# Low energy supersymmetry confronted with current experiments: an overview

Fei Wang1, Wenyu Wang2, Jin Min Yang3,4, Yang Zhang1, Bin Zhu5
1 School of Physics, Zhengzhou University, Zhengzhou 450000, P. R. China
2 School of Science, Beijing University of Technology, Beijing 100124, P. R.
China
3 CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100080, P. R. China
4 School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, P. R. China
5 Department of Physics, Yantai University, Yantai 264005, P. R. China
feiwang@zzu.edu.cnwywang<EMAIL_ADDRESS>zhangyangphy<EMAIL_ADDRESS>
###### Abstract:

This study provides a brief overview of low-energy supersymmetry (SUSY) in
light of current experimental constraints, such as collider searches, dark
matter searches, and muon $g-2$ measurements. In addition, we survey a variety
of low energy supersymmetric models: the phenomenological minimal
supersymmetric model (MSSM); the supersymmetric models with cut-off-scale
boundary conditions, i.e., the minimal supergravity (mSUGRA) or the
constrained MSSM (CMSSM), the gauge mediation of SUSY breaking (GMSB), and the
anomaly mediation of SUSY breaking (AMSB), as well as their extensions. The
conclusion is that the low energy SUSY can survive all current experimental
constraints and remains compelling, albeit suffering from a little fine-tuning
problem. The fancy models like mSUGRA, GMSB, and AMSB need to be extended if
the muon $g-2$ anomaly comes from new physics.

##  1 Introduction

Despite its remarkable phenomenological success, the Standard Model (SM) in
particle physics still has remaining puzzles, such as the origin of the free
parameters, the matter-antimatter asymmetry, the instability of the
electroweak scale, or the divergent quantum correction of Higgs boson mass,
and the nature of cold dark matter. Searching for new physics beyond the SM is
the central theme of today and future particle physics. The low energy
supersymmetry (SUSY) has been the most appealing framework among various new
physics hypotheses. Phenomenologically speaking, the SUSY extension of the SM
could, solve the hierarchy problem, realize gauge couplings unification, adopt
the proper baryogenesis mechanisms, and generate the cold dark matter
candidate.

Most notably, SUSY predicts a neutral CP-even Higgs boson upper bounded
roughly by 135 GeV, corroborated by the Large Hadron Collider (LHC) discovery
of a 125 GeV Higgs boson. From the theoretical view, as a new fundamental
symmetry, SUSY is mathematically charming and, unlike other miscellaneous new
physics models, SUSY is part of a larger vision of physics, not just a
technical solution [1]. SUSY has been in the mainstream of high energy physics
for almost half a century. As depicted in Fig.1 [2], SUSY plays a crucial role
in the map of high energy physics and even the whole tree of the quantum
theory. Indeed, SUSY is needed and predicted by string theory, i.e., the
concept of supersymmetry emerged historically, at least in part because of its
role in string theory [1] . So far, string theory is the most hopeful
candidate for unifying all interactions and such unification is our original
intention since our job in physics is to see things simply, to understand a
great many complicated phenomena in a unified way [3].

Figure 1: The map of high energy physics and the quantum tree taken from [2].

Of course, physics thrives on crisis [4]. Since particle physics is a
discipline relying on experiments, the experimental crises or deviations from
the SM play a crucial role in searching for new physics. Currently, we are
facing two experimental crises: one is the cold dark matter, the other is the
muon $g-2$ anomaly. While the former is a quite robust evidence of new
physics, the latter should be taken with a grain of salt, e.g., 10% chance it
is new physics (much more plausible than other anomalies) [5]. In addition,
the dark matter direct detection experiment has not found any WIMP (Weakly
Interactive Massive Particle) dark matter particle. The LHC has not found any
particles predicted by new physics (albeit found some plausible anomalies in
B-decays). Confronted with these experimental results, what is the status of
low energy SUSY, healthy or needs to be hospitalized ?

In this note, we will briefly review the status of SUSY in light of these
experiments, namely the LHC searches, the dark matter, and muon $g-2$
measurements. The SUSY models investigated in the literature are the MSSM,
mSUGRA or CMSSM, GMSB, and AMSB, as well as respective extensions. We will
conclude that the low energy SUSY can survive all these experimental
constraints, albeit the fancy models like mSUGRA, GMSB and AMSB need to be
extended to accommodate the 125 GeV Higgs boson and the muon $g-2$ anomaly.

