# Singlino-dominated dark matter in $Z_{3}$-NMSSM

Haijing Zhou  Junjie Cao  Jingwei Lian  Di Zhang

###### Abstract

Singlino-dominated dark matter properties are investigated in the $Z_{3}$
Next-to-Minimal Supersymmetric Standard Model, producing superweak
interactions with nucleons involved in dark matter direct detection
experiments. Approximate analytical formulas describing the dark matter
abundance and cross section in the scattering with nucleons are used to
illustrate a dependence on theoretical parameters in neutralino and Higgs
sectors. It is shown that the measured abundance requires a sizable
singlet–doublet Higgs coupling parameter $\lambda$, while the experimental
detection results prefer a small $\lambda$. The parameter space is then
surveyed using a nest sampling technique guided by a likelihood function
containing various observables in dark matter, Higgs, and B physics, such as
the abundance and the scattering cross section. It is demonstrated that dark
matter can achieve the correct abundance through
$\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to t\bar{t}$ or co-annihilation with
higgsinos. The former process provides significantly larger Bayesian evidence
than the latter, but this will be examined by the near-future PandaX-4T
experiment. If the experiment shows no signs of dark matter, it will become
highly disfavored. Furthermore, four cases are summarized to suppress dark
matter scattering with nucleons, namely, a small $\lambda$ and three kinds of
cancellation between different contributions.

##  1 Introduction

The 2012 discovery of the Higgs boson at the Large Hadron Collider (LHC) [1,
2] confirmed the correctness of the Higgs mechanism as the origin of the
masses of subatomic particles. However, the hierarchy problem caused by
sizeable radiative corrections to the Higgs mass term implies there should be
new physics between the electroweak scale and the Planck scale. In addition,
the Standard Model (SM) cannot explain the existence of dark matter (DM).
However, current astronomical observations have confirmed that the universe is
composed of $27\%$ DM [3, 4]. Therefore, new DM particle candidates and new
physics are required beyond the SM. Establishing the structural nature of DM
is one of the most fundamental open questions in cosmology and particle
physics.

Among multiple proposed theories, the most widely accepted are supersymmetry
(SUSY) models with R-parity conservation, i.e., the minimal supersymmetric
standard model (MSSM) [5, 6] and the next-to-minimal supersymmetric standard
model (NMSSM) [8, 7, 9], which provide elegant solutions to the hierarchy
problem by introducing contributions from superpartners to the Higgs mass
term. In addition, R-parity conservation ensures that the lightest neutralino
is a stable neutral particle if it is the lightest supersymmetric particle
(LSP), and it may be an excellent DM candidate. MSSM, the most economic
realization of SUSY, exhibits several attractive features, but also includes
some challenges (e.g., the “$\mu$-problem” [10] and “little hierarchy problem”
[11]) that have become exacerbated in recent years by the first run of the LHC
experiments. This was particularly true for the uncomfortably large mass of
the discovered Higgs boson $m_{h}\simeq 125~{}{\rm GeV}$ [20, 12, 13, 14, 18,
19, 15, 17, 16]. Alternatively, the NMSSM solves the $\mu$-problem by adding a
singlet chiral superfield $\hat{S}$ to the MSSM. In this process, the $\mu$
parameter is replaced by a dynamic quantity $\mu_{eff}=\lambda v_{s}$ when $S$
develops a vacuum expectation value (VEV) $v_{s}$, the magnitude of which is
naturally at the electroweak scale [8, 9]. Furthermore, the SM-like Higgs
squared mass can be enhanced by an additional tree-level contribution
$\lambda^{2}v^{2}\sin^{2}2\beta$ and the singlet-doublet Higgs mixing [12, 13,
15, 17], where $\tan\beta\equiv{v_{u}}/{v_{d}}$ with $v_{u}$ and $v_{d}$
representing the VEVs of the doublet Higgs fields and $v^{2}\equiv
v_{u}^{2}+v_{d}^{2}$. As a byproduct, the neutralino sector includes a
fermionic partner of $S$ (singlino) in addition to neutral electroweak
gauginos (bino and wino) and the neutral fermionic partner of the Higgs
doublets (higgsinos). Since the coupling of a singlino field with SM particles
may be very weak, this study focuses on the case of a singlino-dominated
neutralino as a DM candidate [25, 21, 22, 23, 24, 26]. This scenario is
feasible when the Yukawa-like couplings satisfy $\lambda\geq 2\kappa$ and the
gauginos are assumed to be heavier than the higgsinos.

In the NMSSM, the Z boson mass is related to the higgsino mass $\mu_{eff}$,
and a natural prediction of $m_{Z}$ favors light higgsinos up to several
hundred GeV [27]. Generally, the NMSSM with $\mu_{eff}\lesssim 500~{}\rm{GeV}$
is considered to be a natural NMSSM [18, 27, 19, 28, 29]. However, given the
constraints of recent experiments (e.g., the searches for electroweakinos [30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49],
the WMAP/Planck experiments [3, 4], and the DM direct detection (DD) and
indirect detection (ID) experiments [50, 51, 52, 53, 54]), a large portion of
the parameter space in the natural NMSSM has been strongly constrained. As a
result, the following isolated and narrow parameter spaces with a singlino-
dominated $\tilde{\chi}^{0}_{1}$ are preferred [55, 56, 57, 58]:

  * •

$\lambda\simeq 2\kappa$ with $\lambda\lesssim 0.05$, where
$\tilde{\chi}^{0}_{1}$ mainly co-annihilates with higgsinos to achieve the
measured abundance [55].

  * •

$\kappa\sim 0.01$, $\lambda\lesssim 0.2$, and at least one light singlet-
dominated Higgs boson [56]. Here, $\tilde{\chi}^{0}_{1}$ annihilates in
certain funnel regions and the higgsinos decay in a complex manner to satisfy
the LHC constraints.

These conclusions are applicable for $|\mu_{eff}|\lesssim 500~{}{\rm GeV}$, or
equivalently, the fine-tuning criterion ${{\Delta}m_{Z}}\lesssim 50$, where
$\Delta m_{Z}$ defined in [59] parameterizes the sensitivity of $m_{Z}$ to the
SUSY parameters at the weak scale. Given this situation and the fact that the
fine-tuning criteria lacks a confirmed scientific basis and may reflect
personal prejudice, we update the study by Cao et al. [55] to improve these
conclusions. Specifically, we do not require ${\Delta}m_{Z}$ to be less than
50 but impose the condition that $\mu_{eff}\leq 1~{}{\rm TeV}$. We adopt an
advanced MultiNest algorithm [60, 61] to perform a sophisticated scan over the
parameter space of the $Z_{3}$-invariant NMSSM with a singlino-dominated DM.
This algorithm is much more efficient than the other algorithms (e.g., the
Markov Chain method [62] adopted in [55]) in providing comprehensive
information of the space to reveal the underlying physics, although it usually
involves a tremendous amount of calculation. To the best of our knowledge, few
researchers have used it to study the NMSSM phenomenology [63, 64, 65, 66, 67,
68]. We also present a description of DM annihilation and the mechanisms used
to suppress the DM nucleon scattering cross section through analytical
formulas and numerical analysis. Some of the formulas are new, and some are
consistent with excellent related works [69, 70, 71, 73, 72]. Evidently, such
an analysis is helpful to understand the DM physics.

The remainder of this paper is organized as follows. In Section 2, we briefly
introduce the basic properties of the $Z_{3}$-invariant NMSSM, including the
Higgs and neutralino sections. We then demonstrate DM annihilation and
scattering cross sections for singlino-dominated $\tilde{\chi}^{0}_{1}$ with
nucleons using analytical formulas. In Section 3, we provide a brief
description of our scanning strategy. In Section 4, we investigate predictions
for surviving samples and the properties of singlino-dominated DM scenarios to
understand their distinctive features. Finally, Section 5 includes a
discussion of the results and corresponding conclusions.

##  2  Next-to-minimal supersymmetric standard model model

###  2.1 Fundamental NMSSM properties

As the simplest extension of the MSSM, the NMSSM includes one additional gauge
singlet Higgs field $\hat{S}$. The associated superpotential can be expressed
as follows [9, 8]:

$\displaystyle
W_{\mathrm{NMSSM}}=W_{\mathrm{MSSM}}+\lambda\hat{S}\hat{H_{u}}\hat{H_{d}}+\frac{1}{3}\kappa\hat{S}^{3},$
(2.1)

where $W_{\mathrm{MSSM}}$ is the MSSM superpotential without the $\mu$ term,
$\lambda$ and $\kappa$ are dimensionless parameters, and $\hat{H}_{u}$ and
$\hat{H}_{d}$ are the common Higgs superfields. It is the most general
R-parity-conservation superpotential satisfying a $Z_{3}$ discrete symmetry
given the considered field content.

Assuming CP-conservation, the Higgs sector of the $Z_{3}$-NMSSM is determined
by six parameters at the tree-level [8, 74]:

$\displaystyle\lambda,~{}\kappa,~{}A_{\lambda},~{}A_{\kappa},~{}\mu_{eff},~{}\tan\beta,$
(2.2)

where $A_{\lambda}$ and $A_{\kappa}$ are the soft trilinear coefficients
defined in Eq. (2.5) of Ref. [8]. In the base-vectors, $H_{\rm
SM}\equiv\sin{\beta}Re[H_{u}^{0}]+\cos{\beta}Re[H_{d}^{0}],H_{\rm
NSM}\equiv\cos{\beta}Re[H_{u}^{0}]-\sin{\beta}Re[H_{d}^{0}]$, and $H_{\rm
S}\equiv Re[S]$ for CP-even fields and $A_{\rm
NSM}\equiv\cos{\beta}Im[H_{u}^{0}]+\sin{\beta}Im[H_{d}^{0}]$ and $A_{\rm
S}\equiv Im[S])$ for CP-odd fields111$H_{u}^{0}$, and $H_{d}^{0}$ denote the
neutral component fields of the doublet scalar fields $H_{u}$ and $H_{d}$,
respectively., the three CP-even mass eigenstates $h_{i}=\\{h,H,h_{s}\\}$ and
two CP-odd Higgs mass eigenstates $a_{i}=\\{A_{H},a_{s}\\}$ are given as
follows:

$\displaystyle h_{i}$ $\displaystyle=$ $\displaystyle V_{h_{i}}^{\rm SM}H_{\rm
SM}+V_{h_{i}}^{\rm NSM}H_{\rm NSM}+V_{h_{i}}^{\rm S}H_{\rm S},$ $\displaystyle
a_{i}$ $\displaystyle=$ $\displaystyle V_{a_{i}}^{{}^{\prime}\rm NSM}A_{\rm
NSM}+V_{a_{i}}^{{}^{\prime}\rm S}A_{S},$ (2.3)

where $V$ and $V^{\prime}$ represent the unitary matrices to diagonalize the
corresponding Higgs squared mass matrix. In this work, we denote the physical
Higgs state with the largest $H_{\rm SM}$ component by the symbol $h$, which
is called the SM-like Higgs boson hereafter, and we denote the physical Higgs
state with the largest non-SM doublet (singlet) component $H_{\rm NSM}$
($H_{\rm S}$) by $H$ ($h_{s}$). We also denote the CP-even Higgs bosons by
$h_{1}$, $h_{2}$, and $h_{3}$, with $m_{h_{1}}<m_{h_{2}}<m_{h_{3}}$. The
latter notation is primarily for convenience. To date, the LHC experiments
have measured the couplings of the discovered Higgs boson with about $10\%$
uncertainty, and they revealed that the boson has roughly the same couplings
as the SM Higgs boson [75, 76]. These facts imply that $\sqrt{\left(V_{h}^{\rm
NSM}\right)^{2}+\left(V_{h}^{\rm S}\right)^{2}}\lesssim 0.1$ and $|V_{h}^{\rm
SM}|\sim 1$.

