Supersymmetric Interpretation of the Muon $g-2$ Anomaly

The Fermilab Muon $g-2$ collaboration recently announced the first result of measurement of the muon anomalous magnetic moment ($g-2$), which confirmed the previous result at the Brookhaven National Laboratory and thus the discrepancy with its Standard Model prediction. We revisit low-scale supersymmetric models that are naturally capable to solve the muon $g-2$ anomaly, focusing on two distinct scenarios: chargino-contribution dominated and pure-bino-contribution dominated scenarios. It is shown that the slepton pair-production searches have excluded broad parameter spaces for both two scenarios, but they are not closed yet. For the chargino-dominated scenario, the models with $m_{\tilde{\mu}_{\rm L}}\gtrsim m_{\tilde{\chi}^{\pm}_1}$ are still widely allowed. For the bino-dominated scenario, we find that, although slightly non-trivial, the region with low $\tan \beta$ with heavy higgsinos is preferred. In the case of universal slepton masses, the low mass regions with $m_{\tilde{\mu}}\lesssim 230$ GeV can explain the $g-2$ anomaly while satisfying the LHC constraints. Furthermore, we checked that the stau-bino coannihilation works properly to realize the bino thermal relic dark matter. We also investigate heavy staus case for the bino-dominated scenario, where the parameter region that can explain the muon $g-2$ anomaly is stretched to $m_{\tilde{\mu}}\lesssim 1.3$ TeV.


Introduction
The success of the Standard Model (SM) has been confirmed by the Higgs boson discovery, the Higgs coupling measurements, recent results from the Large Hadron Collider (LHC), and a number of low-energy precision measurements in the quark and lepton flavor physics. However, physics beyond the SM (BSM) is definitely required, for example, in order to explain the dark matter.
The effort on experimental reconfirmation of this anomaly has been made by the Fermilab Muon g − 2 collaboration [15] and the J-PARC muon g − 2/EDM collaboration #1 Recent development of the lattice calculation including QED and isospin-breaking corrections [2] has revealed that the leading-order hadronic vacuum polarization contributions disagree with those calculated by the data-driven approach at 2.3σ level [1,[3][4][5][6] and the resultant ∆aµ is zero-consistent within 2σ level. In order to explain the 2.3σ discrepancy while saving the global fit of the electroweak sector, the cross section σ(e + e − → hadrons) below 1 GeV region has to be changed [7][8][9]. A dedicated study for such the energy region is given in Ref. [10]. #2 This value is calculated with the latest value of the muon-to-proton magnetic ratio [1,14]. [16,17]. These on-going experiments aim to reduce the experimental error at least by a factor of four compared to a BNL µ . Very recently, the Fermilab Muon g − 2 collaboration presented the first result on the measurement [18,19], a FNAL; 2021Apr.
Now the discrepancy between the experimental and theoretical values amounts to ∆a µ ≡ a BNL+FNAL µ − a SM µ = (25.1 ± 5.9) × 10 −10 , whose significance is equivalent to 4.2σ level, and the muon g−2 anomaly is reconfirmed. #3 This discrepancy is as large as the SM electroweak contribution to the muon g − 2, a µ (EW) = (15.4±0.1)×10 −10 [1], which implies that BSM physics around the electroweak scale may be responsible for it. Low-energy supersymmetry (SUSY) is one of such solutions. Its contribution to the muon g − 2, which we denote by a SUSY where M 1 (M 2 ) is the bino (wino) soft mass, µ is the higgsino mass parameter, tan β = v u /v d is the ratio of the vacuum expectation values of the up-and down-type Higgs, and m µ L/R and m νµ are the masses of the left/right-handed smuon and the muon sneutrino, respectively. The loop functions are given by which satisfy 0 ≤ f C/N (x, y) ≤ 1, f C (1, 1) = 1/2 and f N (1, 1) = 1/6. An important fact is that the first three contributions, a WHL µ , a BHL µ , and a BHR µ , are enhanced by tan β but suppressed as µ increases. On the other hand, the last one, a BLR µ , is enhanced by µ tan β. This difference provides us with two completely distinct scenarios.
