-ph].
  * [70] M. Bauer and T. Plehn, Yet Another Introduction to Dark Matter: The Particle Physics Approach, vol. 959 of Lecture Notes in Physics.  Springer, 2019.  arXiv:1705.01987 [hep-ph].
  * [71] P. Minkowski, $\mu\to e\gamma$ at a Rate of One Out of $10^{9}$ Muon Decays?, Phys. Lett. B 67 (1977) 421–428.
  * [72] J. Schechter and J. W. F. Valle, Neutrino Masses in SU(2) x U(1) Theories, Phys. Rev. D 22 (1980) 2227.
  * [73] R. N. Mohapatra and G. Senjanovic, Neutrino Mass and Spontaneous Parity Nonconservation, Phys. Rev. Lett. 44 (1980) 912.
  * [74] J. Schechter and J. W. F. Valle, Neutrino Decay and Spontaneous Violation of Lepton Number, Phys. Rev. D 25 (1982) 774.
  * [75] I. Dorsner and P. Fileviez Perez, Upper Bound on the Mass of the Type III Seesaw Triplet in an SU(5) Model, JHEP 06 (2007) 029, arXiv:hep-ph/0612216.
  * [76] B. Bajc, M. Nemevsek, and G. Senjanovic, Probing seesaw at LHC, Phys. Rev. D 76 (2007) 055011, arXiv:hep-ph/0703080.
  * [77] A. de Gouvea, J. Jenkins, and N. Vasudevan, Neutrino Phenomenology of Very Low-Energy Seesaws, Phys. Rev. D 75 (2007) 013003, arXiv:hep-ph/0608147.
  * [78] A. de Gouvea, GeV seesaw, accidentally small neutrino masses, and Higgs decays to neutrinos, arXiv:0706.1732 [hep-ph].
  * [79] A. Abada and M. Lucente, Looking for the minimal inverse seesaw realisation, Nucl. Phys. B 885 (2014) 651–678, arXiv:1401.1507 [hep-ph].
  * [80] D. Borah and A. Dasgupta, Common origin of neutrino mass, dark matter and dirac leptogenesis, Journal of Cosmology and Astroparticle Physics 2016 no. 12, (Dec, 2016) 034–034. https://doi.org/10.1088/1475-7516/2016/12/034.
  * [81] P. Das and M. K. Das, Phenomenology of $keV$ sterile neutrino in minimal extended seesaw, Int. J. Mod. Phys. A 35 no. 22, (2020) 2050125, arXiv:1908.08417 [hep-ph].
  * [82] A. Merle, keV sterile neutrino Dark Matter, PoS NOW2016 (2017) 082, arXiv:1702.08430 [hep-ph].
  * [83] M. Drewes et al., A White Paper on keV Sterile Neutrino Dark Matter, JCAP 01 (2017) 025, arXiv:1602.04816 [hep-ph].
  * [84] A. Abada, G. Arcadi, and M. Lucente, Dark Matter in the minimal Inverse Seesaw mechanism, JCAP 10 (2014) 001, arXiv:1406.6556 [hep-ph].
  * [85] R. D. Peccei, QCD, strong CP and axions, J. Korean Phys. Soc. 29 (1996) S199–S208, arXiv:hep-ph/9606475.
  * [86] R. D. Peccei, The Strong CP problem and axions, Lect. Notes Phys. 741 (2008) 3–17, arXiv:hep-ph/0607268.
  * [87] J. E. Kim and G. Carosi, Axions and the Strong CP Problem, Rev. Mod. Phys. 82 (2010) 557–602, arXiv:0807.3125 [hep-ph]. [Erratum: Rev.Mod.Phys. 91, 049902 (2019)].
  * [88] A. Hook, TASI Lectures on the Strong CP Problem and Axions, PoS TASI2018 (2019) 004, arXiv:1812.02669 [hep-ph].
  * [89] M. P. Lombardo and A. Trunin, Topology and axions in QCD, Int. J. Mod. Phys. A 35 no. 20, (2020) 2030010, arXiv:2005.06547 [hep-lat].
  * [90] S. Weinberg, A New Light Boson?, Phys. Rev. Lett. 40 (1978) 223–226.
  * [91] F. Wilczek, Problem of Strong $P$ and $T$ Invariance in the Presence of Instantons, Phys. Rev. Lett. 40 (1978) 279–282.
