The new"MUON G-2"Result and Supersymmetry

The electroweak (EW) sector of the Minimal Supersymmetric Standard Model (MSSM), with the lightest neutralino as Dark Matter (DM) candidate, can account for a variety of experimental data. This includes the DM content of the universe, DM direct detection limits, EW SUSY searches at the LHC and in particular the so far persistent $3-4\,\sigma$ discrepancy between the experimental result for the anomalous magnetic moment of the muon, $(g-2)_\mu$, and its Standard Model (SM) prediction. The recently published ``MUON G-2'' result is within $0.8\,\sigma$ in agreement with the older BNL result on $(g-2)_\mu$. The combination of the two results was given as $a_\mu^{\rm exp} = (11 659206.1 \pm 4.1c) \times 10^{-10}$, yielding a new deviation from the SM prediction of $\Delta a_\mu = (25.1 \pm 5.9) \times 10^{-10}$, corresponding to $4.2\,\sigma$. Using this improved bound we update the results presented in [1] and set new upper limits on the allowed parameters space of the EW sector of the MSSM. We find that with the new $(g-2)_\mu$ result the upper limits on the (next-to-) lightest SUSY particle are in the same ballpark as previously, yielding updated upper limits on these masses of $\sim 600$ GeV. In this way, a clear target is confirmed for future (HL-)LHC EW searches, as well as for future high-energy $e^+e^-$ colliders, such as the ILC or CLIC.


Introduction
Searches for Beyond the Standard Model (BSM) particles are performed directly, such as at the LHC, or indirectly in low-energy experiments and via astrophysical measurements. Among the BSM theories under consideration the Minimal Supersymmetric Standard Model (MSSM) [2][3][4][5] is one of the leading candidates. Supersymmetry (SUSY) predicts two scalar partners for all SM fermions as well as fermionic partners to all SM bosons. Contrary to the case of the SM, in the MSSM two Higgs doublets are required. This results in five physical Higgs bosons instead of the single Higgs boson in the SM. These are the light and heavy CP-even Higgs bosons, h and H, the CP-odd Higgs boson, A, and the charged Higgs bosons, H ± . The neutral SUSY partners of the (neutral) Higgs and electroweak (EW) gauge bosons gives rise to the four neutralinos,χ 0 1,2,3,4 . The corresponding charged SUSY partners are the charginos,χ ± 1,2 . The SUSY partners of the SM leptons and quarks are the scalar leptons and quarks (sleptons, squarks), respectively.
In Ref. [1] we performed an analysis taking into account all relevant data for the EW sector of the MSSM, assuming that the lightest SUSY particle (LSP) is given by the lightest neutralino,χ 0 1 , that makes up the full Dark Matter (DM) content of the universe [6,7]. 1 The expermental results comprised the direct searches at the LHC [9, 10], the DM relic abundance [11], the DM direct detection (DD) experiments [12][13][14] and in particular the (then current) deviation of the anomalous magnetic moment of the muon [15,16].
Recently the "MUON G-2" collaboration published the results (referred to as "FNAL" result) of their Run 1 data,which is within 0.8 σ in agreement with the older BNL result on (g − 2) µ . We combine the two results, assuming that they are uncorrelated. We analyze the impact of the combination of the Run 1 FNAL data with the previous BNL result on the allowed MSSM parameter space. The results will be discussed in the context of the upcoming searches for EW particles at the HL-LHC. We will also comment on the discovery prospects for these particles at possible future e + e − colliders, such as the ILC [37,38] or CLIC [38,39].

The model
A detailed description of the EW sector of the MSSM can be found in Ref. [1]. Here we just list the relevant input parameters and masses that are relevant for our analysis. Throughout this paper we assume that all parameters are real, i.e. we have no CP-violation.
