# Impact of recent measurement of $(g-2)_{\mu}$, LHC search for supersymmetry,
and LZ experiment on Minimal Supersymmetric Standard Model

Yangle Hea, Xinglong Jiaa, Lei Menga,∗††*Corresponding author., Yuanfang Yuea,
and Di Zhanga

###### Abstract

Motivated by the recent measurement of muon anomalous magnetic moment at
Fermilab, the rapid progress of the LHC search for supersymmetry, and the
significantly improved sensitivities of dark matter direct detection
experiments, we studied their impacts on the Minimal Supersymmetric Standard
Model (MSSM). We conclude that higgsino mass should be larger than about
$500~{}{\rm GeV}$ for $M_{1}<0$ and $630~{}{\rm GeV}$ for $M_{1}>100~{}{\rm
GeV}$, where $M_{1}$ denotes the bino mass. These improved bounds imply a
tuning of ${\cal{O}}(1\%)$ to predict the $Z$-boson mass and simultaneously
worsen the naturalness of the $Z$\- and $h$-mediated resonant annihilations to
achieve the measured dark matter density. We also concluded that the LHC
restrictions have set lower bounds on the sparticle mass spectra:
$m_{\tilde{\chi}_{1}^{0}}\gtrsim 210~{}{\rm GeV}$,
$m_{\tilde{\chi}_{2}^{0}},m_{\tilde{\chi}_{1}^{\pm}}\gtrsim 235~{}{\rm GeV}$,
$m_{\tilde{\chi}_{3}^{0}}\gtrsim 515~{}{\rm GeV}$,
$m_{\tilde{\chi}_{4}^{0}}\gtrsim 525~{}{\rm GeV}$,
$m_{\tilde{\chi}_{2}^{\pm}}\gtrsim 530~{}{\rm GeV}$,
$m_{\tilde{\nu}_{\mu}}\gtrsim 235~{}{\rm GeV}$, $m_{\tilde{\mu}_{1}}\gtrsim
215~{}{\rm GeV}$, and $m_{\tilde{\mu}_{2}}\gtrsim 250~{}{\rm GeV}$, where
$\tilde{\chi}_{2}^{0}$ and $\tilde{\chi}_{1}^{\pm}$ are wino-dominated when
they are lighter than about $500~{}{\rm GeV}$. These bounds are far beyond the
reach of the LEP experiments in searching for supersymmetry and have not been
acquired before. In addition, we illuminate how some parameter spaces of the
MSSM have been tested at the LHC and provide five scenarios in which the
theory coincides with the LHC restrictions. Once the muon g-2 anomaly is
confirmed to originate from supersymmetry, this research may serve as a guide
to explore the characteristics of the MSSM in future experiments.

Keywords: Supersymmetry, MSSM, muon anomalous magnetic moment, the LHC search
for SUSY, LZ experiment.

##  1 Introduction

As the cornerstone of particle physics, the Standard Model (SM) has
encapsulated our best understanding of fundamental particles and forces.
Although it is well tested by many experimental results, there are still
unsolved puzzles, such as the quadratic divergence in the Higgs squared mass
and the absence of dark matter (DM) candidates. Historically, these puzzles
were viewed as robust evidence of new physics beyond the SM, and looking for
mechanisms to circumvent them was the model-building guideline. Supersymmetry
(SUSY) is the most promising among the new physics theories due to its elegant
structure and remarkable advantages in solving these puzzles [1, 2, 3, 4].

To date, rich information about SUSY has been accumulated due to the rapid
progress of particle physics experiments in recent years. The Run-II data of
the Large Hadron Collider (LHC) enabled scientists to explore the properties
of winos, higgsinos, and scalar leptons (sleptons), which are the SUSY
partners of $W$, Higgs, and lepton fields, respectively. It was found that
wino masses up to about $1060~{}{\rm GeV}$ for
$m_{\tilde{\chi}_{1}^{0}}\lesssim 400~{}{\rm GeV}$ and higgsino masses up to
$900~{}{\rm GeV}$ for $m_{\tilde{\chi}_{1}^{0}}\lesssim 240~{}{\rm GeV}$ have
been excluded in the simplified model of SUSY [5], where
$\tilde{\chi}_{1}^{0}$ denotes the lightest neutralino, acting as the lightest
supersymmetric particle (LSP) and thus a DM candidate under the assumption of
$R$-parity conservation [4], and $m_{\tilde{\chi}_{1}^{0}}$ is its mass. The
data also excluded sleptons lighter than approximately $700~{}{\rm GeV}$ when
the LSP was massless based on statistical methods [5, 6]. Furthermore, the
LUX-ZEPLIN (LZ) experiment just released its first results about the direct
search for DM, where the sensitivities to spin-independent (SI) and spin-
dependent (SD) cross sections of DM-nucleon scattering have reached about
$6.0\times 10^{-48}~{}{\rm cm^{2}}$ and $1.0\times 10^{-42}~{}{\rm cm^{2}}$,
respectively, for the DM mass around $30~{}{\rm GeV}$ [7]. These unprecedented
precision values strongly limit the DM coupling to the SM particles, which are
determined by SUSY parameters. In addition, the combined measurement of the
muon anomalous magnetic moment, $a_{\mu}\equiv(g-2)_{\mu}/2$, by the E821
experiment at the Brookhaven National Laboratory (BNL) [8] and the E989
experiment at Fermilab [9] indicates a 4.2$\sigma$ discrepancy from the SM’s
prediction [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30]. Although this difference may have been induced by the
uncertainties in calculating the hadronic contribution to the moment, as
revealed by the recent lattice simulation of the BMW collaboration [31], it
was widely speculated to arise from new physics (see, e.g., Ref. [32] and the
references therein). Along this direction, it is remarkable that once the
difference is confirmed to originate from SUSY effects, salient features of
the theory, e.g., the mass spectra of the electroweakinos and sleptons, can be
inferred [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68,
69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87,
88, 89].

Given that SUSY predictions on these experimental results rely on different
theoretical inputs, it is essential to collectively study their impacts on the
Minimal Supersymmetric Standard Model (MSSM), which is the most economical
realization of SUSY in particle physics [2, 90, 91]. For this purpose, we
organize this study as follows. In Sec. 2, we briefly introduce the basics of
the MSSM, including its DM physics, the SUSY contribution to $a_{\mu}$, the
signals of SUSY particles (sparticles) at the LHC, and the strategy to search
for them. In Sec. 3, we perform a sophisticated scan over the broad parameter
space of the MSSM and clarify how the MSSM remains consistent with the
experimental results. Finally, we draw conclusions in Sec. 4.

##  2  Theoretical preliminaries of Minimal Supersymmetric Standard Model
(MSSM)

The following superpotential of the MSSM was given in Ref. [2, 90]:

$\displaystyle W=-Y_{d}\hat{q}\cdot\hat{H}_{d}\
\hat{d}-Y_{e}\hat{l}\cdot\hat{H}_{d}\ \hat{e}+Y_{u}\hat{q}\cdot\hat{H}_{u}\
\hat{u}+\mu\hat{H}_{u}\cdot\hat{H}_{d},$ (2.1)

where the superfields $\hat{q}$ and $\hat{l}$ are left-handed SU(2) doublets
for quarks and leptons, respectively, and $\hat{u}$, $\hat{d}$, and $\hat{e}$
are right-handed singlets for fermions. The scalar components of the Higgs
doublet superfields, $\hat{H}_{u}$ and $\hat{H}_{d}$, are given by
$H_{u}=(H_{u}^{+},H_{u}^{0})$ and $H_{d}=(H_{d}^{0},H_{d}^{-})$, respectively,
and their product is defined by $H_{u}\cdot
H_{d}=(H_{u}^{+}H_{d}^{-}-H_{u}^{0}H_{d}^{0})$. The first three terms in the
superpotential represent the Yukawa couplings of the quark and lepton fields,
and the last term is responsible for the higgsino mass.

The MSSM predicts two CP-even Higgs bosons, $h$ and $H$, one CP-odd Higgs
boson $A$, and a pair of charged Higgs boson $H^{\pm}=\cos\beta
H_{u}^{\pm}+\sin\beta H_{d}^{\pm}$ in the Higgs sector [90, 91]. Among these
states, $h$ denotes the SM-like scalar discovered at the LHC with $m_{h}\simeq
125~{}{\rm GeV}$, and the neutral states $H$ and $A$ are approximately
degenerate with $H^{\pm}$ in mass. The LHC search for non-SM-like Higgs bosons
has obtained model-independent upper limits on the production rates of $H$,
$A$, and $H^{\pm}$ (see, e.g., Ref. [92, 93]), indicating that they should be
massive. The electroweakino sector of the MSSM consists of four neutralinos
and two pairs of charginos [90], denoted by $\tilde{\chi}_{i}^{0}$ with
$i=1,2,3,4$ and $\tilde{\chi}_{j}^{\pm}$ with $j=1,2$, respectively, in this
work. The neutralinos are superpositions of bino ($\tilde{B}$), wino
($\tilde{W}^{0}$), and two higgsino fields ($\tilde{H}_{d}^{0}$ and
$\tilde{H}_{u}^{0}$), and they are majorana fermions. By contrast, the left-
handed and right-handed components of the chargino $\tilde{\chi}^{+}_{j}$ come
from the mixing of $\tilde{W}^{+}$ with $\tilde{H}_{u}^{+}$ and
$\tilde{W}^{-}$ with $\tilde{H}_{d}^{-}$, respectively. Thus,
$\tilde{\chi}_{j}^{\pm}$ are Dirac fermions. By convention, the neutralinos as
mass eigenstates are labeled in ascending mass order, and so are the
charginos. In addition, each slepton mass eigenstate in the MSSM is associated
with a definite flavor quantum number if there is no flavor mixing in the
slepton sector [94, 95]. The $\ell$-flavored sleptons $\tilde{\ell}_{i}$
(i=1,2) are mixtures of chiral scalar fields $\tilde{\ell}_{L}$ and
$\tilde{\ell}_{R}$. Given that the mixing is usually small, we also denote
$\tilde{\ell}_{i}$ by its dominant component sometimes to facilitate our
discussion.

###  2.1 Dark matter (DM) physics in MSSM

On the premise of explaining both the measured DM density and the muon g-2
anomaly, the DM candidate in the MSSM must be the bino-dominated lightest
neutralino [96]111In the case that the lightest left-handed sneutrino acts as
a DM candidate, its interaction with $Z$-boson predicts a much smaller density
than its measured value, i.e., $\Omega\hbar^{2}\ll 0.12$, and meanwhile an
unacceptably large DM-nucleon scattering rate [97]. For the wino- or higgsino-
dominated DM case, the density is on the order of $10^{-3}$ by our
calculation. These cases were surveyed in the MSSM to explain the muon g-2
anomaly in Ref. [98].. It achieves the measured density through the co-
annihilation with wino-dominated electroweakinos or sleptons, or through $Z$\-
or $h$-mediated resonant annihilation [99]. In the co-annihilation case, the
reactions $S_{i}S_{j}\to XX^{\prime}$, where $S_{i}S_{j}$ may be LSP-LSP, LSP-
NLSP (next lightest supersymmetric particle), and NLSP-NLSP annihilation
states and $XX^{\prime}$ denotes SM particles, contribute to the density [100,
101]. The effective annihilation rate at a temperature $T$ is then given by
Eq. (3.2) in Ref. [101]. This formula indicates that the annihilation partner
only has a significant effect when the departure of its mass from the DM mass
is less than about $10\%$. The resonant annihilation is distinct in that the
density is very sensitive to the splitting between
$2|m_{\tilde{\chi}_{1}^{0}}|$ and the mediator’s mass [99]. The weaker the DM
coupled to the mediator, the smaller the splitting must be to obtain the
measured density. Evidently, this situation requires the fine-tuning of the
theoretical parameters.

The cross sections of the DM-nucleon scattering are approximated by [102, 103]

$\displaystyle\sigma_{\tilde{\chi}_{1}^{0}-N}^{\rm SI}$ $\displaystyle\simeq$
$\displaystyle 5\times 10^{-45}{\rm
cm^{2}}\left(\frac{C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}h}}{0.1}\right)^{2}\left(\frac{m_{h}}{125{\rm
GeV}}\right)^{2},$ (2.2) $\displaystyle\sigma_{\tilde{\chi}_{1}^{0}-N}^{\rm
SD}$ $\displaystyle\simeq$ $\displaystyle
C_{N}\times\left(\frac{C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}Z}}{0.1}\right)^{2},$
(2.3)

where $C_{p}\simeq 1.8\times 10^{-40}~{}{\rm cm^{2}}$ for protons,
$C_{n}\simeq 1.4\times 10^{-40}~{}{\rm cm^{2}}$ for neutrons, and
$C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}h}$ and
$C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}Z}$ represent DM couplings to the
SM-like Higgs boson and $Z$-boson, respectively. These couplings take the
following form [104, 105, 106]:

$\displaystyle C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}h}$
$\displaystyle\simeq$ $\displaystyle
e\tan\theta_{W}\frac{m_{Z}}{\mu(1-m_{\tilde{\chi}_{1}^{0}}^{2}/\mu^{2})}\left(\cos(\beta+\alpha)+\sin(\beta-\alpha)\frac{m_{\tilde{\chi}_{1}^{0}}}{\mu}\right)$
(2.4) $\displaystyle\simeq$ $\displaystyle
e\tan\theta_{W}\frac{m_{Z}}{\mu(1-m_{\tilde{\chi}_{1}^{0}}^{2}/\mu^{2})}\left(\sin
2\beta+\frac{m_{\tilde{\chi}_{1}^{0}}}{\mu}\right),$ $\displaystyle
C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}Z}$ $\displaystyle\simeq$
$\displaystyle\frac{e\tan\theta_{W}\cos
2\beta}{2}\frac{m_{Z}^{2}}{\mu^{2}-m_{\tilde{\chi}_{1}^{0}}^{2}},$ (2.5)

where $\theta_{W}$ is the weak mixing angle, the DM mass
$m_{\tilde{\chi}_{1}^{0}}$ relates to the bino mass $M_{1}$ by
$m_{\tilde{\chi}_{1}^{0}}\simeq M_{1}$, $\tan\beta=v_{u}/v_{d}$ is the ratio
of Higgs vacuum expectation values, and $\alpha$ is the mixing angle of the
CP-even Higgs states satisfying $\alpha\simeq\beta-\pi/2$ in the large $m_{A}$
limit [91]. The formulae for the SI scattering indicate that if $M_{1}$ and
$\mu$ are of the same sign, $\mu$ must be sufficiently large to be consistent
with the results of the PandaX-4T experiment [107]. These formulas also show
that if $M_{1}$ and $\mu$ are of opposite signs, which can result in the blind
spots of the scattering [108, 109, 110, 111], $|\mu|\sim 100~{}{\rm GeV}$
seems to be experimentally allowed. However, such a possibility has been
limited by the experimental search for spin-dependent (SD) DM-nucleon
scattering because, regardless of the relative sign between $M_{1}$ and $\mu$,
a small $|\mu|$ can enhance the scattering cross section. In summary, the DM
direct detection experiments alone can set a lower bound on the magnitude of
$\mu$. With the improvement of the experimental sensitivity, the bound will
become tightened.