In this paper, we revisit two branches of the SUSY scenarios proposed to explain the muon g − 2 anomaly: #5 one in which the chargino contribution, a WHL µ , is dominant, and the other in which the pure-bino contribution, a BLR µ , is dominant. #6 This work focuses on the minimal setups, where only three or four SUSY multiplets are light to have sizable a WHL µ or a BLR µ , while the other irrelevant particles are decoupled. Although this approach lacks specific SUSY-breaking models and does not cover the whole possible Minimal Supersymmetric Standard Model (MSSM) models, it allows us to clarify the relevant MSSM parameters and tell which LHC (and other) constraints are relevant, compared to studies with scans over the whole parameter space. #7 This paper is organized as follows. In Sec. 2, we revisit the parameter regions in which a SUSY µ ≈ a WHL µ , based on our previous study [48]. Compared with Ref. [48], the new a µ measurements [19], the new theory combination [1], and new results from the LHC [49,50] are implemented. In Sec. 3, we revisit the bino-contribution to the muon g − 2, a BLR µ , taking into account of the latest LHC constraints as well as the vacuum meta-stability. A related analysis was performed in our previous study [51]. Interestingly, a wider region of the parameter space is allowed for smaller tan β, rather than the large tan β case. We study the cases with and without flavor universal slepton masses. We also discuss an implication for the dark matter abundance with the bino-slepton coannihilation. Section 4 is devoted to conclusions and discussion.

Chargino contributions
In this section, we consider the parameter regions where the chargino contribution to the muon g − 2 is dominant, based on our previous study [48]. #5 For recent studies of the muon g − 2 motivated SUSY models based on the LHC Run 2 results, see, e.g., Refs. . #6 The other contributions, a BHL µ and a BHR µ , can also be dominant. See, e.g., Ref. [27]. #7 This strategy has been taken in, e.g., the following works: Refs. [28,29,48] with particular focus on WHL, Refs. [27,41] on BHL and BHR, and Refs. [41,45,47] considering the combination of the contributions.

Setup and Result
The setup in this section is as follows: • Among SUSY particles, neutralinos χ 0 i , charginos χ ± j , and left-handed sleptons l L , ν are within the LHC reach, i.e., with masses of 1 TeV. Here, l denotes e, µ, and τ .
• The right-handed sleptons l R are heavy, so that a BHR µ and a BLR µ are suppressed and our analysis is simplified. The scalar trilinear terms (A e ) ij are neglected for simplicity.
• The soft masses of the left-handed sleptons are flavor universal and diagonal.
• All the colored SUSY particles (gluino and squarks) are decoupled. This assumption makes the LHC constraints more conservative, while their contribution to the muon g − 2 is negligibly small even if they are light because it arises at the two-loop level.
In addition, heavy colored SUSY particles are motivated by the mass of the SM-like Higgs boson as well as by LHC constraints.
• The heavy Higgs bosons, whose contributions to a SUSY µ are also negligible, are decoupled as well.
Then, the following five model parameters are left relevant: where m L stands for the universal soft mass for the left-handed sleptons, and as in Ref. [48], the following four subspace of the parameters are analyzed: #8 The analyses is done in the same manner as in Ref. [48]. In particular, SDECAY 1.5a [53,54] and GM2Calc 1.5.0 [55] are utilized to calculate the decay rates and a SUSY µ , respectively, and the right-handed slepton mass parameter m R is taken to be m R = 3 TeV.
