Testing electroweak SUSY for muon $g-2$ and dark matter at the LHC and beyond

Given that the LHC experiment has produced strong constraints on the colored supersymmetric particles (sparticles), testing the electroweak supersymmetry (EWSUSY) will be the next crucial task at the LHC. On the other hand, the light electroweakinos and sleptons in the EWSUSY can also contribute to the dark matter (DM) and low energy lepton observables. The precision measurements of them will provide the indirect evidence of SUSY. In this work, we confront the EWSUSY with the muon $g-2$ anomaly, the DM relic density, the direct detection limits and the latest LHC Run-2 data. We find that the sneutrino DM or the neutralino DM with sizable higgsino component has been excluded by the direct detections. Then two viable scenarios are pinned down: one has the light compressed bino and sleptons but heavy higgsinos, and the other has the light compressed bino, winos and sleptons. In the former case, the LSP and slepton masses have to be smaller than about 350 GeV. While in the latter case, the LSP and slepton masses have to be smaller than about 700 GeV and 800 GeV, respectively. From investigating the observability of these sparticles in both scenarios at future colliders, it turns out that the HE-LHC with a luminosity of 15 ab$^{-1}$ can exclude the whole BHL and most part of BWL scenarios at $2\sigma$ level. The precision measurement of the Higgs couplings at the lepton colliders could play a complementary role of probing the BWL scenario.


I. INTRODUCTION
In particle physics, the muon anomalous magnetic moment a µ is one of the most precisely measured quantities. Since the first results were reported, there has been a longstanding ∼ 3σ discrepancy between theory and experiment, which triggered numerous studies of new physics explanations. As a successor of the previous E821 experiment performed at BNL, the on-going muon g − 2 experiment E989 at Fermilab is to measure a µ with a relative precision of 140 parts-per-billion (ppb) [1]. This precision is a factor of four improvement from the current experiment [2]. If this anomaly still persists, it would be a clear evidence for new physics beyond the Standard Model (BSM).
Meanwhile, dark matter (DM) that constitutes the majority of matter in the universe has been established by astrophysical and cosmological observations. Understanding its nature and interactions is one of the most important quests of contemporary physics. The Weakly Interacting Massive Particle (WIMP) paradigm provides an attractive solution to the DM issue as it can naturally produce the measured relic density through the robust mechanism of thermal freeze-out. Therefore, various concrete realizations of WIMP models have been proposed, which has been being tested in DM (in)direct detections and collider experiments [3].
Among new physics models for solving these two problems, supersymmetry (SUSY) is one of the most popular candidates, which has a beautiful mathematical structure and is considered as a part of a larger vision of physics. In supersymmetric models, the lightest neutralinoχ 0 1 can serve as a natural WIMP DM candidate if the R-parity is conserved. Meanwhile, the muon g − 2 anomaly can be explained by the contributions of light sleptons and electroweakinos running in the loops [4][5][6][7]. In addition, SUSY can also solve the hierarchy problem and realize the unification of gauge couplings at the GUT scale. Due to its overwhelming virtues and popularity, the low energy SUSY has long been pursued by both theorists and experimentalists.
In this work we perform a comprehensive study of the phenomenology of the EWSUSY scenario for the muon g − 2 and dark matter. Note that in the literature [29][30][31][32][33][34][35][36] such a scenario has been discussed to some extent. However, those studies either did not require the SUSY dark matter to provide the correct abundance or focused on the phenomenology at the LHC. Unlike them, we pin down the viable parameter space of EWSUSY for explaining the dark matter abundance and the muon g − 2 anomaly by a numerical scan. We find that the masses of the electroweakinos and sleptons are bounded in certain ranges, which will guide the search strategies at colliders. In addition to the LHC observability of such a scenario, we will also explore its test at the LHC upgrades and the e + e − Higgs factory. As a precison test machine, the future e + e − Higgs factories, such as CEPC, FCC-ee and ILC, have limited energy to directly search for SUSY particles. However, they can test the low energy SUSY through the precision measurements of Higgs couplings.