###  2.2 Muon g-2

The SUSY source of the muon g-2, $a^{\rm SUSY}_{\mu}$, mainly includes loops
mediated by a smuon and a neutralino and those containing a muon-flavor
sneutrino and a chargino [35, 34, 36, 33]. The full one-loop contributions to
$a^{\rm SUSY}_{\mu}$ in the MSSM are not presented here for brevity. Instead,
we provide the expression of $a_{\mu}^{\rm SUSY}$ in the mass insertion
approximation to reveal its key features [35]. Specifically, at the lowest
order of the approximation, the contributions to $a_{\mu}^{\rm SUSY}$ are
divided into four types: ”WHL,” ”BHL,” ”BHR,” and ”BLR,” where $W$, $B$, $H$,
$L$, and $R$ represent wino, bino, higgsino, and left-handed and right-handed
smuon fields, respectively. They arise from the Feynman diagrams involving
$\tilde{W}-\tilde{H}_{d}$, $\tilde{B}-\tilde{H}_{d}^{0}$,
$\tilde{B}-\tilde{H}_{d}^{0}$, and $\tilde{\mu}_{L}-\tilde{\mu}_{R}$
transitions, respectively, and take the following forms [37, 35, 38]:

$\displaystyle a_{\mu,\rm WHL}^{\rm SUSY}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{2}}{8\pi}\frac{m_{\mu}^{2}M_{2}\mu_{tot}\tan\beta}{m_{\tilde{\nu}_{\mu}}^{4}}\left\\{2f_{C}\left(\frac{M_{2}^{2}}{m_{\tilde{\nu}_{\mu}}^{2}},\frac{\mu_{tot}^{2}}{m_{\tilde{\nu}_{\mu}}^{2}}\right)-\frac{m_{\tilde{\nu}_{\mu}}^{4}}{\tilde{m}_{\tilde{\mu}_{L}}^{4}}f_{N}\left(\frac{M_{2}^{2}}{\tilde{m}_{\tilde{\mu}_{L}}^{2}},\frac{\mu_{tot}^{2}}{\tilde{m}_{\tilde{\mu}_{L}}^{2}}\right)\right\\}\,,\quad\quad$
(2.6) $\displaystyle a_{\mu,\rm BHL}^{\rm SUSY}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{Y}}{8\pi}\frac{m_{\mu}^{2}M_{1}\mu_{tot}\tan\beta}{\tilde{m}_{\tilde{\mu}_{L}}^{4}}f_{N}\left(\frac{M_{1}^{2}}{\tilde{m}_{\tilde{\mu}_{L}}^{2}},\frac{\mu_{tot}^{2}}{\tilde{m}_{\tilde{\mu}_{L}}^{2}}\right)\,,$
(2.7) $\displaystyle a_{\mu,\rm BHR}^{\rm SUSY}$ $\displaystyle=$
$\displaystyle-\frac{\alpha_{Y}}{4\pi}\frac{m_{\mu}^{2}M_{1}\mu_{tot}\tan\beta}{\tilde{m}_{\tilde{\mu}_{R}}^{4}}f_{N}\left(\frac{M_{1}^{2}}{\tilde{m}_{\tilde{\mu}_{R}}^{2}},\frac{\mu_{tot}^{2}}{\tilde{m}_{\tilde{\mu}_{R}}^{2}}\right)\,,$
(2.8) $\displaystyle a_{\mu\rm BLR}^{\rm SUSY}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{Y}}{4\pi}\frac{m_{\mu}^{2}M_{1}\mu_{tot}\tan\beta}{M_{1}^{4}}f_{N}\left(\frac{\tilde{m}_{\tilde{\mu}_{L}}^{2}}{M_{1}^{2}},\frac{\tilde{m}_{\tilde{\mu}_{R}}^{2}}{M_{1}^{2}}\right)\,,$
(2.9)

where $\tilde{m}_{\tilde{\mu}_{L}}$ and $\tilde{m}_{\tilde{\mu}_{R}}$ are
soft-breaking masses for left-handed and right-handed smuon fields,
respectively, at the slepton mass scale. They are approximately equal to
slepton masses. The loop functions are given by

$\displaystyle f_{C}(x,y)$ $\displaystyle=$
$\displaystyle\frac{5-3(x+y)+xy}{(x-1)^{2}(y-1)^{2}}-\frac{2\ln
x}{(x-y)(x-1)^{3}}+\frac{2\ln y}{(x-y)(y-1)^{3}}\,,$ (2.10) $\displaystyle
f_{N}(x,y)$ $\displaystyle=$
$\displaystyle\frac{-3+x+y+xy}{(x-1)^{2}(y-1)^{2}}+\frac{2x\ln
x}{(x-y)(x-1)^{3}}-\frac{2y\ln y}{(x-y)(y-1)^{3}}\,,$ (2.11)

and they satisfy $f_{C}(1,1)=1/2$ and $f_{N}(1,1)=1/6$.

The following points about $a_{\mu}^{\rm SUSY}$ should be noted:

  * •

If all the dimensional SUSY parameters involved in $a_{\mu}^{\rm SUSY}$ take a
common value $M_{\rm SUSY}$, $a_{\mu}^{\rm SUSY}$ is proportional to
$m_{\mu}^{2}\tan\beta/M_{\rm SUSY}^{2}$, indicating that the muon g-2 anomaly
prefers a large $\tan\beta$ and a moderately low SUSY scale.

  * •

The ”WHL” contribution is usually much larger than the other contributions if
$\tilde{\mu}_{L}$ is not significantly heavier than $\tilde{\mu}_{R}$ [72].

  * •

The difference between the $a_{\mu}^{\rm SUSY}$ values calculated by the mass
insertion approximation and the full expression was less than $3\%$. We
verified this conclusion for the green samples in Fig. 1 of this work.

  * •

The two-loop (2L) contributions to $a_{\mu}$, including 2L corrections to SM
one-loop diagrams and those to SUSY one-loop diagrams [39], were about $-5\%$
of the one-loop prediction [89]. These were neglected in this study.

Table 1: Experimental analyses of the electroweakino production processes
considered in this study, which are categorized by the topologies of the
supersymmetry (SUSY) signals.

Scenario | Final State | Name
---|---|---
$\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow WZ\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}$ | $n\ell(n\geq 2)+nj(n\geq 0)+\text{E}_{\text{T}}^{\text{miss}}$ |  CMS-SUS-20-001($137fb^{-1}$) [6]
ATLAS-2106-01676($139fb^{-1}$) [112]
CMS-SUS-17-004($35.9fb^{-1}$) [113]
CMS-SUS-16-039($35.9fb^{-1}$) [114]
ATLAS-1803-02762($36.1fb^{-1}$) [115]
ATLAS-1806-02293($36.1fb^{-1}$) [116]
$\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow\ell\tilde{\nu}\ell\tilde{\ell}$ | $n\ell(n=3)+\text{E}_{\text{T}}^{\text{miss}}$ |  CMS-SUS-16-039($35.9fb^{-1}$) [114]
ATLAS-1803-02762($36.1fb^{-1}$) [115]
$\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow\tilde{\tau}\nu\ell\tilde{\ell}$ | $2\ell+1\tau+\text{E}_{\text{T}}^{\text{miss}}$ |  CMS-SUS-16-039($35.9fb^{-1}$) [114]
$\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow\tilde{\tau}\nu\tilde{\tau}\tau$ | $3\tau+\text{E}_{\text{T}}^{\text{miss}}$ |  CMS-SUS-16-039($35.9fb^{-1}$) [114]
$\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow Wh\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}$ | $n\ell(n\geq 1)+nb(n\geq 0)+nj(n\geq 0)+\text{E}_{\text{T}}^{\text{miss}}$ |  ATLAS-1909-09226($139fb^{-1}$) [117]
CMS-SUS-17-004($35.9fb^{-1}$) [113]
CMS-SUS-16-039($35.9fb^{-1}$) [114]
ATLAS-1812-09432($36.1fb^{-1}$)[118]
CMS-SUS-16-034($35.9fb^{-1}$)[119]
CMS-SUS-16-045($35.9fb^{-1}$) [120]
$\tilde{\chi}_{1}^{\mp}\tilde{\chi}_{1}^{\pm}\rightarrow WW\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}$ | $2\ell+\text{E}_{\text{T}}^{\text{miss}}$ |  ATLAS-1908-08215($139fb^{-1}$) [5]
CMS-SUS-17-010($35.9fb^{-1}$) [121]
$\tilde{\chi}_{1}^{\mp}\tilde{\chi}_{1}^{\pm}\rightarrow 2\tilde{\ell}\nu(\tilde{\nu}\ell)$ | $2\ell+\text{E}_{\text{T}}^{\text{miss}}$ |  ATLAS-1908-08215($139fb^{-1}$) [5]
CMS-SUS-17-010($35.9fb^{-1}$) [121]
$\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\mp}\rightarrow h/ZW\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0},\tilde{\chi}_{1}^{0}\rightarrow\gamma/Z\tilde{G}$ | $2\gamma+n\ell(n\geq 0)+nb(n\geq 0)+nj(n\geq 0)+\text{E}_{\text{T}}^{\text{miss}}$ | ATLAS-1802-03158($36.1fb^{-1}$) [122]
$\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{1}^{\mp}\rightarrow
WW\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0},\tilde{\chi}_{1}^{0}\rightarrow\gamma/Z\tilde{G}$
$\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow ZW\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0},\tilde{\chi}_{1}^{0}\rightarrow h/Z\tilde{G}$ | $n\ell(n\geq 4)+\text{E}_{\text{T}}^{\text{miss}}$ | ATLAS-2103-11684($139fb^{-1}$) [123]
$\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{1}^{\mp}\rightarrow
WW\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0},\tilde{\chi}_{1}^{0}\rightarrow
h/Z\tilde{G}$
$\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{0}\rightarrow
Z\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0},\tilde{\chi}_{1}^{0}\rightarrow
h/Z\tilde{G}$
$\tilde{\chi}_{1}^{\mp}\tilde{\chi}_{1}^{0}\rightarrow
W\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0},\tilde{\chi}_{1}^{0}\rightarrow
h/Z\tilde{G}$
$\tilde{\chi}_{i}^{0,\pm}\tilde{\chi}_{j}^{0,\mp}\rightarrow\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}+\chi_{soft}\rightarrow ZZ/H\tilde{G}\tilde{G}$ | $n\ell(n\geq 2)+nb(n\geq 0)+nj(n\geq 0)+\text{E}_{\text{T}}^{\text{miss}}$ |  CMS-SUS-16-039($35.9fb^{-1}$) [114]
CMS-SUS-17-004($35.9fb^{-1}$) [113]
CMS-SUS-20-001($137fb^{-1}$) [6]
$\tilde{\chi}_{i}^{0,\pm}\tilde{\chi}_{j}^{0,\mp}\rightarrow\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}+\chi_{soft}\rightarrow HH\tilde{G}\tilde{G}$ | $n\ell(n\geq 2)+nb(n\geq 0)+nj(n\geq 0)+\text{E}_{\text{T}}^{\text{miss}}$ |  CMS-SUS-16-039($35.9fb^{-1}$) [114]
CMS-SUS-17-004($35.9fb^{-1}$) [113]
$\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow W^{*}Z^{*}\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}$ | $3\ell+\text{E}_{\text{T}}^{\text{miss}}$ |  ATLAS-2106-01676($139fb^{-1}$) [112]
$\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}\rightarrow Z^{*}W^{*}\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}$ | $2\ell+nj(n\geq 0)+\text{E}_{\text{T}}^{\text{miss}}$ |  ATLAS-1911-12606($139fb^{-1}$) [124]
ATLAS-1712-08119($36.1fb^{-1}$) [125]
|  CMS-SUS-16-048($35.9fb^{-1}$) [126]
$\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}+\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{1}^{\mp}+\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{1}^{0}$ | $2\ell+nj(n\geq 0)+\text{E}_{\text{T}}^{\text{miss}}$ |  ATLAS-1911-12606($139fb^{-1}$) [124]
ATLAS-1712-08119($36.1fb^{-1}$) [125]
CMS-SUS-16-048($35.9fb^{-1}$) [126]

Table 2: Same as Table 1, but for the slepton production processes.

Scenario | Final State | Name
---|---|---
$\tilde{\ell}\tilde{\ell}\rightarrow\ell\ell\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}$ | $2\ell+\text{E}_{\text{T}}^{\text{miss}}$ | ATLAS-1911-12606($139fb^{-1}$) [124]
ATLAS-1712-08119($36.1fb^{-1}$) [125]
ATLAS-1908-08215($139fb^{-1}$) [5]
CMS-SUS-20-001($137fb^{-1}$) [6]
ATLAS-1803-02762($36.1fb^{-1}$) [115]
CMS-SUS-17-009($35.9fb^{-1}$) [127]

###  2.3 LHC search for SUSY

Since some of the electroweakinos and sleptons involved in $a_{\mu}^{\rm
SUSY}$ must be moderately light to account for the anomaly [84], they are
copiously produced at the LHC and thus are subjected to strong constraints
from the SUSY searches at the LHC with $\sqrt{s}=13~{}\rm{TeV}$. These
searches usually concentrate on theories with $R$-parity conservation [128,
129], where the LSP is undetected, leading to missing energy in the final
states. We implement these restrictions by scrutinizing the experimental
analyses in Tables 1 and 2. We find that the following reports are
particularly critical:

  * •

CMS-SUS-20-001 [6]: Search for SUSY signal containing two oppositely charged
same-flavor leptons and missing transverse momentum. This analysis studied not
only strong sparticle productions but also electroweakino productions. The
lepton originates from an on-shell or off-shell $Z$ boson in the decay chain
or from the decay of the produced sleptons. For the electroweakino pair
production, the wino-dominated chargino and neutralino were explored up to
masses of $750~{}{\rm GeV}$ and $800~{}{\rm GeV}$, respectively. For the
slepton pair production, the first two-generation sleptons were explored up to
a mass of $700~{}{\rm GeV}$.

