R-parity Conserving Minimal SUSY U(1)X Model

Satsuki Oda${}^{~{}a,b}$<EMAIL_ADDRESS>Nobuchika Okada${}^{~{}c}$
<EMAIL_ADDRESS>Nathan Papapietro${}^{~{}c}$,
and Dai-suke Takahashi${}^{~{}a,b}$<EMAIL_ADDRESS>
aOkinawa Institute of Science and Technology Graduate University (OIST),
Onna, Okinawa 904-0495, Japan

bInstitute for Pacific Rim Studies, Meio University,
Nago, Okinawa 905-8585, Japan

cDepartment of Physics and Astronomy, University of Alabama,
Tuscaloosa, Alabama 35487, USA

We propose a minimal gauged U(1)X extension of the Minimal Supersymmetric
(SUSY) Standard Model (MSSM) with R-parity conservation. In this model, U(1)X
is a generalization of the well-known U(1) $B-L$ (baryon number minus lepton
number). Apart from the MSSM particle content, the model includes three right-
handed neutrino (RHN) chiral superfields, each carrying a unit U(1)X charge.
In the presence of RHNs, the model is free from all gauge and mixed gauge-
gravitational anomalies. However, there are no U(1)X Higgs chiral superfields
with U(1)X charge $\pm 2$ involved in the model. Two of the RHN superfields
are assigned an odd R-parity, while the last one ($\Psi$) has an even parity.
The U(1)X symmetry is radiatively broken by the vacuum expectation value (VEV)
of the scalar component of $\Psi$ after renormalization group evolution of the
scalar potential. As a consequence of the absence of U(1)X Higgs fields and
the novel R-parity assignment, the three light neutrinos consist of one
massless neutrino and two Dirac neutrinos. In the early universe, the right-
handed components of the Dirac neutrinos are in thermal equilibrium with the
Standard Model (SM) particles through the U(1)X gauge ($Z^{\prime}$) boson.
The extra energy density from the right-handed neutrinos is constrained to
avoid disrupting the success of Big Bang Nucleosynthesis (BBN), leading to a
lower bound on the scale of U(1)X symmetry breaking. In our model, a mixture
of the U(1)X gaugino and the fermionic component of $\Psi$ becomes a new dark
matter (DM) candidate if it is the lightest sparticle mass eigenstate. We
examine this DM phenomenology and identify a parameter region that reproduces
the observed DM relic density. Furthermore, we consider constraints from the
search for $Z^{\prime}$ boson resonance at the Large Hadron Collider (LHC).
The three constraints obtained from the success of BBN, the observed DM relic
density, and the $Z^{\prime}$ resonance search at the LHC complement each
other, narrowing down the allowed parameter region. Additionally, we note that
once the $Z^{\prime}$ boson is discovered at the LHC, it can serve as a probe
to determine the number of left-handed plus right-handed neutrinos through
measurements of its invisible decay width.

##  1 Introduction

The $B-L$ (baryon number minus lepton number) is the unique anomaly-free
global U(1) symmetry in the Standard Model (SM). We can easily gauge this
global symmetry, and the so-called minimal $B-L$ model [1]-[6] is a simple
gauged $B-L$ extension of the Standard Model (SM), where three right-handed
neutrinos (RHNs) and an SM gauge-singlet Higgs field carrying two units of the
$B-L$ charge are introduced. The model is free from all gauge and mixed gauge-
gravitational anomalies in the presence of the RHNs. Associated with a $B-L$
symmetry breaking by a vacuum expectation value (VEV) of the $B-L$ Higgs
field, the $B-L$ gauge ($Z^{\prime}$) boson and the RHNs acquire their masses.
After the electroweak symmetry breaking, the seesaw mechanism [7]-[11] works
to generate tiny SM neutrino masses naturally.

Although the scale of the U(1)B-L gauge symmetry breaking is arbitrary as long
as it is phenomenologically available, the breaking scale of $1-10$ TeV is
probably of the best interest in the viewpoint of the Large Hadron Collider
(LHC) experiment. However, the argument of the gauge hierarchy problem of the
SM is also applicable to the $B-L$ Higgs field and quantum corrections to the
$B-L$ Higgs field mass squared are quadratically sensitive to the scale of
ultraviolet (UV) theory that we expect to take place at high energies. Thus,
the $B-L$ symmetry breaking scale is unstable against quantum corrections. As
is well-known, supersymmetric (SUSY) extension is one of the most promising
ways to make the theory UV insensitive and ensure the stability of the
symmetry breaking scale. Interestingly, we can see that in the SUSY extension
of the minimal $B-L$ extended SM, the $B-L$ asymmetry is also radiatively
broken in the presence of the soft SUSY breaking mass terms at a high energy
[12, 13, 14]. In this radiative symmetry breaking, the soft SUSY breaking
parameters control the $B-L$ symmetry breaking scale, and hence the scale of
$B-L$ symmetry breaking lies at the TeV scale from the naturalness point of
view.

SUSY extension of the $B-L$ model opens a new interesting possibility. As has
been proposed in Ref. [15], it is unnecessary to introduce the $B-L$ Higgs
field in this extension since the scalar partner of a RHN chiral superfield
can play the role of breaking the $B-L$ gauge symmetry. Therefore, we may
define the “minimal SUSY $B-L$ model” by excluding the $B-L$ Higgs chiral
superfields carrying $B-L$ charge $\pm 2$. Interestingly, such a particle
content can be derived from heterotic strings with flux compactifications [16,
17]. In Ref. [15], a negative soft mass squared for a right-handed sneutrino
is assumed to break the $B-L$ gauge symmetry, so that the $B-L$ symmetry
breaking occurs at the TeV scale. Associated with this symmetry breaking,
R-parity is also spontaneously broken. See, for example, Refs. [18, 19, 20,
21] for interesting phenomenology with the R-parity violation. Through the
non-zero VEV of the right-handed sneutrino, neutrinos have mass mixings with
MSSM Higgsinos, MSSM neutralinos, and $B-L$ gaugino. Although the neutrino
mass matrix becomes very complicated, it has enough degrees of freedom to
reproduce the neutrino oscillation data with a characteristic pattern of the
mass spectrum [22, 23]. On the other hand, due to the R-parity violation, the
usual dark matter (DM) candidate in the MSSM, the lightest superpartner (LSP)
neutralino is unstable and no longer a DM candidate. Thus, we need a new DM
candidate to supplement the model.

The U(1)B-L model can be generalized to the U(1)X model [24], where the
particle content is the same as the U(1)B-L model, but the U(1)X charge of a
field in the $B-L$ model is extended to be a linear combination of its $B-L$
change ($Q_{B-L}$) and hypercharge ($Q_{Y}$), namely,
$Q_{X}=x_{H}\,Q_{Y}+Q_{B-L}$ with a common parameter $x_{H}$. The U(1)B-L
model is defined by the limit of $x_{H}\to 0$. Clearly, the U(1)X model is
anomaly free since both $B-L$ and hypercharge gauge symmetries are free from
anomalies in the $B-L$ model.