##  2 A brief description of SUSY

Supersymmetry is an extension of special relativity to include fermionic
symmetries [6]. The anticommutative relation of the supercharges $Q_{\alpha}$
is given by [6]

$Q_{\alpha}Q_{\beta}+Q_{\beta}Q_{\alpha}=\Gamma^{\mu}_{\alpha\beta}P_{\mu},$
(1)

with $\Gamma^{\mu}$ being the Dirac matrix and $P_{\mu}$ the four-momentum.
So, unlike any other internal symmetries which are independent of spacetime
symmetry (no-go theorem of Coleman and Mandula [7]: The most general Lie
algebra of symmetry operators that commute with S-matrix consists of the
generators of the Poincare group and ordinary internal symmetry generators.
The latter act on one-particle states with matrices that are diagonal and
independent of both momentum and spin), supersymmetry is entangled with
spacetime symmetry and is an extension of special relativity (Golfand and
Likhtman [8] found that the S-matrix can have Poincare symmetry extended by
SUSY algebra, while Haag et al. [9] further proved that SUSY algebra is the
only graded Lie algebra of symmetries of the S-matrix consistent with
relativistic QFT, which extends the Poincare group by anti-commutators). In
order words, the spacetime symmetry of QFT is completed by SUSY:

$\displaystyle\left[P_{\mu},P_{\nu}\right]=0,$ (2)
$\displaystyle\left[M_{\mu\nu},P_{\lambda}\right]=i\left(P_{\mu}~{}g_{\nu\lambda}-P_{\nu}~{}g_{\mu\lambda}\right),$
(3)
$\displaystyle\left[M_{\mu\nu},M_{\rho\lambda}\right]=-i\left(M_{\mu\rho}~{}g_{\nu\lambda}+M_{\nu\lambda}~{}g_{\mu\rho}-M_{\mu\lambda}~{}g_{\nu\rho}-M_{\nu\rho}~{}g_{\mu\lambda}\right),$
(4) $\displaystyle\left[P_{\mu},Q_{a}\right]=0,$ (5)
$\displaystyle\left[M_{\mu\nu},Q_{a}\right]=-\left(\sigma_{\mu\nu}^{4}\right)_{ab}Q_{b},$
(6)
$\displaystyle\left\\{Q_{a},Q_{b}\right\\}=-2\left(\gamma^{\mu}C\right)_{ab}P_{\mu},$
(7)
$\displaystyle\left\\{\bar{Q}_{a},\bar{Q}_{b}\right\\}=2\left(C^{-1}\gamma^{\mu}\right)_{ab}P_{\mu},$
(8)
$\displaystyle\left\\{Q_{a},\bar{Q}_{b}\right\\}=2\left(\gamma^{\mu}\right)_{ab}P_{\mu},$
(9)

where $P_{\mu}$ and $M_{\mu\nu}$ are the generators of the Poincare group. At
about the same time as Golfand and Likhtman, a string was introduced that
could be a fermion as well as a boson led to the concept of supersymmetry [10,
11, 12]. When supersymmetry was applied to particle physics (pioneered by Wess
and Zumino [13]), it demonstrated remarkable virtues such as [1]: makes a
“small” Higgs mass natural, survives electroweak tests, makes heavy top mass
as needed. Further, when supersymmetry is localized, we obtain supergravity
[14].

Since any field representation of SUSY algebra involves fields with different
spins and the same mass, SUSY predicts a superpartner for each particle in the
SM. If SUSY is unbroken, we will find light sparticles that were ruled out by
experiments. So SUSY must be spontaneously broken in some hidden sector where
the breaking effects can be mediated to the observable sector via some
interactions. Note that the supertrace sum rule [15, 16]

$\displaystyle StrM^{2}=\sum_{particles}(-1)^{2j}(2j+1)M_{j}^{2}=0~{},$ (10)

with $j$ being the spin, which holds for any renormalizable SUSY Lagrangian,
will lead to the mass relation between electron and selectrons

$\displaystyle m_{\tilde{e}_{1}}^{2}+m_{\tilde{e}_{2}}^{2}=2m_{e}^{2}~{},$
(11)

indicating the existence of a selectron lighter than the electron. Hence SUSY
breaking cannot be merely like the spontaneous breaking of electroweak
symmetry in the Higgs sector, which directly couples to the fermion and gauge
fields to transfer the breaking effects. Suppose the SUSY breaking in the
hidden sector is characterized by the F-term VEV $\langle F\rangle$ and the
mediation scale is $M_{med}$, the superpartners in the visible sector obtain a
soft mass from SUSY breaking :