The results are shown in Fig. 1 under the axes being the physical masses of the lighter chargino χ ± 1 and the left-handed smuon µ L . The SUSY contribution to the muon g − 2 is shown by the black solid contours in terms of a SUSY µ × 10 10 . The parameter spaces where a SUSY µ solves the discrepancy ∆a µ = (25.1 ± 5.9) × 10 −10 at the 1σ (2σ) level are shown by the orange-filled (yellow-filled) regions. The LHC Run 2 constraints are shown #8 We here require that the LSP is the bino-like neutralino with ignorance of the dark matter overabundance. The relic density may be reduced to be consistent with the observed one by an entropy production or considering more generic parameter space of the SUSY models (cf. [47]). Meanwhile, Ref. [52] focuses on a similar scenario in which a WHL µ explains the anomaly with M1 > M2, µ, i.e., a wino-higgsino mixed neutralino as the LSP. . The gray-filled region, where the LSP is ν, and the red-hatched region in (A), which corresponds to a compressed spectrum, are not studied. The red-filled and blue-filled regions are excluded by the LHC experiment [50,56,57]. We also analyzed the results of Refs. [58,59] but only on the model points with x = 0.05, 0.5, and 0.95 [see Eq. (22)]; the excluded ranges are shown by the magenta bars. Detailed description of the LHC constraints is provided in our previous work [48]. by blue-filled regions, red-filled regions, and magenta lines at the 95% confidence level; they are described in the next subsection. We find that, as far as considering the parameter space in Fig. 1, models with m µ L < m χ ± 1 are strongly disfavored as a solution to the muon g − 2 anomaly. Meanwhile, for models with m µ L > m χ ± 1 , LHC constraints are not critical yet; those models may explain the muon g − 2 anomaly and are to be searched for at the future LHC runs. In Table 1, we show the mass spectra and the SUSY contribution to the muon g − 2 (a SUSY µ ) for some viable benchmark points in Fig. 3. The production cross sections and the branching fractions of the SUSY particles for those points can be found in Ref. [48].

LHC constraints
In the present setup, direct pair productions of neutralinos, charginos, and sleptons are the targets at LHC searches. We consider the following channels which have been studied by the ATLAS and CMS collaborations: where = e, µ, L = e L , µ L , l = e, µ, τ , l L = e L , µ L , τ L , and ν = ν e,L , ν µ,L , ν τ,L . For the details of the analysis procedure, we refer the readers to our previous study [48].
In addition, we found that the newly appeared result, #9 • Ref. [50] by CMS collaboration (SLSL, 139 fb −1 ) is also responsible for the exclusion of our parameter spaces; it excludes the blue-filled region with labels "SLSL/C" in Fig. 1. The "SLSL/A" blue-filled region has been excluded by the ATLAS counterpart. The red-filled regions are excluded by the CC/WW and NC/HW channels. For the NC/3L channel, both of the ATLAS and CMS collaborations assume specific mass spectra of electroweakinos and sleptons, which determine the lepton energies. In terms of a mass difference ratio, the CMS collaboration considers three different mass spectra, x = (0.05, 0.5, 0.95), while the ATLAS studies x = 0.5. In Fig. 1, the corresponding model points are displayed by the dashed black lines, and the magenta bars show the regions excluded by the NC/3L channel. Although we have not investigated the NC/3L bounds for arbitrary x value, it is expected that the bounds on x = 0.05, 0.5, 0.95 are continuously connected with a peak around x = 0.5 (cf. Ref. [60]). Several preliminary results on conference papers [61][62][63] are not included because numerical data have not been made public. According to our estimation, they provide #9 We also examined the NC/HW analysis in Ref. [49] and the NC/ZW analysis in Ref. [50] based on the full Run 2 data, but no constraints are obtained on our parameter space. Other fifteen publications from the ATLAS and CMS collaborations are also taken into account; see [48] for the full list of references.
additional constraints based on the NC/3L and NC/HW signatures. The new NC/3L result [62] will confirm that our parameter spaces with m µ L < m χ ± 1 are strongly disfavored, while the new NC/HW result [63] will exclude more region with m µ L > m χ ± 1 , where the impact depends on the LSP mass.

Bino contributions
In the previous section, the wino was assumed to be light to enhance the chargino contributions to the muon g − 2. However, such an assumption is not always necessary for the SUSY contributions to be sizable. In this section, we consider another scenario in which the neutralino contribution is dominant. In particular, as shown in Eq. (9), the bino contribution a BLR µ becomes large when µ tan β is large and the bino-like neutralino, the left-and right-handed sleptons are light.