The structure of this work is organized as follows. In Sec. II, we will recapitulate the studies of muon g − 2 and neutralino DM in the MSSM. In Sec. III, we perform a numerical scan over the parameter space of EWSUSY and discuss the implications for sparticles. In Sec. IV, we investigate the prospects of hunting for the electroweakinos and sleptons in those scenarios at the LHC and future colliders. Finally, we draw our conclusions in Sec. V.

II. NEUTRALINO DARK MATTER AND MUON g − 2 IN THE MSSM
In the MSSM there are four neutralinosχ 0 1,2,3,4 that are the mixtures of bino (B), wino (W 0 ) and neutral higgsinos (H 0 u,d ). The mass matrix is given by [37] where s β , c β , s W and c W stands respectively for sin β, cos β, sin θ W and cos θ W . M 1 and which can be diagonalized by two unitary 2 × 2 matrices U and V .
In the MSSM the lightest sparticle (LSP) can be the neutralinoχ 0 1 , which can play the role of dark matter. It is a mixture of bino, wino and higgsinos. Depending on its dominant component, the LSPχ 0 1 can be bino-like, higgsino-like or wino-like. Whenχ 0 1 is wino-like or higgsino-like, it usually has too large annihilation rates in the early universe to produce sufficient dark matter relic density. If we require them to provide the correct dark matter abundance without other non-SUSY dark matter components like axions, the masses of higgsino-like and wino-like dark matter have to be at TeV scale [38,39], which results in too heavy electroweakino spectrum to generate sizable contributions to muon g − 2. On the other hand, the wino-like or higgsino-like dark matter scattering with nucleon has a sizable cross section and thus subject to stringent limits from dark matter direct detection experiments. Besides, it should be noted that the sneutrino in our study can be dark matter as well, which, however, was excluded by the direct detection. Therefore, in our study, we will focus on the bino-like dark matter, which can give the observed relic density by mixing with higgsino/wino, resonantly annihilating through Z/Higgs bosons or coannihilating with other light sparticles. The first two mechanisms have been tightly constrained by current XENON1T and LHC experiments [40][41][42], while the coannihilation with light sparticles can still be consistent with current data [43].
The SUSY contributions to the muon g − 2 mainly come from the neutralino-smuon and chargino-sneutrino loops. The expressions of one-loop corrections to a µ are given by [44] where with i, m and k being the indices respectively for the neutralinos, smuons and charginos mass eigenstates. y µ = g 2 m µ / √ 2m W cos β being the muon Yukawa coupling. The loop functions F N 1,2 and F C 1,2 , depending on the variables x im = m 2 , which can be found in [44]. The unitary matrix X that diagonalizes the smuon mass matrix M 2 µ is given by, where in the {μ L ,μ R } basis. Assuming all sparticles have an universal mass M SUSY , the SUSY contributions to muon g − 2 can be approximated as [45] δa SUSY It can be seen that the SUSY contributions can be enhanced by a large tan β and suppressed by SUSY mass scale so that heavy SUSY will decouple from such a low energy observable.
To generate sizable contributions to the muon g − 2, the involved charginos and neutralinos as well as the sleptons cannot be too heavy. From Eq. 4, we can find that the contribution of chargino-sneutrino loop usually dominates over that of neutralino-slepton loop. But it should be mentioned that a sizable contribution to g − 2 anomaly can also be from the bino-smuon loop because of the large smuon left-right mixing induced by large µ [46].
Two-loop corrections to the muon g −2 from fermion/sfermion loops in the MSSM are calculated in [47,48]. These corrections are also significant and even logarithmically enhanced for heavy sfermions. For different masses of sparticles running in the loops, a few percent correction for squark masses in the few TeV region can be obtained. Such a non-decoupling behavior is because that the gaugino and higgs couplings can differ from the corresponding gauge and Yukawa couplings when heavy sfermions are integrated out.