  * •

CMS-SUS-16-039 and CMS-SUS-17-004 [114, 113]: Search for electroweakino
productions with two, three, or four leptons and missing transverse momentum
($\rm{E}_{\rm{T}}^{\rm{miss}}$) in the final states. One remarkable strategy
of this analysis was that it included all the possible final states and
defined several categories by the number of leptons in the event, their
flavors, and their charges to enhance the discovery potential. In the context
of simplified models, the observed limit on wino-dominated
$m_{\tilde{\chi}_{1}^{\pm}}$ in the chargino–neutralino production was about
650 GeV for the WZ topology, 480 GeV for the WH topology, and 535 GeV for the
mixed topology.

  * •

ATLAS-2106-01676 [112]: Searched for wino- or higgsino-dominated
chargino–neutralino pair productions. This analysis investigated on-shell
$WZ$, off-shell $WZ$, and $Wh$ categories in the decay chain and focused on
the final state containing exactly three leptons, possible ISR jets, and
$\rm{E}_{\rm{T}}^{\rm{miss}}$. For the wino scenario in the simplified model,
the exclusion bound of $m_{\tilde{\chi}_{2}^{0}}$ was about $640~{}\rm{GeV}$
for a massless $\tilde{\chi}_{1}^{0}$, and it was weakened as the mass
difference between $\tilde{\chi}_{2}^{0}$ and $\tilde{\chi}_{1}^{0}$
diminished. Specifically, $\tilde{\chi}_{2}^{0}$ should be heavier than about
$500~{}\rm{GeV}$ for $m_{\tilde{\chi}_{1}^{0}}=300~{}{\rm GeV}$ (the on-shell
W/Z case), $300~{}\rm{GeV}$ for a positive $m_{\tilde{\chi}_{1}^{0}}$ and
$35~{}{\rm GeV}\lesssim
m_{\tilde{\chi}_{2}^{0}}-m_{\tilde{\chi}_{1}^{0}}\lesssim 90~{}{\rm GeV}$ (the
off-shell W/Z case), and $220~{}\rm{GeV}$ when
$m_{\tilde{\chi}_{2}^{0}}-m_{\tilde{\chi}_{1}^{0}}=15~{}{\rm GeV}$ (the
extreme off-shell W/Z case). By contrast, $\tilde{\chi}_{2}^{0}$ was excluded
only up to a mass of $210~{}\rm{GeV}$ for the off-shell W/Z case of the
higgsino scenario, which occurred when
$m_{\tilde{\chi}_{2}^{0}}-m_{\tilde{\chi}_{1}^{0}}=10~{}{\rm GeV}$ or
$m_{\tilde{\chi}_{2}^{0}}-m_{\tilde{\chi}_{1}^{0}}\gtrsim 35~{}{\rm GeV}$.

  * •

ATLAS-1911-12606 [124]: Concentration on compressed mass spectra case and
search for electroweakino pair or slepton pair production, with two leptons
and missing transverse momentum as the final state. The results were projected
onto the $\Delta m-\tilde{\chi}_{2}^{0}$ plane, where $\Delta m\equiv
m_{\tilde{\chi}_{2}^{0}}-m_{\tilde{\chi}_{1}^{0}}$ for the electroweakino
production. It was found that the tightest bound on the higgsino-dominated
$\tilde{\chi}_{2}^{0}$ was $193~{}{\rm GeV}$ in mass for $\Delta m\simeq
9.3~{}{\rm GeV}$, and the optimum bound on the wino-dominated
$\tilde{\chi}_{2}^{0}$ was $240~{}{\rm GeV}$ in mass when $\Delta m\simeq
7~{}{\rm GeV}$. Similarly, it was found that light-flavor sleptons should be
heavier than about 250 GeV for $\Delta m_{\tilde{\ell}}=10~{}{\rm GeV}$, where
$m_{\tilde{\ell}}\equiv m_{\tilde{\ell}}-m_{\tilde{\chi}_{1}^{0}}$.

Note that all the analyses were based on $139~{}\rm fb^{-1}$ data except for
the second analysis, which studied $36~{}\rm fb^{-1}$ data.

Table 3: Parameter space explored in this study. Other dimensional parameters
not crucial to this study were fixed at 3 TeV, including the SUSY parameters
for the first- and third-generation sleptons, three generation squarks (except
for the soft trilinear coefficients $A_{t}$ and $A_{b}$, which are assumed to
be equal and change freely), and gluinos. All the parameters were defined at
the renormalization scale $Q=1~{}{\rm TeV}$.

Parameter | Prior | Range | Parameter | Prior | Range
---|---|---|---|---|---
$\tan{\beta}$ | Flat | $1\sim 60$ | $A_{t}/{\rm TeV}$ | Flat | $-5.0\sim 5.0$
$\mu/{\rm TeV}$ | Log | $0.1\sim 1.0$ | $m_{\rm A}/{\rm TeV}$ | Log | $0.5\sim 10$
$M_{1}/{\rm TeV}$ | Flat | $-1.5\sim 1.5$ | $M_{2}/{\rm TeV}$ | Log | $0.1\sim 1.5$
$\tilde{m}_{\tilde{\mu}_{L}}/{\rm TeV}$ | Log | $0.1\sim 1.0$ | $\tilde{m}_{\tilde{\mu}_{R}}/{\rm TeV}$ | Log | $0.1\sim 1.0$

##  3 Combined experimental impacts on MSSM

This research utilized the package SARAH-4.14.3 [130, 131, 132, 133] to build
the model file of the MSSM, the codes SPheno-4.0.4 [134, 135] and FlavorKit
[136] to generate particle mass spectra and compute low energy observables,
such as $a_{\mu}^{\rm SUSY}$ and B-physics observables, and the package
MicrOMEGAs-5.0.4 [137, 138, 139, 140, 141, 142] to calculate DM observables,
assuming that the lightest neutralino was the sole DM candidate in the
universe. Bounds from the direct search for extra Higgs bosons at the LEP,
Tevatron, and LHC and the fit of $h$’s property to LHC Higgs data were
implemented by the programs HiggsBounds-5.3.2 [143, 144, 145, 146] and
HiggsSignal-2.2.3 [147, 148, 149, 150], respectively.

###  3.1 Research strategy

The main aims of this research were to explore as many possibilities
(parameter points) of the MSSM as possible, clarify how they remain consistent
with current experimental results, and reveal some distinct characteristics of
the theory. We carried out such research by the following procedures:

  * •

We employed the MultiNest algorithm [151] to comprehensively scan the
parameter space in Table 3. The $n_{\rm live}$ parameter in the algorithm
controlled the number of active points sampled in each iteration of the scan,
and $n_{\rm live}=10000$ was set. The following likelihood function was
constructed to guide the scan:

$\displaystyle\mathcal{L}=\mathcal{L}_{a_{\mu}}\times\mathcal{L}_{const}.$
(3.1)

$\mathcal{L}_{a_{\mu}}$ is the likelihood function of the muon $g-2$ anomaly
given by

$\displaystyle\mathcal{L}_{a_{\mu}}\equiv
Exp\left[-\frac{1}{2}\left(\frac{a_{\mu}^{\rm SUSY}-\Delta a_{\mu}}{\delta
a_{\mu}}\right)^{2}\right]=Exp\left[-\frac{1}{2}\left(\frac{a_{\mu}^{\rm
SUSY}-2.51\times 10^{-9}}{5.9\times 10^{-10}}\right)^{2}\right],$

where $\Delta a_{\mu}\equiv a_{\mu}^{\rm Exp}-a_{\mu}^{\rm SM}$ and $\delta
a_{\mu}$ denote the difference between the experimental central value of
$a_{\mu}$ and its SM prediction and the total uncertainties in determining
$\Delta a_{\mu}$, respectively [9, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. $\mathcal{L}_{const}$ represents
the restrictions of some experiments on the theory. They included $2\sigma$
bounds of the DM relic density [152], upper bounds of the PandaX-4T experiment
on the SI DM-nucleon scattering [107] and the XENON-1T experiment on the SD
scattering [153], consistency of $h$’s properties with the LHC Higgs data at
the $95\%$ confidence level [150], collider searches for extra Higgs bosons
[146], $2\sigma$ bounds on the branching ratios of $B\to X_{s}\gamma$ and
$B_{s}\to\mu^{+}\mu^{-}$ [154], and the vacuum stability of the scalar
potential consisting of the Higgs fields and the last two generations of
slepton fields [155, 156]. We defined $\mathcal{L}_{const}=1$ if the
restrictions were satisfied and $\mathcal{L}_{const}=Exp[-100]$ if they were
not. More details of these restrictions are introduced in Refs. [72, 86].

  * •

We refined the samples obtained in the scan by the criteria
$\mathcal{L}_{const}=1$ and $|a_{\mu}^{\rm SUSY}-\Delta a_{\mu}|/\delta
a_{\mu}\leq 3$, and we projected those passing the selection onto the two-
dimensional planes spanned by any two of the parameters $M_{1}$, $M_{2}$,
$\mu$, $\tilde{m}_{\tilde{\mu}_{L}}$, and $\tilde{m}_{\tilde{\mu}_{R}}$. We
then concentrated on the region of the planes where the samples were sparsely
distributed and performed a new scan by adjusting relevant parameter ranges
and setting $n_{\rm live}=3000$.

  * •

We iterated the last operation with all accumulated samples until the
projected areas on the planes remained unchanged. At this point, we acquired
$2.21\times 10^{5}$ samples surviving the criteria, and about $1.7\times
10^{5}$ of them could further explain the $(g-2)_{\mu}$ anomaly at the
$2\sigma$ level.

  * •

We simplified the study of the restrictions from the LHC search for SUSY.
Specifically, given that the sample number was huge and the Monte Carlo
simulation of each sample introduced below would cost more than one core-hour
for our computing cluster, we selected some representative points and carried
out the simulations. These points were stored in a specially designed sample
database, achieved in the following way:

    1. 1.

We searched for the sample with the greatest likelihood value from a database
storing all the scan results.

    2. 2.

We constructed a hypersphere in the parameter space, which was centered around
the sample with a radius of $10~{}{\rm GeV}$ for $M_{1}$, $M_{2}$, and $\mu$
(note that the simulation results were more sensitive to these three
parameters than the other dimensional parameters), $20~{}{\rm GeV}$ for
$\tilde{m}_{\tilde{\mu}_{L}}$ and $\tilde{m}_{\tilde{\mu}_{R}}$, and one unit
for $\tan\beta$.

    3. 3.

We copied useful information of the central sample to the newly built database
for simulation and sequentially deleted all samples in the hypersphere to
update the initial database.

    4. 4.

We iterated the above operations until all the samples in the initial database
were depleted.

We add that the method to dilute the dense samples was plausible because the
R-value of the simulation relied heavily on both the mass spectra and the
field compositions of sparticles, which determined their production cross
sections and decay branching ratios. All samples in the hypersphere had
similar properties in these two aspects, and the R-value of the studied point
was typical for these samples. In addition, the method paid more attention to
the samples favored by the anomaly, which was the focus of this research.
After requiring the representative points to explain the muon g-2 anomaly at
the $2\sigma$ level, we finally acquired 58242 samples for the simulations.

  * •

We surveyed the LHC restrictions on the representative points by simulating
the following processes:

$\displaystyle pp$ $\displaystyle\to$
$\displaystyle\tilde{\chi}_{i}^{0}\tilde{\chi}_{j}^{\pm},\quad i=2,3,4,5,\quad
j=1,2;$ (3.2) $\displaystyle pp$ $\displaystyle\to$
$\displaystyle\tilde{\chi}_{i}^{\pm}\tilde{\chi}_{j}^{\mp},\quad i,j=1,2;$
(3.3) $\displaystyle pp$ $\displaystyle\to$
$\displaystyle\tilde{\chi}_{i}^{0}\tilde{\chi}_{j}^{0},\quad i,j=2,3,4,5;$
(3.4) $\displaystyle pp$ $\displaystyle\to$
$\displaystyle\tilde{\mu}_{i}^{\ast}\tilde{\mu}_{j},\quad i,j=1,2;$ (3.5)
$\displaystyle pp$ $\displaystyle\to$
$\displaystyle\tilde{\nu}_{\mu}^{\ast}\tilde{\nu}_{\mu}.$ (3.6)

Specifically, the cross sections of these processes at $\sqrt{s}$ = 13 TeV
were calculated at the next-to-leading order (NLO) by the package Prospino2
[157], and 60000 and 40000 events were generated for the electroweakino and
slepton production processes, respectively, by the package MadGraph_aMC@NLO
[158, 159]. A relevant parton shower and hadronization were completed by the
program PYTHIA8 [160]. The resulting event files were then fed into the
package CheckMATE-2.0.29 [161, 162, 163] to calculate the $R$-value defined by
$R\equiv max\\{S_{i}/S_{i,obs}^{95}\\}$, where $S_{i}$ denotes the simulated
event number of the $i$-th SR in the analyses of Tables 1 and 2, and
$S_{i,obs}^{95}$ represents its corresponding $95\%$ confidence level upper
limit. In this process, program Delphes was encoded in CheckMATE for detector
simulation [164].

As an alternative, we also used the program SModelS-2.2.1 [165] to study the
LHC restrictions. We found that this program’s capability to exclude SUSY
points was usually weaker than that of the simulation due to its limited
database and strict working prerequisites.