In the $Z_{3}$-NMSSM, mixtures of bino ($\tilde{B}^{0}$), wino
($\tilde{W}^{0}$), higgsino ($\tilde{H}_{d,u}^{0}$), and singlino
($\tilde{S}^{0}$) fields form neutralinos. Assuming a basis of
$\psi^{0}=(-i\tilde{B},-i\tilde{W}^{0},\tilde{H}_{d}^{0},\tilde{H}_{u}^{0},\tilde{S})$
produces the following neutralino mass matrix [8]:

$\displaystyle{\cal
M}=\left(\begin{array}[]{ccccc}M_{1}&0&-\frac{g_{1}v_{d}}{\sqrt{2}}&\frac{g_{1}v_{u}}{\sqrt{2}}&0\\\
&M_{2}&\frac{g_{2}v_{d}}{\sqrt{2}}&-\frac{g_{2}v_{u}}{\sqrt{2}}&0\\\
&&0&-\mu_{eff}&-\lambda v_{u}\\\ &&&0&-\lambda v_{d}\\\
&&&&\frac{2\kappa}{\lambda}\mu_{eff}\end{array}\right),$ (2.9)

where $M_{1}$, $M_{2}$, and $\mu_{eff}$ denote the soft breaking masses of the
bino, wino, and higgsinos, respectively. Diagonalizing the mass matrix with a
unitary matrix $N$ yields five mass eigenstates (ordered by mass):

$\displaystyle\tilde{\chi}_{i}^{0}=N_{i1}\tilde{B}^{0}+N_{i2}\tilde{W}_{3}^{0}+N_{i3}\tilde{H}_{d}^{0}+N_{i4}\tilde{H}_{u}^{0}+N_{i5}\tilde{S}.$
(2.10)

The lightest neutralino, $\tilde{\chi}_{1}^{0}$, acting as the DM candidate is
the focus of this work.

In the limit that $|M_{1}|$ and $|M_{2}|$ are much larger than $|\mu_{eff}|$
and $v$, $\tilde{\chi}_{1}^{0}$ is approximated by

$\displaystyle\tilde{\chi}_{1}^{0}\simeq
N_{13}\tilde{H}_{d}^{0}+N_{14}\tilde{H}_{u}^{0}+N_{15}\tilde{S}.$ (2.11)

If $|\kappa/\lambda|<1$, the dominant composition of $\tilde{\chi}_{1}^{0}$ is
a singlino. In this case, $\kappa$ is related to $m_{\tilde{\chi}_{1}^{0}}$ as
follows [69, 70]:

$\displaystyle\kappa$ $\displaystyle=$
$\displaystyle\frac{\lambda}{2\mu_{eff}}\left[m_{\tilde{\chi}_{1}^{0}}-\frac{\lambda^{2}v^{2}\left(m_{\tilde{\chi}_{1}^{0}}-\mu_{eff}\sin
2\beta\right)}{m_{\tilde{\chi}_{1}^{0}}^{2}-\mu_{eff}^{2}}\right],$ (2.12)

and the elements of the matrix $N$ exhibit the following relationships [69,
71, 72, 73, 70]:

$\displaystyle\frac{N_{13}}{N_{15}}=\frac{\lambda
v}{\mu_{eff}}\,\frac{(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})\sin\beta-\cos\beta}{1-\left(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff}\right)^{2}}\,,$
(2.13) $\displaystyle\frac{N_{14}}{N_{15}}=\frac{\lambda
v}{\mu_{eff}}\,\frac{(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})\cos\beta-\sin\beta}{1-\left(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff}\right)^{2}}.$
(2.14)

Thus,

$\displaystyle N_{15}^{2}$
$\displaystyle\simeq\left(1+\frac{N^{2}_{13}}{N^{2}_{15}}+\frac{N^{2}_{14}}{N^{2}_{15}}\right)^{-1}$
$\displaystyle\simeq\frac{\left[1-(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})^{2}\right]^{2}}{\left[(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})^{2}-2(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})\sin
2\beta+1\right]\left({\lambda
v}/{\mu_{eff}}\right)^{2}+\left[1-(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})^{2}\right]^{2}}.$

The higgsino and singlino fractions in $\tilde{\chi}_{1}^{0}$ can be defined
as $Z_{h}=N_{13}^{2}+N_{14}^{2}$ and $Z_{s}=N_{15}^{2}$, respectively. The
ratio of $Z_{h}$ to $Z_{s}$ can then be expressed as follows:

$\displaystyle\frac{Z_{h}}{Z_{s}}=\left(\frac{\lambda
v}{\mu_{eff}}\right)^{\\!\\!2}\frac{\left(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff}\right)^{2}-2{(m_{\tilde{\chi}_{1}^{0}}}/{\mu_{eff}})\sin
2\beta+1}{\left[1-\left({m_{\tilde{\chi}_{1}^{0}}}/{\mu_{eff}}\right)^{2}\right]^{2}}\,.$
(2.16)

This expression implies that a small $\lambda$ can suppress the higgsino
fraction in $\tilde{\chi}_{1}^{0}$.

The couplings of DM to scalar Higgs states, the Z boson, and the Goldstone
boson $G^{0}$ are included in the calculation of DM annihilation. They take
the following form [8]:

$\displaystyle{\cal{L}}_{\rm NMSSM}\ni
C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}Z}Z_{\mu}\overline{\tilde{\chi}_{1}^{0}}\gamma^{\mu}\gamma_{5}\tilde{\chi}_{1}^{0}+iC_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}G^{0}}G^{0}\overline{\tilde{\chi}_{1}^{0}}\gamma_{5}\tilde{\chi}_{1}^{0}+C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}h_{i}}h_{i}\overline{\tilde{\chi}_{1}^{0}}\tilde{\chi}_{1}^{0}+iC_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}a_{i}}a_{i}\overline{\tilde{\chi}_{1}^{0}}\gamma_{5}\tilde{\chi}_{1}^{0},$

where the coefficients are given by

$\displaystyle C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}Z}\simeq$
$\displaystyle\frac{m_{Z}}{\sqrt{2}v}\big{(}\frac{\lambda
v}{\mu_{eff}}\big{)}^{2}\,\frac{Z_{s}\cos
2\beta}{1-(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})^{2}},$ (2.17) $\displaystyle
C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}G^{0}}\simeq$
$\displaystyle\frac{\sqrt{2}\mu_{eff}}{v}\left(\frac{\lambda
v}{\mu_{eff}}\right)^{2}\frac{Z_{s}(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})\cos
2\beta}{1-(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})^{2}},$ (2.18) $\displaystyle
C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}h_{i}}\simeq$ $\displaystyle
V_{h_{i}}^{\rm SM}C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}H_{\rm
SM}}+V_{h_{i}}^{\rm NSM}C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}H_{\rm
NSM}}+V_{h_{i}}^{\rm S}C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}H_{\rm S}}$
$\displaystyle\simeq$
$\displaystyle\frac{\sqrt{2}\mu_{eff}}{v}\,\big{(}\frac{\lambda
v}{\mu_{eff}}\big{)}^{2}\,\frac{Z_{s}V_{h_{i}}^{\rm
SM}(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff}-\sin
2\beta)}{1-(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})^{2}}-\frac{\sqrt{2}\mu_{eff}}{v}\,\big{(}\frac{\lambda
v}{\mu_{eff}}\big{)}^{2}\,\frac{Z_{s}V_{h_{i}}^{\rm NSM}\cos
2\beta}{1-(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})^{2}}$
$\displaystyle+\lambda\big{(}\frac{\lambda
v}{\mu_{eff}}\big{)}^{2}\frac{Z_{s}V_{h_{i}}^{\rm S}\sin
2\beta}{\sqrt{2}\big{[}1-(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})^{2}\big{]}}-\sqrt{2}\kappa
Z_{s}V_{h_{i}}^{\rm S}\left[1+\big{(}\frac{\lambda
v}{\mu_{eff}}\big{)}^{2}\frac{2}{1-(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})^{2}}\right],$
(2.19) $\displaystyle C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}a_{i}}\simeq$
$\displaystyle V_{a_{i}}^{{}^{\prime}\rm
NSM}C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}A_{\rm
NSM}}+V_{a_{i}}^{{}^{\prime}\rm
S}C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}A_{\rm S}}$ $\displaystyle\simeq$
$\displaystyle-\frac{\sqrt{2}\mu_{eff}}{v}\,\big{(}\frac{\lambda
v}{\mu_{eff}}\big{)}^{2}\,\frac{Z_{s}V_{a_{i}}^{{}^{\prime}\rm
NSM}(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff}\sin
2\beta-1)}{1-(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})^{2}}$
$\displaystyle+\lambda\big{(}\frac{\lambda
v}{\mu_{eff}}\big{)}^{2}\frac{Z_{s}V_{a_{i}}^{{}^{\prime}\rm S}\sin
2\beta}{\sqrt{2}\big{[}1-(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})^{2}\big{]}}-\sqrt{2}\kappa
Z_{s}V_{a_{i}}^{{}^{\prime}\rm S}\left[1+\big{(}\frac{\lambda
v}{\mu_{eff}}\big{)}^{2}\frac{2}{1-(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})^{2}}\right],$
(2.20)

where the approximation in Eq. (2.11) was applied. In addition, the coupling
of a CP-even Higgs $h_{i}$ to two quasi-pure singlet CP-odd Higgs $a_{s}$,
${\cal{L}}_{\rm NMSSM}\ni C_{h_{i}a_{s}a_{s}}h_{i}a_{s}a_{s}$, is relevant to
our study. Its coefficient $C_{h_{i}a_{s}a_{s}}$ is given as follows [77, 78]:

$\displaystyle C_{h_{i}a_{s}a_{s}}\simeq$ $\displaystyle\sqrt{2}\lambda
v\,V_{h_{i}}^{\rm SM}\,(\lambda+\kappa\sin 2\beta)+\sqrt{2}\lambda\kappa
v\,V_{h_{i}}^{\rm NSM}\cos 2\beta+\sqrt{2}V_{h_{i}}^{\rm
S}(2\kappa^{2}v_{s}-\kappa A_{\kappa}).$ (2.21)

In most cases, $C_{h_{s}a_{s}a_{s}}\gg C_{ha_{s}a_{s}}$, since $v_{s}\gg v$.

###  2.2 Dark matter relic density

In the NMSSM, the abundance of the singlino-dominated DM candidate
($\tilde{\chi}^{0}_{1}$) tends to be unacceptably large, due to small coupling
effects with SM particles. However, such a candidate can still achieve the
measured abundance [3, 4] by a specific mechanism, e.g., via $s$-channel
exchanges of gauge and Higgs bosons and $t$-channel exchanges of
electroweakinos and sfermions. It can also be achieved through co-annihilation
with heavier states, such as sleptons, the next-to-lightest neutralino
($\tilde{\chi}_{2}^{0}$), or the lightest chargino ($\tilde{\chi}_{1}^{\pm}$).

The thermal abundance of the DM at the freeze-out temperature
$T_{F}=m_{\tilde{\chi}_{1}^{0}}/x_{F}$ is given as follows [73]:

$\Omega
h^{2}=0.12\left(\frac{80}{g_{*}}\right)^{1/2}\left(\frac{x_{F}}{25}\right)\left(\frac{2.3\times
10^{-26}\ \mathrm{cm^{3}/s}}{\langle\sigma v\rangle_{x_{F}}}\right)\;,$ (2.22)

with a thermally-averaged annihilation cross section ${\langle\sigma
v\rangle_{x_{F}}}\equiv a+\frac{3b}{x_{F}}$. Dominant contributions to
$\langle\sigma v\rangle_{x_{F}}$ in acquiring the measured abundance are
discussed in Section 2.2 of Baum et al. [73]. The following conclusions were
presented:

1) $\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to t\bar{t}$ is usually the most
crucial channel for the abundance when $m_{\tilde{\chi}_{1}^{0}}>m_{t}$. It
proceeds through the s-channel exchanges of Higgs and Z bosons. Since the top
quark is massive, the contribution from the Z boson’s longitudinal
polarization to $\langle\sigma v\rangle_{x_{F}}$ is important in this process
when the Higgs mediators are far off-shell. $\langle\sigma v\rangle_{x_{F}}$
can then be approximately expressed as follows:

$\langle\sigma v\rangle_{x_{F}}^{t\bar{t}}\sim 2\times 10^{-26}\,\frac{{\rm
cm}^{3}}{{\rm
s}}\left(\frac{\left|C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}G^{0}}\right|}{0.1}\right)^{2}\left(\frac{m_{\tilde{\chi}_{1}^{0}}}{300\,{\rm
GeV}}\right)^{-2}.$ (2.23)

The measured abundance $\Omega h^{2}\sim 0.12$ [3, 4] is achieved through the
coupling of DM pairs to the Goldstone boson
$|C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}G^{0}}|\sim 0.1$, which requires
$\lambda\gtrsim 0.4$ according to Eq. (2.18).