In this paper, we propose the minimal SUSY U(1)X model with R-parity
conservation. The particle content of the model is the same as the one of the
minimal SUSY $B-L$ model discussed above, while we assign an even R-parity to
one RHN chiral superfield ($\Psi$) and an odd R-parity to the other two RHN
chiral superfields. The R-parity assignment for the MSSM fields is as usual.
Because of this parity assignment and the gauge symmetry, the chiral
superfield $\Psi$ has no Dirac Yukawa coupling with the lepton doublet fields.
We consider the case that the $B-L$ symmetry breaking is driven by a VEV of
the scalar component of $\Psi$. Phenomenological consequences in this model
are very different from those of the conventional minimal SUSY $B-L$ model. As
usual in the MSSM, R-parity is conserved, and hence the LSP neutralino is a
candidate for the dark matter (DM). In addition to the neutralino in the MSSM,
a new DM candidate arises in the model, namely, a linear combination of the
fermion component of $\Psi$ and the $B-L$ gaugino, We will investigate this DM
phenomenology and find a parameter region that reproduces the observed DM
relic density. In the absence of the U(1)X Higgs superfields carrying U(1)X
charge $\pm 2$, no Majorana mass term is generated for RHNs relevant to the
mass generation in the SM neutrino sector, and as a result, the SM neutrinos
are Dirac particles. With only the two RHNs involved in the Dirac Yukawa
couplings, the neutrino mass matrix leads to three mass eigenstates: one
massless chiral neutrino and two Dirac neutrinos. The 2-by-3 neutrino mass
matrix includes a sufficient number of free parameters to reproduce the
neutrino oscillation data. This Dirac nature of the SM neutrinos is
distinctive from the neutrinos in the usual $B-L$ model, where the neutrinos
are Majorana particles with a hierarchical mass spectrum through the seesaw
mechanism. In the early universe, the right-handed components of the two Dirac
neutrinos are in thermal equilibrium with the SM particles through the U(1)X
gauge interaction, and they potentially contribute to extra neutrino species
($\Delta N_{\mathrm{eff}}$) at the Big Bang Nucleosynthesis (BBN) era [25,
26]. Applying the cosmological upper bound on $\Delta N_{\mathrm{eff}}$, we
find a lower bound on the scale of U(1)X symmetry breaking. The U(1)X gauge
boson ($Z^{\prime}$) has been searched by the Large Hadron Collider (LHC)
experiment. Using the LHC Run-2 final results, we constrain the model
parameter region. Combining the DM density constraint, the BBN bound and the
$Z^{\prime}$ boson search results, we identify the allowed parameter region.
We also note that if the $Z^{\prime}$ boson is discovered at the LHC, the
number of left-handed plus right-handed neutrinos can be determined by
measuring the invisible decay width of $Z^{\prime}$ boson.

This paper is organized as follows: In the next section, we first define our
minimal SUSY U(1)X model with R-parity conservation. Then, we introduce the
superpotential and soft SUSY breaking terms relevant to our discussion. In
Sec. 3, we consider the renormalization group (RG) equations for the soft SUSY
breaking parameters and examine their RG evolution from high energy to low
energy. We show that the U(1)X symmetry is radiatively broken by the RG
evolution while R-parity remains manifest. With a suitable choice of the soft
SUSY breaking parameters, the LSP neutralino is identified as a linear
combination of the fermion component of $\Psi$ and the U(1)X gaugino. In Sec.
4, we investigate this DM physics and find the parameter region that
reproduces the observed DM relic density. The search for $Z^{\prime}$ boson
resonance at the LHC is discussed in Sec. 5. Using the LHC Run-2 final results
reported by the ATLAS and CMS Collaborations, we find the LHC constraints on
$Z^{\prime}$ boson mass, U(1)X gauge coupling and $x_{H}$. We discuss the BBN
bound in Sec. 6. In Sec. 7, we present the results by combining the
cosmological and collider constraints obtained in Secs. 4, 5 and 6. We discuss
in Sec. 8 how to probe the number of left-handed and right-handed neutrinos in
the model by collider experiments. The last section is devoted to conclusions
and discussions.

##  2 Minimal SUSY U(1)X model with conserved R-parity

| SU(3)c | SU(2)L | U(1)Y | U(1)X |  $R$-parity
---|---|---|---|---|---
$Q_{i}$ | 3 | 2 | $+1/6$ | $x_{Q}$ | = | $+\;\frac{1}{3}x_{H}+\frac{1}{3}x_{\Psi}$ | $-$
$U_{i}^{c}$ | 3∗ | 1 | $-2/3$ | $x_{U}^{c}$ | = | $-\;\frac{4}{3}x_{H}-\frac{1}{3}x_{\Psi}$ | $-$
$D_{i}^{c}$ | 3∗ | 1 | $+1/3$ | $x_{D}^{c}$ | = | $+\;\frac{2}{3}x_{H}-\frac{1}{3}x_{\Psi}$ | $-$
$L_{i}$ | 1 | 2 | $-1/2$ | $x_{L}$ | = | $-\;x_{H}\;\;-x_{\Psi}$ | $-$
$N_{1,2}^{c}$ | 1 | 1 | $0$ | $x_{N}^{c}$ | = | $+\;x_{\Psi}$ | $-$
$\Psi$ | 1 | 1 | $0$ | $x_{\Psi}$ | = | $+\;x_{\Psi}$ | $+$
$E_{i}^{c}$ | 1 | 1 | $+1$ | $x_{E}^{c}$ | = | $+\;2x_{H}\;+x_{\Psi}$ | $-$
$H_{u}$ | 1 | 2 | $+1/2$ | $x_{H_{u}}$ | = | $+\;x_{H}$ | $+$
$H_{d}$ | 1 | 2 | $-1/2$ | $x_{H_{d}}$ | = | $-\;x_{H}$ | $+$
Table 1:  Particle contents of the minimal SUSY U(1)X model with conserved
R-parity. $i=1,2,3$ denotes the generation index. In addition to the MSSM
particle contents, three MSSM singlet superfields ($N_{1,2}^{c}$ and $\Psi$)
are introduced. We consider $x_{\Psi}\neq 0$, and set $x_{\Psi}=1$ without
loss of generality.

The minimal SUSY U(1)X model is based on the gauge group of
SU(3)${}_{c}\times$SU(2)${}_{L}\times$U(1)${}_{Y}\times$U(1)X. In addition to
the MSSM particle content, we introduce three chiral superfields, which are
singlets under the SM gauge groups but have a U(1)X charge, $x_{\Psi}$. Their
existence is essential to make the model free from all gauge and mixed gauge-
gravitational anomalies. Unlike direct supersymmetrization of the minimal
U(1)X model, no U(1)X Higgs superfields carrying U(1)X charge $\pm 2x_{\Psi}$
are included in the model. The particle content is listed in Table 1. The
U(1)X charge of an MSSM field is defined as a linear combination of its
hypercharge and $B-L$ charge such as $Q_{X}=x_{H}\,Q_{Y}+x_{\Psi}\,Q_{B-L}$.
For $x_{\Psi}\neq 0$, we can set $x_{\Psi}=1$ without loss of generality.444
We do not consider $x_{\Psi}=0$. In this case, the phenomenology of the model
is very different from what we discuss in this paper.  The key to our proposal
is that we assign an even R-parity to $\Psi$ while an odd parity to
$N_{1,2}^{c}$, so that $N_{1,2}^{c}$ are identified with the RHN superfields
relevant to the neutrino mass generation. This is in contrast with the minimal
SUSY $B-L$ model proposed in Ref. [15], where all RHN chiral superfields are
R-parity odd states and their scalar components develop VEVs by which the
U(1)B-L symmetry and R-party at the same time are broken. See Ref. [27] for
the $B-L$ version of our model, which corresponds to the limit of $x_{H}\to
0$.

The gauge and R-parity invariant superpotential, which is added to the MSSM
one, is only the neutrino Dirac Yukawa coupling,

$\displaystyle{\cal L}_{\rm Yukawa}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}\sum_{j=1}^{3}Y_{N}^{ij}N_{i}^{c}H_{u}L_{j}.$ (1)

Note that R-parity forbids $\Psi$ to have a direct coupling with the MSSM
fields. After the electroweak symmetry breaking, the neutrino Dirac mass
matrix is generated. Since this is a 2-by-3 matrix, we have one massless
neutrino and two Dirac neutrinos in the model. This mass matrix has a
sufficient number of free parameters to reproduce the neutrino oscillation
data.