The masses and mixings of the charginos and neutralinos are determined by U (1) Y and SU (2) L gaugino masses M 1 and M 2 , the Higgs mixing parameter µ and tan β, the ratio of the two vacuum expectation values (vevs) of the two Higgs doublets of MSSM, tan β = v 2 /v 1 . The four neutralino masses are given as mχ0 Similarly the two chargino-masses are denoted as mχ± As argued in Ref. [1] it is sufficient for our analysis to focus on positive values for M 1 , M 2 and µ.
For the sleptons, as in Ref. [1], we choose common soft SUSY-breaking parameters for all three generations, ml L and ml R . We take the trilinear coupling A l (l = e, µ, τ ) to be zero for all the three generations of leptons. In general we follow the convention thatl 1 (l 2 ) has the large "left-handed" ("right-handed") component. Besides the symbols equal for all three generations, ml 1 and ml 2 , we also explicitly use the scalar electron, muon and tau masses, mẽ 1,2 , mμ 1,2 and mτ 1,2 .
Following the stronger experimental limits from the LHC [9, 10], we assume that the colored sector of the MSSM is sufficiently heavier than the EW sector, and does not play a role in this analysis. For the Higgs-boson sector we assume that the radiative corrections to the light CP-even Higgs boson (largely originating from the top/stop sector) yield a value in agreement with the experimental data, M h ∼ 125 GeV. This naturally yields stop masses in the TeV range [40,41], in agreement with the above assumption. Concerning the Higgsboson mass scale, as given by the CP-odd Higgs-boson mass, M A , we employ the existing experimental bounds from the LHC. In the combination with other data, this results in a non-relevant impact of the heavy Higgs bosons on our analysis, as was discussed in Ref. [1].
Recently a new lattice calculation for the leading order hadronic vacuuum polarization (LO HVP) contribution to a SM µ [65] has been reported, which, however, was not used in the new theory world average, Eq. (2) [44]. Consequently, we also do not take this result into account, see also the discussions in Refs. [1,[65][66][67][68][69]. On the other hand, we are also aware that our conclusions would change substantially if the result presented in [65] turned out to be correct.
All other constraints are taken into account exactly as in Ref. [1]. These comprise • Vacuum stability constraints: All points are checked to possess a stable and correct EW vacuum, e.g. avoiding charge and color breaking minima. This check is performed with the public code Evade [76,77].
• Constraints from the LHC: All relevant EW SUSY searches are taken into account, mostly via CheckMATE [78][79][80], where many analysis had to be implemented newly [1].
• Direct detection constraints of Dark matter: We employ the constraint on the spin-independent DM scattering cross-section σ SI p from XENON1T [12] experiment, evaluating the theoretical prediction for σ SI p using MicrOMEGAs [81][82][83][84]. A combination with other DD experiments would yield only very slightly stronger limits, with a negligible impact on our results.

Parameter scan
We scan the relevant MSSM parameter space to obtain lower and upper limits on the relevant neutralino, chargino and slepton masses. As detailed in Ref. [1] three scan regions cover the relevant parameter space: In all three scans we choose flat priors of the parameter space and generate O(10 7 ) points. M A has also been set to be above the TeV scale. Consequently, we do not include explicitly the possibility of A-pole annihilation, with M A ∼ 2mχ0 1 . As we will briefly discuss below the combination of direct heavy Higgs-boson searches with the other experimental requirements constrain this possibility substantially. Similarly, we do not consider h-or Zpole annihilation (see, e.g., Ref. [36]), as such a light neutralino sector likely overshoots the (g − 2) µ contribution, see the discussion in Ref. [1].

Analysis flow
The data sample is generated by scanning randomly over the input parameter range mentioned above, using a flat prior for all parameters. We use SuSpect [85] as spectrum and SLHA file generator. The points are required to satisfy theχ ± 1 mass limit from LEP [86]. The SLHA output files from SuSpect are then passed as input to GM2Calc and MicrOMEGAs for the calculation of (g − 2) µ and the DM observables, respectively. The parameter points that satisfy the new (g − 2) µ constraint, Eq. (6), the DM relic density and the direct detection constraints and the vacuum stability constraints as checked with Evade are then taken to the final step to be checked against the latest LHC constraints implemented in CheckMATE, as described in detail in Ref. [1]. The branching ratios of the relevant SUSY particles are computed using SDECAY [87] and given as input to CheckMATE.