  * [92] Z. G. Berezhiani and M. Y. Khlopov, Cosmology of Spontaneously Broken Gauge Family Symmetry, Z. Phys. C 49 (1991) 73–78.
  * [93] J. E. Kim, Weak Interaction Singlet and Strong CP Invariance, Phys. Rev. Lett. 43 (1979) 103.
  * [94] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Can Confinement Ensure Natural CP Invariance of Strong Interactions?, Nucl. Phys. B 166 (1980) 493–506.
  * [95] M. Dine, W. Fischler, and M. Srednicki, A Simple Solution to the Strong CP Problem with a Harmless Axion, Phys. Lett. B 104 (1981) 199–202.
  * [96] A. Hook, S. Kumar, Z. Liu, and R. Sundrum, High Quality QCD Axion and the LHC, Phys. Rev. Lett. 124 no. 22, (2020) 221801, arXiv:1911.12364 [hep-ph].
  * [97] K. J. Kelly, S. Kumar, and Z. Liu, Heavy Axion Opportunities at the DUNE Near Detector, arXiv:2011.05995 [hep-ph].
  * [98] H. Georgi, D. B. Kaplan, and L. Randall, Manifesting the Invisible Axion at Low-energies, Phys. Lett. B 169 (1986) 73–78.
  * [99] I. Brivio, M. Gavela, L. Merlo, K. Mimasu, J. No, R. del Rey, and V. Sanz, ALPs Effective Field Theory and Collider Signatures, Eur. Phys. J. C 77 no. 8, (2017) 572, arXiv:1701.05379 [hep-ph].
  * [100] A. Salvio, A. Strumia, and W. Xue, Thermal axion production, JCAP 01 (2014) 011, arXiv:1310.6982 [hep-ph].
  * [101] Y. Hochberg, E. Kuflik, R. Mcgehee, H. Murayama, and K. Schutz, Strongly interacting massive particles through the axion portal, Phys. Rev. D 98 no. 11, (2018) 115031, arXiv:1806.10139 [hep-ph].
  * [102] K. Mimasu and V. Sanz, ALPs at Colliders, JHEP 06 (2015) 173, arXiv:1409.4792 [hep-ph].
  * [103] J. Jaeckel and M. Spannowsky, Probing MeV to 90 GeV axion-like particles with LEP and LHC, Phys. Lett. B 753 (2016) 482–487, arXiv:1509.00476 [hep-ph].
  * [104] A. Alves, A. G. Dias, and K. Sinha, Diphotons at the $Z$-pole in Models of the 750 GeV Resonance Decaying to Axion-Like Particles, JHEP 08 (2016) 060, arXiv:1606.06375 [hep-ph].
  * [105] M. J. Dolan, F. Kahlhoefer, C. McCabe, and K. Schmidt-Hoberg, A taste of dark matter: Flavour constraints on pseudoscalar mediators, JHEP 03 (2015) 171, arXiv:1412.5174 [hep-ph]. [Erratum: JHEP 07, 103 (2015)].
  * [106] E. Izaguirre, T. Lin, and B. Shuve, Searching for Axionlike Particles in Flavor-Changing Neutral Current Processes, Phys. Rev. Lett. 118 no. 11, (2017) 111802, arXiv:1611.09355 [hep-ph].
  * [107] K. Choi, K. Kang, and J. E. Kim, Effects of $\eta^{\prime}$ in low-energy axion physics, Physics Letters B 181 no. 1, (1986) 145–149. https://www.sciencedirect.com/science/article/pii/0370269386912736.
  * [108] A. Salvio and S. Scollo, Axion-Sterile-Neutrino Dark Matter, arXiv:2104.01334 [hep-ph].
  * [109] A. Salvio, A Simple Motivated Completion of the Standard Model below the Planck Scale: Axions and Right-Handed Neutrinos, Phys. Lett. B 743 (2015) 428–434, arXiv:1501.03781 [hep-ph].
  * [110] A. Alves, A. G. Dias, and D. D. Lopes, Probing alp-sterile neutrino couplings at the lhc, arXiv:1911.12394 [hep-ph].
  * [111] A. Atre, T. Han, S. Pascoli, and B. Zhang, The Search for Heavy Majorana Neutrinos, JHEP 05 (2009) 030, arXiv:0901.3589 [hep-ph].
  * [112] Particle Data Group Collaboration, K. A. Olive et al., Review of Particle Physics, Chin. Phys. C 38 (2014) 090001.