Next, we introduce soft SUSY breaking terms for the fields in the U(1)X
sector:

$\displaystyle{\cal L}_{\rm soft}$ $\displaystyle=$
$\displaystyle-\left(\frac{1}{2}M_{X}\lambda_{X}\lambda_{X}+{\rm
h.c.}\right)-\left(\sum_{i=1}^{2}m_{\tilde{n}_{i}^{c}}^{2}|\tilde{n}_{i}^{c}|^{2}+m_{\phi}^{2}|\phi|^{2}\right),$
(2)

where $\lambda_{X}$ is the U(1)X gaugino, and $\tilde{n}_{i}^{c}$ and $\phi$
are the scalar components of $N_{i}^{c}$ and $\Psi$, respectively. Since the
Dirac Yukawa couplings are very small to reproduce the neutrino oscillation
data, we omit terms relevant to the couplings. In the next section, we analyze
the renormalization group (RG) evolution of the soft SUSY-breaking masses. We
find that $m_{\phi}^{2}$ is driven to be negative while
$m_{\tilde{n}_{i}^{c}}^{2}$ remain positive at low energy, and as result only
$\phi$ develops non-zero VEV. Thus, the U(1)X symmetry is radiatively broken,
but R-parity is conserved at the vacuum.

##  3 Radiative U(1)X symmetry breaking

Figure 1:  The RG evolution of the soft SUSY breaking mass $m_{\phi}^{2}$ from
$M_{U}$ to low energies.

It is well-known that the electroweak symmetry breaking in the MSSM is
triggered by radiative corrections which drive the mass squared of the up-type
Higgs doublet negative. In this mechanism, the electroweak symmetry breaking
scale is controlled by the soft SUSY breaking mass scale. Thus, the SUSY
breaking scale at the TeV naturally results in the electroweak scale of ${\cal
O}$(100 GeV). In this section, we show that similarly to the MSSM, a radiative
U(1)X symmetry breaking occurs by the RG evolution of the soft SUSY breaking
parameter $m_{\phi}^{2}$ from high energy to low energy. However, note that
the mechanism that drives $m_{\phi}^{2}$ negative differs from the one in the
MSSM where the large top Yukawa coupling plays a crucial role.

For simplicity, we consider the RG equations only for the U(1)X sector.555 See
Refs. [28, 29] for more elaborate analysis and parameter scans to identify
parameter regions. At the one-loop level, the RG equations relevant to our
discussion are given by

$\displaystyle\mu\frac{dm_{\phi}^{2}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{g_{X}^{2}}{16\pi^{2}}\left[2\left(\sum_{j=1}^{2}m_{\tilde{n}_{j}^{c}}^{2}+m_{\phi}^{2}\right)-8M_{X}^{2}\right],$
(3) $\displaystyle\mu\frac{dm_{\tilde{n}_{i}^{c}}^{2}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{g_{X}^{2}}{16\pi^{2}}\left[2\left(\sum_{j=1}^{2}m_{\tilde{n}_{j}^{c}}^{2}+m_{\phi}^{2}\right)-8M_{X}^{2}\right],$
(4) $\displaystyle\mu\frac{dM_{X}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{g_{X}^{2}}{16\pi^{2}}\,32M_{X},$ (5)

where the U(1)X gauge coupling ($g_{X}$) obeys

$\displaystyle\mu\frac{dg_{X}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\,16g_{X}^{3}.$ (6)

Here, we have neglected the contributions from neutrino Dirac Yukawa couplings
which are very small. The first term inside the bracket on the right-hand side
of Eq. (3), which originates from the U(1)X $D$-term interaction, plays an
essential role in driving $m_{\phi}^{2}$ negative at low energy. Since squarks
and sleptons have U(1)X charges, their soft squared masses also appear in the
RG equations through their $D$-term interactions. However, for simplicity, we
have omitted them by assuming their squared masses are much smaller than
$m_{\tilde{n}^{c}_{i}}^{2}$.

To illustrate the radiative U(1)X symmetry breaking, we numerically solve the
RG equations from $M_{U}=2\times 10^{16}$ GeV to low energy with the following
boundary conditions:

$\displaystyle\alpha_{g_{X}}=0.0189,\;\;M_{X}=16.0\;{\rm
TeV},\;\;m_{\tilde{n}^{c}_{1}}=m_{\tilde{n}^{c}_{2}}=25.0\;{\rm
TeV},\;\;m_{\phi}=5.46\;{\rm TeV},$ (7)

where $\alpha_{g_{X}}=\frac{g_{X}^{2}}{4\pi}$. Although we do not consider the
grand unification of our model in this paper, we took a typical scale of the
gauge coupling unification in the MSSM, $M_{U}=2\times 10^{16}$ GeV, as a
reference scale at which the boundary conditions are imposed. We show in Fig.
1 the RG evolution of $m_{\phi}^{2}$. We can see that $m_{\phi}^{2}$ becomes
negative at low energies, while we find the other squared masses remain
positive. To arrive at this result, the hierarchy
$m_{\tilde{n}^{c}_{i}}^{2}\gg m_{\phi}^{2},M_{X}^{2}$ is crucial, which drives
only $m_{\phi}^{2}$ negative.

After solving the RG equations, we evaluate the soft SUSY breaking parameters
at the U(1)X symmetry breaking scale, $\mu=v_{X}=\sqrt{2}\langle\phi\rangle$.
For $v_{X}=14$ TeV as a reference, we find

$\displaystyle\alpha_{g_{X}}=0.008,\;\;M_{X}=6.75\;{\rm
TeV},\;\;m_{\tilde{n}^{c}_{1}}=m_{\tilde{n}^{c}_{2}}=24.2\;{\rm
TeV},\;\;|m_{\phi}|=3.18\;{\rm TeV}.$ (8)

Solving the stationary conditions for the scalar potential,

$\displaystyle
V=m_{\tilde{n}^{c}_{1}}^{2}|\tilde{n}^{c}_{1}|^{2}+m_{\tilde{n}^{c}_{2}}^{2}|\tilde{n}^{c}_{2}|^{2}+m_{\phi}^{2}|\phi|^{2}+\frac{g_{X}^{2}}{2}\left(|\tilde{n}^{c}_{1}|^{2}+|\tilde{n}^{c}_{2}|^{2}+|\phi|^{2}\right)^{2},$
(9)

with the above inputs, we find in units of TeV

$\displaystyle\langle\tilde{n}^{c}_{1}\rangle=\langle\tilde{n}^{c}_{2}\rangle=0,\;\;v_{X}=\sqrt{2}\langle\phi\rangle=\sqrt{2}\frac{|m_{\phi}(\mu=v_{X})|}{g_{X}(\mu=v_{X})}\simeq
14.$ (10)

The last equation indicates the consistency of our analysis. For a given
choice of the input parameters of $g_{X}$, $M_{X}$ and $m_{\phi}$, we can
arrange $m_{\tilde{n}^{c}_{1,2}}$ to make our analysis consistent. We now see
that the U(1)X gauge symmetry is radiatively broken by the VEV of $\phi$.
Since $\phi$ is R-parity even, R-parity is manifest at the vacuum.

##  4 New Neutralino Dark Matter

Since R-parity is manifest at the vacuum as usual in our model, the LSP
neutralino is stable and play the DM role. In addition to the MSSM
neutralinos, a new dark matter candidate arises in our model, namely, a linear
combination of the fermion component of $\Psi$ ($\psi$) and the U(1)X gaugino.
In this section, assuming this particle is the LSP, we investigate the DM
phsyics.