Figure 1:  Projection of the obtained samples onto $M_{1}-M_{2}$ plane (left
panel) and $M_{1}-\mu$ plane (right panel). The gray triangles denote the
samples that satisfy all restrictions listed in the text, in particular those
from the DM relic density measured by the Planck experiment [152] and the DM
direct detection of the PandaX-4T experiment [107]. The blue circles represent
those that could further explain the muon g-2 anomaly at the $2\sigma$ level,
and the green stars are the part of the blue circles that agree with the
results from the LHC search for SUSY. Although only about one-third of the
blue circles were studied by the simulations in this work (see the research
strategy in the last sub-section), the shape of the green areas is unlikely to
significantly change. We verified this point by randomly selecting several
thousands of samples from the blue areas and simulating the LHC restrictions.

###  3.2 Key features of the results

First, we studied the DM physics of the MSSM by projecting the samples
acquired by the scans onto the $M_{1}-M_{2}$ and $M_{1}-\mu$ planes to obtain
Fig. 1. This figure reveals the following facts:

  * •

If only the restrictions from the DM physics are considered, the DM candidate
is bino-dominated when $|M_{1}|\leq 800~{}{\rm GeV}$. It may achieve the
measured relic density by Z-mediated resonant annihilation, $h$-mediated
resonant annihilation, or co-annihilation with wino-like electroweakinos. In
addition, we will show later in Figs. 5 and 6 that it may also acquire the
density by co-annihilating with $\tilde{\mu}_{L}/\tilde{\nu}_{\mu}$ or
$\tilde{\mu}_{R}$.

  * •

If the samples are further required to explain the muon g-2 anomaly at the
$2\sigma$ level, $|M_{1}|$ is upper bounded by about $620~{}{\rm GeV}$.
Furthermore, if the LHC restrictions are included, it is upper bounded by
about $570~{}{\rm GeV}$.

  * •

There is a vacant region on the $M_{1}-M_{2}$ plane, located in the ranges of
$350~{}{\rm GeV}\lesssim M_{1}\lesssim 570~{}{\rm GeV}$, $400~{}{\rm
GeV}\lesssim M_{2}\lesssim 600~{}{\rm GeV}$, and $M_{1}+30~{}{\rm GeV}\lesssim
M_{2}\lesssim M_{1}+100~{}{\rm GeV}$. This region is distinct because winos
can significantly affect the mass of $\tilde{\nu}_{\mu}$ by radiative
corrections, and given the values of $M_{1}$ and $M_{2}$, one needs to fine-
tune the soft-breaking parameter $\tilde{m}_{\tilde{\mu}_{L}}$ to predict the
measured DM density by the co-annihilation with
$\tilde{\nu}_{\mu}/\tilde{\mu}_{L}$. This situation is challenging in the
scans since it easily results in $\tilde{\nu}_{\mu}$ as the LSP.

We add that this vacant region corresponds to the void on the bottom right
corner of the $|m_{\tilde{\chi}_{1}^{0}}|-m_{\tilde{\chi}_{1}^{\pm}}$ plane in
Fig. 4.

  * •

The PandaX-4T experiment has required $\mu\gtrsim 300~{}{\rm GeV}$ when
$M_{1}$ is negative and $\mu\gtrsim 500~{}{\rm GeV}$ if $M_{1}>100~{}{\rm
GeV}$. This conclusion originates from the facts that
$m_{\tilde{\chi}_{1}^{0}}\simeq M_{1}$ and the formulae of
$\sigma_{\tilde{\chi}_{1}^{0}-N}^{\rm SI}$ in Eqs. 2.2 and 2.4. This explains
why $\mu$ at $M_{1}\simeq-m_{Z}/2$ may be much smaller than $\mu$ at
$M_{1}\simeq m_{Z}/2$.

  * •

The observables in the DM physics and the muon g-2 anomaly may prefer
different parameter spaces of the MSSM, even though broad parameter regions
can still accommodate both. For example, a negative $M_{1}$ is disfavored by
the muon g-2 anomaly in the large $\mu$ and $M_{2}$ region, because the ‘BLR’
contribution to $a_{\mu}^{\rm SUSY}$ is negatively sizable. This case,
however, can easily reproduce the results of DM experiments.

Figure 2:  Similar to Fig. 1, but showing the correlations of the parameters
that $a_{\mu}^{\rm SUSY}$ is sensitive to. $m_{\tilde{\mu}_{L}}$ is the mass
of the left-handed-dominated smuon, which is approximately equal to the soft-
breaking parameter $\tilde{m}_{\tilde{\mu}_{L}}$ at the slepton mass scale.
Samples surviving the LHC restrictions are classified into five types, marked
by A, B, C, D, and E in this figure. They are distinguished by different DM
annihilation mechanisms and locations in the SUSY parameter space (see Table
4).  Table 4:  Parameter spaces for the five types of samples in Fig. 2, where
the theoretical inputs in the last five columns are in units of GeV. The
second column denotes which particles $\tilde{\chi}_{1}^{0}$ will co-
annihilate with to acquire the measured DM density. In this aspect, Type-A and
Type-B samples are different in that $m_{\tilde{\mu}_{L}}\gtrsim
M_{2}+30~{}{\rm GeV}$ for the former and $m_{\tilde{\mu}_{L}}\simeq M_{2}$ for
the latter.

Sample type | Annihilation partner | $M_{1}$ | $M_{2}$ | $\mu$ | $m_{\tilde{\mu}_{L}}$ | $m_{\tilde{\mu}_{R}}$
---|---|---|---|---|---|---
Type-A | $\tilde{W}$ | (220, 560) | (230, 600) | (430, 1000) | (350, 1000) | (300, 1000)
Type-B |  $\tilde{\mu}_{L}$ and $\tilde{W}$ | (210, 550) | (230, 600) | (430, 1000) | (240, 600) | (300, 1000)
Type-C | $\tilde{\mu}_{L}$ | (230, 540) | (600, 1400) | (540, 1000) | (240, 550) | (230, 1000)
Type-D | $\tilde{\mu}_{R}$ | (210, 520) | (300, 700) | (440, 1000) | (400, 1000) | (210, 1000)
Type-E | $\tilde{\mu}_{R}$ | (230, 320) | (950, 1300) | (570, 1000) | (240, 600) | (230, 1000)

Second, we concentrate on the interplay between the muon g-2 anomaly and the
LHC restrictions. As introduced in the last section, explaining the muon g-2
anomaly requires more than one sparticle to be moderately light [84]. In
particular, $M_{2}$, $\mu$, and $m_{\tilde{\mu}_{L}}$ can not be very large
simultaneously since the ‘WHL’ contribution is usually dominant. This
situation leads to sizable SUSY signals and thus strengthens the LHC
restrictions. In Fig. 2, we show the correlations of any two of the three
parameters, $M_{2}$, $\mu$, and $m_{\tilde{\mu}_{L}}$, and also the
correlation between $\mu$ and $\tan\beta$. The following distinct features are
shown:

  * •

The LHC restrictions have set lower bounds on the SUSY parameters:
$\tan\beta\gtrsim 12$, $\mu\gtrsim 400~{}{\rm GeV}$, $M_{2}\gtrsim 230~{}{\rm
GeV}$, and $m_{\tilde{\mu}_{L}}\gtrsim 240~{}{\rm GeV}$.

  * •

As indicated by the top left panel, the LHC restrictions are particularly
strong for $\tan\beta\lesssim 25$. The underlying reason is the muon g-2
anomaly prefers light winos, higgsinos, and left-handed-dominant smuon as
$\tan\beta$ becomes small.

  * •

In the case that $\tilde{\mu}_{L}$ is lighter than winos and/or higgsinos, the
LHC restrictions are also strong, which is reflected by the wedge-shaped
excluded regions on the $M_{2}-m_{\tilde{\mu}_{L}}$ and $\mu-
m_{\tilde{\mu}_{L}}$ planes. This was because the heavy electroweakinos could
decay into the slepton first and thus enhance the leptonic signal of the
electroweakino pair production processes (compared with the case where
$\tilde{\mu}_{L}$ is heavier than the electroweakinos). We elaborate on this
point by fixing $m_{\tilde{\mu}_{L}}=300~{}{\rm GeV}$ and varying $M_{2}$. For
$M_{2}\simeq 300~{}{\rm GeV}$, although the total wino pair production cross
sections exceeded $550~{}{\rm fb}$ [166, 167], there were some samples
surviving the LHC constraints due to the small mass splittings between the
wino-like particles and $\tilde{\chi}_{1}^{0}$. In this case, the wino-like
particles decayed into $\tilde{\chi}_{1}^{0}$ and a soft virtual Z or W, which
made the signal detection difficult. With the increase in $M_{2}$, all the
samples were excluded because the wino-like particles were copiously produced
at the LHC due to their moderate lightness, and simultaneously the branching
ratios of their decays into the slepton were sizable. Specifically, we found
that the ratios were always larger than $20\%$. With the further increase in
$M_{2}$, the wino pair production rates rapidly decreased so that the LHC
constraints were weakened, reflected by the appearance of the green areas at
$M_{2}\simeq 900~{}{\rm GeV}$ in the bottom right panel. We add that this
discussion can be applied to the $\mu-m_{\tilde{\mu}_{L}}$ plane in the bottom
left panel.

  * •

Samples consistent with the LHC restrictions can be classified into five
types, distinguished by their DM annihilation mechanisms and locations in the
parameter space. They are marked as A, B, C, D, and E in the figure. In Table
4, we provide the criteria of this classification. We will present benchmark
points to reveal their properties and study the LHC restrictions later.

Figure 3:  Left panel: same as the upper right panel of Fig. 2, except that all the samples are required further to satisfy the LZ restrictions. Right panel: same as the left panel of this figure, except that the samples are projected onto the $\mu-m_{\tilde{\mu}_{L}}$ plane.  Table 5:  Numbers of the samples studied by the simulations. They are categorized by DM-dominant annihilation mechanisms, where a DM direct detection experiment is used to set the limits, and whether the LHC restrictions are included in the research.  Annihilation Mechanisms | Before LHC Constraints | After LHC Constraints
---|---|---
| PandaX | LZ | PandaX | LZ
Total Samples | 58242 | 39657 | 11204 | 5656
$\tilde{B}-\tilde{W}$ co-annihilation | 31108 | 20123 | 7927 | 3435
$\tilde{B}-\tilde{\mu}_{L}-\tilde{W}$ co-annihilation | 10052 | 6622 | 287 | 174
$\tilde{B}-\tilde{\mu}_{L}$ co-annihilation | 13445 | 11052 | 2737 | 1884
$\tilde{B}-\tilde{\mu}_{R}$ co-annihilation | 2869 | 1860 | 253 | 163
$Z-$ funnel | 408 | 0 | 0 | 0
$h-$ funnel | 360 | 0 | 0 | 0

Third, we studied the impact of the LZ experiment on the MSSM. In Figs. 1 and
2, the PandaX-4T results were used to set upper bounds on the SI cross
sections of the DM-nucleon scattering [107]. We utilized the LZ limits to
refine the samples further, project the selected samples onto various panels,
and compare the resulting figures with their correspondence plotted with the
samples in Fig. 1. The most remarkable change was that $\mu$ was more strongly
limited, which was reflected in the following aspects.

  * •

Given the measured DM density, the LZ experiment alone required $\mu\gtrsim
380~{}{\rm GeV}$ for $M_{1}<0$ and $\mu\gtrsim 600~{}{\rm GeV}$ for
$M_{1}>100~{}{\rm GeV}$. If the restrictions from the muon g-2 anomaly and the
LHC experiment were also included, the lower bounds became about $500~{}{\rm
GeV}$ and $630~{}{\rm GeV}$, respectively, indicating that the theory needs a
tuning of ${\cal{O}}(1\%)$ to predict the $Z$-boson mass [168]. Compared with
the restriction from the PandaX-4T experiment, these bounds were improved by
about $100~{}{\rm GeV}$.

  * •

The $Z$-mediated resonant annihilation becomes less favored because an
enhanced $\mu$ reduced $C_{\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}Z}$ in Eq.
2.5, and $2|m_{\tilde{\chi}_{1}^{0}}|$ must be closer to $m_{Z}$ to obtain the
measured density. This situation requires the fine-tuning quantity defined in
Eq. (19) of Ref. [169] to be larger than about 150 to achieve the measured
density. This conclusion also applies to the $h$-mediated resonant
annihilation. We point out that, due to the tuning, these resonant
annihilation scenarios were usually missed in the scans (see, e.g., the
results in Fig. 4 of this study). This point was discussed with Bayesian
statistics in footnote 6 of Ref. [89].

  * •

Since the LZ constraint and the LHC restriction are sensitive to different
SUSY parameters, they complement each other in exploring the features of the
MSSM. This is particularly so if one intends to explain the muon g-2 anomaly
at the $2\sigma$ level. The basic reason is that $\mu$ is correlated with the
other parameters by the anomaly, and any enhancement of $\mu$ in a massive
higgsino scenario will make winos and $\tilde{\mu}_{L}$ lighter to keep
$a_{\mu}^{\rm SUSY}$ unchanged. This situation usually improves the LHC
restrictions.

To show the combined effects, we projected the samples passing the LZ
restrictions onto the $M_{2}-\mu$ and $\mu-m_{\tilde{\mu}_{L}}$ planes in Fig.
3 and compared the resulting panels with their corresponding ones in Fig. 2.
We also focused on the samples studied by the Monte Carlo simulations. We
classified them by DM-dominant annihilation mechanisms, where the DM
experiment was used to set limits, and whether the LHC restrictions are
included in this research, and the results are presented in Table 5. Both the
figure and the table show that the two experiments promoted each other to
limit the parameter space of the MSSM.

Figure 4:  Distribution of $m_{\tilde{\chi}_{1}^{\pm}}$ versus
$|m_{\tilde{\chi}_{1}^{0}}|$. The left panel studies the samples in Figs. 1,
while the right panel focuses on those in Fig. 3. The classification in Table
4 is applied to the samples of this figure to illuminate the underlying
physics.

Figure 5:  Same as Fig. 4 except that it shows the distribution on the
$|m_{\tilde{\chi}_{1}^{0}}|-m_{\tilde{\mu}_{L}}$ plane.