  * [113] N. Vinyoles, A. Serenelli, F. L. Villante, S. Basu, J. Redondo, and J. Isern, New axion and hidden photon constraints from a solar data global fit, JCAP 10 (2015) 015, arXiv:1501.01639 [astro-ph.SR].
  * [114] G. G. Raffelt, Astrophysical axion bounds, Lect. Notes Phys. 741 (2008) 51–71, arXiv:hep-ph/0611350.
  * [115] A. Friedland, M. Giannotti, and M. Wise, Constraining the Axion-Photon Coupling with Massive Stars, Phys. Rev. Lett. 110 no. 6, (2013) 061101, arXiv:1210.1271 [hep-ph].
  * [116] A. Ayala, I. Domínguez, M. Giannotti, A. Mirizzi, and O. Straniero, Revisiting the bound on axion-photon coupling from Globular Clusters, Phys. Rev. Lett. 113 no. 19, (2014) 191302, arXiv:1406.6053 [astro-ph.SR].
  * [117] CMS Collaboration, V. Khachatryan et al., Search for dark matter, extra dimensions, and unparticles in monojet events in proton–proton collisions at $\sqrt{s}=8$ TeV, Eur. Phys. J. C 75 no. 5, (2015) 235, arXiv:1408.3583 [hep-ex].
  * [118] ATLAS Collaboration, G. Aad et al., Search for new phenomena in final states with an energetic jet and large missing transverse momentum in pp collisions at $\sqrt{s}=$8 TeV with the ATLAS detector, Eur. Phys. J. C 75 no. 7, (2015) 299, arXiv:1502.01518 [hep-ex]. [Erratum: Eur.Phys.J.C 75, 408 (2015)].
  * [119] G. Krnjaic, Probing Light Thermal Dark-Matter With a Higgs Portal Mediator, Phys. Rev. D 94 no. 7, (2016) 073009, arXiv:1512.04119 [hep-ph].
  * [120] J. D. Clarke, R. Foot, and R. R. Volkas, Phenomenology of a very light scalar (100 MeV $<$ $m_{h}$ $<$ 10 GeV) mixing with the SM Higgs, JHEP 02 (2014) 123, arXiv:1310.8042 [hep-ph].
  * [121] XENON100 Collaboration, E. Aprile et al., First Axion Results from the XENON100 Experiment, Phys. Rev. D 90 no. 6, (2014) 062009, arXiv:1404.1455 [astro-ph.CO]. [Erratum: Phys.Rev.D 95, 029904 (2017)].
  * [122] N. Viaux, M. Catelan, P. B. Stetson, G. Raffelt, J. Redondo, A. A. R. Valcarce, and A. Weiss, Neutrino and axion bounds from the globular cluster M5 (NGC 5904), Phys. Rev. Lett. 111 (2013) 231301, arXiv:1311.1669 [astro-ph.SR].
  * [123] O. Rodríguez-Tzompantzi, Conserved laws and dynamical structure of axions coupled to photons, Int. J. Mod. Phys. A 36 no. 33, (2021) 2150259, arXiv:2001.07101 [hep-th].
  * [124] CAST Collaboration, V. Anastassopoulos et al., New CAST Limit on the Axion-Photon Interaction, Nature Phys. 13 (2017) 584–590, arXiv:1705.02290 [hep-ex].
  * [125] M. Bauer, M. Heiles, M. Neubert, and A. Thamm, Axion-Like Particles at Future Colliders, Eur. Phys. J. C 79 no. 1, (2019) 74, arXiv:1808.10323 [hep-ph].
  * [126] N. Vinyoles, A. Serenelli, F. L. Villante, S. Basu, J. Redondo, and J. Isern, New axion and hidden photon constraints from a solar data global fit, Journal of Cosmology and Astroparticle Physics 2015 no. 10, (2015) 015.
  * [127] BaBar Collaboration, J. P. Lees et al., Search for an Axionlike Particle in $B$ Meson Decays, Phys. Rev. Lett. 128 no. 13, (2022) 131802, arXiv:2111.01800 [hep-ex].
  * [128] E787 Collaboration, S. Adler et al., Further search for the decay K+ —$>$ pi+ nu anti-nu in the momentum region P $<$ 195-MeV/c, Phys. Rev. D 70 (2004) 037102, arXiv:hep-ex/0403034.