$\displaystyle M_{soft}\sim\frac{\langle F\rangle}{M_{med}}~{},$ (12)

up to some numerical factor such as the loop factor for gauge mediation.
$M_{med}$ can be identified to be the Planck scale $M_{Pl}$ for gravity
mediation and the messenger scale $M_{mess}$ for gauge mediation, with
$M_{mess}$ possibly being much smaller than the Planck scale $M_{mess}\ll
M_{Pl}$. Superpartners of the standard model particles get masses both from
electroweak symmetry breaking and SUSY breaking. Therefore it is natural for
them to be a bit above the $Z$, which gets mass only from electroweak symmetry
breaking [1].

Although SUSY predicts numerous sparticles, only the lightest sparticle (LSP)
is stable assuming R-parity (without R-parity no sparticles are stable), which
is usually assumed to be the lightest neutralino. This lightest neutralino is
a good candidate for the WIMP cold dark matter, and at the LHC it appears as
missing energy at the end of the decay chain of each produced sparticle.

Another remarkable feature of SUSY is that to give masses for both up and down
type quarks, two Higgs doublets with opposite hypercharges are needed since
SUSY forbids the appearance of the complex conjugate of the Higgs field in the
superpotential. So SUSY predicts five Higgs bosons: two CP-even ones, one CP-
odd one, and a pair of charged ones. The Higgs quartic coupling $\lambda$
arises from D-terms [17]

$\displaystyle
V_{D}=\frac{1}{2}\left[D^{a}D^{a}+(D^{\prime})^{2}\right]=\frac{1}{8}(g^{2}+g^{\prime
2})\left(H_{d}^{\dagger}H_{d}-H_{u}^{\dagger}H_{u}\right)^{2}+\frac{1}{2}g^{2}\left(H_{d}^{\dagger}H_{u}\right)^{2},$
(13)

which is gauge coupling $\lambda\sim g^{2}$ instead of a free parameter as in
the SM. Thus in the MSSM, the mass of the lightest CP-even Higgs boson must be
light $m_{h}^{2}\sim\lambda v^{2}\sim g^{2}v^{2}$. In detail, its mass is
upper bounded by $Z$-boson mass at tree level and by about 135 GeV at loop
level:

$\displaystyle m_{h}<m_{Z}|\cos 2\beta|<m_{Z}~{}~{}~{}~{}~{}~{}~{}~{}({\rm
at~{}tree~{}level}),$ (14) $\displaystyle
m_{h}\leq\sqrt{m_{Z}^{2}+\epsilon}\leq 135~{}{\rm GeV}~{}~{}~{}~{}({\rm
at~{}loop~{}level}),$ (15)

where $\epsilon$ is the one-loop effects from top quarks and top squarks given
by [18]

$\displaystyle\epsilon=\frac{3m_{t}^{4}}{4\pi^{2}v^{2}}\left[\log\frac{M_{S}^{2}}{m_{t}^{2}}+\frac{X_{t}^{2}}{M_{S}^{2}}\left(1-\frac{X_{t}^{2}}{12M_{S}^{2}}\right)\right],$
(16)

with $v=174{\rm~{}GeV}$, $X_{t}\equiv A_{t}-\mu\cot\beta$ ($A_{t}$ is the
trilinear Higgs-stop coupling and $\mu$ is the Higgsino mass parameter), and
$M_{S}=\sqrt{m_{\tilde{t}_{1}}m_{\tilde{t}_{2}}}$. As shown in Fig.2, the SM-
like Higgs boson mass in the MSSM is more restricted than the SM Higgs boson
mass. Since SM Higgs boson mass is bounded by about 800 GeV if its cut-off
scale is 1 TeV.

Figure 2: The SM-like Higgs boson mass range in the MSSM taken from [19],
compared with the SM Higgs boson mass range taken from [20] which considered
the requirements of non-triviality [21] and vacuum stability [22, 23].