A scenario of the parity-odd right-handed Majorana neutrino dark matter was
first proposed in Ref. [30] in the context of the non-SUSY minimal $B-L$
model, where a $Z_{2}$ parity is introduced and an odd parity is assigned to
one right-handed neutrino while the other fields are all parity-even. No new
particle is introduced other than the particle content in the minimal $B-L$
model. Thanks to the conservation of $Z_{2}$-parity, the parity-odd right-
handed neutrino is stable, and hence the DM candidate in the model. The
phenomenology of this DM particle along with the LHC physics and astrophysics
has been investigated in Refs. [30, 31, 32, 33] (see Ref. [34] for a concise
review). For the U(1)X extension of the scenario, see, for example, Refs. [35,
36, 37, 38]. A SUSY version of the minimal $B-L$ model with the right-handed
neutrino dark matter has been proposed in [14]. The extension of this SUSY
model to a U(1)X model is discussed in Ref. [39].

Although the dark matter scenario in this section shares similar properties
with the scenario discussed in Refs. [14, 39], there is a crucial difference,
namely, $\psi$ has no Majorana mass on its own, but the DM particle acquires a
Majorana mass through the mixing with the U(1)X gaugino ($\lambda_{X}$). With
the U(1)X symmetry breaking by $\langle\phi\rangle=v_{X}/\sqrt{2}$, the mass
matrix for $\psi$ and $\lambda_{X}$ is given by

$\displaystyle M_{\chi}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&m_{Z^{\prime}}\\\
m_{Z^{\prime}}&M_{X}\end{pmatrix},$ (11)

which is diagonalized as

$\displaystyle\begin{pmatrix}\chi_{\ell}\\\ \chi_{h}\end{pmatrix}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}\psi\\\
\lambda_{X}\end{pmatrix},$ (12)

with $\tan 2\theta=2m_{Z^{\prime}}/M_{X}$. Here, $m_{Z^{\prime}}=g_{X}v_{X}$
is the $Z^{\prime}$ boson mass. Let us assume that the lighter mass eigenstate
($\chi_{\ell}$) is the LSP neutralino. Since $\psi$ and $\lambda_{X}$ are the
MSSM gauge singlets, possible annihilation processes of a pair of DM particles
are very limited. Furthermore, given a small U(1)X gauge coupling and the
Majorana nature of the dark matter particle, the annihilation processes via
sfermion exchanges are not efficient. We find that the most important
annihilation process for a pair of DM particles is through the $Z^{\prime}$
boson resonance in the $s$-channel with the DM mass being close to half of the
$Z^{\prime}$ boson mass. To realize this situation, we set
$M_{X}\simeq(3/2)m_{Z^{\prime}}$, so that the lightest mass eigenvalue is
given by $m_{DM}\simeq m_{Z^{\prime}}/2$ and $\cos^{2}\theta\simeq 0.8$. Our
parameter choice in the previous section is suitable for this setup,
$M_{X}=(3/2)m_{Z^{\prime}}=6.75$ TeV for $m_{Z^{\prime}}=4.5$ TeV.

We now calculate the DM relic density by integrating out the Boltzmann
equation,

$\displaystyle\frac{dY}{dx}$ $\displaystyle=$
$\displaystyle-\frac{xs\langle\sigma v\rangle}{H(m_{\rm DM})}(Y^{2}-Y_{\rm
EQ}^{2}),$ (13)

where the temperature of the universe is normalized by the DM mass
$x=m_{DM}/T$, $H(m_{DM})$ is the Hubble parameter at $T=m_{DM}$, $Y$ is the
yield (the ratio of the dark matter number density to the entropy density $s$)
of the DM particle, $Y_{EQ}$ is the yield of the DM particle in thermal
equilibrium, and $\langle\sigma v\rangle$ is the thermal average of the dark
matter annihilation cross section times relative velocity. Explicit formulas
of the quantities involved in the Boltzmann equation are as follows:

$\displaystyle s$ $\displaystyle=$
$\displaystyle\frac{2\pi^{2}}{45}g_{\star}\frac{m_{DM}^{3}}{x^{3}},$
$\displaystyle H(m_{DM})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\pi^{2}}{90}g_{\star}}\frac{m_{DM}^{2}}{M_{P}},$
$\displaystyle sY_{EQ}$ $\displaystyle=$
$\displaystyle\frac{g_{DM}}{2\pi^{2}}\frac{m_{DM}^{3}}{x}K_{2}(x),$ (14)

where $M_{P}=2.43\times 10^{18}$ GeV is the reduced Planck mass, $g_{DM}=2$ is
the number of degrees of freedom for the Majorana dark matter particle,
$g_{\star}$ is the effective total number of degrees of freedom for particles
in thermal equilibrium (in the following analysis, we use $g_{\star}=106.75$
for the SM particles), and $K_{2}$ is the modified Bessel function of the
second kind. In our scenario, a DM pair annihilates into the SM particles
dominantly through the $Z^{\prime}$ boson exchange in the $s$-channel. The
thermal average of the annihilation cross section is given by

$\displaystyle\langle\sigma v\rangle$ $\displaystyle=$ $\displaystyle(sY_{\rm
EQ})^{-2}g_{\rm DM}^{2}\frac{m_{\rm DM}}{64\pi^{4}x}\int_{4m_{\rm
DM}^{2}}^{\infty}ds\,2(s-4m_{DM}^{2})\sigma(s)\sqrt{s}K_{1}\left(\frac{x\sqrt{s}}{m_{\rm
DM}}\right),$ (15)

where $\sigma(s)$ is the total annihilation cross section, and $K_{1}$ is the
modified Bessel function of the first kind. The total cross section of the
dark matter annihilation process $\chi_{\ell}\chi_{\ell}\to Z^{\prime}\to
f{\bar{f}}$ ($f$ denotes the SM fermions plus two right-handed neutrinos) is
calculated as

$\displaystyle\sigma(s)$ $\displaystyle=$
$\displaystyle\frac{\pi}{3}\alpha_{g_{X}}^{2}\cos^{4}\theta\frac{\sqrt{s(s-4m_{\rm
DM}^{2})}}{(s-m_{Z^{\prime}}^{2})^{2}+m_{Z^{\prime}}^{2}\Gamma_{Z^{\prime}}^{2}}$
(16) $\displaystyle\times$
$\displaystyle\left[\frac{103x_{H}^{2}+86x_{H}+37}{3}+\frac{17x_{H}^{2}+10x_{H}+2+(7x_{H}^{2}+20x_{H}+4)\frac{m_{t}^{2}}{s}}{3}\sqrt{1-\frac{4m_{t}^{2}}{s}}\right.$
$\displaystyle\hskip
56.9055pt\left.+18x_{H}^{2}\frac{(s-m_{Z^{\prime}}^{2})^{2}}{s(s-4m_{\rm
DM}^{2})}\frac{m_{\rm
DM}^{2}m_{t}^{2}}{m_{Z^{\prime}}^{4}}\sqrt{1-\frac{4m_{t}^{2}}{s}}\right],$

where all final state fermion masses have been neglected except for the top
quark mass $m_{t}$, and we have assumed that all MSSM particles are heavier
than the DM particle. The total decay width of $Z^{\prime}$ boson is given by

$\displaystyle\Gamma_{Z^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{g_{X}}m_{Z^{\prime}}}{6}\\!\\!\left[\frac{103x_{H}^{2}+86x_{H}+43}{3}+\frac{17x_{H}^{2}+10x_{H}+2+(7x_{H}^{2}+20x_{H}+4)\frac{m_{t}^{2}}{m_{Z^{\prime}}^{2}}}{3}\sqrt{1-\frac{4m_{t}^{2}}{m_{Z^{\prime}}^{2}}}\right.$
(17) $\displaystyle\hskip 42.67912pt\left.+\cos^{4}\theta\left(1-\frac{4m_{\rm
DM}^{2}}{m_{Z^{\prime}}^{2}}\right)^{\frac{3}{2}}\Theta\left(\frac{m_{Z^{\prime}}^{2}}{m_{\rm
DM}^{2}}-4\right)+2x_{H}^{2}\sqrt{\lambda[a,b,c]}\left(\lambda[a,b,c]+12b\right)\right].$

where $\Theta$ is the Heaviside step function, and the last term in the
bracket on the right-hand side corresponds to $Z^{\prime}\to Z\,h$ with
$\lambda[x,y,z]=x^{2}+y^{2}+z^{2}-2xy-2yz-2zx$, $a=1$,
$b=\left(\frac{m_{Z}}{m_{Z^{\prime}}}\right)^{2}\ll 1$ and
$c=\left(\frac{m_{h}}{m_{Z^{\prime}}}\right)^{2}\ll 1$ [40], respectively,
with the $Z$ boson mass $m_{Z}=91.2$ GeV and the Higgs boson mass $m_{h}=125$
GeV.