2) $\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to\Phi_{i}\Phi_{j}$ is another
crucial annihilation process for the abundance, where $\Phi_{i}$ denotes a
scalar or pseudo-scalar Higgs mass eigenstate. Such processes occur via
$s$-channel Higgs or $Z$ boson exchange and $t$-channel neutralino exchange.
Since $\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to h_{i}h_{j}$ and
$\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to a_{i}a_{j}$ are $p$-wave
suppressed [73], and because
$C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}h_{i}}$ with $h_{i}=h,H$ in Eq.
(2.19) and $C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}A_{H}}$ in Eq. (2.20)
are usually smaller than 0.1, here we consider only the contribution from
$\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to h_{s}a_{s}$ to $\langle\sigma
v\rangle_{x_{F}}$. $\langle\sigma v\rangle_{x_{F}}^{h_{s}a_{s}}$ is then given
as follows [73, 79]:

$\displaystyle\left\langle\sigma v\right\rangle_{x_{F}}^{h_{s}a_{s}}\simeq$
$\displaystyle\frac{1}{64\pi
m_{\tilde{\chi}_{1}^{0}}^{2}}\left\\{\left[1-\frac{\left(m_{h_{s}}+m_{a_{s}}\right)^{2}}{4m_{\tilde{\chi}_{1}^{0}}^{2}}\right]\left[1-\frac{\left(m_{h_{s}}-m_{a_{s}}\right)^{2}}{4m_{\tilde{\chi}_{1}^{0}}^{2}}\right]\right\\}^{1/2}|{\cal{A}}_{s}+{\cal{A}}_{t}|^{2},\quad$
(2.24)

where the $s$\- and $t$-channel contributions are approximated as

$\displaystyle{\cal{A}}_{s}$ $\displaystyle\simeq$
$\displaystyle\frac{-2m_{\tilde{\chi}_{1}^{0}}C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}a_{s}}C_{h_{s}a_{s}a_{s}}}{m_{a_{s}}^{2}-4m_{\tilde{\chi}_{1}^{0}}^{2}},$
$\displaystyle{\cal{A}}_{t}$ $\displaystyle\simeq$
$\displaystyle-2C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}h_{s}}\,C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}a_{s}}\left[1+\frac{2m_{a_{s}}^{2}}{4m_{\tilde{\chi}_{1}^{0}}^{2}-\left(m_{h_{s}}^{2}+m_{a_{s}}^{2}\right)}\right],$
(2.25)

if there are no resonant contributions222Note that the
$\tilde{\chi}_{i}^{0}$-mediated ($i\neq 1$) contribution to $\langle\sigma
v\rangle_{x_{F}}^{h_{s}a_{s}}$ is less important than the
$\tilde{\chi}_{1}^{0}$-mediated contribution for two reasons. One is that, if
$|\kappa|$ is comparable to $\lambda$,
$|C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{i}^{0}h_{s}}|$ and
$|C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{i}^{0}a_{s}}|$ are significantly
smaller than $|C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}h_{s}}|$ and
$|C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}a_{s}}|$, respectively. The other
is that, since $m_{\tilde{\chi}_{i}^{0}}>m_{\tilde{\chi}_{1}^{0}}$, the former
contribution is relatively suppressed by the propagator.. According to Eqs.
(2.19) and (2.20), if $h_{s}$, $a_{s}$, and $\tilde{\chi}_{1}^{0}$ are pure
singlet states, then
$|C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}h_{i}}|=|C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}a_{i}}|\sim\sqrt{2}|\kappa|$.
The measured abundance then requires $\kappa\sim
0.15\left(\frac{m_{\tilde{\chi}_{1}^{0}}}{300\,{\rm GeV}}\right)^{1/2}$ in the
case of $|{\cal{A}}_{t}|\gg|{\cal{A}}_{s}|$ [73]. Hence, it is evident that
once the involved particle masses are fixed, the density is primarily
determined by the parameter $\kappa$. However, because $\lambda>2|\kappa|$ to
ensure a singlino-dominated $\tilde{\chi}_{1}^{0}$, the measured abundance can
set a lower bound on $\lambda$.

To date, the sensitivities of the XENON-1T experiments have reached the
precision of $10^{-47}~{}{\rm cm^{2}}$ for the SI cross section [50] and
$10^{-42}~{}{\rm cm^{2}}$ for the SD cross section [51]. They have strongly
restricted the $\lambda\gtrsim 0.3$ case (see the discussion about DM-nucleon
scattering), which is preferred by the annihilations
$\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to t\bar{t},h_{s}a_{s}$ to account
for the measured abundance. In addition, the LHC searches for new particles
[30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,
49], and its precise measurement of the discovered scalar’s properties [75,
76] have limited the theoretical prediction on light sparticles and Higgs
bosons. This has a significant impact on the DM annihilation channels since
they are usually accompanied with light particles to account for the abundance
(see the discussion in [55]). Thus, the scenario preferred by the scan of Ref.
[55] involves a singlino-dominated DM with $\lambda\lesssim 0.05$. In this
case, an effective mechanism to obtain the measured abundance includes co-
annihilation with higgsinos. The corresponding reaction is
$\tilde{\chi}_{i}\tilde{\chi}_{j}\rightarrow XX^{\prime}$, in which
$XX^{\prime}$ denotes SM particles and $\tilde{\chi}_{i}\tilde{\chi}_{j}$ may
be an LSP-NLSP or NLSP-NLSP annihilation state (e.g.,
$\tilde{\chi}_{1}^{0}\tilde{\chi}^{0}_{2}$,
$\tilde{\chi}_{1}^{0}\tilde{\chi}^{+}_{1}$, or
$\tilde{\chi}_{2}^{0}\tilde{\chi}^{+}_{1}$). This mechanism is distinct in
that the effective annihilation rate at a temperature $T$ is very sensitive to
the $\tilde{\chi}_{1}^{0}$-higgsino mass splitting [79, 80], and even for a
small $\lambda$ and $\kappa$, it can still explain the measured
abundance333Note that the co-annihilation mechanism applies under the premise
that $\tilde{\chi}_{1}^{0}$ and the higgsinos remained in thermal equilibrium
in the early universe [79, 80]. In the $Z_{3}$-NMSSM, many processes, such as
$\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\leftrightarrow\tilde{\chi}_{i}\tilde{\chi}_{j}$,
$\tilde{\chi}_{1}^{0}X\leftrightarrow\tilde{\chi}_{i}X^{\prime}$, and
$\tilde{\chi}_{i}\leftrightarrow\tilde{\chi}_{1}^{0}XX^{\prime}$, could keep
$\tilde{\chi}_{1}^{0}$ in chemical equilibrium with $\tilde{\chi}_{i}$, and
the conversion rates of some of them might be enhanced if the mediator were
around its mass-shell. We add that maintaining the thermal equilibrium does
not necessarily require the involved couplings to be moderately large. For
example, the equilibrium condition was discussed in Eqs. (4.3) and (4.4) of
Ref. [80] in the framework of the DM model ST11. It was found that the
involved coupling may be as low as $10^{-4}$ to maintain the equilibrium..

###  2.3 DM-nucleon cross sections

Serving for a WIMP, $\tilde{\chi}^{0}_{1}$ might be detected by measuring the
recoil of a nucleus after an elastic scattering of $\tilde{\chi}^{0}_{1}$ on a
nucleus taking place. In the non-relativistic limit, only two different kinds
of interactions between a neutralino and a nucleon need to be considered [81]:
the spin-dependent interaction (SD) where the WIMP couples to the spin of the
nucleus, and the spin-independent interaction (SI) where the WIMP couples to
the mass of the nucleus.

When $m_{\tilde{q}}\gtrsim 2~{}{\rm TeV}$, only the $t$-channel $Z$ exchange
diagram contributes significantly to the spin-dependent (SD) scattering cross
section at the tree level, which is approximated by [82, 83]

$\displaystyle\sigma_{\tilde{\chi}_{1}^{0}-N}^{\rm SD}\simeq
C_{N}\times\left(\frac{C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}Z}}{0.01}\right)^{2},$
(2.26)

with $N=p(n)$ denoting protons (neutrons) and $C_{p}\simeq 2.9\times
10^{-41}~{}{\rm cm^{2}}$ ($C_{n}\simeq 2.3\times 10^{-41}~{}{\rm cm^{2}}$)
[71, 72]. From the expression for
$C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}Z}$ in Eq. (2.17), it is evident
that $\sigma_{\tilde{\chi}_{1}^{0}-N}^{\rm SD}$ is proportional to $(\lambda
v/\mu_{eff})^{4}$. Furthermore, in the co-annihilation case, the degeneracy of
$m_{\tilde{\chi}_{1}^{0}}$ and $\mu_{eff}$ leads to a minuscule denominator in
Eq. (2.17), which requires a small value of $\lambda v/\mu_{eff}$ to satisfy
the DM-DD experimental constraints.

In contrast, the spin-independent (SI) scattering cross section in the heavy
squark limit is dominated by a $t$-channel exchange of CP-even Higgs bosons
$h_{i}$ [86, 87, 84, 85] and can be expressed as [88]

$\sigma_{\tilde{\chi}^{0}_{1}-{N}}^{\rm SI}=\frac{m_{N}^{2}}{2\pi
v^{2}}\left(\frac{m_{N}m_{\tilde{\chi}_{1}^{0}}}{m_{N}+m_{\tilde{\chi}_{1}^{0}}}\right)^{2}{\left(\frac{1}{125~{}\rm
GeV}\right)^{4}}\left\\{\sum_{h_{i}}\left[F^{N}_{u}\left({a_{u}}\right)_{h_{i}}+F^{N}_{d}\left({a_{d}}\right)_{h_{i}}\right]\right\\}^{2},$
(2.27)

where $m_{N}$ is the nucleon mass,
$F^{(N)}_{d}=f^{(N)}_{d}+f^{(N)}_{s}+\frac{2}{27}f^{(N)}_{G}$ and
$F^{(N)}_{u}=f^{(N)}_{u}+\frac{4}{27}f^{(N)}_{G}$ with
$f^{(N)}_{q}=m_{N}^{-1}\left<N|m_{q}q\bar{q}|N\right>$ ($q=u,d,s$) represent
the normalized light quark contribution to the nucleon mass, and
$f^{(N)}_{G}=1-\sum_{q=u,d,s}f^{(N)}_{q}$ influences other heavy quark mass
fractions in nucleons [86, 87]. In this study, the default settings for
$f_{q}^{N}$ were used in the micrOMEGAs package [85], and they predict
$F_{u}^{p}\simeq F_{u}^{n}\simeq 0.15$ and $F_{d}^{p}\simeq F_{d}^{n}\simeq
0.13$. Hence, SI cross sections for DM-proton scattering and DM-neutron
scattering are approximately equal (i.e.,
$\sigma^{SI}_{{\tilde{\chi}^{0}_{1}}-p}\simeq\sigma^{SI}_{{\tilde{\chi}^{0}_{1}}-n}$)
[89]. The quantities $\left(a_{u}\right)_{h_{i}}$ and
$\left(a_{d}\right)_{h_{i}}$ are defined by