  * [129] CHARM Collaboration, F. Bergsma et al., Search for Axion Like Particle Production in 400-GeV Proton - Copper Interactions, Phys. Lett. B 157 (1985) 458–462.
  * [130] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, and B. Fuks, FeynRules 2.0 - A complete toolbox for tree-level phenomenology, Comput. Phys. Commun. 185 (2014) 2250–2300, arXiv:1310.1921 [hep-ph].
  * [131] A. Belyaev, N. D. Christensen, and A. Pukhov, CalcHEP 3.4 for collider physics within and beyond the Standard Model, Comput. Phys. Commun. 184 (2013) 1729–1769, arXiv:1207.6082 [hep-ph].
  * [132] C. Boehm, M. J. Dolan, C. McCabe, M. Spannowsky, and C. J. Wallace, Extended gamma-ray emission from Coy Dark Matter, JCAP 05 (2014) 009, arXiv:1401.6458 [hep-ph].
  * [133] M. Freytsis and Z. Ligeti, On dark matter models with uniquely spin-dependent detection possibilities, Phys. Rev. D 83 (2011) 115009, arXiv:1012.5317 [hep-ph].
  * [134] H.-Y. Cheng and C.-W. Chiang, Revisiting Scalar and Pseudoscalar Couplings with Nucleons, JHEP 07 (2012) 009, arXiv:1202.1292 [hep-ph].
  * [135] S. Banerjee, D. Barducci, G. Bélanger, B. Fuks, A. Goudelis, and B. Zaldivar, Cornering pseudoscalar-mediated dark matter with the LHC and cosmology, JHEP 07 (2017) 080, arXiv:1705.02327 [hep-ph].
  * [136] G. Jungman, M. Kamionkowski, and K. Griest, Supersymmetric dark matter, Phys. Rept. 267 (1996) 195–373, arXiv:hep-ph/9506380.
  * [137] G. Bertone, D. Hooper, and J. Silk, Particle dark matter: Evidence, candidates and constraints, Phys. Rept. 405 (2005) 279–390, arXiv:hep-ph/0404175.
  * [138] D. Hooper and S. Profumo, Dark Matter and Collider Phenomenology of Universal Extra Dimensions, Phys. Rept. 453 (2007) 29–115, arXiv:hep-ph/0701197.
  * [139] J. McDonald, Gauge singlet scalars as cold dark matter, Phys. Rev. D 50 (1994) 3637–3649, arXiv:hep-ph/0702143.
  * [140] C. P. Burgess, M. Pospelov, and T. ter Veldhuis, The Minimal model of nonbaryonic dark matter: A Singlet scalar, Nucl. Phys. B 619 (2001) 709–728, arXiv:hep-ph/0011335.
  * [141] L. Lopez Honorez, E. Nezri, J. F. Oliver, and M. H. G. Tytgat, The Inert Doublet Model: An Archetype for Dark Matter, JCAP 02 (2007) 028, arXiv:hep-ph/0612275.
  * [142] R. Barbieri, L. J. Hall, and V. S. Rychkov, Improved naturalness with a heavy Higgs: An Alternative road to LHC physics, Phys. Rev. D 74 (2006) 015007, arXiv:hep-ph/0603188.
  * [143] L. Lopez-Honorez, T. Schwetz, and J. Zupan, Higgs portal, fermionic dark matter, and a Standard Model like Higgs at 125 GeV, Phys. Lett. B 716 (2012) 179–185, arXiv:1203.2064 [hep-ph].
  * [144] N. Okada and S. Okada, $Z^{\prime}$-portal right-handed neutrino dark matter in the minimal U(1)X extended Standard Model, Phys. Rev. D 95 no. 3, (2017) 035025, arXiv:1611.02672 [hep-ph].
  * [145] P. Bandyopadhyay, E. J. Chun, and R. Mandal, Implications of right-handed neutrinos in $B-L$ extended standard model with scalar dark matter, Phys. Rev. D 97 no. 1, (2018) 015001, arXiv:1707.00874 [hep-ph].
  * [146] A. Das, S. Goswami, K. N. Vishnudath, and T. Nomura, Constraining a general U(1)′ inverse seesaw model from vacuum stability, dark matter and collider, Phys. Rev. D 101 no. 5, (2020) 055026, arXiv:1905.00201 [hep-ph].