Figure 6:  Same as Fig. 4 except that the samples are displayed on the
$|m_{\tilde{\chi}_{1}^{0}}|-m_{\tilde{\mu}_{R}}$ plane.  Figure 7:  Sparticle
mass spectra preferred by different experiments. The cyan band is acquired
from the gray samples in Fig. 1, and the yellow, orange, and red bands are
from the gray, blue, and green samples in Fig. 3, respectively.

Fourth, we surveyed the influences of these experiments on the particle
properties. We projected the samples of Fig. 1 (Fig. 3) onto the
$|m_{\tilde{\chi}_{1}^{0}}|-m_{\tilde{\chi}_{1}^{\pm}}$,
$|m_{\tilde{\chi}_{1}^{0}}|-m_{\tilde{\mu}_{L}}$, and
$|m_{\tilde{\chi}_{1}^{0}}|-m_{\tilde{\mu}_{R}}$ planes to acquire the left
(right) panels of Figs. 4, 5, and 6, respectively. The classification of the
samples was the same as before. From these plots, we obtained the following
points:

  * •

With the increase in $|m_{\tilde{\chi}_{1}^{0}}|$, the upper bounds of
$m_{\tilde{\chi}_{1}^{\pm}}$ and $m_{\tilde{\mu}_{L}}$ decrease, and they
terminate at $m_{\tilde{\chi}_{1}^{\pm}}\lesssim 600~{}{\rm GeV}$ and
$m_{\tilde{\mu}_{L}}\lesssim 700~{}{\rm GeV}$ for
$m_{\tilde{\chi}_{1}^{\pm}}\simeq 570~{}{\rm GeV}$. This tendency is not
evident for $m_{\tilde{\mu}_{R}}$ in the case of $m_{\tilde{\mu}_{R}}\leq
1~{}{\rm TeV}$ because $a_{\mu}^{\rm SUSY}$ is more sensitive to $M_{2}$,
$\mu$, and $m_{\tilde{\mu}_{L}}$ than to $m_{\tilde{\mu}_{R}}$, as indicated
by Eqs. 2.6–2.9, and it decouples in the heavy sparticles limit.

  * •

The LHC restrictions have set lower bounds on the sparticle mass spectra,
which were $m_{\tilde{\chi}_{1}^{0}}\gtrsim 210~{}{\rm GeV}$,
$m_{\tilde{\chi}_{1}^{\pm}}\gtrsim 235~{}{\rm GeV}$,
$m_{\tilde{\mu}_{L}}\gtrsim 240~{}{\rm GeV}$, and $m_{\tilde{\mu}_{R}}\gtrsim
215~{}{\rm GeV}$ in this research. The basic reason for this phenomenon is as
follows: if $\tilde{\chi}_{1}^{0}$ is lighter, more missing momentum will be
emitted in the sparticle production processes at the LHC, which can improve
the sensitivities of the experimental analyses; however, if sparticles other
than $\tilde{\chi}_{1}^{0}$ are lighter, they will be more copiously produced
at the LHC to increase the number of events containing multiple leptons. We
emphasize that these bounds should be regarded as rough estimates, instead of
accurate values, since far from enough samples were studied, given the broad
parameter space of the MSSM.

Finally, we summarize the sparticle mass spectra preferred by different
experiments in Fig. 7. This figure reveals that $570~{}{\rm GeV}\gtrsim
m_{\tilde{\chi}_{1}^{0}}\gtrsim 210~{}{\rm GeV}$,
$m_{\tilde{\chi}_{2}^{0}},m_{\tilde{\chi}_{1}^{\pm}}\gtrsim 235~{}{\rm GeV}$,
$m_{\tilde{\chi}_{3}^{0}}\gtrsim 515~{}{\rm GeV}$,
$m_{\tilde{\chi}_{4}^{0}}\gtrsim 525~{}{\rm GeV}$,
$m_{\tilde{\chi}_{2}^{\pm}}\gtrsim 530~{}{\rm GeV}$,
$m_{\tilde{\nu}_{\mu}}\gtrsim 235~{}{\rm GeV}$, $950~{}{\rm GeV}\gtrsim
m_{\tilde{\mu}_{1}}\gtrsim 215~{}{\rm GeV}$, and $m_{\tilde{\mu}_{2}}\gtrsim
250~{}{\rm GeV}$ if all the latest restrictions are considered. The lower
bounds come from the LHC restrictions, and the upper bounds arise from the
explanation of the muon g-2 anomaly at the $2\sigma$ level. In addition, it is
verified that $\tilde{\chi}_{2}^{0}$ and $\tilde{\chi}_{1}^{\pm}$ are wino-
dominated when they are lighter than about $500~{}{\rm GeV}$ and
$m_{\tilde{\mu}_{1}}$ may be either $m_{\tilde{\mu}_{L}}$ or
$m_{\tilde{\mu}_{R}}$. It was also verified that $600~{}{\rm GeV}\gtrsim
m_{\rm NLSP}\gtrsim 220~{}{\rm GeV}$ and $700~{}{\rm GeV}\gtrsim m_{\rm
NNLSP}\gtrsim 250~{}{\rm GeV}$, where the NLSP and NNLSP may be either
electroweakinos or sleptons. In principle, the sparticles other than
$\tilde{\chi}_{1}^{0}$ and $\tilde{\mu}_{1}^{0}$ are also upper bounded in
mass by the explanation of the anomaly. We do not provide these bounds since
the parameter space in Table 3 is limited.

Table 6:  Supplement to Table 5 with the SRs that contribute to the largest
R-values and their capability to exclude the samples, expressed by the
percentage of the total numbers in the second column. This table only counts
the samples satisfying the LZ restrictions and studied by the simulations. The
second and third columns denote the sample numbers before and after
implementing the LHC restrictions, respectively, which are also presented in
Table 5. They reflect that the LHC restrictions on the
$\tilde{B}-\tilde{\mu}_{L}-\tilde{W}$ and $\tilde{B}-\tilde{\mu}_{R}$ co-
annihilation cases are very strong. One can understand this feature from the
previous discussions in this research and the benchmark points listed below.
The fourth column reveals that only SR-1 from the experimental analyses in
Ref. [6] and SR-2 from Ref. [114, 113] played a role in excluding the
$\tilde{B}-\tilde{\mu}_{L}$ co-annihilation case, which was different from the
other cases.

Annihilation mechanism | Before | After | SRs and their exclusion percentages
---|---|---|---
Total samples | 39657 | 5656 | SR-1(23.7%),SR-2(18.5%),SR-3(12.7%),SR-4(11.6%)
$\tilde{B}-\tilde{W}$ co-annihilation | 20123 | 3435 | SR-3(23.8%), SR-4(21.1%), SR-5(6.3%), SR-1(0.1%)
$\tilde{B}-\tilde{\mu}_{L}-\tilde{W}$ co-annihilation | 6622 | 174 | SR-3(28.6%),SR-4(16.4%),SR-2(14.4%),SR-5(11.7%)
$\tilde{B}-\tilde{\mu}_{L}$ co-annihilation | 11052 | 1884 | SR-1(52.0%),SR-2(30.4%)
$\tilde{B}-\tilde{\mu}_{R}$ co-annihilation | 1860 | 163 | SR-2(44.0%), SR-4(32.3%), SR-3(6.2%), SR-1(0.8%)

###  3.3 More details of LHC restrictions

Even in the simple realization of SUSY, such as the MSSM, the decay products
of heavy sparticles are complex (see the benchmark points presented below). As
a result, the pair productions of winos, higgsinos, $\tilde{\mu}_{L}$, and
$\tilde{\mu}_{R}$ at the LHC,
$pp\to\bar{\tilde{W}}\tilde{W},\bar{\tilde{H}}\tilde{H},\tilde{\mu}_{L}^{\ast}\tilde{\mu}_{L},\tilde{\mu}_{R}^{\ast}\tilde{\mu}_{R}$,
may contribute to the same SR of the analyses in Tables 1 and 2. All these
contributions must be suppressed for any parameter point to circumvent the LHC
restrictions, which may occur in the following situations:

  1. 1.

The DM candidate is massive, compared with the results of pertinent
experimental analyses. In this case, all SM particles in the final state are
not energetic enough, and the missing momentum emitted in the sparticle
production processes tends to be small.

  2. 2.

The mass splitting between the decaying sparticle and $\tilde{\chi}_{1}^{0}$
is less than several tens of GeV. In this compressed spectra case, the SM
particles as the decay product are soft and hard to detect without a
deliberate search strategy.

  3. 3.

Heavy sparticles decay by several channels with comparable branching ratios in
terms of size. This situation usually leads to complicated final decay states.

  4. 4.

Sparticles are sufficiently heavy, compared with their experimental exclusion
bounds in simplified models, so that their production cross sections are
negligibly small.

We refer to these situations as survival mechanisms I, II, III, and IV in the
following discussion. An illuminating example of situations 1 and 2 is
presented in Fig. 16 of Ref. [112], which concludes that there are no LHC
restrictions on winos in the $\tilde{B}-\tilde{W}$ co-annihilation case if
$m_{\tilde{\chi}_{1}^{0}}\gtrsim 220~{}{\rm GeV}$. Evidently, this bound is
significantly weaker than the other searches for winos at the LHC.

Table 7:  Two benchmark points, P1 and P2, for Type-A and -B samples in Table
4, respectively. Both points satisfy all the restrictions listed in the text.
The R-value of each point and its corresponding SR are presented in the last
line of this table.