Results
We present the results of the allowed parameter ranges in the three scenarios defined above. We follow the analysis flow as described above and denote the points surviving certain constraints with different colors: • grey (round): all scan points.
• green (round): all points that are in agreement with (g − 2) µ , taking into account the new limit as given in Eq. (6).
• blue (triangle): points that additionally give the correct relic density, see Sect. 2.2.

Results in the three DM scenarios
We start in Fig. 1 with the results in theχ ± 1 -coannihilation scenario. In the mχ0 1 -mχ± 1 plane, shown in the upper left plot, by definition ofχ ± 1 -coannihilation the points are clustered in the diagonal of the plane. One observes a clear upper limit on the (green) points allowed by the new (g − 2) µ result of about 700 GeV, which is similar to the previously obtained one in Ref. [1]. Applying the CDM constraints reduce the upper limit further to about 600 GeV, again similar as for the old (g − 2) µ result. Applying the LHC constraints, corresponding to the "surviving" red points (stars), does not yield a further reduction from above, but cuts always only points in the lower mass range. Thus, the new experimental data set an upper as well as a lower bound, yielding a clear search target for the upcoming LHC runs, and in particular for future e + e − colliders, as will be briefly discussed in Sect. 4.2.
The distribution of the lighter slepton mass (where it should be kept in mind that we have chosen the same masses for all three generations, see Sect. 2.1) is presented in the mχ0 1ml 1 plane, shown in the upper right plot of Fig. 1. The (g − 2) µ constraint places important constraints in this mass plane, since both types of masses enter into the contributing SUSY diagrams, see Sect. 2.2. The constraint is satisfied in a triangular region with its tip around (mχ0 1 , ml 1 ) ∼ (700 GeV, 800 GeV), compatible with the old limits. This is slightly reduced to ∼ (600 GeV, 700 GeV) when the DM constraints are taken in to account, as can be seen in the distribution of the blue, cyan and red points (triangle/diamond/star). The LHC constraints cut out lower slepton masses, going up to ml 1 < ∼ 400 GeV, as well as part of the very low mχ0 1 points nearly independent of ml 1 . Details on these cuts can be found in Ref. [1].
We finish our analysis of theχ ± 1 -coannihilation case with the mχ0 1 -tan β plane presented in the lower plot of Fig. 1. The (g − 2) µ constraint is fulfilled in a triangular region with largest neutralino masses allowed for the largest tan β values (where we stopped our scan at tan β = 60). In agreement with the previous plots, the largest values for the lightest neutralino masses are ∼ 600 GeV, as before compatible with the old (g − 2) µ limit [1]. The LHC constraints cut out points at low mχ0 1 , but nearly independent on tan β. In this plot we also show as a black line the current bound from LHC searches for heavy neutral Higgs bosons [88] in the channel pp → H/A → τ τ in the M 125 h (χ) benchmark scenario (based on the search data published in Ref. [89] using 139 fb −1 .) 3 . The black line corresponds to mχ0 1 = M A /2, i.e. roughly to the requirement for A-pole annihilation, where points above the black lines are experimentally excluded. There are a very few passing the current (g − 2) µ constraint below the black A-pole line, reaching up to mχ0 1 ∼ 250 GeV, for which the A-pole annihilation could provide the correct DM relic density. It can be expected that with the improved limits as given in [89] this possibility is further restricted. These effects makea the A-pole annihilation in this scenario marginal.