Setup
Let us summarize our setup in this section.
• The bino-like neutralino χ 0 1 , left-handed sleptons l L , ν and right-handed sleptons l R are assumed to be light. In addition, the higgsinos may be relatively light when tan β is large.
• The wino is assumed to be decoupled. Then, the chargino contributions a WHL µ discussed in Sec. 2 are suppressed even when µ is relatively small.
• As in the previous section, all the colored SUSY particles and the heavy Higgs bosons are decoupled.
• We assume the scalar trilinear terms (A e ) ij to be zero for simplicity. #10 Then, the following five model parameters are left relevant: where m L and m R represent the soft masses for the left-and right-handed sleptons, respectively, with the flavor index i = e, µ, τ . More concretely, we consider the following two subspace of the parameters: • Universal slepton mass: Although a BLR µ is independent of the stau mass, some of the constraints discussed below depend on it. It will be shown that much wider parameter space is allowed for the heavy stau scenario.
In addition to the one-loop contribution a BLR µ , there are non-negligible two-loop corrections. Among them, QED corrections [64] and those to the Yukawa couplings [65,66] are taken into account in this section. The latter corrections are also included in calculating the slepton mass spectra and mixing matrices. In order to treat them consistently when the corrections are large, we do not use GM2Calc in the following analysis. Meanwhile, we neglect radiative corrections to the neutralino couplings, which are not decoupled by heavy SUSY particles (see Refs. [51,[67][68][69][70][71][72]). Although the Bino coupling is equal to the gauge coupling at the tree level, this equality is violated by the SUSY-breaking effects, and the deviations are enlarged when the soft SUSY-breaking scale is large. According to the renormalization group approach [51], they can amount to 5-10% corrections to a SUSY µ when the soft masses, especially the wino mass, are as large as 10-100 TeV. Other higher-order contributions are not-yet-estimated or expected to be smaller.

Constraints
When the bino and sleptons are light, there are several phenomena which are potentially correlated with the muon g − 2. We consider the following four constraints: • LHC (and LEP) searches for the SUSY particles, • Higgs coupling measurements, • neutralino LSP as a candidate of the dark matter, • stability of the electroweak vacuum.
Let us discuss them in turn. First, similarly to the previous section, the LHC experiment has studied signatures of the direct slepton productions, From this SLSL channel, where events with energetic two electrons or muons are studied, lower bounds on the lightest neutralino mass are obtained as a function of the slepton mass by the ATLAS collaboration (8 TeV, 20.3 fb −1 ) [73], (13 TeV, 139 fb −1 ) [56], and the CMS collaboration (13 TeV, 139 fb −1 ) [50]. Furthermore, the ATLAS collaboration has studied the compressed mass spectrum of the neutralino and sleptons. From the SLSL-soft channel, where events with low-transverse momentum electrons or muons are studied, bounds on the slepton masses are obtained for the mass difference 30 GeV (13 TeV, 139 fb −1 ) [74]. Furthermore, we impose the LEP bound on the stau mass, m τ 1 > 95.7 GeV [75].
In addition, events with two energetic hadronically-decaying taus have been studied to search for the direct stau pair productions. Although we investigated the latest results by the ATLAS collaboration (13 TeV, 139 fb −1 ) [76] and the CMS collaboration (13 TeV, 77.2 fb −1 ) [77], their constraints were found to be weaker than the above. Also, the smuon/selectron masses are limited by the LEP results [75,78]. The constraints are weaker than 100 GeV and out of the range of our plots.
The decays of the SM-like Higgs boson have been measured well at the LHC experiment. Among the various decay channels, the staus may affect the branching ratio of h → γγ especially when µ tan β is large and the staus are light [79][80][81][82]. #11 Currently, the #11 Effects of smuons or selectrons are much weaker because the Yukawa couplings are tiny.