$\displaystyle\left({a_{u}}\right)_{h_{i}}$ $\displaystyle=$
$\displaystyle\left(\frac{125~{}\rm
GeV}{m_{h_{i}}}\right)^{2}\left(V_{h_{i}}^{\rm
SM}+\frac{1}{\tan\beta}V_{h_{i}}^{\rm
NSM}\right)C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}h_{i}}~{},$ (2.28)
$\displaystyle\left({a_{d}}\right)_{h_{i}}$ $\displaystyle=$
$\displaystyle\left(\frac{125~{}\rm
GeV}{m_{h_{i}}}\right)^{2}\left(V_{h_{i}}^{\rm SM}-\tan\beta V_{h_{i}}^{\rm
NSM}\right)C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}h_{i}}~{}.$ (2.29)

Currently, non-SM doublet Higgs bosons $H$ are preferred to be heavier than
several hundreds of GeV in LHC experiments. In this case, the contribution
from $H$ to the SI cross section will be suppressed by $(a_{q})_{H}^{2}\propto
1/m_{H}^{4}$, and it is much smaller than that from $h$ for a not exceedingly
large $\tan\beta$. As such, the primary contribution to SI scattering comes
from the t-channel exchange of SM-like Higgs bosons $h$ and the singlet Higgs
boson $h_{s}$. The latter contribution may be crucial when $h_{s}$ is much
lighter than $h$. Since the non-SM doublet components of $h$ and $h_{s}$ are
approximately zero,
$\left({a_{u}}\right)_{h}+\left({a_{u}}\right)_{h_{s}}\simeq\left({a_{d}}\right)_{h}+\left({a_{d}}\right)_{h_{s}}\equiv\mathcal{A}$,
which can be expressed as

$\displaystyle\mathcal{A}\simeq\left(\frac{125\rm
GeV}{m_{h}}\right)^{2}V_{h}^{\rm
SM}C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}h}+\left(\frac{125\rm
GeV}{m_{h_{s}}}\right)^{2}V_{h_{s}}^{\rm
SM}C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}h_{s}}~{},$ (2.30)

where $C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}h}$ and
$C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}h_{s}}$ are given by Eq. (2.19).
Thus, the SI scattering cross section in Eq. (2.27) can be rewritten as

$\displaystyle\sigma_{\tilde{\chi}^{0}_{1}-{N}}^{\rm SI}$
$\displaystyle\simeq\frac{m_{N}^{2}}{2v^{2}\pi}\left(\frac{m_{N}m_{\tilde{\chi}_{1}^{0}}}{m_{N}+m_{\tilde{\chi}_{1}^{0}}}\right)^{2}{\left(\frac{1}{125\rm
GeV}\right)^{4}}\left(F^{N}_{u}+F^{N}_{d}\right)^{2}\mathcal{A}^{2}\sim
5\times 10^{-45}{\rm cm^{2}}\times\left(\frac{\mathcal{A}}{0.1}\right)^{2}.$
(2.31)

In these expressions, if only the contribution from a pure SM Higgs state is
considered, $\mathcal{A}$ can be simply expressed as

$\displaystyle\mathcal{A}\sim\left(\frac{125\rm
GeV}{m_{h}}\right)^{2}\frac{\sqrt{2}\lambda^{2}v}{\mu_{eff}}\,\frac{Z_{s}(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff}-\sin
2\beta)}{1-(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})^{2}}.$ (2.32)

It is immediately evident that $\sigma_{\tilde{\chi}_{1}^{0}-N}^{\rm SI}$ will
vanish for $m_{\tilde{\chi}_{1}^{0}}/\mu_{eff}=\sin 2\beta$, which corresponds
to a blind spot (BS) condition in [71, 90, 73].

The above analytical formulas for the SD and SI scattering cross sections
suggest that $\sigma^{SD}\varpropto\big{(}\frac{\lambda
v}{\mu_{eff}}\big{)}^{2}\frac{1}{1-(m_{\tilde{\chi}_{1}^{0}}/\mu_{eff})^{2}}$
and (by contrast) $\sigma^{SI}$ depends on $\lambda$, $\mu_{eff}$, and
$m_{\tilde{\chi}^{0}_{1}}$ in a complex way. In general, a larger $\lambda$, a
smaller $\mu_{eff}$, and $m_{\tilde{\chi}_{1}^{0}}/\mu_{eff}\rightarrow 1$
will increase $\sigma^{SI}$ and thus strengthen the DM-DD experimental
constraints.

Table 1: Signal of final state for the electroweakino pair production
processes considered in this work. Relevant experimental analyses were
performed using a simplified model by ATLAS and CMS collaborations, and their
results have been encoded in the SmodelS-1.2.3 [110].

Name | Simplified Scenario | Signal of Final State | Luminosity ($\bm{fb^{-1}}$)
---|---|---|---
13 TeV
| CMS-SUS-17-010 [30]
---
(arXiv:1807.07799)
| $\bm{\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{1}^{\mp}\rightarrow
W^{\pm}\tilde{\chi}_{1}^{0}W^{\mp}\tilde{\chi}_{1}^{0}}$
---
$\bm{\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{1}^{\mp}\rightarrow\nu\tilde{\ell}\ell\tilde{\nu}}$
2$\bm{\ell}$ \+ $\bm{E_{\rm T}^{\rm miss}}$ | 35.9
| CMS-SUS-17-009 [31]
---
(arXiv:1806.05264)
$\bm{\tilde{\ell}\tilde{\ell}}$ | 2$\bm{\ell}$ \+ $\bm{E_{\rm T}^{\rm miss}}$ | 35.9
| CMS-SUS-17-004 [32]
---
(arXiv:1801.03957)
| n$\bm{\ell}$(n>=0) + nj(n>=0) + $\bm{E}_{\rm T}^{\rm miss}$ | 35.9
| CMS-SUS-16-045 [33]
---
(arXiv:1709.00384)
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow W^{\pm}\tilde{\chi}_{1}^{0}h\tilde{\chi}_{1}^{0}}$ | 1$\bm{\ell}$2b + $\bm{E}_{\rm T}^{\rm miss}$ | 35.9
| CMS-SUSY-16-039 [34]
---
(arxiv:1709.05406)
|
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow\ell\tilde{\nu}\ell\tilde{\ell}}$
---
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow\tilde{\tau}\nu\tilde{\ell}\ell}$
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow\tilde{\tau}\nu\tilde{\tau}\tau}$
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow
WZ\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}}$
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow
WH\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}}$
n$\bm{\ell(n\textgreater{}0)}$($\tau$$)\bf+$E_T^miss | 35.9
|  CMS-SUS-16-034 [35]
---
(arXiv:1709.08908)
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow W\tilde{\chi}_{1}^{0}Z(h)\tilde{\chi}_{1}^{0}}$ | n$\bm{\ell}$(n>=2) + nj(n>=1) $\bm{E_{\rm T}^{\rm miss}}$ | 35.9
|  ATLAS-1803-02762 [36]
---
(ATLAS-SUSY-2016-24)
| $\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow
WZ\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}}$
---
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow\nu\tilde{\ell}l\tilde{\ell}}$
$\bm{\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{1}^{\mp}\rightarrow\nu\tilde{\ell}\nu\tilde{\ell}}$
$\bm{\tilde{\ell}\tilde{\ell}}$
n$\bm{\ell}$(n>=2) + $\bm{E}_{\rm T}^{\rm miss}$ | 36.1
|  ATLAS-1812-09432 [37]
---
(ATLAS-SUSY-2017-01)
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow Wh\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}}$ | n$\bm{\ell}$(n>=0) + nj(n>=0) + nb(n>=0) + n$\bm{\gamma}$(n>=0) + $\bm{E}_{\rm T}^{\rm miss}$ | 36.1
|  ATLAS-1806-02293 [38]
---
(ATLAS-SUSY-2017-03)
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow WZ\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}}$ | n$\bm{\ell}$(n>=2) + nj(n>=0) + $\bm{E}_{T}^{miss}$ | 36.1
|  ATLAS-1912-08479 [39]
---
(ATLAS-SUSY-2018-06)
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow W(\rightarrow l\nu)\tilde{\chi}_{1}^{0}Z(\rightarrow\ell\ell)\tilde{\chi}_{1}^{0}}$ | 3$\bm{\ell}$ \+ $\bm{E}_{\rm T}^{\rm miss}$ | 139
|  ATLAS-1908-08215 [40]
---
(ATLAS-SUSY-2018-32)
| $\bm{\tilde{\ell}\tilde{\ell}}$
---
$\bm{\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{1}^{\mp}}$
2$\bm{\ell}$ \+ $\bm{E}_{\rm T}^{\rm miss}$ | 139
|  ATLAS-1909-09226 [41]
---
(ATLAS-SUSY-2019-08)
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow Wh\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}}$ | 1$\bm{\ell}$ \+ h($\bm{\rightarrow}$bb) + $\bm{E}_{\rm T}^{\rm miss}$ | 139
8 TeV
|  ATLAS-CONF-2013-035 [42]
---
| $\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow
Z^{(*)}\tilde{\chi}_{1}^{0}W^{(*)}\tilde{\chi}_{1}^{0}}$
---
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow\tilde{l}(\tilde{\nu})\ell(\nu)\tilde{\ell}(\tilde{\nu})\nu(\tilde{\ell})}$
n$\bm{\ell}$(n>=2) + $\bm{E}_{\rm T}^{\rm miss}$ | 20.3
|  ATLAS-CONF-2013-049 [43]
---
(arxiv:1801.01846)
| $\bm{\tilde{\ell}\tilde{\ell}}$
---
$\bm{\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{1}^{\mp}\rightarrow\tilde{\ell}\nu(\tilde{\nu}\ell)}$
2$\bm{\ell}$ \+ $\bm{E}_{\rm T}^{\rm miss}$ | 20.3
|  ATLAS-CONF-2013-093 [44]
---
| $\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow
W\tilde{\chi}_{1}^{0}h\tilde{\chi}_{1}^{0}}$
---
1$\bm{\ell}$ \+ 2$\bm{b}$ \+ $\bm{E}_{\rm T}^{\rm miss}$ | 20.3
|  ATLAS-1403-5294 [45]
---
(ATLAS-SUSY-2013-11)
|
$\bm{\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{1}^{\mp}\rightarrow\tilde{l}(\tilde{\nu})\nu(l)}$
---
$\bm{\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{1}^{\mp}\rightarrow
W\tilde{\chi}_{1}^{0}W\tilde{\chi}_{1}^{0}}$
$\bm{\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{2}^{0}\rightarrow
W\tilde{\chi}_{1}^{0}Z\tilde{\chi}_{1}^{0}}$
$\bm{\tilde{\ell}\tilde{\ell}}$
n$\bm{\ell}$(n>=2) + $\bm{E_{\rm T}^{\rm miss}}$ | 20.3
|  ATLAS-1402-7029 [46]
---
(ATLAS-SUSY-2013-12)
| $\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow
W\tilde{\chi}_{1}^{0}Z\tilde{\chi}_{1}^{0}}$
---
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow
W\tilde{\chi}_{1}^{0}h\tilde{\chi}_{1}^{0}}$
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow\ell\tilde{\nu}\ell\tilde{\ell}(\tilde{\nu}\nu)}$,$\bm{\ell\tilde{\ell}\nu\tilde{\ell}(\tilde{\nu}\nu)}$
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow\tilde{\tau}\nu\tilde{\tau}\tau(\tilde{\nu}\nu)}$,$\bm{\tau\tilde{\nu}\tilde{\tau}\tau(\tilde{\nu}\nu)}$
3$\bm{\ell}(\tau)$ \+ $\bm{E}_{\rm T}^{\rm miss}$ | 20.3
|  ATLAS-1501-07110 [47]
---
(ATLAS-SUSY-2013-23)
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow W\tilde{\chi}_{1}^{0}h\tilde{\chi}_{1}^{0}}$ | n$\bm{\ell}$ (n$\bm{\textgreater}{}0$) + n$\bm{\gamma}$(n>=0) + n$\bm{b}$(n>=0) + $\bm{E}_{\rm T}^{\rm miss}$ | 20.3
|  CMS-PAS-SUSY-12-022[48]
---
|
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow\tilde{\ell}\ell\tilde{\nu}\ell}$,$\bm{\ell\tilde{\ell}\nu\tilde{\ell}}$
---
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow
Z\tilde{\chi}_{1}^{0}W\tilde{\chi}_{1}^{0}}$
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{3}^{0}\rightarrow
Z\tilde{\chi}_{1}^{0}Z\tilde{\chi}_{1}^{0}}$
$\bm{\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{1}^{\mp}\rightarrow\ell\tilde{\nu}\nu\tilde{\ell}}$
$\bm{\tilde{\ell}\tilde{\ell}}$
n$\bm{\ell}$(n$\bm{\textgreater}{}=2$) + $\bm{E}_{\rm T}^{\rm miss}$ | 9.2
|  CMS-SUSY-13-006 [49]
---
|
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow\tilde{\ell}\ell\tilde{\nu}\ell}$,$\bm{\ell\tilde{\ell}\nu\tilde{\ell}}$
---
$\bm{\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow
Z\tilde{\chi}_{1}^{0}W\tilde{\chi}_{1}^{0}}$
$\bm{\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{1}^{\mp}\rightarrow\ell\tilde{\nu}\nu\tilde{\ell}}$
$\bm{\tilde{\ell}\tilde{\ell}}$
n$\bm{\ell}$(n>=2) + $\bm{E}_{\rm T}^{\rm miss}$ | 19.5