Figure 2:  The relic abundance of the dark matter particle as a function of
the dark matter mass ($m_{DM}$) for $x_{H}=-0.5$, $m_{Z^{\prime}}=4.5$ TeV,
$\cos^{2}\theta=0.8$ and various values of the gauge coupling,
$\alpha_{g_{X}}=0.007$, $0.00793$, $0.009$, $0.010$ and $0.011$ (solid lines
from top to bottom). The region between two horizontal dashed lines denote the
range of the observed dark matter relic abundance,
$0.119\leq\Omega_{DM}h^{2}\leq 0.121$ (68% limit) [41].

We solve the Boltzmann equation numerically to find the asymptotic value of
the yield $Y(\infty)$. The dark matter relic abundance is evaluated as

$\displaystyle\Omega_{DM}h^{2}=\frac{m_{DM}s_{0}Y(\infty)}{\rho_{c}/h^{2}},$
(18)

where $s_{0}=2890$ cm-3 is the entropy density of the present universe, and
$\rho_{c}/h^{2}=1.05\times 10^{-5}$ GeV/cm3 is the critical density. In our
analysis, only three parameters, namely $\alpha_{g_{X}}$, $m_{Z^{\prime}}$ and
$m_{DM}$, are involved. Note that the mixing angle $\theta$ is not a free
parameter since it is determined once $m_{Z^{\prime}}$ and $m_{DM}$ are fixed.
We identify the parameter region so as to reproduce the observed DM relic
density of $\Omega_{DM}h^{2}=0.12$ from the Planck results [41].

As mentioned above, a sufficiently large annihilation cross section is
achieved only if $m_{DM}\simeq m_{Z^{\prime}}/2$. Thus, we focus on the DM
mass in this region and in this case $\cos^{2}\theta\simeq 0.8$. For
$x_{H}=-0.5$, $m_{Z^{\prime}}=4.5$ TeV and $\cos^{2}\theta=0.8$, Fig. 2 shows
the resultant DM relic density as a function of its mass $m_{DM}$ for various
values of the gauge coupling, $\alpha_{g_{X}}=0.007$, $0.00793$, $0.009$,
$0.010$ and $0.011$. The region between two horizontal dashed lines is the
allowed region, $0.119\leq\Omega_{DM}h^{2}\leq 0.121$ (68% limit), from the
Planck satellite experiment [41]. For each contour, the DM masses reproducing
the observed relic abundance are marked as green dots. We have confirmed that
the observed relic abundance can be reproduced when the dark matter mass is
close to half of the $Z^{\prime}$ boson mass. Note that we find the lower
bound on $\alpha_{g_{X}}$ once the values of $x_{H}$ and $m_{Z^{\prime}}$ are
fixed. In Fig. 2, we can see $\alpha_{g_{X}}\leq 0.00793$.

##  5 LHC Run-2 constraints

The ATLAS and the CMS Collaborations have searched for $Z^{\prime}$ boson
resonance at the LHC Run-2 with $\sqrt{s}=13$ TeV. The most stringent bounds
on the $Z^{\prime}$ boson production cross section times branching ratio have
been obtained by using the dilepton final state. For the so-called sequential
SM $Z^{\prime}$ ($Z^{\prime}_{SSM}$) [42], which has exactly the same
couplings with the SM fermions as those of the SM $Z$ boson, we have
$m_{Z^{\prime}_{SSM}}\geq 5.1$ TeV from the ATLAS 2019 results [43] and
$m_{Z^{\prime}_{SSM}}\geq 5.15$ TeV from the CMS 2019 results [44],
respectively. We interpret these ATLAS and CMS results into the U(1)X
$Z^{\prime}$ boson case and derive the LHC Run-2 constraints on $x_{H}$,
$\alpha_{g_{X}}$ and $m_{Z^{\prime}}$.

###  5.1 Bounds from CMS 2019 results

We calculate the dilepton production cross section for the process $pp\to
Z^{\prime}+X\to\ell^{+}\ell^{-}+X$. The differential cross section with
respect to the invariant mass $M_{\ell\ell}$ of the final state dilepton is
described as

$\displaystyle\frac{d\sigma}{dM_{\ell\ell}}=\sum_{a,b}\int^{1}_{\frac{M^{2}_{\ell\ell}}{E^{2}_{\rm
CM}}}dx_{1}\frac{2M_{\ell\ell}}{x_{1}E^{2}_{\rm
CM}}f_{a}(x_{1},M^{2}_{\ell\ell})f_{b}\left(\frac{M^{2}_{\ell\ell}}{x_{1}E^{2}_{\rm
CM}},M^{2}_{\ell\ell}\right)\hat{\sigma}(\bar{q}q\to
Z^{\prime}\to\ell^{+}\ell^{-}),$ (19)

where $f_{a}$ is the parton distribution function (PDF) for a parton $a$, and
$E_{\rm CM}=13$ TeV is the center-of-mass energy of the LHC Run-2. In our
numerical analysis, we employ CTEQ6L [45] for the PDFs. In the U(1)X model,
the cross sections for the colliding partons are given by

$\displaystyle\hat{\sigma}(\bar{u}u\rightarrow Z^{\prime}\to\ell^{+}\ell^{-})$
$\displaystyle=$
$\displaystyle\frac{\pi\alpha_{g_{X}}^{2}}{81}\frac{M_{\ell\ell}^{2}}{(M_{\ell\ell}^{2}-m_{Z^{\prime}}^{2})^{2}+m_{Z^{\prime}}^{2}\Gamma_{Z^{\prime}}^{2}}(85x_{H}^{4}+152x_{H}^{3}+104x_{H}^{2}+32x_{H}+4),$
$\displaystyle\hat{\sigma}(\bar{d}d\rightarrow Z^{\prime}\to\ell^{+}\ell^{-})$
$\displaystyle=$
$\displaystyle\frac{\pi\alpha_{g_{X}}^{2}}{81}\frac{M_{\ell\ell}^{2}}{(M_{\ell\ell}^{2}-m_{Z^{\prime}}^{2})^{2}+m_{Z^{\prime}}^{2}\Gamma_{Z^{\prime}}^{2}}(25x_{H}^{4}+20x_{H}^{3}+8x_{H}^{2}+8x_{H}+4),$
(20)

where the total decay width of the $Z^{\prime}$ boson is given in Eq. (17).