In the analysis, we require that the lightest neutralino is the LSP. #14 Although it is one of the best-known candidates of the dark matter, if it is almost composed of the bino, its thermal relic abundance easily exceeds the measured value [83,85], In the present setup, the neutralino relic abundance can be consistent with this result by the slepton coannihilation, i.e., when the slepton mass is close to the lightest neutralino appropriately. We estimate the relic abundance of the LSP neutralino by using the public package micrOMEGAs 5.2.7.a [86][87][88][89]. Such a dark matter may be detected by the direct detection. The LUX [90], PandaX-II [91] and XENON1T [92] experiments have provided stringent constraints on the spinindependent cross section of the dark matter scattering off the nuclei. In the analysis, the cross section is estimated by using micrOMEGAs.
The last constraint is the vacuum meta-stability condition. The trilinear coupling of the sleptons and the SM-like Higgs boson is given by where v 174 GeV is the vacuum expectation value of the SM-like Higgs, and ∆ represents the radiative corrections to the lepton Yukawa couplings mentioned above [66]. As µ tan β increases, the trilinear coupling is enhanced, charge-breaking minima become deeper, and eventually, the stability of the electroweak vacuum is spoiled. We require the lifetime of the vacuum being longer than the age of the Universe, restricting |µ tan β| from #12 To be exact, the production cross section and the total width of the Higgs boson may also be modified by the SUSY particles. However, the staus barely affect it, and we ignore such contributions. #13 The CMS collaboration recently reported a preliminary result, µγγ = 1.12 ± 0.09, using the LHC Run 2 full data. Although it has not been accounted in the above PDG average, the result is consistent with Eq. (25), and the following results are almost unchanged. #14 The lightest charged slepton is likely to be lighter than sneutrinos when µ tan β is large enough.
above. For the following analysis, we adopt the fitting formula explored in Refs. [51,93], with = e, µ, τ . The numerical evaluation was done by the bounce method [94] via CosmoTransitions 1.0.2 [95]. #15 A factor η ∼ 1 provides a mild tan β dependence coming from the Yukawa contributions to the quartic terms in the scalar potential. The explicit value of η is found in Refs. [51,93].

Result in universal slepton mass case
Let us first consider the slepton universal mass case. The left-handed (also right-handed) selectron, smuon and stau have a common soft SUSY-breaking mass, Such a spectrum is motivated to avoid dangerous lepton flavor or CP violations (see Ref. [51]). The SUSY contribution to the muon g − 2 cannot be arbitrarily large because µ tan β is bounded from above; due to the stau left-right mixing, a too large µ tan β can violate one (or more) of the following constraints, (i) the vacuum stability in the stau-Higgs potential, Eq.  100 GeV, the LEP bound on the stau mass also affects the boundary. Meanwhile, without the LHC constraints, there would be #15 Note that we ignore thermal effects and radiative corrections to the formula, which may affect the following result [96,97] and will be discussed elsewhere.  [50,56,73,74].
lower boundaries (i.e., lower bounds on m χ 0 1 , or equivalently upper bounds on m µ 1 ) due to the vacuum meta-stability condition, though they are hidden by the LHC constraints from the SLSL channel (lower blue-filled regions). Here, the region with smaller m µ 1 is constrained by the ATLAS 8 TeV result, while the larger one is by the ATLAS 13 TeV result. Besides, the SLSL-soft channel restricts the regions of degenerate neutralino-smuon masses (upper blue-filled region).