##  3 SUSY confronted with the LHC searches

At the LHC the typical signature of SUSY with R-parity is multi-jets or/and
multi-leptons plus missing energy since the sparticles must be produced in
pairs and each produced sparticle has a cascade decay with the final states
containing an odd number of LSP (the stable neutral LSPs just escape the
detector) plus some jets or/and leptons [24, 25, 26, 27]. So far the CMS and
ATLAS collaborations at the LHC have intensively searched for sparticle
productions and have failed to find any evidence. For simplified models with
significant mass splittings between the LSP and other sparticles, the searches
have pushed gluino and first generation squarks above about 2 TeV [28, 29],
pushed top squarks above about 1 TeV [30, 31], as shown in Fig.3, while for
uncolored sparticles (electroweakinos and sleptons) the bounds are much
weaker, only a few hundred GeV due to their small production cross-sections
[33, 34, 35]. As we will discuss later, these relatively light uncolored
sparticles are just needed to make sizable contributions to the muon $g-2$.

Figure 3: The current LHC bounds on gluino and stop masses, taken from [32].
Figure 4: The Higgs mass ranges predicted by the MSSM and SM compared with the
discovered 125 GeV Higgs mass, taken from [36].  Figure 5: The scatter plots
of the parameter space in the MSSM giving $125\pm 2$ GeV Higgs mass and
satisfying other constraints, taken from [37].

On the other hand, the LHC discovered a 125 GeV Higgs boson, which is within
the range $<135$ GeV predicted by SUSY as shown in Fig.4 [36], and requires
sizable loop effects of top squarks heavier than about TeV for $X_{t}\equiv
A_{t}-\mu\cot\beta$ not much larger than
$M_{S}\equiv\sqrt{m_{\tilde{t}_{1}}m_{\tilde{t}_{2}}}$, as shown in Fig.5
[37]. If we believe the stops to be the lightest colored sparticles from the
view of renormalization-group equation (RGE) runnings, the 125 GeV Higgs mass
then requires all colored sparticles above TeV. So the absence of any colored
sparticles and the discovery of a 125 GeV Higgs boson is consistent in the
framework of SUSY, both pushing the colored sparticles above TeV. Whereas,
since the uncolored sparticles are subject to extremely weak constraints from
the LHC searches, their allowed low masses below TeV are just required by the
muon $g-2$ discrepancy, as discussed later.

Note that the LHC has not yet discovered any particles predicted by new
physics, not just the sparticles predicted by SUSY. Among the new physics
models, SUSY remains the most compelling one. Another point we should note is
that the LHC bounds on the sparticle masses are usually valid for simplified
SUSY models with significant mass splittings between the LSP and other
sparticles. If the LSP mass is very close to the relevant sparticles like the
higgsinos [38] or stops [46, 47], the LHC bounds on the masses of these
sparticles will become much weaker. For example, in the natural SUSY scenario
[39, 40, 41, 42, 43, 44] the higgsinos are light 100-300 GeV while the
gauginos are heavy, and thus the LSP $\chi^{0}_{1}$ is higgsino-like and
nearly degenerate with the higgsino-like neutralino $\chi^{0}_{2}$ and the
higgsino-like chargino $\chi^{\pm}_{1}$. In this case, the productions of
these sparticles $\chi^{0}_{1,2}$ and $\chi^{\pm}_{1}$ at the LHC give missing
energy and can only be searched by requiring a hard jet radiated from initial
partons, i.e., the signal of monojet plus missing energy [38]. Moreover,
global likelihood analysis of the electroweakino sector shows that no range of
neutralino or chargino masses can be robustly excluded by current LHC
searches[45].

##  4 SUSY confronted with the dark matter

The cold dark matter can be explained in SUSY by the neutral and stable LSP
which can be a weakly interacting massive particle (WIMP)-the lightest
neutralino or a superweakly interacting massive particles (superWIMP) such as
the pseudo-goldstino or gravitino:

  * (i)

For the lightest neutralino case, it can be bino-like, higgsino-like, or wino-
like, depending on its dominant components. The bino-like LSP can easily
satisfy the direct detection limits since its scattering with the nucleon is
very weak and also can give the correct thermal relic density through the
coannihilation with other sparticles like winos [48] or sfermions. For the
bino-like dark matter in the MSSM, the parameter space satisfying the relic
density at $2\sigma$ level (and also explaining the muon $g-2$ at $2\sigma$
level) is displayed in Fig.6 which is taken from [50] (see also [51, 52, 53,
54, 55, 56]). It has to be heavier than 100 GeV, owing to the LHC direct
searches of electroweakino and the DM direct searches[57]. For the bino-like
dark matter in the CMSSM/mSUGRA, the $2\sigma$ region from a global fit
considering various measurements including the dark matter relic density is
shown in Fig.7 taken from [58] (also see [59] for the most updated global fit
of CMSSM).