Benchmark Point P1 for Type-A samples | Benchmark Point P2 for Type-B samples
---|---
$\mu$ | 750.7 GeV | $\text{m}_{\text{H}}$ | 10359.8 GeV | $\mu$ | 636.8 GeV | $\text{m}_{\text{H}}$ | 7359.7 GeV
$\text{tan}\beta$ | 53.2 | $\text{m}_{\text{A}}$ | 10359.8 GeV | $\text{tan}\beta$ | 35.5 | $\text{m}_{\text{A}}$ | 7359.7 GeV
$\text{A}_{t}$ | 2496.8 GeV | $\text{m}_{\text{h}}$ | 125.5 GeV | $\text{A}_{t}$ | 2409.8 GeV | $\text{m}_{\text{h}}$ | 125.2 GeV
$\text{M}_{1}$ |  $-380.0$ GeV | $\text{m}_{\text{H}^{\pm}}$ | 10337.7 GeV | $\text{M}_{1}$ |  $-292.9$ GeV | $\text{m}_{\text{H}^{\pm}}$ | 7363.2 GeV
$\text{M}_{2}$ | 386.9 GeV | $\text{m}_{\tilde{\nu}_{\mu}}$ | 678.0 GeV | $\text{M}_{2}$ | 304.9 GeV | $\text{m}_{\tilde{\nu}_{\mu}}$ | 316.2 GeV
$\tilde{\text{m}}_{\tilde{\mu}_{\text{L}}}$ | 550.2 GeV | $\text{m}_{\tilde{\chi}_{1}^{0}}$ |  $-380.7$ GeV | $\tilde{\text{m}}_{\tilde{\mu}_{\text{L}}}$ | 188.4 GeV | $\text{m}_{\tilde{\chi}_{1}^{0}}$ |  $-292.7$ GeV
$\tilde{\text{m}}_{\tilde{\mu}_{\text{R}}}$ | 972.5 GeV | $\text{m}_{\tilde{\chi}_{2}^{0}}$ | 404.8 GeV | $\tilde{\text{m}}_{\tilde{\mu}_{\text{R}}}$ | 708.0 GeV | $\text{m}_{\tilde{\chi}_{2}^{0}}$ | 317.4 GeV
$\text{m}_{\tilde{\mu}_{\text{L}}}$ | 682.2 GeV | $\text{m}_{\tilde{\chi}_{3}^{0}}$ |  $-772.6$ GeV | $\text{m}_{\tilde{\mu}_{\text{L}}}$ | 325.6 GeV | $\text{m}_{\tilde{\chi}_{3}^{0}}$ |  $-656.4$ GeV
$\text{m}_{\tilde{\mu}_{\text{R}}}$ | 806.5 GeV | $\text{m}_{\tilde{\chi}_{4}^{0}}$ | 777.7 GeV | $\text{m}_{\tilde{\mu}_{\text{R}}}$ | 612.9 GeV | $\text{m}_{\tilde{\chi}_{4}^{0}}$ | 662.0 GeV
$a_{\mu}^{\text{SUSY}}$ | $1.95\times 10^{-9}$ | $\text{m}_{\tilde{\chi}_{1}^{\pm}}$ | 405.0 GeV | $a_{\mu}^{\text{SUSY}}$ | $2.43\times 10^{-9}$ | $\text{m}_{\tilde{\chi}_{1}^{\pm}}$ | 317.6 GeV
${\Omega h}^{2}$ | 0.10 | $\text{m}_{\tilde{\chi}_{2}^{\pm}}$ | 779.6 GeV | ${\Omega h}^{2}$ | 0.12 | $\text{m}_{\tilde{\chi}_{2}^{\pm}}$ | 664.2 GeV
$\sigma_{p}^{\text{SI}}$ | $4.50\times 10^{-47}\text{cm}^{2}$ | $\sigma_{p}^{\text{SD}}$ | $5.38\times 10^{-43}\text{cm}^{2}$ | $\sigma_{p}^{\text{SI}}$ | $4.34\times 10^{-47}\text{cm}^{2}$ | $\sigma_{p}^{\text{SD}}$ | $9.21\times 10^{-43}\text{cm}^{2}$
$\text{N}_{11},\text{N}_{12},\text{N}_{13},\text{N}_{14}$ |  $-0.997$, $-0.004$, $-0.075$, $-0.036$ | $\text{N}_{11},\text{N}_{12},\text{N}_{13},\text{N}_{14}$ |  $-0.996$, $-0.004$, $-0.083$, $-0.036$
$\text{N}_{21},\text{N}_{22},\text{N}_{23},\text{N}_{24}$ | 0.004, 0.987, $-0.140$, 0.075 | $\text{N}_{21},\text{N}_{22},\text{N}_{23},\text{N}_{24}$ | 0.006, 0.985, $-0.157$, 0.079
$\text{N}_{31},\text{N}_{32},\text{N}_{33},\text{N}_{34}$ | 0.079, $-0.046$, $-0.702$, $-0.706$ | $\text{N}_{31},\text{N}_{32},\text{N}_{33},\text{N}_{34}$ | 0.084, $-0.055$, $-0.700$, $-0.707$
$\text{N}_{41},\text{N}_{42},\text{N}_{43},\text{N}_{44}$ |  $-0.027$, 0.152, 0.694, $-0.703$ | $\text{N}_{41},\text{N}_{42},\text{N}_{43},\text{N}_{44}$ |  $-0.033$, 0.166, 0.691, $-0.702$
Annihilations | Fractions[%] | Annihilations | Fractions[%]
$\tilde{B}-\tilde{W}$ Co-annihilation | 89.9 |  $\tilde{B}-\tilde{W}/\tilde{B}-\tilde{\mu}_{L}$ Co-annihilation | 73.0/14.5
Decays | Branching ratios[%] | Decays | Branching ratios[%]
$\tilde{\chi}_{2}^{0}\to\tilde{\chi}_{1}^{0}Z^{*}/\tilde{\chi}_{1}^{0}h^{*}$ | 100 | $\tilde{\chi}_{2}^{0}\to\tilde{\nu}_{\mu}\nu_{\mu}$ | 100
$\tilde{\chi}_{3}^{0}\to\tilde{\chi}_{1}^{\pm}\text{W}^{\mp}/\tilde{\chi}_{2}^{0}Z/\tilde{\chi}_{1}^{0}h/\tilde{\chi}_{2}^{0}h/\tilde{\chi}_{1}^{0}Z$ | 61.0/27.0/8.3/2.3/1.1 | $\tilde{\chi}_{3}^{0}\to\tilde{\chi}_{1}^{\pm}\text{W}^{\mp}/\tilde{\chi}_{2}^{0}Z/\tilde{\chi}_{1}^{0}h/\tilde{\chi}_{2}^{0}h/\tilde{\chi}_{1}^{0}Z$ | 61.2/26.4/7.7/2.6/1.4
$\tilde{\chi}_{4}^{0}\to\tilde{\chi}_{1}^{\pm}W^{\mp}/\tilde{\chi}_{2}^{0}h/\tilde{\chi}_{1}^{0}Z/\tilde{\chi}_{2}^{0}Z/\tilde{\chi}_{1}^{0}h$ | 61.3/26.2/8.3/2.8/1.0 | $\tilde{\chi}_{4}^{0}\to\tilde{\chi}_{1}^{\pm}W^{\mp}/\tilde{\chi}_{2}^{0}h/\tilde{\chi}_{1}^{0}Z/\tilde{\chi}_{2}^{0}Z/\tilde{\nu}_{\mu}\nu_{\mu}/\tilde{\chi}_{1}^{0}h/\tilde{\mu}^{\pm}_{\text{L}}\mu^{\mp}$ | 60.9/24.7/7.6/3.3/1.4/1.2/0.9
$\tilde{\chi}_{1}^{\pm}\to\tilde{\chi}_{1}^{0}(W^{\pm})^{*}$ | 100 | $\tilde{\chi}_{1}^{\pm}\to\tilde{\nu}_{\mu}\mu^{\pm}$ | 99.9
$\tilde{\chi}_{2}^{\pm}\to\tilde{\chi}_{2}^{0}W^{\pm}/\tilde{\chi}_{1}^{\pm}Z/\tilde{\chi}_{1}^{\pm}h/\tilde{\chi}_{1}^{0}W^{\pm}$ | 31.0/30.1/28.6/9.9 | $\tilde{\chi}_{2}^{\pm}\to\tilde{\chi}_{2}^{0}W^{\pm}/\tilde{\chi}_{1}^{\pm}Z/\tilde{\chi}_{1}^{\pm}h/\tilde{\chi}_{1}^{0}W^{\pm}/\tilde{\mu}^{\pm}_{\text{L}}\nu_{\mu}/\tilde{\nu}_{\mu}\mu^{\pm}$ | 30.8/29.7/27.4/9.6/1.9/0.6
$\tilde{\mu}_{\text{L}}^{\pm}\to\tilde{\chi}_{1}^{\pm}\nu_{\mu}/\tilde{\chi}_{2}^{0}\mu^{\pm}/\tilde{\chi}_{1}^{0}\mu^{\pm}$ | 58.8/30.0/11.2 | $\tilde{\mu}_{\text{L}}^{\pm}\to\tilde{\chi}_{1}^{0}\mu^{\pm}/\tilde{\chi}_{1}^{\pm}\nu_{\mu}/\tilde{\chi}_{2}^{0}\mu^{\pm}$ | 63.4/23.7/12.9
$\tilde{\mu}_{\text{R}}^{\pm}\to\tilde{\chi}_{1}^{0}\mu^{\pm}/\tilde{\nu}_{\mu}\text{W}^{\pm}$ | 99.8/0.1 | $\tilde{\mu}_{\text{R}}^{\pm}\to\tilde{\chi}_{1}^{0}\mu^{\pm}/\tilde{\nu}_{\mu}W^{\pm}/\tilde{\mu}^{\pm}_{\text{L}}h/\tilde{\mu}^{\pm}_{\text{L}}Z$ | 99.5/0.3/0.1/0.1
$\tilde{\nu}_{\mu}\to\tilde{\chi}_{1}^{\pm}\mu^{\mp}/\tilde{\chi}_{2}^{0}\nu_{\mu}/\tilde{\chi}_{1}^{0}\nu_{\mu}$ | 59.8/29.4/10.8 | $\tilde{\nu}_{\mu}\to\tilde{\chi}_{1}^{0}\nu_{\mu}$ | 100
R value | 0.37, S-high-mm-05 in SR-3 | R value | 0.30, SRG07_0j_mll in SR-1

Table 8:  Same as Table 7, but for the benchmark points of Type-C and -D
samples in Table 4, labeled as P3 and P4, respectively.

Benchmark Point P3 for Type-C samples | Benchmark Point P4 for Type-D samples
---|---
$\mu$ | 773.0 GeV | $\text{m}_{\text{H}}$ | 9252.5 GeV | $\mu$ | 751.5 GeV | $\text{m}_{\text{H}}$ | 2738.6 GeV
$\text{tan}\beta$ | 54.1 | $\text{m}_{\text{A}}$ | 9252.5 GeV | $\text{tan}\beta$ | 58.2 | $\text{m}_{\text{A}}$ | 2738.6 GeV
$\text{A}_{t}$ | 2653.7 GeV | $\text{m}_{\text{h}}$ | 125.6 GeV | $\text{A}_{t}$ | 2485.9 GeV | $\text{m}_{\text{h}}$ | 125.4 GeV
$\text{M}_{1}$ | 360.6 GeV | $\text{m}_{\text{H}^{\pm}}$ | 9229.6 GeV | $\text{M}_{1}$ | 264.8 GeV | $\text{m}_{\text{H}^{\pm}}$ | 2766.4 GeV
$\text{M}_{2}$ | 1091.0 GeV | $\text{m}_{\tilde{\nu}_{\mu}}$ | 363.9 GeV | $\text{M}_{2}$ | 530.4 GeV | $\text{m}_{\tilde{\nu}_{\mu}}$ | 593.6 GeV
$\tilde{\text{m}}_{\tilde{\mu}_{\text{L}}}$ | 113.7 GeV | $\text{m}_{\tilde{\chi}_{1}^{0}}$ | 360.7 GeV | $\tilde{\text{m}}_{\tilde{\mu}_{\text{L}}}$ | 592.0 GeV | $\text{m}_{\tilde{\chi}_{1}^{0}}$ | 264.1 GeV
$\tilde{\text{m}}_{\tilde{\mu}_{\text{R}}}$ | 1274.6 GeV | $\text{m}_{\tilde{\chi}_{2}^{0}}$ | 785.0 GeV | $\tilde{\text{m}}_{\tilde{\mu}_{\text{R}}}$ | 263.0 GeV | $\text{m}_{\tilde{\chi}_{2}^{0}}$ | 545.0 GeV
$\text{m}_{\tilde{\mu}_{\text{L}}}$ | 372.0 GeV | $\text{m}_{\tilde{\chi}_{3}^{0}}$ |  $-793.7$ GeV | $\text{m}_{\tilde{\mu}_{\text{L}}}$ | 599.2 GeV | $\text{m}_{\tilde{\chi}_{3}^{0}}$ |  $-769.1$ GeV
$\text{m}_{\tilde{\mu}_{\text{R}}}$ | 1187.7 GeV | $\text{m}_{\tilde{\chi}_{4}^{0}}$ | 1136.1 GeV | $\text{m}_{\tilde{\mu}_{\text{R}}}$ | 268.5 GeV | $\text{m}_{\tilde{\chi}_{4}^{0}}$ | 782.4 GeV
$a_{\mu}^{\text{SUSY}}$ | $1.99\times 10^{-9}$ | $\text{m}_{\tilde{\chi}_{1}^{\pm}}$ | 784.3 GeV | $a_{\mu}^{\text{SUSY}}$ | $2.33\times 10^{-9}$ | $\text{m}_{\tilde{\chi}_{1}^{\pm}}$ | 545.2 GeV
${\Omega h}^{2}$ | 0.13 | $\text{m}_{\tilde{\chi}_{2}^{\pm}}$ | 1136.2 GeV | ${\Omega h}^{2}$ | 0.14 | $\text{m}_{\tilde{\chi}_{2}^{\pm}}$ | 782.9 GeV
$\sigma_{p}^{\text{SI}}$ | $6.78\times 10^{-47}\text{cm}^{2}$ | $\sigma_{p}^{\text{SD}}$ | $4.54\times 10^{-43}\text{cm}^{2}$ | $\sigma_{p}^{\text{SI}}$ | $4.64\times 10^{-47}\text{cm}^{2}$ | $\sigma_{p}^{\text{SD}}$ | $4.16\times 10^{-43}\text{cm}^{2}$
$\text{N}_{11},\text{N}_{12},\text{N}_{13},\text{N}_{14}$ |  $-0.997$, 0.003, $-0.071$, 0.034 | $\text{N}_{11},\text{N}_{12},\text{N}_{13},\text{N}_{14}$ |  $-0.998$, 0.006, $-0.066$, 0.024
$\text{N}_{21},\text{N}_{22},\text{N}_{23},\text{N}_{24}$ | 0.074, 0.164, $-0.698$, 0.693 | $\text{N}_{21},\text{N}_{22},\text{N}_{23},\text{N}_{24}$ |  $-0.023$, $-0.970$, 0.197, $-0.140$
$\text{N}_{31},\text{N}_{32},\text{N}_{33},\text{N}_{34}$ | 0.027, $-0.029$, $-0.706$, $-0.708$ | $\text{N}_{31},\text{N}_{32},\text{N}_{33},\text{N}_{34}$ |  $-0.030$, 0.041, 0.705, 0.708
$\text{N}_{41},\text{N}_{42},\text{N}_{43},\text{N}_{44}$ |  $-0.008$, 0.986, 0.096, $-0.136$ | $\text{N}_{41},\text{N}_{42},\text{N}_{43},\text{N}_{44}$ |  $-0.060$, 0.239, 0.678, $-0.692$
Annihilations | Fractions[%] | Annihilations | Fractions[%]
$\tilde{B}-\tilde{\mu}_{L}$ Co-annihilation | 92.6 |  $\tilde{B}-\tilde{\mu}_{R}$ Co-annihilation | 95.2
Decays | Branching ratios[%] | Decays | Branching ratios[%]
$\tilde{\chi}_{2}^{0}\to\tilde{\chi}_{1}^{0}h/\tilde{\mu}^{\pm}_{\text{L}}\mu^{\mp}/\tilde{\chi}_{1}^{0}Z/\tilde{\nu}_{\mu}\nu_{\mu}$ | 70.2/15.1/9.7/5.0 | $\tilde{\chi}_{2}^{0}\to\tilde{\chi}_{1}^{0}h/\tilde{\chi}_{1}^{0}Z/\tilde{\mu}^{\pm}_{\text{R}}\mu^{\mp}$ | 78.6/10.7/10.2
$\tilde{\chi}_{3}^{0}\to\tilde{\chi}_{1}^{0}Z/\tilde{\chi}_{1}^{0}h/\tilde{\mu}^{\pm}_{\text{L}}\mu^{\mp}/\tilde{\nu}_{\mu}\nu_{\mu}$ | 86.7/10.8/1.6/0.8 | $\tilde{\chi}_{3}^{0}\to\tilde{\chi}_{1}^{\pm}W^{\mp}/\tilde{\chi}_{2}^{0}Z/\tilde{\chi}_{1}^{0}Z/\tilde{\chi}_{1}^{0}h/\tilde{\chi}_{2}^{0}h/\tilde{\mu}^{\pm}_{\text{R}}\mu^{\mp}$ | 58.2/25.3/12.6/2.8/0.5/0.4
$\tilde{\chi}_{4}^{0}\to\tilde{\nu}_{\mu}\nu_{\mu}/\tilde{\mu}^{\pm}_{\text{L}}\mu^{\mp}/\tilde{\chi}_{1}^{\pm}W^{\mp}/\tilde{\chi}_{3}^{0}Z/\tilde{\chi}_{2}^{0}h/\tilde{\chi}_{2}^{0}Z/\tilde{\chi}_{3}^{0}h$ | 26.8/26.0/24.1/11.3/10.9/0.4/0.3 | $\tilde{\chi}_{4}^{0}\to\tilde{\chi}_{1}^{\pm}W^{\mp}/\tilde{\chi}_{2}^{0}h/\tilde{\chi}_{1}^{0}h/\tilde{\chi}_{1}^{0}Z/\tilde{\nu}_{\mu}\nu_{\mu}/\tilde{\chi}_{2}^{0}Z/\tilde{\mu}^{\pm}_{\text{L}}\mu^{\mp}/\tilde{\mu}^{\pm}_{\text{R}}\mu^{\mp}$ | 58.6/23.8/11.2/2.7/1.3/1.0/0.7/0.6
$\tilde{\chi}_{1}^{\pm}\to\tilde{\chi}_{1}^{0}W^{\pm}/\tilde{\nu}_{\mu}\mu^{\pm}/\tilde{\mu}^{\pm}_{\text{L}}\nu_{\mu}$ | 79.9/13.7/6.0 | $\tilde{\chi}_{1}^{\pm}\to\tilde{\chi}_{1}^{0}W^{\pm}/\tilde{\mu}^{\pm}_{\text{R}}\nu_{\mu}$ | 92.5/6.9
$\tilde{\chi}_{2}^{\pm}\to\tilde{\mu}^{\pm}_{\text{L}}\nu_{\mu}/\tilde{\nu}_{\mu}\mu^{\pm}/\tilde{\chi}_{2}^{0}W^{\pm}/\tilde{\chi}_{1}^{\pm}Z/\tilde{\chi}_{3}^{0}W^{\pm}/\tilde{\chi}_{1}^{\pm}h$ | 26.3/26.1/12.8/11.9/11.7/11.2 | $\tilde{\chi}_{2}^{\pm}\to\tilde{\chi}_{2}^{0}W^{\pm}/\tilde{\chi}_{1}^{\pm}Z/\tilde{\chi}_{1}^{\pm}h/\tilde{\chi}_{1}^{0}W^{\pm}/\tilde{\mu}^{\pm}_{\text{R}}\nu_{\mu}/\tilde{\nu}_{\mu}\mu^{\pm}$ | 31.4/28.4/24.6/13.3/1.3/0.8
$\tilde{\mu}_{\text{L}}^{\pm}\to\tilde{\chi}_{1}^{0}\mu^{\pm}$ | 100 | $\tilde{\mu}_{\text{L}}^{\pm}\to\tilde{\chi}_{1}^{0}\mu^{\pm}/\tilde{\chi}_{1}^{\pm}\nu_{\mu}/\tilde{\chi}_{2}^{0}\mu^{\pm}/\tilde{\mu}_{\text{R}}^{\pm}h/\tilde{\mu}_{\text{R}}^{\pm}Z$ | 68.8/18.5/9.8/1.5/1.4
$\tilde{\mu}_{\text{R}}^{\pm}\to\tilde{\chi}_{1}^{0}\mu^{\pm}/\tilde{\chi}_{2}^{0}\mu^{\pm}/\tilde{\nu}_{\mu}W^{\pm}/\tilde{\chi}_{1}^{\pm}\nu_{\mu}$ | 98.7/0.3/0.3/0.2 | $\tilde{\mu}_{\text{R}}^{\pm}\to\tilde{\chi}_{1}^{0}\mu^{\pm}$ | 100
$\tilde{\nu}_{\mu}\to\tilde{\chi}_{1}^{0}\nu_{\mu}$ | 100 | $\tilde{\nu}_{\mu}\to\tilde{\chi}_{1}^{0}\nu_{\mu}/\tilde{\chi}_{1}^{\pm}\mu^{\mp}/\tilde{\chi}_{2}^{0}\nu_{\mu}/\tilde{\mu}^{\pm}_{\text{R}}W^{\mp}$ | 73.0/16.3/7.8/2.8
R value | 0.38, SR_A44 in SR-2 | R value | 0.50, SS15 in SR-2