In Fig. 2 (A), (B), which correspond to tan β = 5, 10 respectively, µ 1 is required to be m µ 1 230 (270) GeV to explain the muon g − 2 discrepancy at the 1σ (2σ) level. It is noticed that the regions become narrower as tan β increases (see Fig. 2 (C), (D) for tan β = 30, 50). The reason is as follows. At each set of (M 1 , m L = m R ), the upper bound on µ tan β is almost unchanged. Then, since larger tan β leads to smaller µ, the BHR contribution to a SUSY µ , which destructively interferes with a BLR µ , becomes enhanced when µ is smaller, i.e., tan β is large. Note that the BHL contribution is weaker than that of BHR for m L ∼ m R [see Eqs. (7) and (8) The branching ratio of h → γγ is affected by light staus with large µ tan β. The signal strength is likely to be enhanced in the vicinity of the boundary of the SLSL constraint, while it is weaker for larger m χ 0 1 as the upper bound on µ tan β is tighter. #17 Around the muon g − 2 regions in the figures, it can be deviated from the SM value µ γγ by 2-5% at most, which satisfies the current limit (25) and is accessible in future by HL-LHC, ILC, or FCC-ee/eh/hh.
In Fig. 2, the thermal relic abundance of χ 0 1 is not tuned to reproduce the dark matter value (26). After imposing the LHC constraints, we found that the abundances in the muon g − 2 region satisfy Ω χ 0 1 < Ω DM because the stau masses are well close to that of χ 0 1 . Let us next consider the case where µ tan β is determined by requiring that the thermal relic abundance of the neutralino is equal to Eq. (26). It is almost determined by the coannihilations with the lightest stau. #18 In Fig. 3, the muon g − 2 discrepancy is explained at the 1σ (2σ) level in the orange-filled (yellow-filled) regions while satisfying Ω χ 0 1 h 2 = 0.12. Here, tan β = 5 (A) and 10 (B) with m L = m R . It is found that µ 1 is required to be m µ 1 220 (270) GeV to explain the muon g − 2 discrepancy at the 1σ (2σ) level.
In the figures, the left boundaries of the orange-filled (yellow-filled) regions for small m χ 0 1 are determined by the XENON1T constraint of the dark matter direct detection [92]. Here, µ cannot be so large, and thus, the dark matter-nucleon elastic scattering crosssection is enhanced. For smaller tan β, wider parameter regions are allowed because the coannihilation works efficiently by larger µ, i.e., without the well-tempered contributions. #17 Note that contributions to µγγ coming from light chargino loops may amount to additional few % at most. Such contributions, however, are scaled by ≈ 1/(µM2 tan β) [98,99]. Hence, they are certainly suppressed by M2 in the present setup. #18 There can be more than one values of the µ parameter that realize the desired dark matter density; in such a case, we take the largest µ as it gives the largest contribution to the muon g − 2. The bino-dominated SUSY scenario for the muon g − 2 anomaly with proper dark matter relic abundance. In the same way as Fig. 2, the universal slepton mass with m L = m R are used, but the µ parameter is tuned (µ DM ) to realize Ω DM h 2 = 0.12. We show cases of tan β = 5 and 10. In the orange-filled (yellow-filled) regions, the muon g − 2 anomaly can be solved at the 1σ (2σ) level while satisfying the XENON1T constraint.
In the near future, the sensitivity of the direct detection is planned to be improved by orders of magnitudes. We have checked that most of the muon g −2 parameter regions can be probed by the XENONnT experiment [100] if the slepton mass spectrum is universal.
We also found that the branching ratio of the SM-like Higgs boson decay to two photons, µ γγ , can deviate from the SM prediction by at most 2% in the muon g − 2 parameter region.