III. PARAMETER SCAN FOR MUON g − 2 AND DARK MATTER
In conjuncture with the requirements of the dark matter relic density and LHC data, we perform our study in the EWSUSY framework, where only electroweakinos and sleptons are light and colored sparticles are heavy. Such a framework allows us to remain agnostic of the detailed UV-physics, yet still capture the features of models for muon g − 2 and dark matter in the MSSM. We will focus on two promising scenarios: one has bino, winos and sleptons (BWL), and the other has bino, higgsinos and sleptons (BHL). This will narrow down the parameter space of the MSSM and provide a guidance of hunting for electroweakinos and sleptons at the LHC and future colliders. The relevant parameters are scanned in the following ranges: Other SUSY parameters in both scenarios are taken as In our scan we consider the following experimental constraints: (1) We use SUSY-HIT [49] to calculate the mass spectrum and branching ratios of the particles. We require the Higgs boson h to be SM-like and in the range of 122 < m H < 128 GeV.
(2) We impose the constraint of meta-stability of the vacuum state by demanding |A t | < ∼ (3) The sleptons must be above 100 GeV, as required by the LEP2 constraints.
(5) We require the SUSY contribution to explain the current value of muon g − 2 data within the 2σ range. In Fig. 1, we present the contributions of sparticles to the muon g −2 for samples survived the constraints (1)-(4) on the plane of δa SUSY µ versus mχ0 1 and m˜ . The red shaded areas are the 2σ ranges of explaining current muon g − 2 anomaly. We find that the dark matter abundance in BWL scenario is achieved mainly through the co-annihilation of the bino-likẽ χ 0 1 and wino-likeχ 0 2 andχ ± 1 . The slepton coannihilation also contributes to the relic density in the relatively heavy mass range. While in the BHL scenario, the correct dark matter abundance is obtained through the co-annihilation of the LSP with sleptons.
In order to interpret the muon g − 2 deviation, we can see that the masses ofχ 0 1 and˜ have to be lighter than about 700 GeV and 800 GeV in BWL scenario, respectively. But in the BHL scenario, the masses ofχ 0 1 and˜ have to be less than about 350 GeV. On the other hand, we find that the smuon with a mass less than about 200 GeV have been excluded in BWL scenario because of the over-enhancement of g −2. On the other hand, a lighter smuon can exist in BHL scenario. This is because that the right-handed smuon in BHL scenario will lead to a negative contribution to g − 2 so that a lighter smuon is needed to compensate for such a suppression.
By assuming the expected central value same as the current result of g − 2, we also show the projected 2σ sensitivity of the E989 experiment at Fermilab that are the regions between the dotted lines. It will further constrain the viable mass ranges of sparticles. To be specific, χ 0 1 and˜ have to be lighter than about 500 GeV and 600 GeV in BWL scenario, respectively, while in the BHL scenario, the masses ofχ 0 1 and˜ have to be less than about 200 GeV. If these turn out to be true, several popular high scale SUSY models, such as the CMSSM, mSUGRA, GMSB and AMSB, have to be extended because their sfermion spectrum that needs to explain the 125 GeV Higgs mass is too heavy to accommodate muon g − 2.
In Fig. 2, we plot the spin-independent and spin-dependent LSP-nucleon scattering cross sections of the samples survived the constraints (1)- (5). Since in BWL scenario the higgsinos are rather heavy and the LSPχ 0 1 is extremely bino-like, it scatters with nucleon very weakly and thus the SI and SD LSP-nucleon scattering cross sections are very small, which can be much below the LZ-projected sensitivities. Those samples may be probed at colliders [53,54].
On the other hand, the LSPχ 0 1 in BHW scenario can have certain higgsino component so that it can scatter with the nucleons sizably and are tightly constrained by current direct detection limits.