##  3  Model scans and constraints

The NMSSMTools-5.4.1 package [77, 91] was used to produce samples of singlino-
dominated DM scenarios in the $\rm Z_{3}$-NMSSM and to model the corresponding
features in detail444The NMSSMTools package includes codes to compute various
observables in Higgs physics, DM physics, B physics, and sparticle physics. In
this sector, we only briefly introduce the calculation of the observables we
are interested in.. A sophisticated scan was first performed over the
following ranges in the parameter space:

$\displaystyle 0<\lambda\leq 0.7,~{}~{}|\kappa|\leq
0.7,~{}~{}1\leq\tan\beta\leq 60,~{}~{}100~{}{\rm GeV}\leq\mu_{eff}\leq
1000~{}{\rm GeV},$ (3.1) $\displaystyle|A_{\kappa}|<1~{}{\rm
TeV},~{}~{}0<A_{\lambda}\leq 5~{}{\rm TeV},~{}~{}|A_{t}|\leq 5~{}{\rm
TeV},~{}~{}|M_{1}|\leq 500~{}{\rm GeV},$

in which all parameters were defined at the scale $Q=1~{}{\rm TeV}$. Upper
bounds of $0.7$ were imposed on $\lambda$ and $|\kappa|$ to maintain a
perturbable theory up to the grand unification scale. A lower bound of
$100~{}{\rm GeV}$ was placed on $\mu_{eff}$ by the LEP search for
electroweakinos [92], and an upper bound of 1000 GeV for $\mu_{eff}$ is large
enough to allow us to consider various possibilities (see the discussion
presented below). In addition, noting that the LHC search for SUSY prefers
massive charged sparticles, the following assumptions were made concerning
unimportant SUSY parameters. The electroweak gaugino masses were set to
$M_{2}=2~{}{\rm TeV}$, and the gluino masses were set to $M_{3}=5~{}{\rm
TeV}$. Soft SUSY-breaking parameters in the squarks sector were fixed at
$2~{}{\rm TeV}$, excluding trilinear couplings $A_{t}=A_{b}$ used as free
parameters to adjust the Higgs mass spectrum to coincide with relevant
experimental measurements at the LHC. In addition, all slepton soft parameters
were set to $2~{}{\rm TeV}$, as we did not want to explain the muon $g$-2
anomaly. We also required $\lambda\geq 2|\kappa|$ in the scan to achieve a
singlino-dominated $\tilde{\chi}^{0}_{1}$.

Specifically, the MultiNest algorithm [60, 61] with flat distributions for all
the parameters in Eq. (3.1) and nlive = 20000 were adopted during the scan to
ensure that the conclusions were as complete as possible, and more than 20
thousands CPU hours were spent on the calculations555The multi-nest sampling
algorithm explores a high-dimensional parameter space by determining the iso-
likelihood contour in each iteration with nlive active points (the integer
nlive is an input parameter of the algorithm, and it usually takes a value
larger than 1000. In general, the larger value nlive adopts, the more accurate
the scan result becomes.) [60, 61]. It is good at dealing with the case in
which the samples’ posterior distribution is multi-modal or degenerate, which
is frequently encountered in new physics studies. In contrast, the Markov
Chain method [62] is highly inefficient for such a situation, and thus, it
usually provides incomplete information about the distribution.. Several
constraints were imposed by constructing the following corresponding
likelihood function to guide the process:

$\displaystyle\mathcal{L}=\mathcal{L}_{\rm{m_{h}}}\times\mathcal{L}_{h,{\rm
coupling}}\times{\mathcal{L}_{B}}\times{\mathcal{L}_{EW}}\times{\mathcal{L}_{\Omega
h^{2}}}\times{\mathcal{L}_{DD}},$ (3.2)

where $\mathcal{L}_{\rm{m_{h}}}$ and $\mathcal{L}_{h,{\rm coupling}}$ are
likelihood functions for the experimentally measured SM-like Higgs boson mass
and couplings, respectively. The computation of $m_{h}$ included leading
electroweak corrections, two loop terms, and propagator corrections, as in
Ref. [93]. Its experimental central value was taken as $m_{h}=125.09~{}{\rm
GeV}$ [20], and a total experimental and theoretical uncertainty of $3~{}{\rm
GeV}$ was assumed. $\mathcal{L}_{h,{\rm coupling}}$ works in a seven-parameter
$\kappa$-framework with related experimental measurements, such as the central
values and uncertainties of the Higgs couplings and their correlation
coefficients, taken from the ATLAS analysis, using 80${\rm fb}^{-1}$ data
collected during the LHC Run-II [75]. Some knowledge about probability and
statistics was used in constructing $\mathcal{L}_{h,{\rm coupling}}$ (see the
introduction in Ref. [92]). ${\mathcal{L}_{B}}$ is the likelihood function for
the measurement of the branching ratio for the decays $B\to X_{s}\gamma$ and
$B_{s}\to\mu^{+}\mu^{-}$. These ratios were calculated by the formulae in
Refs. [94, 95], and their experimental values were taken from Ref. [92].
${\mathcal{L}_{\Omega h^{2}}}$ and ${\mathcal{L}_{DD}}$ are likelihood
functions for the measured abundance from the WMAP/Planck experiments [3, 4]
and the detection of both spin-independent (SI) and spin-dependent (SD) DM-
nucleon scattering in the XENON-1T experiment [50, 51]. Relevant quantities
were calculated using the micrOMEGAs package [96, 78, 97, 98, 99]. In
addition, ${\mathcal{L}_{EW}}$ denotes a likelihood function for precision
electroweak observables of $\epsilon_{i}$ (i=1,2,3) [100, 101, 102] or,
equivalently, $S$, $T$, and $U$ parameters [103, 104] calculated using the
formulas from Cao and Yang [105] and fitted to corresponding measurements by
the procedure presented in Ref. [106]. Each of these likelihood functions was
assumed to follow a Gaussian distribution, with explicit representations
provided in Cao et al. [107].

The acquired samples were further refined using the following criteria: the
SM-like Higgs mass was within the range of 122–128 GeV, the observed DM relic
abundance was within $\pm 10\%$ of the measured central value $\Omega
h^{2}=0.1187$ [4], the upper bound was a $90\%$ confidence level of the
XENON-1T experimental results of the SI cross section [50] and the SD cross
section [51], and all other constraints implemented in NMSSMTools, including
various B-physics observables in corresponding experimentally allowed ranges,
were at the $2\sigma$ level. We also required the samples to satisfy
$\chi^{2}_{EW}\leq 7.8$ and $\chi^{2}_{h,{\rm coupling}}\leq 14.1$, which
corresponded to $95\%$ confidence level exclusion limits for three and seven
degrees of freedom, respectively. Since a more significant deviation of the
electroweak precision observables (the Higgs couplings) from their measured
values would enhance $\chi^{2}_{EW}$ ($\chi^{2}_{h,{\rm coupling}}$), this
requirement delineated the experimentally allowed range of these observables
to further limit the $Z_{3}$-NMSSM.

Constraints were also implemented from the LHC search for electroweakinos
using the SModelS-1.2.3 code [108, 109, 110]. The final states listed in Table
2.3 from all the electroweakino pair production processes were considered
during this process666The concrete procedure to determine the limitation is as
follows: we first determined the signal region (SR) with the largest expected
sensitivity for a given sample, then we checked its $R$ value defined by
$R\equiv S/S_{95}^{OBS}$, where $S$ stands for the number of signal events in
the SR with the statistical uncertainty considered, and $S_{95}^{OBS}$ denotes
the observed limit at 95% confidence level for the SR. Evidently, $R$
represents the capability of the LHC in exploring a point. $R>1$ implies that
the point is excluded; otherwise, it is allowed.. It is worth noting that a
large portion of the samples satisfying these constraints were characterized
by a small mass splitting between $\tilde{\chi}_{1}^{0}$ and
$\tilde{\chi}_{2}^{0}$. However, the latest ATLAS analysis of the compressed
mass spectra, acquired by searching soft di-lepton signals [111], was not
included in the SModelS-1.2.3 code. As such, the analysis constraints are
validated in the Appendix, and they were applied to each sample by elaborate
Monte Carlo simulations. We verified that they were very effective in
excluding the co-annihilation case. We illustrated this point in our recent
publication [88].

We add that we did not consider constraints from indirect DM detection
experiments (i.e., the Fermi-LAT search for DM annihilation from dwarf
spheroidal galaxies) as they become loose for
$m_{\tilde{\chi}^{0}_{1}}>100~{}\rm GeV$ [54]. In addition, in the co-
annihilation case encountered in this work, the annihilation rate of singlino-
dominated DM in present day is very small, which weakens the constraints.

Table 2: Benchmark points satisfying various experimental constraints. Mass
parameters are in units of GeV, and the DM-nucleon scattering cross sections
are in units of ${\rm cm^{2}}$. The number preceding each annihilation process
represents its fraction of contributions to the total DM annihilation cross
section at the freeze-out temperature. Dots include the information of the
decay modes with smaller branching ratios.