In Table 2, we show the mass spectra, the SUSY contribution to the muon g − 2 (a SUSY µ ), the thermal relic abundance of χ 0 1 (Ω χ 0 1 h 2 ), the spin-independent cross section of the neutralino for the dark matter direct detection (σ SI p ), and the branching ratio of h → γγ (µ γγ ) for some benchmark points in Fig. 3. The points BLR1 and BLR2 can be probed by the direct stau searches at the ILC 250 GeV, while all the four points are accessible by the ILC 500 GeV as well as the XENONnT experiment. Furthermore, all the points could be tested by measuring the branching ratio of the SM-like Higgs boson decay at FCC-ee/eh/hh. So far, we have assumed that the soft masses of the left-and right-handed sleptons have a common value, m L = m R . According to the loop function of a BLR µ , for a fixed value of µ 1 the SUSY contributions to the muon g − 2 is maximized when m L = m R , i.e., the muon g −2 regions in Fig. 2 [51]. Meanwhile, the stau coannihilation is realized by larger µ for m L = m R , which enhances a BLR µ simultaneously. Consequently, the muon g − 2 regions in Fig. 3 are enlarged for m L = m R , though we need detailed studies for the decay of the

Result in heavy stau case
Next, we discuss the case that the staus are heavier than selectrons and smuons, #19 Although the stau masses are irrelevant for the SUSY contribution to the muon g − 2, the vacuum meta-stability condition of the stau-Higgs potential as well as the condition m τ 1 > m χ 0 1 are relaxed as they increase. For m τ m with = e, µ, the vacuum meta-stability constraint from the smuon-Higgs potential becomes severer, and the latter condition is replaced by m µ 1 > m χ 0 1 .
#19 It is assumed that dangerous flavor or CP violations are suppressed by some mechanisms.  In the orange-filled (yellowfilled) region, the SUSY contribution can explain the muon g − 2 discrepancy at the 1σ (2σ) level. If µ tan β is maximized under the above conditions, a SUSY µ becomes too large compared with the observed discrepancy in most of the orange-filled region. Therefore, µ is understood to be optimized at each point of these figures. The upper boundaries of the regions (the upper bound on m χ 0 1 ) is determined by m µ 1 > m χ 0 1 , and the right boundaries (the upper bound on m µ 1 ) is given by the vacuum meta-stability condition of the stau-Higgs potential in (A) and the smuon-Higgs potential in (B). As a result, it is found that µ 1 is required to be m µ 1 390 (490) GeV to explain the muon g − 2 discrepancy at the 1σ (2σ) level for (m L,R ) τ = 1.5(m L,R ) . As the stau becomes heavier, these values become larger. In the limit of (m L,R ) τ (m L,R ) , one obtains m µ 1 1.3 (1.6) TeV. The blue-filled regions are excluded by the SLSL and SLSL-soft channels on the lighter smuon, and the magenta-filled region is excluded by h → γγ [Eq. (25)] in Fig. 4 (A). In addition, the heavier smuon is potentially excluded by the SLSL search. As the staus become heavier than the smuons, µ tan β is allowed to become larger, and thus, a mass splitting between the lighter and heavier smuons is enlarged especially when m L = m R . In the red-hatched region, the decay of the heavier smuon to the lightest neutralino is excluded if we suppose that BR( µ 2 → µ χ 0 1 ) = 1 and that µ tan β is maximized under the above conditions. #20 Although a smaller value is enough to explain the muon g − 2 discrepancy, detailed LHC analysis is necessary to check the viability of this region. In Fig. 4, the relic abundance of χ 0 1 is not tuned to reproduce the dark matter value (26), but can exceed it. In order to obtain Ω χ 0 1 = Ω DM the smuon is required to be degenerate with the LSP appropriately for the annihilation to work. We have checked that the dark matter abundance can be reproduced for m µ 1 − m χ 0 1 0.1m χ 0 1 while keeping the explanation to the muon g − 2 discrepancy, where the coannihilation works with the charged smuons, selectrons, and/or sneutrinos. We also found that the dark matternucleon elastic scattering cross-sections are much smaller than the XENONnT sensitivity in the heavy stau scenario.

Conclusions and Discussion
The first result of the measurement of the muon anomalous magnetic moment (g − 2) by the Fermilab Muon g − 2 collaboration confirmed the previous result at the Brookhaven National Laboratory and thus the long-standing discrepancy with the Standard Model prediction. In this paper, we revisited low-scale supersymmetric models as a solution to this anomaly, focusing on two distinct scenarios; one in which the chargino contribution, a WHL µ , is dominant, and the other in which the pure-bino contribution, a BLR µ , is dominant. In the chargino-contribution dominated scenario, we revisited our previous study [48], taking account of the latest LHC results as well as the new a µ measurements [19] and the new theory combination [1]. It was found that models with m µ L < m χ ± 1 in this scenario are disfavored as a solution to the muon g − 2 anomaly, while models with m µ L > m χ ± 1 are still widely allowed. Several benchmark points are listed in Table 1, for which a SUSY µ is sizable and the LHC constraints are still allowed.