IV. OBSERVABILITIES AT LHC UPGRADES AND HIGGS FACTORY
In Fig. 3, we display the samples survived the constraints (1)-(5) and the dark matter direct detection. The BWL scenario is shown in the upper panel of Fig. 3, whereχ 0 1 is bino-like andχ 0 2 is wino-like. For most samples the mass difference between the wino-likeχ 0 2 and the bino-likeχ 0 1 is rather small, while the smuon mass can be quite near or significantly heavier than the mass ofχ 0 1 . The BHL scenario is shown in the lower panel of Fig. 3, which has a light spectrum of bino and sleptons but with heavy higgsinos. In this scenarioχ 0 1 is also rather bino-like, albeit with small higgsino admixture, while theχ 0 2 is higgsino-like. The mass difference between the LSP and sleptons is quite small as well.
We also present the latest exclusion limits from the null results of searching for slepton   larger than that of slepton pair, we perform a detailed Monte Carlo simulation of the process bino-wino in BWL scenario. While since the sleptons are much lighter than the higgsinos in the BHL, we will analyze the process pp → j˜ (→ −χ0 1 )˜ * (→ +χ0 1 ) → j + + − + / E T for the compressed bino-slepton in the BHL scenario. The schematic diagram of those two process are shown in Fig. 4. So in both signal processes, there are a pair of soft opposite-sign sameflavor leptons plus jets plus large missing transverse energy. We will utilize these features to enhance the sensitivity of our signals. The main SM backgrounds come from the Drell-Yan processes, dibosons and the leptonic tt events. We generate parton-level events by using MadGraph5_aMC@NLO [60] and then the events are passed to Pythia [61] for showering and hadronization. The detector effects are simulated by Delphes [62]. We perform the analysis of events in the framework of CheckMATE2 [63], and evaluate the significance by where β stands for the expected systematic uncertainty. It has to be revisited with the real performance of the upgraded LHC detectors. As a theoretical estimation, we take β = 10% in our calculations.
In Fig. 5, we show the normalized distributions of the missing transverse energy / E T and the dilepton invariant mass m of the signal and background events. We find that both signals have more events in the range of the large / E T , which can highly suppress the Drell-Yan and tt backgrounds. In additional, due to two soft leptons decaying from the sleptons, both signals predict a small value of m . According to the kinematical features, we impose the following event selection criteria: • We require the missing transverse energy / E T > 200 GeV.
• Two opposite-sign same-flavor (OSSF) leptons are required. The leading and subleading leptons should have the transverse momentum p T ( 1 ) > 5 GeV and p T ( 2 ) > 4 GeV.
• We require at least one jet and the leading jet p T (j 1 ) > 100 GeV. The angular separations have to be ∆φ(j 1 , P miss T ) > 2 and ∆φ(j, P miss T ) > 0.4. Also we veto b-jets to reduce tt background.
• We require the dilepton invariant mass 1 GeV < m < 60 GeV and m / ∈ (3, 3.2) GeV to suppress contributions from J/ψ decays and on-shell Z boson decays.
• The scalar sum of the lepton transverse momenta H lep for slepton pair, where the stransverse mass is defined in [64].  In Tables I and II, we demonstrate the cut flows for the benchmark points in two scenarios.
We can see that the soft OSSF leptons cut will significantly reduce all backgrounds, in  GeV < m < 60 GeV can further suppress tt and diboson backgrounds by about one order.
As pointed in [59], the observable / E T /H lep T is useful for reducing the tt background. In Fig. 6, we display the significances of the processes pp → jχ 0 2χ ± 1 and pp → j˜ ˜ * at the HL-LHC and HE-LHC. It can be seen that a portion of the samples in both scenarios will be excluded by the search for soft lepton pair plus missing energy events at the HL-LHC. The future HE-LHC is able to further exclude the whole parameter space of BHL and most part of BWL scenarios for satisfying muon g − 2 and DM experimental results within 2σ level.