In interpreting the CMS results into the U(1)X $Z^{\prime}$ boson case, we
follow the strategy in Refs. [33, 36]. In order to take into account the
difference between the parton distribution functions used in the CMS analysis
and our analysis, and QCD corrections of the process, we scale our resultant
cross section by a $k$-factor as a function of $m_{Z^{\prime}}$, which is
determined so as to fit our cross section calculation for the
$Z^{\prime}_{SSM}$ boson to the theoretical line presented in Ref. [44].
According to the analysis by the CMS Collaboration, we integrate the
differential cross section for the range of $0.97\;m_{Z^{\prime}}\leq
M_{\ell\ell}\leq 1.03\;m_{Z^{\prime}}$ [44] and obtain the cross section as a
function of $x_{H}$, $\alpha_{g_{X}}$ and $m_{Z^{\prime}}$. In the CMS
analysis, the limits are set on the ratio of the cross sections mediated by
$Z^{\prime}$ and $Z$ bosons,

$\displaystyle R_{\sigma}$ $\displaystyle=$ $\displaystyle\frac{\sigma(pp\to
Z^{\prime}+X\to\ell\ell+X)}{\sigma(pp\to Z+X\to\ell\ell+X)},$ (21)

as a function of $m_{Z^{\prime}}$, where the cross section $\sigma(pp\to
Z+X\to\ell\ell+X)$ is evaluated in the mass window of $60$ GeV$\leq
M_{\ell\ell}\leq 120$ GeV which is predicted to be $1928$ pb at the LHC Run-2
[46].

###  5.2 Bounds from ATLAS 2019 results

To interpret the ATLAS 2019 results into the U(1)X model, we use the narrow
width approximation to evaluate the $Z^{\prime}$ boson production cross
section. In this approximation, the total $Z^{\prime}$ boson production cross
section is given by

$\displaystyle\sigma(pp\to Z^{\prime})$ $\displaystyle=$ $\displaystyle
2\sum_{q,\bar{q}}\int dx\int
dy\>f_{q}(x,Q)\>f_{\bar{q}}(y,Q)\>\hat{\sigma}(\hat{s}),$ (22)

where $\hat{s}=xys$ is the invariant mass squared of the colliding partons
(quarks) with a center-of-mass energy $\sqrt{s}=13$ TeV, and the $Z^{\prime}$
boson production cross section at the parton level is expressed as

$\displaystyle\hat{\sigma}(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{4\pi^{2}}{3}\frac{\Gamma(Z^{\prime}\to
q\bar{q})}{m_{Z^{\prime}}}\>\delta(\hat{s}-m_{Z^{\prime}}^{2}).$ (23)

Here, $\Gamma(Z^{\prime}\to q\bar{q})$ is the $Z^{\prime}$ boson partial decay
width to $q\bar{q}$. For the up-type and down-type quarks, respectively,

$\displaystyle\Gamma(Z^{\prime}\to u\bar{u})$ $\displaystyle=$
$\displaystyle\frac{g_{X}^{2}}{72\pi}(2+10x_{H}+17x_{H}^{2})m_{Z^{\prime}},$
$\displaystyle\Gamma(Z^{\prime}\to d\bar{d})$ $\displaystyle=$
$\displaystyle\frac{g_{X}^{2}}{72\pi}(2-2x_{H}+5x_{H}^{2})m_{Z^{\prime}}.$
(24)

We then simply express the dilepton production cross section as

$\displaystyle\sigma(pp\to Z^{\prime}\to\ell\bar{\ell})$ $\displaystyle\simeq$
$\displaystyle\sigma(pp\to Z^{\prime})\times{\rm
BR}(Z^{\prime}\to\ell\bar{\ell})$ (25)

with the branching ratio to $Z^{\prime}\to\ell\ell$, where $\ell=e$ or $\mu$.
In order to take into account the difference of the parton distribution
functions used in the ATLAS analysis and our analysis, and QCD corrections of
the process, we scale our resultant cross section by a $k$-factor as a
function of $m_{Z^{\prime}}$, which is determined so as to fit our cross
section calculation for $Z^{\prime}_{SSM}$ boson to the theoretical line
presented in [43].

##  6 BBN Bound

The right-handed neutrinos ($\nu_{R}$) in our model are in thermal equilibrium
with the SM particles in the early universe through $Z^{\prime}$ boson
interaction. The thermal averaged cross section at the temperature $T\ll
m_{Z^{\prime}}$ is expressed as

$\displaystyle\langle\sigma v\rangle$ $\displaystyle=$
$\displaystyle\sum_{f}\langle\sigma
v(f\bar{f}\leftrightarrow\nu_{R}\overline{\nu_{R}})\rangle,$ (26)

with

$\displaystyle\langle\sigma
v(u\bar{u}\leftrightarrow\nu_{R}\overline{\nu_{R}})\rangle$
$\displaystyle\simeq$ $\displaystyle\frac{g_{X}^{4}N_{c}T^{2}}{18\pi
m_{Z^{\prime}}^{4}}(2+10x_{H}+17x_{H}^{2}),$ (27) $\displaystyle\langle\sigma
v(d\bar{d}\leftrightarrow\nu_{R}\overline{\nu_{R}})\rangle$
$\displaystyle\simeq$ $\displaystyle\frac{g_{X}^{4}N_{c}T^{2}}{18\pi
m_{Z^{\prime}}^{4}}(2-2x_{H}+5x_{H}^{2}),$ (28) $\displaystyle\langle\sigma
v(\ell\bar{\ell}\leftrightarrow\nu_{R}\overline{\nu_{R}})\rangle$
$\displaystyle\simeq$ $\displaystyle\frac{g_{X}^{4}T^{2}}{2\pi
m_{Z^{\prime}}^{4}}(2+6x_{H}+5x_{H}^{2}),$ (29) $\displaystyle\langle\sigma
v(\nu_{L}\overline{\nu_{L}}\leftrightarrow\nu_{R}\overline{\nu_{R}})\rangle$
$\displaystyle\simeq$ $\displaystyle\frac{g_{X}^{4}T^{2}}{2\pi
m_{Z^{\prime}}^{4}}(1+x_{H})^{2},$ (30)

and the color factor $N_{c}=3$ for quarks. The decoupling temperature $T_{\rm
dec}$ of $\nu_{R}$ from the thermal bath is evaluated by

$\displaystyle\langle\sigma v\rangle n_{\nu_{R}}|_{T=T_{\rm dec}}$
$\displaystyle=$ $\displaystyle H(T_{\rm dec}),$ (31)

where $H=\sqrt{\frac{\rho_{\rm rad}}{3M_{\rm P}^{2}}}$, and $\rho_{\rm
rad}=\frac{\pi^{2}}{30}g_{\star}T^{4}$.

After the decoupling, we parameterize the energy density of $\nu_{R}$ at the
BBN era as

$\displaystyle\rho_{\nu_{R}}=\Delta
N_{\mathrm{eff}}\,\frac{7}{4}\,\frac{\pi^{2}}{30}\,T_{\rm BBN}^{4},$ (32)

where $T_{\rm BBN}\simeq 1$ MeV is the photon temperature. The total effective
number of neutrinos is expressed as

$\displaystyle N_{\mathrm{eff}}=N_{\mathrm{eff}}^{\mathrm{SM}}+\Delta
N_{\mathrm{eff}},$ (33)

Using the Planck result [41],

$\displaystyle N_{\mathrm{eff}}=2.99^{+0.34}_{-0.33}$ (34)

at $95$% C.L. ($2.99\pm 0.17$ at 68% C.L.), and the SM prediction
$N_{\mathrm{eff}}^{\mathrm{SM}}=3.046$ (see e.g., Refs. [47, 48, 49, 50, 51]),
we obtain the upper bound on $\Delta N_{\mathrm{eff}}\leq 0.284$ at $95$% C.L.
($0.114$ at $68$% C.L.). The contribution of the two right-handed neutrinos is
estimated as

$\displaystyle\Delta
N_{\mathrm{eff}}=2\times\left(\frac{g_{\star}(T_{BBN})}{g_{\star}(T_{\rm
dec})}\right)^{4/3},$ (35)

where $g_{\star}(T_{BBN})=10.75$, and $g_{\star}(T_{\rm dec})$ is the total
relativistic degrees of freedom of SM particles at $T=T_{\rm dec}$. To satisfy
the upper bound on $\Delta N_{\mathrm{eff}}$, we find a lower bound on $T_{\rm
dec}$, which leads to a lower bound on $v_{X}=m_{Z^{\prime}}/g_{X}$ for a
fixed $x_{H}$ value. Note that in our model, the minimum value of $\Delta
N_{\mathrm{eff}}$ is found to be $\Delta N_{\mathrm{eff}}\geq 0.0937$ by using
$g_{\star}(T_{\rm dec}>m_{t})=106.75$. This value can be tested in the
upcoming the CMB-S4 experiment [52].