The higgsino-like or wino-like LSP has relatively strong interactions and
freeze out rather late from the thermal bath in the early universe, leading to
an under-abundance of dark matter if they are light below TeV. To give the
sufficient aboundance for the dark matter, the thermal freeze-out higgsino-
like (wino-like) LSP must be around $1.0\pm 0.1$ TeV [61, 62] ($2.9\pm 0.1$
TeV [62, 63, 64]), which is so far allowed by the direct or indirect
detections and will be sensitively probed by the indirect detections like the
upcoming Cherenkov Telescope Array [60].

The light higgsino-like LSP with an under-abundance can also satisfy the
direct detection limits. For example, in natural SUSY the higgsinos are quite
light, around 100-300 GeV, and thus the thermal relic density of the higgsino-
like LSP is far below the dark matter abundance. If we want to enhance its
thermal relic density to the required abundance by mixing bino with higgsino,
it may lead to a too large scattering cross-section with the nucleon [49].

Figure 6: The scatter plots of the bino-like dark matter parameter space in
the MSSM satisfying the relic density at $2\sigma$ level (and also explaining
the muon $g-2$ at $2\sigma$ level), displayed in the plane of the direct
detection limits on the spin-independent scattering cross section with the
nucleon, taken from [50]. Figure 7: The $2\sigma$ region of the bino-like dark
matter parameter space in the CMSSM/mSUGRA satisfying the relic density and
other observables at $2\sigma$ level (not considering the muon $g-2$),
displayed in the plane of the direct detection limits on the spin-independent
scattering cross section with the nucleon, colored by DM annihilation
mechanisms, taken from [58].

  * (ii)

For the superWIMP pseudo-goldstino or gravitino, it can be produced from the
late decay [65, 66, 67] of the freeze-out lightest neutralino and thus can
provide the correct dark matter relic density while easily satisfying the
direct detection limits. The pseudo-goldstinos are predicted in multi-sector
SUSY breaking with different scales, in which amount of goldstinos is
generated, with one linear combination of these goldstinos being massless and
eaten by the gravitino while the orthogonal combinations acquire masses and
become the physical states (depending on the messenger masses in the context
of gauge mediation, the pseudo-goldstino mass can be from 0.1 GeV to hundred
GeV) [68, 69]. Note that if the superWIMP is thermally produced in the
reheating period after inflation, its relic density could be overabundant and
hence set an upper bound on the reheating temperature [70]. Such superWIMP
dark matter is usually not accessible in direct or indirect detections, but
may cause exotic phenomenology at the colliders and may help to alleviate some
cosmic problem, such as the Hubble tension [71] or Xenon1T excess [72]. The
produced lightest ordinary supersymmetric particle (LOSP) at the colliders,
which can possibly be charged, could decay to the superWIMP plus visible
particles (photon, Higgs boson, Z-boson etc) inside or outside the detectors
[73, 74, 75, 76, 77], depending on the interaction strength of the superWIMP
particle.

Therefore, SUSY (both the phenomenological MSSM and the constrained frameworks
like the mSUGRA) can explain the cold dark matter relic density and satisfy
the tightened direct detection limits on the WIMP.

##  5 SUSY confronted with the muon $g-2$

The muon $g-2$ from the Fermilab [78] measurement combined with the BNL result
[79] shows a $4.2\sigma$ deviation from the SM prediction [80, 81] (however,
if the lattice simulation result of the hadronic contribution from the BMW
group is taken, the deviation can be reduced to $1.5\sigma$ [82]), which can
be readily explained in the phenomenological minimal supersymmetric model
(MSSM). The explanation requires relatively light uncolored sparticles
(sleptons and electroweakinos). While for the models with boundary conditions
at some UV scales for the soft parameters, such as the mSUGRA, GMSB, they need
to be extended to account for the muon $g-2$ and the 125 GeV Higgs boson mass.

In SUSY, the smuons, muon sneutrino, and electroweakinos can contribute to
muon $g-2$ at the one-loop level. Since the SUSY contribution to muon $g-2$
[83, 84] is enhanced by a large $\tan\beta$ and suppressed by heavy sparticle
masses involved in the loops, to generate the required contribution to
explaining the muon $g-2$ deviation, a low SUSY mass and a large $\tan\beta$
are favored.