Table 9:  Benchmark points P5 for Type-E samples in Table 4 and P6 for the
curved blue area on the $|m_{\tilde{\chi}_{1}^{0}}|-m_{\tilde{\mu}_{L}}$ plane
in Fig. 5. P5 satisfies all the restrictions, while P6 is excluded by the LHC
search for SUSY.

Benchmark Point P5 for Type-E samples | Benchmark Point P6 for the curved blue area in Fig. 5
---|---
$\mu$ | 719.2 GeV | $\text{m}_{\text{H}}$ | 3235.4 GeV | $\mu$ | 551.2 GeV | $\text{m}_{\text{H}}$ | 6279.6 GeV
$\text{tan}\beta$ | 38.7 | $\text{m}_{\text{A}}$ | 3235.4 GeV | $\text{tan}\beta$ | 31.7 | $\text{m}_{\text{A}}$ | 6279.6 GeV
$\text{A}_{t}$ | 2697.6 GeV | $\text{m}_{\text{h}}$ | 125.7 GeV | $\text{A}_{t}$ | 2398.8 GeV | $\text{m}_{\text{h}}$ | 125.2 GeV
$\text{M}_{1}$ | 277.0 GeV | $\text{m}_{\text{H}^{\pm}}$ | 3253.2 GeV | $\text{M}_{1}$ |  $-304.9$ GeV | $\text{m}_{\text{H}^{\pm}}$ | 6284.5 GeV
$\text{M}_{2}$ | 1263.2 GeV | $\text{m}_{\tilde{\nu}_{\mu}}$ | 310.2 GeV | $\text{M}_{2}$ | 317.5 GeV | $\text{m}_{\tilde{\nu}_{\mu}}$ | 457.4 GeV
$\tilde{\text{m}}_{\tilde{\mu}_{\text{L}}}$ | 297.4 GeV | $\text{m}_{\tilde{\chi}_{1}^{0}}$ | 276.0 GeV | $\tilde{\text{m}}_{\tilde{\mu}_{\text{L}}}$ | 409.6 GeV | $\text{m}_{\tilde{\chi}_{1}^{0}}$ |  $-304.1$ GeV
$\tilde{\text{m}}_{\tilde{\mu}_{\text{R}}}$ | 288.2 GeV | $\text{m}_{\tilde{\chi}_{2}^{0}}$ | 731.5 GeV | $\tilde{\text{m}}_{\tilde{\mu}_{\text{R}}}$ | 755.9 GeV | $\text{m}_{\tilde{\chi}_{2}^{0}}$ | 327.2 GeV
$\text{m}_{\tilde{\mu}_{\text{L}}}$ | 321.0 GeV | $\text{m}_{\tilde{\chi}_{3}^{0}}$ |  $-736.8$ GeV | $\text{m}_{\tilde{\mu}_{\text{L}}}$ | 464.5 GeV | $\text{m}_{\tilde{\chi}_{3}^{0}}$ |  $-570.1$ GeV
$\text{m}_{\tilde{\mu}_{\text{R}}}$ | 279.6 GeV | $\text{m}_{\tilde{\chi}_{4}^{0}}$ | 1301.3 GeV | $\text{m}_{\tilde{\mu}_{\text{R}}}$ | 700.3 GeV | $\text{m}_{\tilde{\chi}_{4}^{0}}$ | 578.7 GeV
$a_{\mu}^{\text{SUSY}}$ | $2.11\times 10^{-9}$ | $\text{m}_{\tilde{\chi}_{1}^{\pm}}$ | 730.8 GeV | $a_{\mu}^{\text{SUSY}}$ | $2.03\times 10^{-9}$ | $\text{m}_{\tilde{\chi}_{1}^{\pm}}$ | 327.5 GeV
${\Omega h}^{2}$ | 0.13 | $\text{m}_{\tilde{\chi}_{2}^{\pm}}$ | 1301.5 GeV | ${\Omega h}^{2}$ | 0.10 | $\text{m}_{\tilde{\chi}_{2}^{\pm}}$ | 581.0 GeV
$\sigma_{p}^{\text{SI}}$ | $5.44\times 10^{-47}\text{cm}^{2}$ | $\sigma_{p}^{\text{SD}}$ | $5.12\times 10^{-43}\text{cm}^{2}$ | $\sigma_{p}^{\text{SI}}$ | $1.09\times 10^{-46}\text{cm}^{2}$ | $\sigma_{p}^{\text{SD}}$ | $2.04\times 10^{-42}\text{cm}^{2}$
$\text{N}_{11},\text{N}_{12},\text{N}_{13},\text{N}_{14}$ |  $-0.997$, 0.002, $-0.071$, 0.028 | $\text{N}_{11},\text{N}_{12},\text{N}_{13},\text{N}_{14}$ |  $-0.993$, $-0.007$, $-0.106$, $-0.055$
$\text{N}_{21},\text{N}_{22},\text{N}_{23},\text{N}_{24}$ | 0.070, 0.100, $-0.703$, 0.700 | $\text{N}_{21},\text{N}_{22},\text{N}_{23},\text{N}_{24}$ | 0.009, 0.971, $-0.206$, 0.122
$\text{N}_{31},\text{N}_{32},\text{N}_{33},\text{N}_{34}$ | 0.030, $-0.027$, $-0.706$, $-0.708$ | $\text{N}_{31},\text{N}_{32},\text{N}_{33},\text{N}_{34}$ |  $-0.114$, 0.060, 0.697, 0.705
$\text{N}_{41},\text{N}_{42},\text{N}_{43},\text{N}_{44}$ |  $-0.004$, 0.995, 0.052, $-0.090$ | $\text{N}_{41},\text{N}_{42},\text{N}_{43},\text{N}_{44}$ | 0.036, $-0.232$, $-0.679$, 0.696
Annihilations | Fractions[%] | Annihilations | Fractions[%]
$\tilde{B}-\tilde{\mu}_{R}$ Co-annihilation | 94.2 |  $\tilde{B}-\tilde{W}$ Co-annihilation | 91.4
Decays | Branching ratios[%] | Decays | Branching ratios[%]
$\tilde{\chi}_{2}^{0}\to\tilde{\chi}_{1}^{0}h/\tilde{\chi}_{1}^{0}Z/\tilde{\mu}^{\pm}_{\text{L}}\mu^{\mp}/\tilde{\mu}^{\pm}_{\text{R}}\mu^{\mp}/\tilde{\nu}_{\mu}\nu_{\mu}$ | 73.1/14.4/7.5/3.6/1.4 | $\tilde{\chi}_{2}^{0}\to\tilde{\chi}_{1}^{0}Z^{*}/\tilde{\chi}_{1}^{0}h^{*}$ | 100
$\tilde{\chi}_{3}^{0}\to\tilde{\chi}_{1}^{0}Z/\tilde{\chi}_{1}^{0}h/\tilde{\mu}^{\pm}_{\text{R}}\mu^{\mp}/\tilde{\mu}^{\pm}_{\text{L}}\mu^{\mp}/\tilde{\nu}_{\mu}\nu_{\mu}$ | 82.2/14.8/1.1/0.9/0.8 | $\tilde{\chi}_{3}^{0}\to\tilde{\chi}_{1}^{\pm}W^{\mp}/\tilde{\chi}_{2}^{0}Z/\tilde{\chi}_{1}^{0}h/\tilde{\chi}_{2}^{0}h/\tilde{\chi}_{1}^{0}Z$ | 62.1/27.0/8.1/1.4/1.1
$\tilde{\chi}_{4}^{0}\to\tilde{\chi}_{1}^{\pm}W^{\mp}/\tilde{\nu}_{\mu}\nu_{\mu}/\tilde{\mu}^{\pm}_{\text{L}}\mu^{\mp}/\tilde{\chi}_{2}^{0}h/\tilde{\chi}_{3}^{0}Z/\tilde{\chi}_{2}^{0}Z/\tilde{\chi}_{3}^{0}h/\tilde{\mu}^{\pm}_{\text{R}}\mu^{\mp}$ | 25.2/25.1/24.2/11.6/11.5/0.9/0.8/0.4 | $\tilde{\chi}_{4}^{0}\to\tilde{\chi}_{1}^{\pm}W^{\mp}/\tilde{\chi}_{2}^{0}h/\tilde{\chi}_{1}^{0}Z/\tilde{\chi}_{2}^{0}Z/\tilde{\chi}_{1}^{0}h/\tilde{\nu}_{\mu}\nu_{\mu}/\tilde{\mu}^{\pm}_{\text{L}}\mu^{\mp}$ | 63.6/24.0/8.2/2.2/0.8/0.7/0.5
$\tilde{\chi}_{1}^{\pm}\to\tilde{\chi}_{1}^{0}W^{\pm}/\tilde{\nu}_{\mu}\mu^{\pm}/\tilde{\mu}^{\pm}_{\text{L}}\nu_{\mu}/\tilde{\mu}^{\pm}_{\text{R}}\nu_{\mu}$ | 89.7/7.0/1.7/1.1 | $\tilde{\chi}_{1}^{\pm}\to\tilde{\chi}_{1}^{0}(W^{\pm})^{*}$ | 100
$\tilde{\chi}_{2}^{\pm}\to\tilde{\nu}_{\mu}\mu^{\pm}/\tilde{\mu}^{\pm}_{\text{L}}\nu_{\mu}/\tilde{\chi}_{2}^{0}W^{\pm}/\tilde{\chi}_{1}^{\pm}Z/\tilde{\chi}_{3}^{0}W^{\pm}/\tilde{\chi}_{1}^{\pm}h$ | 24.8/24.4/12.8/12.5/12.5/12.4 | $\tilde{\chi}_{2}^{\pm}\to\tilde{\chi}_{2}^{0}W^{\pm}/\tilde{\chi}_{1}^{\pm}Z/\tilde{\chi}_{1}^{\pm}h/\tilde{\chi}_{1}^{0}W^{\pm}/\tilde{\mu}^{\pm}_{\text{L}}\nu_{\mu}/\tilde{\nu}_{\mu}\mu^{\pm}$ | 32.3/30.4/25.7/10.3/0.9/0.4
$\tilde{\mu}_{\text{L}}^{\pm}\to\tilde{\chi}_{1}^{0}\mu^{\pm}$ | 100 | $\tilde{\mu}_{\text{L}}^{\pm}\to\tilde{\chi}_{1}^{\pm}\nu_{\mu}/\tilde{\chi}_{2}^{0}\mu^{\pm}/\tilde{\chi}_{1}^{0}\mu^{\pm}$ | 57.2/29.8/13.0
$\tilde{\mu}_{\text{R}}^{\pm}\to\tilde{\chi}_{1}^{0}\mu^{\pm}$ | 100 | $\tilde{\mu}_{\text{R}}^{\pm}\to\tilde{\chi}_{1}^{0}\mu^{\pm}/\tilde{\chi}_{3}^{0}\mu^{\pm}$ | 99.5/0.2
$\tilde{\nu}_{\mu}\to\tilde{\chi}_{1}^{0}\nu_{\mu}$ | 100 | $\tilde{\nu}_{\mu}\to\tilde{\chi}_{1}^{\pm}\mu^{\mp}/\tilde{\chi}_{2}^{0}\nu_{\mu}/\tilde{\chi}_{1}^{0}\nu_{\mu}$ | 59.2/28.5/12.3
R value | 0.34, SR_A44 in SR-2 | R value | 1.22, SR_WZoff_high_njd in SR-4

Table 10:  Survival mechanisms for the six benchmark points listed in Tables
7–9. These mechanisms rely on both the properties of the points and
experimental search strategies. They are acquired by comparing the dominant
signals of the productions with relevant experimental data.