##  7 Combined results

Figure 3:  The solid blue line is the lower bound on $\alpha_{g_{X}}$ as a
function of $m_{Z^{\prime}}$ to reproduce the observed DM relic density of the
Planck result [41]. The solid (dashed) red line shows the upper bound on
$\alpha_{g_{X}}$ obtained by the ATLAS 2019 results [43] (the CMS 2019 results
[44]). The lighter red horizontal solid line denotes the upper bound for the
narrow width approximation to be valid, where we have set
$\Gamma_{Z^{\prime}}/m_{Z^{\prime}}<3\%$. The black solid (dashed) line shows
the 95% C.L. (68% C.L.) BBN bound on $\Delta N_{\mathrm{eff}}$. The green-
shaded region satisfies all constraints.

(a)

(b)

(c)

(d)

Figure 4:  Allowed parameter regions in the ($x_{H}$, $\alpha_{g_{X}}$)-plain
for various $m_{Z^{\prime}}$ values. 4 is for $m_{Z^{\prime}}=5.25$ TeV. The
(blue) convex-downward solid line shows the cosmological lower bound on
$\alpha_{g_{X}}$ as a function of $x_{H}$. The (red) convex-upward solid
(dashed) line shows the upper bound on $\alpha_{g_{X}}$ obtained from the
$Z^{\prime}$ boson search by the ATLAS [43] (CMS [44]) Collaboration, along
with the upper bound (lighter red horizontal solid line) for which the narrow
width approximation is valid (we have set
$\Gamma_{Z^{\prime}}/m_{Z^{\prime}}\leq 3\%$). The black solid (dashed) line
shows the 95% C.L. (68% C.L.) BBN bound on $\Delta N_{\mathrm{eff}}$. The
green shaded region satisfies all constraints. 4, 4 and 4 are the same as 4,
but $m_{Z^{\prime}}=5$ TeV, 4.75 TeV and 4.5 TeV, respectively.

Now we combine all the constraints that we have obtained in the previous
sections from the DM relic density, the LHC constraints, and the BBN bound. In
Fig. 3, we show the allowed region in the ($m_{Z^{\prime}}$,
$\alpha_{g_{X}}$)-plain for fixed $x_{H}=-0.5$, as an example. The solid blue
line is the lower bound on $\alpha_{g_{X}}$ as a function of $m_{Z^{\prime}}$
to reproduce the observed DM relic density of the Planck result [41]. The
solid (dashed) red line shows the upper bound on $\alpha_{g_{X}}$ obtained by
the ATLAS 2019 results [43] (the CMS 2019 results [44]). We also show the
upper bound (lighter red horizontal solid line) for the narrow width
approximation to be valid, where we have set
$\Gamma_{Z^{\prime}}/m_{Z^{\prime}}<3\%$. The black solid (dashed) line shows
the 95% C.L. (68% C.L.) BBN bound on $\Delta N_{\mathrm{eff}}$. The green-
shaded region satisfies all constraints. These three constraints are
complementary to narrow down the allowed region to be $m_{Z^{\prime}}\gtrsim
4.5$ TeV and $0.005\lesssim\alpha_{g_{X}}\lesssim 0.05$.

In Fig. 4, we show allowed parameter regions in the ($x_{H}$,
$\alpha_{g_{X}}$)-plain for various $m_{Z^{\prime}}$ values. Fig. 44 is for
$m_{Z^{\prime}}=5.25$ TeV. The (blue) convex-downward solid line shows the
lower bound on $\alpha_{g_{X}}$ as a function of $x_{H}$ to reproduce the
observed DM relic density. The (red) convex-upward solid (dashed) line shows
the upper bound on $\alpha_{g_{X}}$ obtained from the search results for
$Z^{\prime}$ boson resonance by the ATLAS [43] (CMS [44]) Collaboration, along
with the upper bound (lighter red horizontal solid line) for which the narrow
width approximation is valid (we have set
$\Gamma_{Z^{\prime}}/m_{Z^{\prime}}\leq 3\%$). The green shaded region
satisfies all constraints. These three constraints are complementary to narrow
down the allowed region to be $-0.8\lesssim x_{H}\lesssim-0.2$ and
$0.01\lesssim\alpha_{g_{X}}\lesssim 0.03$. Figs. 44, 44 and 44 are the same as
Fig. 44, but $m_{Z^{\prime}}=5$ TeV, 4.75 TeV and 4.5 TeV, respectively. From
Fig. 44, the allowed region to satisfy these three constraints indicates
$-0.65\lesssim x_{H}\lesssim-0.45$ and $0.008\lesssim\alpha_{g_{X}}\lesssim
0.009$ for fixed $m_{Z^{\prime}}=4.5$ TeV. As $m_{Z^{\prime}}$ decreases, the
LHC upper bound lines are shifted downward, while the DM lower bound line
remains almost the same (it slightly moves to downward). Therefore, the
allowed region between the LHC upper bounds and the DM lower bound narrows. On
the other hand, the BBN bound remains almost the same, so that the green
shaded region disappears for $m_{Z^{\prime}}<4.5$ TeV, as shown in Fig. 44.

##  8 Determining the number of RHNs at High-Luminosity LHC

The SM neutrinos are Dirac particles in our model. This is in sharp contrast
with usual U(1)X extension of the SM, where RHNs are heavy Majorana states.
Since the RHNs are singlet under the SM gauge groups and the Dirac Yukawa
coupling constants are very small to reproduce the neutirno oscillation data,
the $Z^{\prime}$ boson interaction is the only way for RHNs to communicate
with the SM particles. The ATLAS and CMS Collaborations continue to search for
$Z^{\prime}$ boson resonance. Once discovered, the $Z^{\prime}$ boson will
allows us to investigate physics of the RHNs through precise measurements of
$Z^{\prime}$ boson properties. In this section, we discuss an implication of
the Dirac neutrinos in our model to LHC physics.

If the RHNs are heavy Majorana particles as in the minimal U(1)X model, a pair
of RHNs, if kinematically allowed, can be produced by $Z^{\prime}$ boson decay
once $Z^{\prime}$ boson is produced at the LHC. A produced RHN subsequently
decays to weak gauge bosons/Higgs boson plus lepton. Because of the Majorana
nature of the RHN, the final states include same-sign leptons. This is the
“smoking-gun” signature of the lepton number violation, and we can expect a
high possibility to detect such final states with much less SM background. For
studies of such a signature at the LHC, see, for example, Refs. [53, 54, 55,
56, 57, 58]. See also Refs. [59, 60] for studies of RHN production at the
International Linear Collider.

The Majorana RHNs are heavy and can be produced only if they are kinematically
allowed, while the Dirac neutrinos in our model are always included in the
$Z^{\prime}$ boson decay products. Although, like the SM left-handed
neutrinos, RHNs produced at colliders are hard to detect directly, there is a
possibility to measure them indirectly. Recall the great success of the LEP
experiments that precisely measured the $Z$ boson decay width and determined
the number of the SM (left-handed) neutrinos to be three [61]. We notice that
the $Z^{\prime}$ production is analogous to the $Z$ production at the LEP.
Although the RHNs produced by the $Z^{\prime}$ boson decay will be
undetectable, the total $Z^{\prime}$ boson decay width carries the information
of the invisible decay width. A precise measurement of the $Z^{\prime}$ boson
cross section at the LHC may reveal the existence of RHNs. To illustrate this
idea, we calculate the differential the cross section for the process with the
dilepton final state, $pp\to\ell^{+}\ell^{-}+X$ ($\ell=e$ or $\mu$) mediated
by photon, $Z$ boson and $Z^{\prime}$ boson at the LHC with a collider energy
$\sqrt{s}=14$ TeV.