  * (i)

In the low energy effective MSSM, the masses of bino, winos, higgsinos, smuons
and sneutrino are all independent parameters. As shown in Fig.8 taken from
[85] (see also [86, 87]), under other constraints including the dark matter,
the vacuum stability and the LHC search for sleptons, there still remains a
part of the MSSM parameter space which can explain the muon $g-2$ at $2\sigma$
level [85, 86, 87, 50].

Figure 8:  The MSSM parameter space for the explanation of the muon $g-2$ at
$2\sigma$ level,taken from [85]. The upper-left green region above the dashed
line is allowed by all constraints and can explain the muon $g-2$ at $2\sigma$
level. Figure 9: The tension between the muon $g-2$ explanation and the 125
GeV SM-like Higgs mass as well as the LHC searches for the CMSSM/mSUGRA, taken
from [85].  Figure 10: The tension between the muon $g-2$ and the 125 GeV SM-
like Higgs mass for the CMSSM/mSUGRA, plotted from a global fit in [58]. The
regions encircled by the curves correspond to the $1\sigma$, $2\sigma$ and
$3\sigma$ levels, respectively.

  * (ii)

In the constrained models with certain UV boundary conditions for the soft
breaking parameters, such as mSUGRA/CMSSM, GMSB, the soft masses at the weak
scale are correlated. To give a 125 GeV SM-like Higgs mass, the top squarks
must be heavy and the correlated slepton masses cannot be as light as required
by the explanation of the muon $g-2$ anomaly. Fig.9 and Fig.10 show the
tension between the muon $g-2$ explanation and the 125 GeV SM-like Higgs mass
as well as the LHC searches for the CMSSM/mSUGRA [85] with inputs at the GUT
scale. To solve such a tension, these models need to be extended, e.g., couple
the messengers with the Higgs doublets to raise the tree-level SM-like Higgs
mass [88, 89] in gauge mediation or make the colored sparticles much heavier
than uncolored sparticles at the weak scale [90, 91, 92, 93, 94, 95]. In
superGUT [96] or subGUT [97] scenarios for SUGRA, which adopts the UV input
upon or below the GUT scale, there are still survived parameter spaces that
can account for both the muon g-2 anomaly and other constraints, such as the
dark matter and collider searches. In the deflected AMSB, which is an elegant
extension of AMSB, the muon g-2 anomaly can easily be explained [98, 99, 100,
101, 102]. The mixed mediation scenarios, such as the mirage mediation, can
also successfully account for the muon g-2 anomaly [103, 104].

  * (ii)

The non-minimal frameworks of SUSY with more free parameters than the MSSM can
easily accommodate the 125 GeV SM-like Higgs mass and the muon $g-2$ at
$2\sigma$ level, e.g., the next-to-minimal SUSY model (NMSSM) extends the MSSM
by a singlet Higgs superfield and can satisfy all current constraints [105,
106, 107, 108].

We should also note the electron $g-2$ which shows a slight deviation
according to the Berkeley measurement [109]. A joint explanation of such an
electron $g-2$ and the muon $g-2$ is feasible in the MSSM [110, 111, 112, 113,
114, 115]. If we further consider the measurement of electron EDM, its
correlation with the muon $g-2$ in the MSSM can set stringent constraints on
the CP-phases of the soft parameters [116, 117].

In this review, we did not discuss the plausible B-decay anomalies, which are
hard to be explained in SUSY with minimal flavor violation and the explanation
needs non-minimal flavor violation [118].

##  6 What is the problem of SUSY

The most obvious drawback is simply that SUSY has not been found yet, though
we have been hoping for a long time. It is disappointing that we have not
found SUSY yet, but for the most part, it is perhaps not too surprising [1].
Although the undiscovery of sparticles at the LHC is not too surprising
because sparticles obtain masses from both electroweak symmetry breaking and
SUSY breaking and should be significantly heavier than their SM partners, the
top squarks which have been pushed above TeV by the direct LHC searches and by
the 125 GeV Higgs boson mass will unpleasantly enlarge the logarithmic
divergence in the Higgs boson mass and lead to a little fine-tuning [119,
120]:

$\displaystyle\Delta_{\rm HS}\equiv\frac{\delta
m_{h}^{2}}{m_{h}^{2}}=\frac{3y_{t}^{2}}{4\pi^{2}m_{h}^{2}}(m_{Q_{3}}^{2}+m_{U_{3}}^{2}+A_{t}^{2})\log\frac{\Lambda}{M_{\rm
SUSY}},$ (17)

with $M_{\rm SUSY}=\sqrt{m_{\tilde{t}_{1}}m_{\tilde{t}_{2}}}$, $\Lambda$ being
the cut-off scale, $Q_{3}=(\tilde{t}_{L},\tilde{b}_{L})$ and
$U_{3}=\tilde{t}_{R}$.