SM-like Higgs: $h_{1}$ | SM-like Higgs: $h_{2}$
---|---
| $P_{1}$ | $P_{2}$ |  | $P_{3}$ | $P_{4}$
$\lambda$ | 0.696 | 0.028 |  | 0.108 | $9.64\times 10^{-3}$
$\kappa$ | 0.208 |  $-$0.013 |  | 0.048 |  $-$$4.2\times 10^{-3}$
$\tan\beta$ | 2.15 | 7.2 |  | 12.8 | 9.9
$\mu_{eff}$ | 683.7 | 227.8 |  | 206.1 | 164.4
$A_{t}$ |  $-$2734 | 2754 |  |  $-$3394 |  $-$3510
$A_{\lambda}$ | 1128 | 3908 |  | 3125 | 1142
$A_{\kappa}$ |  $-$28.3 | 24.4 |  |  $-$323.1 | 147.3
$M_{1}$ |  $-$471.3 |  $-$418.2 |  |  $-$484.7 | 211.8
$m_{\tilde{\chi}^{0}_{1}}$ | 422.4 |  $-$216.5 |  | 181.9 |  $-$148.6
$m_{\tilde{\chi}^{0}_{2}}$ |  $-$473.7 |  $-$233.0 |  | 213.2 | 151.9
$m_{\tilde{\chi}^{0}_{3}}$ | 702.3 | 234.5 |  |  $-$215.1 |  $-$174.5
$m_{\tilde{\chi}^{\pm}_{1}}$ | 694.9 | 233.2 |  | 211.4 | 170.4
$N_{13}$ |  $-$0.035 |  $-$0.164 |  |  $-$0.32 |  $-$0.032
$N_{14}$ | 0.176 |  $-$0.171 |  |  $-$0.35 |  $-$0.038
$N_{15}$ | 0.984 | 0.971 |  | 0.87 | 0.999
$Z_{h}$ | 0.032 | 0.056 |  | 0.23 | 0.003
$Z_{S}$ | 0.968 | 0.943 |  | 0.76 | 0.997
$m_{h_{1}}$ | 124.6 | 125.2 |  | 64.2 | 100.7
$m_{h_{2}}$ | 440.4 | 206.5 |  | 126.7 | 125.3
$m_{h_{3}}$ | 1536 | 2532 |  | 2905 | 1311
$m_{a_{1}}$ | 178.0 | 88.1 |  | 298.0 | 178.4
$m_{a_{2}}$ | 1536 | 2532 |  | 2905 | 1311
$V_{h_{1}}^{\rm NSM},V_{h_{2}}^{\rm NSM}$ | 0.0, $-$0.05 |  $-$0.0, $-$0.0 |  |  $-$0.0, $-$0.0 |  $-$0.0, $-$0.0
$V_{h_{1}}^{\rm SM},V_{h_{2}}^{\rm SM}$ | 0.99, 0.15 | 0.99, $-$0.11 |  | 0.147, 0.989 | 0.04, 0.999
$V_{h_{1}}^{\rm S},V_{h_{2}}^{\rm S}$ |  $-$0.15, 0.99 | 0.11, 0.99 |  | 0.989, $-$0.147 | 0.999, $-$0.04
$V_{a_{1}}^{{}^{\prime}\rm NSM},V_{a_{1}}^{{}^{\prime}\rm S}$ |  $-$0.03, 1.0 |  $-$0.0, 1.0 |  |  $-$0.0, 1.0 |  $-$0.0, 1.0
$\sigma^{SI}_{\tilde{\chi}^{0}_{1}-p}$ | $4.2\times 10^{-49}$ | $3.1\times 10^{-47}$ |  | $8.89\times 10^{-47}$ | $2.67\times 10^{-49}$
$\sigma^{SD}_{\tilde{\chi}^{0}_{1}-n}$ | $3.0\times 10^{-41}$ | $2.94\times 10^{-43}$ |  | $2.23\times 10^{-41}$ | $6.57\times 10^{-45}$
$\Omega h^{2}$ | 0.118 | 0.109 |  | 0.12 | 0.108
| $68.4\%$ | $\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}\rightarrow t\bar{t}$ | $14.0\%$ | $\tilde{\chi}^{0}_{2}\tilde{\chi}^{+}_{1}\rightarrow u\bar{d}$ |  | $28.7\%$ | $\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}\rightarrow W^{+}W^{-}$ | $35.0\%$ | $\tilde{\chi}^{0}_{2}\tilde{\chi}^{0}_{2}\rightarrow W^{+}W^{-}$
| $19.4\%$ | $~{}~{}~{}~{}~{}~{}\rightarrow h_{s}a_{s}$ | $5.4\%$ | $~{}~{}~{}~{}~{}~{}\rightarrow\nu_{l}\bar{l}$ |  | $22.0\%$ | $~{}~{}~{}~{}~{}~{}\rightarrow ZZ$ | $25.4\%$ | $~{}~{}~{}~{}~{}~{}\rightarrow ZZ$
| $5.0\%$ | $~{}~{}~{}~{}~{}~{}\rightarrow h_{sm}a_{s}$ | $13.0\%$ | $\tilde{\chi}^{0}_{3}\tilde{\chi}^{+}_{1}\rightarrow u\bar{d}$ |  | $17.6\%$ | $~{}~{}~{}~{}~{}~{}\rightarrow t\bar{t}$ | $16.2\%$ | $\tilde{\chi}^{0}_{2}\tilde{\chi}^{+}_{1}\rightarrow u\bar{d}$
| $2.2\%$ | $~{}~{}~{}~{}~{}~{}\rightarrow W^{+}W^{-}$ | $4.8\%$ | $~{}~{}~{}~{}~{}~{}\rightarrow\nu_{l}\bar{l}$ |  | $13.0\%$ | $\tilde{\chi}^{0}_{1}\tilde{\chi}^{+}_{1}\rightarrow u\bar{d}$ | $6.6\%$ | $~{}~{}~{}~{}~{}~{}\rightarrow\nu_{l}\bar{l}$
annihilation | … |  … | $11.4\%$ | $\tilde{\chi}^{0}_{1}\tilde{\chi}^{+}_{1}\rightarrow u\bar{d}$ |  | $4.8\%$ | $~{}~{}~{}~{}~{}~{}\rightarrow\nu_{l}\bar{l}$ | 4.1% | $\tilde{\chi}^{0}_{2}\tilde{\chi}^{0}_{3}\rightarrow q\bar{q}$
channels |  |  | 4.2% | $~{}~{}~{}~{}~{}~{}\rightarrow\nu_{l}\bar{l}$ |  | $3.8\%$ | $\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{3}\rightarrow q\bar{q}$ | 1.2% | $~{}~{}~{}~{}~{}~{}\rightarrow\nu_{l}\bar{\nu}_{l}$
|  |  | 6.8% | $\tilde{\chi}^{0}_{2}\tilde{\chi}^{0}_{3}\rightarrow q\bar{q}$ |  | $...$ | $~{}~{}~{}~{}~{}~{}...$ | 3.6% | $\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{2}\rightarrow q\bar{q}$
|  |  | 1.8% | $~{}~{}~{}~{}~{}~{}\rightarrow\nu_{l}\bar{\nu}_{l}$ |  |  |  | … |  …
|  |  | 7.8% | $\tilde{\chi}^{+}_{1}\tilde{\chi}^{-}_{1}\rightarrow q\bar{q}$ |  |  |  |  |
|  |  | 2.7% | $~{}~{}~{}~{}~{}~{}\rightarrow l\bar{l}$ |  |  |  |  |
|  |  | 2.2% |  $\rightarrow W^{+}W^{-}$ |  |  |  |  |
|  |  | 5.7% | $\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{3}\rightarrow q\bar{q}$ |  |  |  |  |
|  |  | 1.5% |  $\rightarrow\nu_{l}\bar{\nu}_{l}$ |  |  |  |  |
|  |  | … | $~{}~{}~{}~{}~{}~{}...$ |  |  |  |  |

Figure 1: Type-I, -II, and -III samples projected on the $\sin 2\beta-
m_{\tilde{\chi}_{1}^{0}}/\mu_{eff}$ and $\lambda v/\mu_{eff}-Z_{h}$ planes.

Figure 2: Type-I samples projected on the
$\sigma_{\tilde{\chi}_{1}^{0}-p}^{SI}-m_{\tilde{\chi}_{1}^{0}}$ and
$\sigma_{\tilde{\chi}_{1}^{0}-n}^{SD}-m_{\tilde{\chi}_{1}^{0}}$ planes. The
color bar represents the values of $\lambda v/\mu_{eff}$. The solid line
represents the current exclusion bound of the XENON-1T experiment on the cross
section and the dashed line denotes the projected sensitivity of the near-
future PandaX-4T experiment.

##  4 Numerical Results

A closer analysis suggested that the eventual surviving parameter space could
be classified into the following three types:

  * •

Type-I samples: the lightest $\rm{CP}$-even Higgs boson $h_{1}$ as the SM-like
Higgs boson, $0.4\lesssim\lambda\lesssim 0.7$, $0.13\lesssim\kappa\lesssim
0.23$, $1.5\lesssim\tan\beta\lesssim 6$, $450~{}{\rm
GeV}\lesssim\mu_{eff}\lesssim 720~{}{\rm GeV}$, and the Bayesian evidence $\ln
Z_{1}=-24.2$. Due to the limited capability of the Markov Chain algorithm,
this type of samples were neglected in [55].

  * •

Type-II samples: the lightest $\rm{CP}$-even Higgs boson $h_{1}$ as the SM-
like Higgs boson, $\lambda\lesssim 0.08$, $-0.04\lesssim\kappa<0$ with
$2|\kappa|/\lambda\simeq 1$, $4\lesssim\tan\beta\lesssim 24$, $170~{}{\rm
GeV}\lesssim\mu_{eff}\lesssim 420~{}{\rm GeV}$, and the Bayesian evidence $\ln
Z_{2}=-27.5$.

  * •

Type-III samples: the second lightest $\rm{CP}$-even Higgs boson $h_{2}$ as
the SM-like Higgs boson, $\lambda\lesssim 0.15$, $|\kappa|\lesssim 0.06$ with
$2|\kappa|/\lambda\simeq 1$, $4.5\lesssim\tan\beta\lesssim 32$, $135~{}{\rm
GeV}\lesssim\mu_{eff}\lesssim 260~{}{\rm GeV}$, and the Bayesian evidence $\ln
Z_{3}=-27.0$.

It is evident that the three types of parameter spaces are extremely narrow.
The Bayesian evidence $Z_{i}$ ($i=1,2,3$) produced the Jeffreys’ scales [112,
113] $\delta_{12}\equiv\ln Z_{1}-\ln Z_{2}=3.3$ and $\delta_{13}\equiv\ln
Z_{1}-\ln Z_{3}=2.8$. These results suggest that the considered experiments
slightly preferred Type-I samples to Type-II and -III samples. However, as
discussed below, Type-I samples will be examined by the near-future PandaX-4T
experiment [114]. Furthermore, Type-II and -III samples share the following
features in the parameter space: both $\lambda$ and $|\kappa|$ are small,
$2|\kappa|/\lambda\simeq 1$, and the higgsinos are moderately light. These
facts lead to the similarities of the Type-II and -III samples in many aspects
of DM physics (see the discussions below for detailed similarities and
differences). Nevertheless, they are still distinguished from each other in
Higgs physics [13, 14, 18, 15, 17, 16].

In the following, we investigate the characteristics of the singlino-dominated
DM based on numerical results. In Figure 1, Type-I, -II, and -III samples are
projected on the $\sin 2\beta-m_{\tilde{\chi}_{1}^{0}}/\mu_{eff}$ and $\lambda
v/\mu_{eff}-Z_{h}$ planes. In Figures 2, 3, and 4, Type-I, -II, and -III
samples are projected on the
$\sigma_{\tilde{\chi}_{1}^{0}-p}^{SI}-m_{\tilde{\chi}_{1}^{0}}$ and
$\sigma_{\tilde{\chi}_{1}^{0}-n}^{SD}-m_{\tilde{\chi}_{1}^{0}}$ planes,
respectively, with different colors indicating the value of $\lambda
v/\mu_{eff}$. In Table 2, a selection of benchmark points are shown to further
clarify features in each of the three scenarios.

The following points about Type-I samples can be determined from Figures 1 and
2 and the point $P_{1}$ in Table 2.

  * •

Type-I samples are characterized by a relatively large $\lambda v/\mu_{eff}$
ranging from $0.14$ to $0.21$ (see right panel of Figure 1). This will
increase the higgsino composition in $\tilde{\chi}_{1}^{0}$ through Eq. (2.16)
and the $\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}Z$ coupling through Eq.
(2.17). As indicated in the right panel of Figure 2, the SD scattering rates
are thus larger than $7\times 10^{-42}\ {\rm cm^{2}}$, which are exceedingly
close to the near-future PandaX-4T exclusion limit.