  * [147] A. Das, S. Oda, N. Okada, and D.-s. Takahashi, Classically conformal U(1)’ extended standard model, electroweak vacuum stability, and LHC Run-2 bounds, Phys. Rev. D 93 no. 11, (2016) 115038, arXiv:1605.01157 [hep-ph].
  * [148] E. Ma, Verifiable radiative seesaw mechanism of neutrino mass and dark matter, Phys. Rev. D 73 (2006) 077301, arXiv:hep-ph/0601225.
  * [149] M. Hirsch, R. A. Lineros, S. Morisi, J. Palacio, N. Rojas, and J. W. F. Valle, WIMP dark matter as radiative neutrino mass messenger, JHEP 10 (2013) 149, arXiv:1307.8134 [hep-ph].
  * [150] A. Merle, M. Platscher, N. Rojas, J. W. F. Valle, and A. Vicente, Consistency of WIMP Dark Matter as radiative neutrino mass messenger, JHEP 07 (2016) 013, arXiv:1603.05685 [hep-ph].
  * [151] I. M. Ávila, V. De Romeri, L. Duarte, and J. W. F. Valle, Phenomenology of scotogenic scalar dark matter, Eur. Phys. J. C 80 no. 10, (2020) 908, arXiv:1910.08422 [hep-ph].
  * [152] S. Mandal, R. Srivastava, and J. W. F. Valle, The simplest scoto-seesaw model: WIMP dark matter phenomenology and Higgs vacuum stability, Phys. Lett. B 819 (2021) 136458, arXiv:2104.13401 [hep-ph].
  * [153] S. Mandal, N. Rojas, R. Srivastava, and J. W. F. Valle, Dark matter as the origin of neutrino mass in the inverse seesaw mechanism, Phys. Lett. B 821 (2021) 136609, arXiv:1907.07728 [hep-ph].
  * [154] D. Feldman, Z. Liu, P. Nath, and G. Peim, Multicomponent Dark Matter in Supersymmetric Hidden Sector Extensions, Phys. Rev. D 81 (2010) 095017, arXiv:1004.0649 [hep-ph].
  * [155] H. Baer, A. Lessa, S. Rajagopalan, and W. Sreethawong, Mixed axion/neutralino cold dark matter in supersymmetric models, JCAP 06 (2011) 031, arXiv:1103.5413 [hep-ph].
  * [156] M. Aoki, M. Duerr, J. Kubo, and H. Takano, Multi-Component Dark Matter Systems and Their Observation Prospects, Phys. Rev. D 86 (2012) 076015, arXiv:1207.3318 [hep-ph].
  * [157] S. Bhattacharya, A. Drozd, B. Grzadkowski, and J. Wudka, Two-Component Dark Matter, JHEP 10 (2013) 158, arXiv:1309.2986 [hep-ph].
  * [158] L. Bian, R. Ding, and B. Zhu, Two Component Higgs-Portal Dark Matter, Phys. Lett. B 728 (2014) 105–113, arXiv:1308.3851 [hep-ph].
  * [159] Y. Kajiyama, H. Okada, and T. Toma, Multicomponent dark matter particles in a two-loop neutrino model, Phys. Rev. D 88 no. 1, (2013) 015029, arXiv:1303.7356 [hep-ph].
  * [160] S. Esch, M. Klasen, and C. E. Yaguna, A minimal model for two-component dark matter, JHEP 09 (2014) 108, arXiv:1406.0617 [hep-ph].
  * [161] S. Bhattacharya, P. Poulose, and P. Ghosh, Multipartite Interacting Scalar Dark Matter in the light of updated LUX data, JCAP 04 (2017) 043, arXiv:1607.08461 [hep-ph].
  * [162] S. Bhattacharya, P. Ghosh, A. K. Saha, and A. Sil, Two component dark matter with inert Higgs doublet: neutrino mass, high scale validity and collider searches, JHEP 03 (2020) 090, arXiv:1905.12583 [hep-ph].
  * [163] S. Bhattacharya, N. Chakrabarty, R. Roshan, and A. Sil, Multicomponent dark matter in extended $U(1)_{B-L}$: neutrino mass and high scale validity, JCAP 04 (2020) 013, arXiv:1910.00612 [hep-ph].
  * [164] S. Bhattacharya, P. Ghosh, T. N. Maity, and T. S. Ray, Mitigating Direct Detection Bounds in Non-minimal Higgs Portal Scalar Dark Matter Models, JHEP 10 (2017) 088, arXiv:1706.04699 [hep-ph].