Figure 11: The marginal posterior probability distribution function of the
CMSSM showing the tuning extent versus the stop mass from a Bayesian analysis
plotted from the analysis in [58] where the muon $g-2$ data is not considered.

The more traditional measure of tuning extent is defined by [121, 122]

$\Delta_{\rm BG}\equiv max_{i}\left|\frac{p_{i}}{m_{Z}^{2}}\frac{\partial
m_{Z}^{2}}{\partial p_{i}}\right|,$ (18)

with $p_{i}$ being the independent SUSY parameters at the cut-off scale. In
terms of this measure, in the CMSSM the fine-tuning extent is about per mille,
as shown in Fig.11 [58]. The fine-tuning can be lower [36] if we use another
measure $\Delta_{\rm EW}$ [43, 123]:

$\Delta_{\rm EW}\equiv max_{i}\left|C_{i}\right|/(m_{Z}^{2}/2),$ (19)

with [124]

$\displaystyle C_{H_{u}}$ $\displaystyle=$
$\displaystyle\left|-\frac{m^{2}_{H_{u}}\tan^{2}\beta}{\tan^{2}\beta-1}\right|~{},~{}C_{H_{d}}=\left|\frac{m^{2}_{H_{d}}}{\tan^{2}\beta-1}\right|~{},~{}C_{\mu}=\left|-\mu_{eff}^{2}\right|~{},~{}$
$\displaystyle C_{\Sigma_{u}^{u}(\tilde{t}_{1,2})}$ $\displaystyle=$
$\displaystyle\frac{\tan^{2}\beta}{\tan^{2}\beta-1}\left|\frac{3}{16\pi^{2}}F(m_{\tilde{t}_{1,2}}^{2})\left[y_{t}^{2}-g_{Z}^{2}\pm\frac{y_{t}^{2}A_{t}^{2}-8g_{Z}^{2}(\frac{1}{4}-\frac{2}{3}x_{w})\Delta_{t}}{m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}\right]\right|~{},$
$\displaystyle C_{\Sigma_{d}^{d}(\tilde{t}_{1,2})}$ $\displaystyle=$
$\displaystyle\frac{1}{\tan^{2}\beta-1}\left|\frac{3}{16\pi^{2}}F(m_{\tilde{t}_{1,2}}^{2})\left[g_{Z}^{2}\pm\frac{y_{t}^{2}\mu_{eff}^{2}+8g_{Z}^{2}(\frac{1}{4}-\frac{2}{3}x_{w})\Delta_{t}}{m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}\right]\right|~{},$
(20)

where $x_{w}=\sin^{2}\theta_{W}$ and

$\displaystyle\Delta_{t}$ $\displaystyle=$
$\displaystyle\frac{(m_{\tilde{t}_{L}}^{2}-m_{\tilde{t}_{R}}^{2})}{2}+M_{Z}^{2}\cos
2\beta(\frac{1}{4}-\frac{2}{3}x_{w})~{},$ $\displaystyle F(m^{2})$
$\displaystyle=$ $\displaystyle
m^{2}\left(\log\frac{m^{2}}{m_{\tilde{t}_{1}}m_{\tilde{t}_{2}}}-1\right)~{}.$
(21)

Anyway, we admit that SUSY has a little fine-tuning. Note that a solution has
been developed to solve such a little fine-tuning problem [125], in which a
framework of supersoft top-squarks is proposed to soften the logarithms to
screen the UV-sensitive logs.

Another unpleasant point caused by top-squarks above TeV is that the MSSM can
no longer realize the first order electroweak phase transition. However, the
NMSSM can do this job (see [127] and refs therein).

Figure 12: The parameter space of split-SUSY satisfying dark matter relic
density ($2\sigma$ range of Planck data) and the $125\pm 2$ GeV Higgs mass,
taken from [133]. The green (red) samples can (cannot) achieve the gauge
coupling unification at GUT scale.