In the pure-bino-contribution dominated scenario, we first studied the universal sleptonmass case where the µ parameter is maximized under the constraints of the vacuum stability, the neutralino being the LSP, and the LEP bound on the stau mass. Although the LHC slepton searches are tight, there remain viable parameter spaces for m χ 0 1 m µ 1 300 GeV: The muon g − 2 anomaly can be explained within 1σ (2σ) for m µ 1 230 (270) GeV. Interestingly, the region with low tan β with heavy higgsinos is preferred, because the destructive BHR contribution is suppressed.
Furthermore, the thermal relic of the bino-like neutralino can become the dominant component of the dark matter, if the stau-bino coannihilation works properly. It was shown that the relic abundance as well as the muon g − 2 anomaly can be explained simultaneously for m µ 1 220 (270) GeV. Besides, such parameter regions satisfy the XENON1T constraint on the dark matter scattering cross section. We list several benchmark points in Table 2, in which the SUSY contribution to the muon g − 2 is sizable, the neutralino relic abundance is consistent with the measured dark matter value, and all the constraints are satisfied. #20 In the universal slepton mass case, the smuon mass splitting is small enough.
We also investigated the setup that the staus are (much) heavier than the selectrons and smuons. Because the conditions of the vacuum meta-stability in the scalar potential and that the sleptons are heavier than the lightest neutralino are alleviated drastically, the parameter regions that explain the muon g − 2 anomaly are stretched. The muon g − 2 anomaly can be explained within 1σ (2σ) when m µ 1 390 (490) GeV for (m L,R ) τ = 1.5 (m L,R ) and when m µ 1 1.3 (1.6) TeV for (m L,R ) τ (m L,R ) . Similarly to the universal slepton mass case, the neutralino relic abundance can be consistent with that of the dark matter, if the smuon mass is within ∼ 10% of the neutralino mass. We checked that the masses can become as large as ∼ 1 TeV while keeping the sizable contribution to the muon g − 2 as well as the correct relic abundance of the dark matter. Note that the dark matter scattering cross sections are tiny and it is challenging for the dark matter direct detection.
Meanwhile, the ILC can play essential roles in testing the pure-bino-contribution dominated scenario. In the universal slepton mass case, wide parameter regions that explain the muon g − 2 discrepancy are accessible by the slepton searches if the collision energy is as large as 500 GeV. The benchmark points BLR1 and BLR2 in Table 2 can be probed even at ILC 250 GeV. Furthermore, if the SUSY particles responsible for the muon g − 2 are within the kinematical reach, it is possible to experimentally reconstruct the SUSY contribution to the muon g − 2 by utilizing the precise ILC measurements [109]. It is noticed that the muon g − 2 parameter space falls in the region in which sleptons and the LSP are degenerate. Such spectra could also be a good target of the future LHC Runs, e.g., by investigating photon collisions [110]. Furthermore, higgsinos tend to be light when the lightest neutralino is light and/or tan β is large. Then, their production can be probed by analyzing events with SM bosons or taus with large missing transverse momentum. Such signals will be studied elsewhere.
In the scenario where the thermal relic bino dark matter abundance is consistent with the observed one, such as the benchmark points in Table 2, most of the viable parameter space will also be probed by the XENONnT experiment. In addition, the muon g − 2 parameter regions can be tested by measuring the branching ratio of the SM-like Higgs boson decay at future HL-LHC, ILC and/or FCC-ee/eh/hh especially when the lightest neutralino as well as the staus are light.
The Fermilab Muon g − 2 collaboration has already completed Run-3 and the new results are anticipated with the full data set, which might shed further light on the SUSY scenarios.