The final parameter constrained in this scenario is the Higgs-mixing parameter µ. Here in particular the DD bounds are important. Following the analysis in Ref. [1] we find a lower limit of µ/M 1 > ∼ 1.8 in theχ ± 1 -coannihilation scenario. We now turn to the case ofl ± -coannihilation Case-L, as shown in Fig. 2. We start with the mμ 1 − mχ0 1 plane in the upper left plot. By definition of the scenario, the points are located along the diagonal of the plane. The new constraint from (g − 2) µ puts an upper bound of ∼ 650 GeV on the masses, which is in the same ballpark as for the old (g − 2) µ results [1]. Including the DM and LHC constraints, the bound is reduced to ∼ 540 GeV, again compatible with Ref. [1]. As in the case ofχ ± 1 -coannihilation the LHC constraints cut away only low mass points. The corresponding implications for the searches at future colliders are briefly discussed in Sect. 4.2.
In upper right plot of Fig. 2 we show the results in the mχ0 1 -mχ± 1 plane. The (g − 2) µ limits on mχ0 1 become slightly stronger for larger chargino masses, and upper limits on the chargino mass are set at ∼ 2.8 TeV. The LHC limits cut away a lower wedge going up to mχ± 1 < ∼ 600 GeV. The results for thel ± -coannihilation Case-L in the mχ0 1 -tan β plane are presented in the lower plot of Fig. 2. The overall picture is similar to theχ ± 1 -coannhiliation case shown above in Fig. 1. Larger LSP masses are allowed for larger tan β values. On the other hand the combination of small mχ0 1 and large tan β leads to a too large contribution to a SUSY µ and is thus excluded. As in Fig. 1 we also show the limits from H/A searches at the LHC, where we set (as above) mχ0 1 = M A /2, i.e. roughly to the requirement for A-pole annihilation, where points above the black lines are experimentally excluded. In this case for the current (g − 2) µ limit substantially more points passing the (g − 2) µ constraint "survive" below the black line, i.e. they are potential candidates for A-pole annihilation. The masses reach up to ∼ 240 GeV. Together with the already stronger bounds on H/A → τ τ [89] this does not fully exclude A-pole annihilation, but leaves it as a rather remote possibility.
The limits on µ/M 1 (not shown) in thel ± -coannihilation Case-L are again mainly driven by the DD-experiments. Given both CDM constraints and the LHC constraints, the smallest µ/M 1 value we find is 1.7.
We now turn to our third scenario,l ± -coannihilation Case-R, where in the scan we require the "right-handed" sleptons to be close in mass with the LSP. It should be kept in mind that in our notation we do not mass-order the sleptons: for negligible mixing as it is given for selectrons and smuons the "left-handed" ("right-handed") slepton corresponds tol 1 (l 2 ). We start in Fig. 3 with the mχ0 1 -mμ 2 plane in the upper left plot. By definition of the scenario the points are concentrated on the diagonal. The new (g − 2) µ bound yields an upper limit on the LSP of ∼ 690 GeV, a small reduction w.r.t. the previous results [1]. The new (g − 2) µ bound also places an upper limit on mμ 2 (which is close in mass to theẽ 2 andτ 2 ) of ∼ 800 GeV, again in the same ballpark as for the old (g − 2) µ result. Including the CDM and LHC constraints, these limits reduce to ∼ 520 GeV for the LSP, and correspondingly to ∼ 600 GeV for mμ 2 , and ∼ 530 GeV for mτ 2 . The LHC constraints cut out some, but not all lower-mass points.
The distribution of the heavier slepton is displayed in the mχ0 1 -mμ 1 plane in the upper right plot of Fig. 3. Although the "left-handed" sleptons are allowed to be much heavier, the (g − 2) µ constraint imposes an upper limit of ∼ 950 GeV, about ∼ 50 GeV less than previously. This effect is discussed in detail in Ref. [1]. The DM and LHC constraints to not yield a further reduction in this case, which cut away only lower mass points and set a lower limit of ∼ 300 GeV for the heavier sleptons in the Case-R.