  * •

As shown in the left panel of Figure 1, $m_{\tilde{\chi}^{0}_{1}}/\mu_{eff}$
and $\sin 2\beta$ are correlated by
$m_{\tilde{\chi}^{0}_{1}}/\mu_{eff}\simeq\sin 2\beta$. In this case,
$C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}H_{\rm SM}}$ in Eq. (2.19) is
suppressed by $m_{\tilde{\chi}^{0}_{1}}/\mu_{eff}$ and $\sin 2\beta$
cancellation, favored by the stringent bound of the XENON-1T experiment on the
SI cross section. We further explore its implication by focusing on the point
$P_{1}$ in Table 2, which predicts the following four terms in Eq. (2.19):

$\displaystyle C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}h}$
$\displaystyle\simeq$ $\displaystyle
0.2702\times(0.6178-0.7647)+0.0-0.0029+0.0465\sim 0.0027,$ (4.1)
$\displaystyle C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}h_{s}}$
$\displaystyle\simeq$ $\displaystyle-0.006-0.0+0.0183-0.312\sim-0.2997.$ (4.2)

The two contributions in Eq. (2.30) are as follows:

$\displaystyle\mathcal{A}$
$\displaystyle\simeq~{}0.0027-0.0036\sim-0.0009~{}.$ (4.3)

These results show that, besides the mentioned cancellation, there is a strong
offsetting effect between the first and fourth terms within the
$\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}h$ coupling itself (i.e.,
cancellation between $C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}H_{\rm SM}}$
and $C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}H_{\rm S}}$ terms in
$C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}h}$) and a strong cancellation
between the two contributions to the SI cross section from $h$ and $h_{s}$.
These accidental cancellations result in
$\sigma^{SI}_{\tilde{\chi}_{1}^{0}-p}\sim 10^{-49}\ {\rm cm^{2}}$. In
contrast, $\sigma^{SI}_{\tilde{\chi}_{1}^{0}-p}$ would be around $10^{-42}\
{\rm cm^{2}}$ without them. This feature explains the SI cross section for
Type-I samples possibly being as low as $10^{-50}\ {\rm cm^{2}}$, as shown in
the left panel of Figure 2.

  * •

Based on $m_{\tilde{\chi}_{1}^{0}}/\mu$, $\lambda v/\mu$, $\sin 2\beta$, and
$Z_{h}$ in Figure 1, it is usually predicted that
$C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}G^{0}}\sim 0.1$ by Eq. (2.18) and
$\langle\sigma v\rangle_{x_{F}}^{t\bar{t}}\sim 10^{-26}\ {\rm cm}^{3}{\rm
s^{-1}}$ by Eq. (2.23). This implies that
$\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to t\bar{t}$ played a significant
role in determining the abundance. Concerning the point P1 in Table 2, we
found $C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}G^{0}}\simeq-0.108$ and
$\langle\sigma v\rangle_{x_{F}}^{t\bar{t}}\simeq 1.2\times 10^{-26}\ {\rm
cm}^{3}{\rm s^{-1}}$, indicating that the annihilation contributed to the
total annihilation rate by about $52\%$. We also obtained $\langle\sigma
v\rangle_{x_{F}}^{h_{s}a_{s}}\simeq 3.4\times 10^{-27}\ {\rm cm}^{3}{\rm
s^{-1}}$ by Eq. (2.24), which means that
$\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to h_{s}a_{s}$ contributed to the
total rate by $15\%$. We add that our estimation roughly agrees with the
results in Table 2, calculated by the micrOMEGAs package.

Figure 3: Same plots as in Fig.2, but for the results of the Type-II samples.

Next we consider Type-II samples. Figures 1 and 3 demonstrate following
features:

  * •

$\lambda v/\mu_{eff}\lesssim 0.06$ and $-0.95\lesssim
m_{\tilde{\chi}^{0}_{1}}/\mu_{eff}\lesssim-0.90$, which imply that

$\displaystyle|C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}Z}|\lesssim
2.4\times(\lambda v/\mu_{eff})^{2}<0.009,$
$\displaystyle\sigma^{SD}_{\tilde{\chi}_{1}^{0}-n}/{\rm cm^{2}}\lesssim
1.3\times 10^{-36}\times(\lambda v/\mu_{eff})^{4}<1.7\times 10^{-41}$ (4.4)

by Eq. (2.17) and (2.26), respectively. This feature is shown on the right
panel of Figure 3.

  * •

As indicated in the left panel of Figure 1,
$m_{\tilde{\chi}^{0}_{1}}/\mu_{eff}$ and $\sin 2\beta$ are of opposite sign,
and thus, their contributions to
$C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}H_{\rm SM}}$ in Eq. (2.19) do not
cancel each other. This implies that both
$C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}h}$ and $\mathcal{A}$ in Eq.
(2.30) are mainly contributed by
$C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}H_{\rm SM}}$. Consequently,
$\sigma^{SI}_{\tilde{\chi}_{1}^{0}-p}$ in Eq.(2.31) is approximated by

$\displaystyle\sigma_{\tilde{\chi}^{0}_{1}-{p}}^{\rm SI}$
$\displaystyle\simeq$ $\displaystyle 5\times 10^{-45}{\rm
cm^{2}}\times\left(\frac{C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}H_{\rm
SM}}}{0.1}\right)^{2},$ (4.5)

where $C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}H_{\rm SM}}\simeq(4--
10)\times\sqrt{2}\lambda(\lambda v/\mu_{eff})$777It is noticeable that the
enhancement coefficient $4--10$ comes from the factor
$1/\\{1-(m_{\tilde{\chi}_{1}}/\mu_{eff})^{2}\\}$ in Eq. (2.19). This is a
common characteristic for the Type-II and -III samples.. This approximation
reflects the relation $\sigma_{\tilde{\chi}^{0}_{1}-{p}}^{\rm
SI}\propto\lambda^{4}$. For a small $\lambda$, the cross section may be as low
as $10^{-48}\ {\rm cm^{2}}$, as shown on the left panel of Figure 3.

We add that the SI cross sections in Figure 3 are larger than $3\times
10^{-49}\ {\rm cm^{2}}$ because smaller cross sections require smaller values
of $\lambda$, which are not readily available since Bayesian evidence is
suppressed significantly. We also add that the characteristics of
$\sigma_{\tilde{\chi}^{0}_{1}-{p}}^{\rm SI}$ for the Type-I and -II samples
are different because $\sigma_{\tilde{\chi}^{0}_{1}-{p}}^{\rm SI}$ has no
significant cancellation effect in the latter case. Furthermore, we verified
the approximation for $\sigma_{\tilde{\chi}^{0}_{1}-{p}}^{\rm SI}$ by
considering the point $P_{2}$ in Table 2. The four terms in Eq. (2.19) and the
two contributions in Eq. (2.30) were as follows:

$\displaystyle C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}h}$
$\displaystyle\simeq$ $\displaystyle-0.01+0.0+0.0+0.002\sim-0.008,$
$\displaystyle\mathcal{A}$ $\displaystyle\simeq$
$\displaystyle~{}-0.008-0.0008\sim-0.0088.$ (4.6)

  * •

Since $2|\kappa|/\lambda\simeq 1$, the singlino-dominated
$\tilde{\chi}_{1}^{0}$ co-annihilated with the higgsino-dominated neutralinos
and charginos to provide the measured abundance. In addition, because
$\lambda$ and $\kappa$ must be small to suppress
$C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}G^{0}}$ and
$C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}a_{s}}$, the channels
$\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to t\bar{t},h_{s}a_{s}$ never have a
crucial effect on the abundance, even if they are kinematically accessible.

Figure 4: Same plot as in Fig. 2, but for the results of the Type-III samples.

Finally, we investigate the Type-III samples. Figure 1 indicates that some
samples correspond to the same parameter space as the Type-II samples, and
thus, they predict similar DM physics. This conclusion was verified by
studying some points in the two scenarios, e.g., points P2 and P4 in Table 2.
In the following, we only concentrate on the samples with same sign
$m_{\tilde{\chi}_{1}^{0}}/\mu$ and $\sin 2\beta$. The following results were
obtained from Figures 1 and 4.

  * •

Compared with the Type-II samples, $\lambda v/\mu_{eff}$ may have a greater
value of up to 0.12. The reason is there are accidental cancellations
contributing to $\sigma_{\tilde{\chi}^{0}_{1}-{p}}^{\rm SI}$, similar to the
Type-I samples. This will relax the XENON-1T constraint. We show this
characteristic by considering the point $P_{3}$ in Table 2 and finding

$\displaystyle C_{\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}h}$
$\displaystyle\simeq$ $\displaystyle 0.047\times(0.88-0.16)+0.0-0.0+0.0085\sim
0.0437,$ $\displaystyle\mathcal{A}$ $\displaystyle\simeq$ $\displaystyle
0.042-0.0285\sim 0.0135~{}.$

The cancellation also explains why $\sigma_{\tilde{\chi}^{0}_{1}-{p}}^{\rm
SI}$ was as low as $10^{-50}{\rm cm^{2}}$, as shown in the left panel of
Figure 4.

  * •

Due to a relatively large $\lambda v/\mu_{eff}$ and
$|m_{\tilde{\chi}^{0}_{1}}/\mu_{eff}|\rightarrow 1$ in Eq. (2.17),
$\sigma_{\tilde{\chi}^{0}_{1}-{n}}^{\rm SD}$ may be on the border of
$10^{-41}\ {\rm cm^{2}}$ (see right panel of Figure 4). Furthermore, similar
to the Type-II samples, $\tilde{\chi}_{1}^{0}$ co-annihilated with the
higgsino-dominated neutralinos and charginos to achieve the measured
abundance.

We add that the point $P_{3}$ predicts a significantly larger $\lambda$ than
points $P_{2}$ and $P_{4}$. Consequently, this will be readily tested in the
near-future PandaX-4T experiments for both SI and SD scattering.

##  5 Conclusion

This work provided updates to previous studies of singlino-dominated DM mainly
in four aspects:

  * (1)

We no longer require the fine-tuning measurement ${\Delta}m_{Z}$ as a model
selection criterion, since it may only reflect personal prejudice.

  * (2)

We adopted an advanced MultiNest algorithm to perform a sophisticated scan
over the $Z_{3}$-NMSSM parameter space to ensure that the obtained conclusions
were as complete as possible.

  * (3)

We utilized the latest experimental results to restrict singlino-dominated DM
scenarios, including the XENON-1T search for both SI and SD DM-nucleon
scattering since 2018, ATLAS analyses of sparticle signals with 139-${\rm
fb^{-1}}$ data, and measurements of the Higgs couplings with 80-${\rm
fb^{-1}}$ data.

  * (4)

We provided simplified analytical formulas for both the DM annihilation cross
sections and the SI and SD cross sections of DM-nucleon scattering, and we
numerically scrutinized each contribution in these formulas.

As a result, a new singlino-dominated DM scenario (Type-I samples) was found.
More model information, such as its Bayesian evidence, was obtained, and the
current and future statuses of the scenarios were presented. More importantly,
this study provided clear insight into singlino-dominated DM scenarios and
explained why they had been tightly limited in the $Z_{3}$-NMSSM.

Specifically, this study indicated that the surviving samples can be
categorized into three types:

  * •

For Type I samples, $0.4\lesssim\lambda\lesssim 0.7$,
$0.13\lesssim\kappa\lesssim 0.23$, $1.5\lesssim\tan\beta\lesssim 6$,
$450~{}{\rm GeV}\lesssim\mu_{eff}\lesssim 720~{}{\rm GeV}$, and the
annihilation $\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to t\bar{t}$ is
primarily responsible for the DM abundance.

  * •

For Type II and III samples, $0<\lambda\lesssim 0.15$, $\lambda\simeq
2|\kappa|$, and the dominant annihilation involves a co-annihilation with
higgsinos.