We also checked that the 100 TeV proton-proton collider SPPC with the same luminosty cannot do much better than the HE-LHC due to the enhanced backgrounds. On the other hand, it should be mentioned that the heavy higgsinos decaying to light bino in the BHL scenario will provide 3 + / E T signature at a 100 TeV hadron collider, which can exclude the higgsino mass up to about 3 TeV at 95% C.L.. Besides conventional cut-based analysis, the machine learning methods have been recently proposed to enhance the sensitivity in the search of sparticles at the LHC [65][66][67][68][69]. We expect that our result may be improved by using those advanced analysis approaches.
Since the LHC experiment has been continuously pushing up the new physics scale, the future e + e − Higgs factory, either CEPC, FCC-ee or ILC, has limited energy to directly produce new particles. However, due to its clean environment, such a Higgs factory is a precision test machine and can measure the Higgs couplings at one percent level or better, which may reveal the new physics effects through the Higgs couplings (see examples, [74][75][76][77][78][79]). As shown in the above section, the electroweakinos and sleptons cannot be too heavy in order to explain the muon g − 2 and provide the correct dark matter abundance. These light uncolored SUSY particles may cause some indirect effects in the Higgs couplings. Among the Higgs couplings, the hbb and hτ + τ − couplings can still deviate from the SM predictions sizably [80]. In the following we demonstrate the hbb coupling as an illustration.
In Fig. 7, we display the hbb coupling for the samples in Fig. 6   at the HE-LHC. At tree level the hbb coupling is given by g(m b /2m W )(sin α/cos β) and the one-loop corrections are presented in [81]. In our calculations we use the package FeynHiggs-2.11.3 [82] which includes the one-loop effects and also various two-loop contributions. The sensitivities of the HL-LHC (14 TeV, 3 ab −1 ), ILC (250 GeV, 2 ab −1 ), FCC-ee (240 GeV, 5 ab −1 ) and CEPC (240 GeV, 5 ab −1 ) to the Higgs couplings are also shown. We can see that in the BWL scenario the hbb coupling can still be enhanced by about two percent, which is below the HL-LHC sensitivity but can be readily covered by the Higgs factory ILC, FCC-ee, or CEPC. Therefore, the precision measurement of the Higgs couplings could play a complementary role of probing such a scenario at future high energy lepton collider.

V. CONCLUSIONS
Since the colored sparticles have been excluded up to TeV scale, searching for the electroweak supersymmetry will be one of the major tasks in future experiments. Besides the LHC, the on-going muon g − 2 and dark matter experiments provide another good place to hunt for electroweakinos and sleptons in EWSUSY. In this work, we examined the parameter space of EWSUSY under the constraints of the muon g − 2 anomaly, the DM relic density, the DM direct detections and the LHC data. By analyzing the survived samples, we obtained the following observations: (1) There are two viable scenarios for explaining the muon g − 2 anomaly. One has the light compressed bino, winos and sleptons (BWL), and the other has light compressed bino and sleptons but heavy higgsinos (BHL). In the BHL scenario, the masses ofχ 0 1 and˜ have to be smaller than about 350 GeV. In the BWL scenario, the masses ofχ 0 1 and˜ have to be smaller than about 700 GeV and 800 GeV, respectively. If this anomaly persists in the on-going E989 experiment, the allowed parameter space will be further narrowed. (2) In both scenarios, the dark matter has to be the binolike neutralino and the dominant annihilation mechanism to achieve the correct dark matter abundance is through the bino-wino or bino-slepton coannihilation. Also, we found that the sneutrino DM or the neutralino DM with sizable higgsino component has been excluded by direct detections, due to the large scattering cross section of dark matter and nucleus. (3) The BWL scenario has been tightly constrained by the latest LHC Run-2 results of searches for soft + − + / E T events from slepton pair, which implies a compressed spectrum of bino, winos and sleptons. In contrast, the BHL scenario can escape the current LHC limits. We explored the observability of these sparticles in both scenarios at future colliders. We found that the HE-LHC with the luminosity L = 15 ab −1 can exclude the whole BHL scenario and most part of BWL scenarios at 2σ level. The rest of samples that alter the Higgs coupling by two percent level may be excluded by the precision measurement of the Higgs couplings at a future Higgs factory.