  * [165] B. Díaz Sáez, P. Escalona, S. Norero, and A. R. Zerwekh, Fermion singlet dark matter in a pseudoscalar dark matter portal, JHEP 10 (2021) 233, arXiv:2105.04255 [hep-ph].
  * [166] A. Mohamadnejad, Electroweak phase transition and gravitational waves in a two-component dark matter model, arXiv:2111.04342 [hep-ph].
  * [167] CMS Collaboration, A. M. Sirunyan et al., Search for invisible decays of a Higgs boson produced through vector boson fusion in proton-proton collisions at $\sqrt{s}=$ 13 TeV, Phys. Lett. B 793 (2019) 520–551, arXiv:1809.05937 [hep-ex].
  * [168] ATLAS Collaboration, M. Aaboud et al., Combination of searches for invisible Higgs boson decays with the ATLAS experiment, Phys. Rev. Lett. 122 no. 23, (2019) 231801, arXiv:1904.05105 [hep-ex].
  * [169] V. D. Barger, W.-Y. Keung, and E. Ma, Doubling of Weak Gauge Bosons in an Extension of the Standard Model, Phys. Rev. Lett. 44 (1980) 1169.
  * [170] ATLAS Collaboration, A combination of measurements of Higgs boson production and decay using up to $139\,\text{fb}^{-1}$ of $pp$ collision data at $\sqrt{s}=$ 13 TeV collected with the ATLAS experiment, 8, 2020.
  * [171] CMS Collaboration, Combined Higgs boson production and decay measurements with up to 137 fb-1 of proton-proton collision data at $\sqrt{s}$ = 13 TeV, 2020.
  * [172] D. Buttazzo, D. Redigolo, F. Sala, and A. Tesi, Fusing Vectors into Scalars at High Energy Lepton Colliders, JHEP 11 (2018) 144, arXiv:1807.04743 [hep-ph].
  * [173] K. Mekala, A. F. Zarnecki, B. Grzadkowski, and M. Iglicki, Searches for invisible scalar decays at CLIC, in 28th International Workshop on Deep Inelastic Scattering and Related Subjects.  7, 2021.  arXiv:2107.13903 [hep-ex].
  * [174] PandaX-II Collaboration, A. Tan et al., Dark Matter Results from First 98.7 Days of Data from the PandaX-II Experiment, Phys. Rev. Lett. 117 no. 12, (2016) 121303, arXiv:1607.07400 [hep-ex].
  * [175] F. Staub, Exploring new models in all detail with SARAH, Adv. High Energy Phys. 2015 (2015) 840780, arXiv:1503.04200 [hep-ph].
  * [176] G. Belanger, A. Mjallal, and A. Pukhov, Recasting direct detection limits within micrOMEGAs and implication for non-standard Dark Matter scenarios, Eur. Phys. J. C 81 no. 3, (2021) 239, arXiv:2003.08621 [hep-ph].
  * [177] G. Belanger, K. Kannike, A. Pukhov, and M. Raidal, Impact of semi-annihilations on dark matter phenomenology - an example of $Z_{N}$ symmetric scalar dark matter, JCAP 04 (2012) 010, arXiv:1202.2962 [hep-ph].
  * [178] Q.-H. Cao, E. Ma, J. Wudka, and C. P. Yuan, Multipartite dark matter, arXiv:0711.3881 [hep-ph].
  * [179] W. Wang and Z.-L. Han, Radiative linear seesaw model, dark matter, and $U(1)_{B-L}$, Phys. Rev. D 92 (2015) 095001, arXiv:1508.00706 [hep-ph].
  * [180] J. M. Cline, K. Kainulainen, P. Scott, and C. Weniger, Update on scalar singlet dark matter, Phys. Rev. D 88 (2013) 055025, arXiv:1306.4710 [hep-ph]. [Erratum: Phys.Rev.D 92, 039906 (2015)].
  * [181] G. Arcadi, S. Profumo, F. S. Queiroz, and C. Siqueira, Right-handed Neutrino Dark Matter, Neutrino Masses, and non-Standard Cosmology in a 2HDM, JCAP 12 (2020) 030, arXiv:2007.07920 [hep-ph].
  * [182] LUX Collaboration, D. S. Akerib et al., Results from a search for dark matter in the complete LUX exposure, Phys. Rev. Lett. 118 no. 2, (2017) 021303, arXiv:1608.07648 [astro-ph.CO].