The Bayesian evidence ($Z$) for the three sample types showed that the
experiments slightly preferred Type-I samples to Type-II and -III. However,
Type-I samples will be examined in the near-future PandaX-4T experiment. They
will become highly disfavored if the experiment shows no signs of DM. It
should be emphasized that DM-annihilation by a singlet scalar or pseudo-scalar
funnel, a Z-boson funnel, and the SM-like Higgs funnel presented by Abdallah
et al. [56] were not observed in this study, due to the small Bayesian
evidence. In addition, both analytical formulas and numerical results were
used to summarize the theory’s four primary cases in significantly suppressing
the SI scattering cross section for DM-nucleons. This included 1) a small
$\lambda v/\mu_{eff}$, 2) cancellation between
$m_{|tilde{\chi}_{1}^{0}}/\mu_{eff}$ and $\sin 2\beta$, 3) cancellation
between two contributions from the SM doublet component ($H_{\rm SM}$) and
singlet component ($H_{s}$) within the
$\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}h$ coupling strength itself, and 4)
cancellation between two contributions from $h$ and $h_{s}$.

In summary, the interaction of the singlino-dominated DM with nucleons in the
$Z_{3}$-NMSSM has been tightly restricted in current DM-DD experiments, while
the measured abundance favors its involvement in weak interactions. Given the
theory’s natural preference for electroweak symmetry breaking, it has become
increasingly difficult to represent these two seemingly paradoxical features
using neutralino DM, due to the limited theoretical structure. Thus, the
benefits of singlino-dominated DM are waning unless one extends the
$Z_{3}$-invariant theory. Recent studies on the MSSM and NMSSM imply that DM
candidates should be gauge singlet fields or that singlet components should at
least be naturally dominant over other components [58]. This requirement
motivates us to extend the $Z_{3}$-NMSSM to a general NMSSM to increase the
Bayesian evidence of the scenarios significantly [88]. It also motivates us to
augment the $Z_{3}$-NMSSM with a seesaw mechanism to generate neutrino masses
and select the lightest sneutrino as a DM candidate [115, 107, 58, 116]888The
properties of the sneutrino DM in the $Z_{3}$-NMSSM were first studied one
decade ago [117, 118, 119, 120]. However, these studies considered the cases
where the cross section of the sneutrino-nucleon scattering was much larger
than current DD experimental bounds, so they obtained different conclusions
from our work..

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
(NNSFC) under Grant No. 11575053 and No. 12075076.

## Appendix

Recently, the ATLAS collaboration aimed to limit the compressed mass spectra
case. They analyzed 139 ${\rm fb^{-1}}$ of $\sqrt{s}=13\ {\rm TeV}$
proton–proton collision data collected at the LHC, focusing on the events with
missing transverse momentum and two same-flavor, oppositely charged, low-
transverse-momentum leptons, further categorizing them by the presence of
hadronic activity from initial-state radiation [111]. We repeated this
analysis using the simulation tools<EMAIL_ADDRESS>[121, 122] to
generate the parton level events, Pythia-8.2 [123] for parton fragmentation
and hadronization, Delphes-3.4.2 [124] for fast simulation of the performance
of the ATLAS detector, and CheckMATE-2.0.26 [125, 126, 127] to implement the
analysis cut selections.

Below, we validate our code for all signal regions (SRs) [111]. We considered
$\tilde{l}^{+}\tilde{l}^{-}$ production in the MSSM and set the masses of all
charginos and neutralinos other than the bino-like $\tilde{\chi}_{1}^{0}$ to
be 2.5 TeV. Thus, the sleptons will decay by
$\tilde{l}^{\pm}\rightarrow\l^{\pm}\tilde{\chi}_{1}^{0}$. We consider the
benchmark point $m_{\tilde{l}}=150~{}{\rm GeV}$ and
$m_{\tilde{\chi}_{1}^{0}}=140~{}{\rm GeV}$. As a result, the cross section at
the next-leading order is $126.62~{}{\rm fb}$ for $\tilde{l}_{L}\tilde{l}_{L}$
production and $47.62~{}{\rm fb}$ for $\tilde{l}_{R}\tilde{l}_{R}$ production.
The involved cards were set as follows:

⬇

import model MSSM_SLHA2 –modelname

generate p p > sl1+ sl1-, (sl1- > e- n1), (sl1+ > e+ n1)

add process p p > sl1+ sl1- j, (sl1- > e- n1), (sl1+ > e+ n1)

generate p p > sl2+ sl2-, (sl2- > mu- n1), (sl2+ > mu+ n1)

add process p p > sl2+ sl2- j, (sl1- > mu- n1), (sl2+ > mu+ n1).

For the proc_card.dat:

⬇

100000 = nevents ! Number of unweighted events requested.

0 = ickkw ! 0 no matching, 1 MLM

37.5 = ktdurham,

For the run_card.dat:

⬇

Block mass

1000011 1.500000e+02 # Msl1

1000013 1.500000e+02 # Msl2

1000022 1.400000e+02 # Mneu1

1000023 2.500000e+03 # Mneu2

1000024 2.500000e+03 # Mch1

1000025 -2.50000e+03 # Mneu3

1000035 2.500000e+03 # Mneu4

1000037 2.500000e+03 # Mch2

Block selmix

1 1 1.000000e+00 # RRl1x1

2 2 1.000000e+00 # RRl2x2

4 4 1.000000e+00 # RRl4x4

5 5 1.000000e+00 # RRl5x5

For the param_card.dat and the pythia8_card.dat:

⬇

Merging:Process = pp>{sl1-,1000011}{sl1+,-1000015}{sl2-,1000013}

{sl2+,-1000013}

Merging:mayRemoveDecayProducts=on

In our simulation, we generated 100,000 events for the production process. The
results, shown in Table 3, indicate that we can reproduce the ATLAS analysis
at the $20\%$ level for most cases.

Table 3: Cut flow for the analysis in [111]. We considered the point
$m(\tilde{\ell},\tilde{\chi}_{1}^{0})=(150,140)~{}{\rm GeV}$ in the
calculations.

Process | Production of $\tilde{\ell}\tilde{\ell}$
---|---
Point |  $m_{\tilde{\ell}}$ = 150 GeV; $m_{\tilde{\chi}_{1}^{0}}$ = 140 GeV
Generated Events | 100000
Selection | ATLAS | CheckMATE
Events | Efficiency | Events | Efficiency
Total events | 24069 | $\cdots$ | 24069 | $\cdots$
$E_{T}^{\rm miss}$ trigger | 2355.37 | $\cdots$ | 2355.37 | $\cdots$
Two leptons | 1014.55 | 43.07$\%$ | 1079.07 | 45.81$\%$
veto 3 GeV $<m_{\ell\ell}<3.2~{}{\rm GeV}$ | 1013.21 | 99.87$\%$ | 1077.69 | 99.87$\%$
lepton author 16 veto | 1009.48 | 99.63$\%$ | 1077.69 | 100.00$\%$
min$(\Delta\phi(\rm anyjet,p_{\rm T}^{\rm miss}))>$ 0.4 | 970.36 | 96.12$\%$ | 1049.11 | 97.35$\%$
$\Delta\phi(\rm j_{1},p_{\rm T}^{\rm miss})>$ 2.0 | 961.15 | 99.05$\%$ | 1027.05 | 97.90$\%$
lepton truth matching | 958.99 | 99.78$\%$ | 1027.05 | 100.00$\%$
1 $<m_{\ell\ell}<$ 60 GeV | 827.86 | 86.33$\%$ | 883.55 | 86.03$\%$
$\Delta R_{\rm ee}>$ 0.3, $\Delta R_{\rm\mu\mu}>$ 0.05, $\Delta R_{\rm e\rm\mu}>$ 0.2 | 826.19 | 99.80$\%$ | 883.48 | 99.99$\%$
$p_{\rm T}^{j_{1}}>$ 5 GeV | 823.70 | 99.70$\%$ | 880.95 | 99.71$\%$
$n_{\rm jet}\geq$ 1 | 810.59 | 98.41$\%$ | 880.95 | 100.00$\%$
$p_{\rm T}^{\rm j_{1}}>$ 100 GeV | 705.86 | 87.08$\%$ | 702.58 | 79.75$\%$
$n_{b-{\rm jet}}=0$ | 611.05 | 86.57$\%$ | 643.78 | 91.63$\%$
$m_{\rm\tau\tau}<0$ or $>$ 160 GeV | 533.29 | 87.27$\%$ | 569.78 | 88.51$\%$
ee or $\rm\mu\mu$ | 532.33 | 99.82$\%$ | 569.01 | 99.86$\%$
SR-highMass |  |  |  |
$E_{\rm T}^{\rm miss}>200~{}{\rm GeV}$ | 229.81 | 43.17$\%$ | 265.83 | 46.72$\%$
max(0.85, 0.98 - 0.02 $\times m_{\rm T_{2}}^{\rm 100}$) $<R_{\rm ISR}<$ 1.0 | 160.30 | 69.75$\%$ | 165.78 | 62.36$\%$
$p_{\rm T}^{\ell_{2}}>$ min(20.0, 2.5 + 2.5$\times(m_{\rm T_{2}}^{\rm 100}-\rm 100))$ | 70.71 | 44.11$\%$ | 72.51 | 43.74$\%$
$m_{\rm T2}^{\rm 100}<$ 140 GeV | 70.71 | 100.00$\%$ | 72.51 | 100.00$\%$
$m_{\rm T2}^{\rm 100}<$ 130 GeV | 70.71 | 100.00$\%$ | 72.51 | 100.00$\%$
$m_{\rm T2}^{\rm 100}<$ 120 GeV | 70.71 | 100.00$\%$ | 72.31 | 99.73$\%$
$m_{\rm T2}^{\rm 100}<$ 110 GeV | 70.71 | 100.00$\%$ | 72.23 | 99.90$\%$
$m_{\rm T2}^{\rm 100}<$ 105 GeV | 53.72 | 75.97$\%$ | 57.10 | 79.05$\%$
$m_{\rm T2}^{\rm 100}<$ 102 GeV | 20.21 | 37.62$\%$ | 23.77 | 41.63$\%$
$m_{\rm T2}^{\rm 100}<$ 101 GeV | 9.38 | 46.41$\%$ | 9.90 | 41.62$\%$
$m_{\rm T2}^{\rm 100}<$ 100.5 GeV | 4.68 | 49.89$\%$ | 4.86 | 49.11$\%$
SR-lowMass |  |  |  |
150 $<E_{\rm T}^{\rm miss}<$ 200 GeV | 146.36 | 27.49$\%$ | 167.63 | 29.46$\%$
0.8 $<R_{\rm ISR}<$ 1.0 | 107.82 | 73.67$\%$ | 93.17 | 55.58$\%$
$p_{\rm T}^{\ell_{2}}>$ min(15.0, 7.5 + 0.75$\times(m_{\rm T2}^{\rm 100}-\rm 100))$ | 52.74 | 48.91$\%$ | 42.29 | 45.39$\%$
$m_{\rm T2}^{\rm 100}<$ 140 GeV | 52.74 | 100.00$\%$ | 42.29 | 100.00$\%$
$m_{\rm T2}^{\rm 100}<$ 130 GeV | 52.74 | 100.00$\%$ | 42.29 | 100.00$\%$
$m_{\rm T2}^{\rm 100}<$ 120 GeV | 52.74 | 100.00$\%$ | 42.29 | 100.00$\%$
$m_{\rm T2}^{\rm 100}<$ 110 GeV | 52.64 | 99.81$\%$ | 41.65 | 98.49$\%$
$m_{\rm T2}^{\rm 100}<$ 105 GeV | 38.05 | 72.28$\%$ | 29.09 | 69.85$\%$
$m_{\rm T2}^{\rm 100}<$ 102 GeV | 16.66 | 43.78$\%$ | 11.24 | 38.62$\%$
$m_{\rm T2}^{\rm 100}<$ 101 GeV | 8.70 | 52.22$\%$ | 5.60 | 49.82$\%$
$m_{\rm T2}^{\rm 100}<$ 100.5 GeV | 4.39 | 50.46$\%$ | 2.29 | 40.88$\%$

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