Figure 5:  The differential cross section for $pp\to\ell^{+}\ell^{-}+X$ at the
High-Luminosity LHC with $\sqrt{s}=14$ TeV for $m_{Z^{\prime}}=4.5$ TeV,
$\alpha_{g_{X}}=0.008$, and $x_{H}=-0.5$. The solid and dashed curves
correspond to the results for $N(\nu_{R})=2$ and $0$, respectively. The
horizontal long-dashed line represents the SM cross section, which is
negligibly small compared with the $Z^{\prime}$ boson mediated process.

For the $Z^{\prime}$ boson mediated process, we consider two cases,
$N(\nu_{R})=0$ and $N(\nu_{R})=2$, where $N(\nu_{R})$ is the number of light
RHNs. The case $N(\nu_{R})=0$ corresponds to the minimal U(1)X model with all
RHN Majorana neutrino masses $>m_{Z^{\prime}}/2$. The total $Z^{\prime}$ boson
decay width given in Eq. (17) corresponds to $N(\nu_{R})=2$ while we add $-2$
to the inside of the bracket on the right-hand side for $N(\nu_{R})=0$. Fig. 5
shows the differential cross sections for $pp\to\ell^{+}\ell^{-}+X$ ($\ell=e$
or $\mu$) for $m_{Z^{\prime}}=4.5$ TeV, $\alpha_{g_{X}}=0.008$, and
$x_{H}=-0.5$, along with the SM cross section mediated by the $Z$-boson and
photon (horizontal long-dashed line). The solid and dashed curves correspond
to the results for $N(\nu_{R})=2$ and $0$, respectively. The dependence of the
total decay width on $N(\nu_{R})$ reflects the resultant cross sections (width
and height at the peak of the cross section). When we choose a kinematical
region for the invariant mass of the dilepton final state to be in the range
of $m_{Z^{\prime}}-100\leq m_{ll}[{\rm GeV}]\leq M_{Z^{\prime}}+100$, for
example, the signal events of $60.1$ and $80.3$ for $N(\nu_{R})=2$ and $0$,
respectively, would be observed with the goal integrated luminosity of 3000/fb
at the High-Luminosity LHC. The difference between $N(\nu_{R})=2$ and $0$ are
distinguishable with about $2.6\sigma$ significance. If we simply combine the
events for the final di-electon and di-muon, the signal events become $120.2$
and $160.6$ for $N(\nu_{R})=2$ and $0$, respectively, the significant
increases to $3.7\sigma$.

##  9 Conclusions and Discussions

We have proposed a minimal SUSY U(1)X model with conserved R-parity. The gauge
group U(1)X is a generalization of the well-known U(1)B-L, and the U(1)X
charge of an MSSM field is assigned as a linear combination of its $B-L$
charge and U(1) hypercharge, $Q^{f}_{X}=x_{H}\,Q_{Y}^{f}+Q_{B-L}^{f}$, with a
common parameter $x_{H}$. Three right-handed neutrino chiral superfields are
introduced, which make the model free from all gauge and mixed gauge-
gravitational anomalies, but no U(1)X Higgs fields with U(1)X charge $\pm 2$
are included in the particle content. We assign an even R-parity to one right-
handed neutrino superfield $\Psi$, while an odd parity for the other two
right-handed neutrino superfields, as usual. The scalar component of $\Psi$
($\phi$) plays a role of the U(1)X Higgs field to break the U(1)X gauge
symmetry. We have shown that the negative mass squared for $\phi$ is
radiatively generated by the RG evolution, and hence the U(1)X symmetry is
radiatively broken. Due to the absence of U(1)X Higgs fields with U(1)X charge
$\pm 2$ and our novel R-parity assignment, three light neutrinos consist of
one massless neutrino and two massive Dirac neutrinos. Since R-parity is
conserved, the LSP neutralino is a prime candidate for the dark matter. Our
model offers a new dark matter candidate ($\chi_{\ell}$) which is a linear
combination of the fermion component of $\Psi$ and the U(1)X gaugino. Assuming
this $\chi_{\ell}$ is the LSP, we have investigated the dark matter
phenomenology to find the parameter region that reproduces the observed dark
matter relic density. We have found that an enhancement of dark matter pair
annihilation process through $Z^{\prime}$ boson resonance in the $s$-channel
is crucial for reproducing the observed dark matter density and hence the dark
matter mass is close to half of the $Z^{\prime}$ boson mass. As a result, the
dark matter relic density constrains three parameters of the model,
$\alpha_{g_{X}}$, $m_{Z^{\prime}}$ and $x_{H}$. The $Z^{\prime}$ boson
resonance has been searched for at the LHC Ran-2. Using the final LHC Run-2
results reported by the ATLAS and CMS Collaborations, we have derived the
constraints on the three parameters, $\alpha_{g_{X}}$, $m_{Z^{\prime}}$ and
$x_{H}$. In the early universe, two right-handed components of the light Dirac
neutrinos are in thermal equilibrium with the Standard Model particle plasma
through the U(1)X gauge interaction. Their energy density increases the
expansion rate of the universe at the Big Bang Nucleosynthesis era. To avoid
disrupting the success of the Big Bang Nucleosynthesis, we have found the
lower bound on the scale of U(1)X symmetry breaking,
$v_{X}=m_{Z^{\prime}}/g_{X}$, for a fixed value of $x_{H}$. Combining three
constraints obtained from the observed dark matter relic density, the
$Z^{\prime}$ boson resonance search at the LHC Run-2 and the success of Big
Bang Nucleosynthesis, we have narrowed down the allowed parameter region. We
have also discussed a possibility of determining the number of light right-
handed neutrinos at the High-Luminosity LHC. The $Z^{\prime}$ boson, once
discovered at the LHC, can be a probe for this purpose, since the shape of the
$Z^{\prime}$ boson resonance as a function of the invariant mass of the final
state dilepton carries the information of the number of light right-handed
neutrinos.

Since the light neutrinos are Dirac particles in our model, their Dirac Yukawa
couplings must be extremely small (${\cal O}(10^{-12})$) to fit the neutrino
oscillation data. We need to consider a natural way of realizing such
extremely small Yukawa couplings. To solve this problem, we may extend our
model in 4-dimension to the brane-world framework with 5-dimensional warped
space-time [62]. Arranging the bulk mass parameters for the bulk
hypermultiplets corresponding to matter and Higgs fields in our model, we can
obtain a large hierarchy among parameters in 4-dimensional effective theory
with mildly hierarchical model parameters in the original 5-dimensional
theory. For example, we may arrange the bulk mass parameters for R-parity even
right-handed neutrino superfields to localize around one brane, while for the
lepton doublet superfileds to localize around the other brane. In this way,
the 4-dimensional effective Yukawa coupling can be exponentially suppressed.
For such an idea, see, for example, Ref. [63].

## Acknowledgements

The work of N.O. is supported in part by the United States Department of
Energy grant (DE-SC0012447). S.O. and D.-s.T. are supported by the
mathematical and theoretical physics unit (Hikami unit) of the Okinawa
Institute of Science and Technology Graduate University. S.O. and D.-s.T. are
also supported by Japan Society for the Promotion of Science (JSPS), Grant-in-
Aid for Scientific Research (C), Grant Number JP18K03661.

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