  * [183] PandaX-II Collaboration, A. Tan et al., Dark Matter Results from First 98.7 Days of Data from the PandaX-II Experiment, Phys. Rev. Lett. 117 no. 12, (2016) 121303, arXiv:1607.07400 [hep-ex].
  * [184] M. Schumann, Direct detection of WIMP dark matter: concepts and status, Journal of Physics G: Nuclear and Particle Physics 46 no. 10, (Aug, 2019) 103003. https://doi.org/10.1088%2F1361-6471%2Fab2ea5.
  * [185] C. Gross, O. Lebedev, and T. Toma, Cancellation mechanism for dark-matter–nucleon interaction, Physical Review Letters 119 no. 19, (Nov, 2017) . https://doi.org/10.1103%2Fphysrevlett.119.191801.
  * [186] Y. Abe, T. Toma, and K. Tsumura, Pseudo-nambu-goldstone dark matter from gauged u(1)b-l symmetry, Journal of High Energy Physics 2020 no. 5, (May, 2020) . https://doi.org/10.1007%2Fjhep05%282020%29057.
  * [187] Y. Abe, T. Toma, K. Tsumura, and N. Yamatsu, Pseudo-nambu-goldstone dark matter model inspired by grand unification, Physical Review D 104 no. 3, (Aug, 2021) . https://doi.org/10.1103%2Fphysrevd.104.035011.
  * [188] S. Gola, S. Mandal, and N. Sinha, ALP-portal majorana dark matter, Int. J. Mod. Phys. A 37 no. 22, (2022) 2250131, arXiv:2106.00547 [hep-ph].
  * [189] N. Okada, D. Raut, and Q. Shafi, Pseudo-goldstone dark matter in a gauged $b-l$ extended standard model, Physical Review D 103 no. 5, (Mar, 2021) . https://doi.org/10.1103%2Fphysrevd.103.055024.
  * [190] S. Oda, N. Okada, and D. suke Takahashi, Classically conformal u(1)′ extended standard model and higgs vacuum stability, Physical Review D 92 no. 1, (Jul, 2015) . https://doi.org/10.1103%2Fphysrevd.92.015026.
  * [191] A. Das, N. Okada, S. Okada, and D. Raut, Probing the seesaw mechanism at the 250 GeV ILC, Physics Letters B 797 (Oct, 2019) 134849. https://doi.org/10.1016%2Fj.physletb.2019.134849.
  * [192] A. Das, S. Mandal, T. Nomura, and S. Shil, Heavy majorana neutrino pair production from z‘ at hadron and lepton colliders, Physical Review D 105 no. 9, (May, 2022) . https://doi.org/10.1103%2Fphysrevd.105.095031.
  * [193] N. Darvishi, M. Masouminia, and A. Pilaftsis, Maximally symmetric three-higgs-doublet model, Physical Review D 104 no. 11, (Dec, 2021) . https://doi.org/10.1103%2Fphysrevd.104.115017.
  * [194] T. Robens, T. Stefaniak, and J. Wittbrodt, Two-real-scalar-singlet extension of the SM: LHC phenomenology and benchmark scenarios, The European Physical Journal C 80 no. 2, (Feb, 2020) . https://doi.org/10.1140%2Fepjc%2Fs10052-020-7655-x.
  * [195] K. Kannike, Vacuum Stability Conditions From Copositivity Criteria, Eur. Phys. J. C 72 (2012) 2093, arXiv:1205.3781 [hep-ph].
  * [196] A. Djouadi, The anatomy of electroweak symmetry breaking, Physics Reports 457 no. 1-4, (Feb, 2008) 1–216. https://doi.org/10.1016%2Fj.physrep.2007.10.004.
  * [197] ATLAS Collaboration, A. et.al., Combination of searches for invisible Higgs boson decays with the ATLAS experiment, .
  * [198] K. Ishiwata and T. Toma, Probing pseudo nambu-goldstone boson dark matter at loop level, Journal of High Energy Physics 2018 no. 12, (Dec, 2018) . https://doi.org/10.1007%2Fjhep12%282018%29089.
  * [199] S. L. Glashow, Partial Symmetries of Weak Interactions, Nucl. Phys. 22 (1961) 579–588.
  * [200] S. Weinberg, A Model of Leptons, Phys. Rev. Lett. 19 (1967) 1264–1266.