Production Points Survival mechanism | P1 | P2 | P3 | P4 | P5 | P6
---|---|---|---|---|---|---
$\bar{\tilde{W}}\tilde{W}$ | I, II [112] | I, II [124] | III, IV [113] | I, IV [113] | IV [113] | I, II [112]
$\bar{\tilde{H}}\tilde{H}$ | I, III, IV [112] | III, IV [124] | IV [113] | III, IV [113] | IV [113] | III [112]
$\tilde{\mu}_{L}^{*}\tilde{\mu}_{L}$ | I, III, IV [6] | I, II [124] | I, II [124] | IV [6] | I, II [124] | III [6]
$\tilde{\mu}_{R}^{*}\tilde{\mu}_{R}$ | I, IV [6] | IV [6] | IV [6] | I, II [124] | I, II [124] | IV [6]

To simplify the discussion of the LHC restrictions, we first classify the blue
samples in Fig. 3 by DM annihilation mechanisms, similar to what we did in
Table 5. Then, we only focused on the samples studied by the Monte Carlo
simulations. We show in Table 6 the numbers before and after simulating the
LHC restrictions, the SRs that contributed to the largest R-values, and their
capability to exclude the samples of each mechanism, expressed by the
percentage of the total numbers in the second column. The following SRs are
involved:

  * •

SR-1: Signal regions SRG08_0j_mll and SRG07_0j_mll defined in Ref. [6]. These
come from the LHC search for slepton pair production by the final state
containing two opposite-sign-same-flavor (OSSF) leptons and missing transverse
momentum.

  * •

SR-2: Signal regions SR_A44, SS15, and SR_A08 in Ref. [114, 113]. They arise
from the LHC search for electroweakinos with the final state containing
missing transverse momentum, no jets, and two same-sign (SS) dileptons for the
SS15, three electrons, or muons that form at least one OSSF pair for the
SR_A44 and SR_A08.

  * •

SR-3: Signal regions E - high - mm - 30, S - high - mm - 05, S - high - mm -
10, and so on proposed in Ref. [124]. Their study concentrated on the
electroweakinos with compressed mass spectra and investigated their production
at the LHC by the final state containing two leptons and missing transverse
momentum.

  * •

SR-4: Signal regions SR_incWZoff_high_njc1, SR_WZoff_high_njd, SR_W-
Z_off_high_njc, and so on defined in Ref. [112]. These authors studied the
chargino–neutralino associated production at the LHC by the final state
containing three leptons and missing transverse momentum, where the chargino
and neutralino decayed into off-shell W and Z bosons, respectively.

  * •

SR-5: Signal regions SR1_weakino_3high_mll_2, SR2_stop_3high_pt_1, and
SR1_weakino_2media_mll_2 defined in Ref. [126]. These arise from the LHC
search for new physics by the signal containing two soft oppositely charged
leptons and missing transverse momentum.

These revealed how these annihilation mechanisms have been tested at the LHC.
In particular, Table 6 shows that different SRs complement each other in doing
so, and a single SR never plays a dominant role in this regard. This
conclusion depends not only on the intrinsic physics of the MSSM but also on
the details of these SRs. It arises from the fact that we include as many
experimental analyses as possible to study the LHC restrictions, and each of
them usually defines several signal regions. This situation allows us to make
good use of the experimental data to explore the parameter space of the MSSM.

Next, we illuminate how the MSSM manages to survive the LHC restriction. We
looked for benchmark points from the Type-A, B, C, D, and E samples specified
in Table 4 and labeled them P1, P2, P3, P4, and P5, respectively. We present
their details in Tables 7, 8, and 9, and provide corresponding survival
mechanisms in Table 10. These mechanisms are acquired by comparing the
dominant signals of the productions with pertinent experimental data. For
example, the DM in P1 achieved the measured density by co-annihilating with
wino-like particles. The applicable restriction to the wino productions cames
from the search for the chargino–neutralino associated production, where the
chargino and neutralino decayed into off-shell W and Z bosons, respectively.
The most robust results came from the analysis in Ref. [112], and they served
as the guidelines of our judgment. We also provided a benchmark point, P6, for
the curved blue area on the $|m_{\tilde{\chi}_{1}^{0}}|-m_{\tilde{\mu}_{L}}$
plane in Fig. 5, characterized by $230~{}{\rm
GeV}\lesssim|m_{\tilde{\chi}_{1}^{0}}|\lesssim 350~{}{\rm GeV}$ and
$300~{}{\rm GeV}\lesssim m_{\tilde{\mu}_{L}}\lesssim 500~{}{\rm GeV}$. Our
simulation indicated that it is excluded by SR_WZoff_high_njd in SR-4, which
is designed for the tri-lepton signal from off-shell $W$ and $Z$ bosons. This
point is distinct because both the wino pair production and the
$\tilde{\mu}_{L}$ pair production contribute to this signal, and none of these
processes alone can exclude this point. In addition, the program SModelS-2.2.1
is also unable to exclude it since it misses the $\tilde{\mu}_{L}$
contribution.

At this stage, we clarify the following points about the LHC restrictions:

  * •

Throughout this study, both the theoretical uncertainties incurred in the
simulations and the experimental (systematic and statistic) uncertainties were
not included. Although these effects can relax the LHC restrictions, it is
expected that much stronger restrictions on the MSSM will be acquired given
the advent of high-luminosity LHC in the near future.

  * •

We did not include the search for charginos and neutralinos by the fully
hadronic final states of W/Z and Higgs bosons with $139~{}{\rm fb}^{-1}$ data
[170] in this study. This search rejects large SM backgrounds by identifying
high-$p_{T}$ bosons with large-radius jets and jet substructure information.
Thus, it is more efficient than the leptonic signal search in Ref. [112] only
when winos are heavier than $600~{}{\rm GeV}$ [170]. This conclusion holds in
simplified models of SUSY but was never applied to this research. The reason
is that the DM must co-annihilate with $\tilde{\mu}_{L}$ or $\tilde{\mu}_{R}$
to achieve the measured density for $M_{2}\gtrsim 600~{}{\rm GeV}$ (see the
results in Fig. 2), and the wino-like particles may decay into the slepton
first to enhance their leptonic signal. As a result, it is the leptonic
signals that are more powerful in excluding SUSY points. This conclusion is
the same as that of the General Next-to-Minimal Supersymmetric Standard Model
(GNMSSM) studied in Ref. [89].

In the latest version of SModelS, namely SModelS-2.2.1 [165], the cut
efficiencies of the hadronic analysis and relevant signal topologies have been
implemented in its database. We utilized this code to limit the green samples
in Fig. 1. We did not find that it has exclusion capabilities.

  * •

In some high-energy SUSY-breaking theories, $\tilde{\tau}$ may be the NLSP due
to its larger Yukawa coupling than those of the first- and second-generation
sleptons. In this case, heavy sparticles may decay into $\tilde{\tau}$ to
change the $e/\mu$ signals of this research, and the LHC restrictions may be
relaxed [46]. We will discuss such a possibility in our future work.

###  3.4 Related issues

We stress the following issues related to this research:

  * •

Throughout this study, we assumed that $\tilde{\chi}_{1}^{0}$ was fully
responsible for the measured density. This assumption determined that the DM
is bino-dominated and approximately degenerate with winos or sleptons in mass.
This has profound implications on the phenomenology of the MSSM. Relaxing this
assumption usually complicates this kind of research and makes the obtained
conclusions untenable [98].

  * •

As pointed out by the recent lattice simulation of the BMW collaboration on
the hadronic vacuum polarization contribution to $a_{\mu}$ [31], the muon g-2
anomaly may arise from the uncertainties in calculating the hadronic
contribution to the moment. If this speculation is corroborated, $a_{\mu}^{\rm
SUSY}$ should be much smaller than its currently favored size, and any of the
electroweakinos and $\tilde{\mu}_{L/R}$ are not necessarily light. In this
case, the LHC restrictions will be relaxed significantly. For example, we
recently updated the results of Ref. [171], which only studied the DM physics
in GNMSSM, by including the recent LZ restrictions. We found that the analyses
in Table 1 only excluded about $4\%$ of the remaining samples in Fig. 2 of
Ref. [171].

  * •

It is remarkable that the explanations of the anomaly in the MSSM are to be
tested at future colliders, given that it predicts some moderately light
sparticles, such as $m_{\tilde{\chi}_{1}^{0}}$, $m_{\rm NLSP}$, and $m_{\rm
NNLSP}$. It was found that although only parts of the preferred parameter
space can be covered in the high-luminosity LHC, exhaustive coverage of the
parameter space is possible at a high-energy $e^{+}e^{-}$ collider with
$\sqrt{s}\gtrsim 1~{}{\rm TeV}$, such as ILC with $\sqrt{s}=1~{}{\rm TeV}$
[172] and CLIC with $\sqrt{s}=1~{}{\rm TeV}$ [173, 174]. This conclusion was
acquired from Fig. 4 of Ref. [175], where the capabilities of different
colliders to survey the explanation were compared for the
$\tilde{B}-\tilde{W}$ co-annihilation case.

##  4 Summary

Inspired by the rapid progress of particle physics experiments in recent
years, we studied their impacts on the MSSM and how the theory kept consistent
with them. We are particularly interested in the recent measurement of the
muon g-2 at Fermilab, the LHC search for SUSY, and the DM direct detection by
the LZ experiment since they are sensitive to different parameters and
complement each other to provide valuable information of the MSSM. In
surveying the status of the MSSM, we utilized the MultiNest algorithm to
comprehensively scan its parameter space. We adopted the muon g-2 observable
to guide the scans and include the restrictions from the LHC Higgs data, DM
experiments, B-physics measurements, and vacuum stability. We also examined
the samples acquired from the scans by the restrictions from the LHC search
for SUSY and the latest LZ experiment. The main conclusions of this research
are as follows:

  * •

The bino-dominated DM achieved the measured relic density by co-annihilating
with wino-like electroweakinos, $\tilde{\mu}_{L}$, or $\tilde{\mu}_{R}$ if one
intends the theory to explain the muon g-2 anomaly.

  * •

Given the measured DM density, the LZ experiment alone has required
$\mu\gtrsim 380~{}{\rm GeV}$ for $M_{1}<0$ and $\mu\gtrsim 600~{}{\rm GeV}$
for $M_{1}>100~{}{\rm GeV}$. If the restrictions from the muon g-2 anomaly and
the LHC experiment are also included, the lower bounds become about
$500~{}{\rm GeV}$ and $630~{}{\rm GeV}$, respectively, indicating that the
theory needs a tuning of ${\cal{O}}(1\%)$ to predict $Z$-boson mass [168].
Compared with the restriction from the PandaX-4T experiment on the SI
scattering cross section, these bounds are improved by about $100~{}{\rm
GeV}$. This situation makes the $Z$\- and $h$-mediated resonant annihilations
even more unnatural so that they are usually missed in the scans by the
MultiNest algorithm. The fundamental reason of such phenomena is the
correlation between the DM physics and the electroweak symmetry breaking in
the MSSM.

  * •

Based on the premise of explaining the muon g-2 anomaly at the $2\sigma$
level, the LHC restrictions have set bounds on the sparticle mass spectra:
$m_{\tilde{\chi}_{1}^{0}}\gtrsim 210~{}{\rm GeV}$,
$m_{\tilde{\chi}_{2}^{0}},m_{\tilde{\chi}_{1}^{\pm}}\gtrsim 235~{}{\rm GeV}$,
$m_{\tilde{\chi}_{3}^{0}}\gtrsim 515~{}{\rm GeV}$,
$m_{\tilde{\chi}_{4}^{0}}\gtrsim 525~{}{\rm GeV}$,
$m_{\tilde{\chi}_{2}^{\pm}}\gtrsim 530~{}{\rm GeV}$,
$m_{\tilde{\nu}_{\mu}}\gtrsim 235~{}{\rm GeV}$, $m_{\tilde{\mu}_{1}}\gtrsim
215~{}{\rm GeV}$, and $m_{\tilde{\mu}_{2}}\gtrsim 250~{}{\rm GeV}$, where
$\tilde{\chi}_{2}^{0}$ and $\tilde{\chi}_{1}^{\pm}$ are wino-dominated when
they are lighter than about $500~{}{\rm GeV}$ and $m_{\tilde{\mu}_{1}}$ may be
either $m_{\tilde{\mu}_{L}}$ or $m_{\tilde{\mu}_{R}}$. These bounds should be
regarded as rough estimates, instead of accurate values, since the samples
studied in this research are far from sufficient given the broad parameter
space of the MSSM. In addition, these bounds are far beyond the reach of the
LEP experiments in searching for SUSY and have not been acquired before.

The results can be interpreted as follows: if $\tilde{\chi}_{1}^{0}$ is
lighter, more missing transverse energy will be emitted in the sparticle
production processes at the LHC, which can improve the sensitivities of the
experimental analyses. However, the sparticles other than
$\tilde{\chi}_{1}^{0}$ are lighter, and they will be more copiously produced
at the LHC to increase the events containing multiple leptons.

  * •

We illuminate how some parameter spaces of the MSSM have been tested at the
LHC in Table 6. We also list five scenarios that are consistent with the LHC
restrictions in Table 4 and explain why they can do so by presenting benchmark
points in Tables 7–9, and their survival mechanisms are shown in Table 10.

This work extends the previous studies of the muon g-2 anomaly in the MSSM, in
particular those of Refs. [84] and [85], by utilizing more sophisticated
research strategies and surveying the LHC restrictions comprehensively. As a
result, the conclusions acquired in this research are more robust than those
in previous works. They exhibit the most essential characteristics of the
MSSM.

##  5 Acknowledgement

We sincerely thank Prof. Junjie Cao for numerous helpful discussions and his
great efforts to improve the manuscript. This work is supported by the
National Natural Science Foundation of China (NNSFC) under grant Nos. 12075076
and 11905044.

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