Note that in this review we focused on low energy SUSY which helps to solve
the naturalness problem in particle physics. It is clear that even with low
energy SUSY our particle theory still has a little tuning, which means we
cannot have perfect naturalness. If we give up naturalness and believe
simplicity, we may have the fascinating split-SUSY [128, 129, 130, 131], which
is emphasized in the recent Witten Reflects [132]. Split-SUSY gives up
naturalness and retains the original motivation to explain the dark matter and
realize successful gauge coupling unification. Actually, as shown in Fig.12,
the dark matter relic density and direct detection limits as well as the gauge
coupling unification impose stringent constraints on the parameter space of
split-SUSY, i.e., only the higgsino-like dark matter around 1.2 TeV can
survive the XENON1T limits assuming the universal GUT input for gaugino masses
(which leads to $M_{1}:M_{2}:M_{3}\sim~{}1:~{}2:~{}6$ at the weak scale).
Since in split-SUSY the gauginos and higgsinos are not so heavy, they may be
accessible at future colliders [133, 134]. This is that the electroweak scale
is not natural in the customary sense, but additional particles and forces
that would help us understand what is going on exist at an energy not too much
above LHC energies [132].

##  7 The continual search of SUSY at HL-LHC

Despite the fact that no sparticles have been discovered so far, SUSY will be
actively probed at the High-Luminosity Large Hadron Collider (HL-LHC) as the
leading contender for new physics beyond the SM.

Figure 13:  The HL-LHC coverage of the MSSM parameter space for the
explanation of the muon $g-2$ at $2\sigma$ level, taken from [50].  Figure 14:
The $2\sigma$ parameter space of the CMSSM from a global fit, showing the
coverages of the current and future LHC runs, plotted from the analysis in
[58].

As shown above, the MSSM can allow for an explanation for the muon $g-2$ at
$2\sigma$ level while satisfying all other constraints. The light
electroweakinos and sleptons required for the muon $g-2$ explanation may be
accessible at the HL-LHC. For example, in the scenario where bino-like dark
matter annihilates with winos to give the correct dark matter abundance, the
light winos required by the muon $g-2$ may be fairly produced and detected
through the signal of soft leptons plus missing energy at the LHC, as shown in
Fig.13 [50]. The $2\sigma$ parameter space is shown in this figure was allowed
by the muon $g-2$, the dark matter relic density, and direct detection limits
as well as the LHC Run-2 data [50]. We see that the HL-LHC can cover a sizable
part of the MSSM parameter space for the explanation of muon $g-2$ at
$2\sigma$ level. In addition, the precision measurement of the Higgs
properties at a lepton collider as a Higgs factory could play a complementary
role via detecting the residual effects of SUSY in Higgs couplings [50, 126].

Since the muon $g-2$ may not be a real hint of new physics, we consider the
CMSSM without explaining the muon $g-2$. As shown in Fig.14, the HL-LHC will
continue to narrow down the CMSSM parameter space. The $2\sigma$ parameter
space shown in this figure was obtained from a likelihood analysis, taking
into account various data including the electroweak precision observables, the
$B$-physics measurements, the LHC Run-1 and Run-2 data of SUSY direct
searches, the Planck observation of the dark matter relic density and the
combined LUX Run-3 and Run-4 detection limits [58].

##  8 Conclusion

So we see that the low energy SUSY can generally explain the muon $g-2$ at
$2\sigma$ level and also satisfy other constraints like the 125 GeV Higgs
mass, the dark matter and the LHC searches. We conclude that low energy SUSY
is still a compelling candidate for new physics, though suffering from a
little fine-tuning. The future runs of the LHC will continue to explore SUSY
and could bring surprise at any moment.

###### Acknowledgments.

Thanks to Lei Wu for discussions and comments. This work was supported by the
National Natural Science Foundation of China (NNSFC) under grant Nos.
12105248, 12075213, 11821505, 12075300, 11775012 and 11805161, by the Key
Research Project of Henan Education Department for colleges and universities
under grant number 21A140025, by Peng-Huan-Wu Theoretical Physics Innovation
Center (12047503), by the CAS Center for Excellence in Particle Physics
(CCEPP), by the CAS Key Research Program of Frontier Sciences, and by a Key
R&D Program of Ministry of Science and Technology of China under number
2017YFA0402204, by the Key Research Program of the Chinese Academy of
Sciences, grant No. XDPB15, and by Korea Research Fellowship Program through
the National Research Foundation of Korea (NRF) funded by the Ministry of
Science and ICT (2019H1D3A1A01070937).

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