In the lower left plot of Fig. 3 we show the results in the mχ0 1 -mχ± 1 plane. As in the Case-L the (g − 2) µ limits on mχ0 1 become slightly stronger for larger chargino masses. The upper limits on the chargino mass, however, are substantially stronger as in the Case-L. They are reached at ∼ 900 GeV using the new (g − 2) µ result, similar to the limits for the old (g − 2) µ limit [1].
We finish our analysis of thel ± -coannihilation Case-R with the results in the mχ0 1 -tan β plane, presented in the lower right plot of Fig. 3. The overall picture is similar to the previous cases shown above. Larger LSP masses are allowed for larger tan β values. On the other hand the combination of small mχ0 1 and very large tan β values, tan β > ∼ 40 leads to stau masses below the LSP mass, which we exclude for the CDM constraints. The LHC searches mainly affect parameter points with tan β < ∼ 20. Larger tan β values induce a larger mixing in the third slepton generation, enhancing the probability for charginos to decay via staus and thus evading the LHC constraints. As above we also show the limits from H/A searches at the LHC, where we set (as above) mχ0 1 = M A /2, i.e. roughly to the requirement for A-pole annihilation, where points above the black lines are experimentally excluded. Comparing Case-R and Case-L substantially less points are passing the new (g − 2) µ constraint below the black line, i.e. are potential candidates for A-pole annihilation. The masses reach only up to ∼ 150 GeV, about ∼ 50 GeV less than with the old (g − 2) µ result. Together with the already stronger bounds on H/A → τ τ [89] this leaves A-pole annihilation as a quite remote possibility in this scenario.
The limits on µ/M 1 (not shown) in thel ± -coannihilation Case-R are as before mainly driven by the DD-experiments. Given both CDM constraints and the LHC constraints, the smallest µ/M 1 value we find is 1.7.

Implications for future colliders
In Ref. [1] we had evaluated the prospects for EW searches at the HL-LHC [95] and at a hypothetical future e + e − collider such as ILC [37,38] or CLIC [38,39].
The prospects for BSM phenomenology at the HL-LHC have been summarized in [95] for a 14 TeV run with 3 ab −1 of integrated luminosity. For the direct production of charginos and neutralinos through EW interaction, the projected 95% exclusion reach as well as a 5σ discovery reach have been presented. Following the discussion in Ref. [1] we conclude that via these searches the updated (g − 2) µ limit together with DM constraints can conclusively probe "almost" the entire allowed parameter region ofl ± -coannihilation Case-R scenario and a significant part of the same parameter space for Case-L scenario at the HL-LHC. On the other hand, the analysis for compressed higgsino-like spectra at the HL-LHC, see the discussion in Ref. [8], may exclude mχ0 2 ∼ mχ± 1 ∼ 350 GeV with mass gap as low as 2 GeV for mχ± 1 around 100 GeV. Hence, a substantial parameter region can be curbed for theχ ± 1coannihilation case in the absence of a signal in the compressed scenario analysis with soft leptons at the final state. However, higher energies in pp collisions or an e + e − collider with energies up to √ s ∼ 1.5 TeV will be needed to probe this scenario completely [96,97]).
Direct production of EW particles at e + e − colliders clearly profits from a higher center-ofmass energy, √ s. Consequently, we focus here on the two proposals for linear e + e − colliders, ILC [37,38] and CLIC [38,39], which can reach energies up to 1 TeV, and 3 TeV, respectively. In Ref. [1] we had evaluated the cross-sections for the various SUSY pair production modes (based on Refs. [98,99]) for the energies currently foreseen in the run plans of the two colliders.
Taking into account the results for the cross-sections evaluated in Ref. [1], one can conclude that the new accuracy on (g −2) µ , yielding similar upper limits on EW SUSY particles, guarantees the discovery at the higher-energy stages of the ILC and/or CLIC. This holds in particular for the LSP and the NLSP. The improved (g − 2) µ constraint, confirming the deviation of a exp µ from the SM prediction, clearly strengthens the case for future e + e − colliders. As discussed in Sect. 3 we have not considered the possibility of Z or h pole annihilation to find agreement of the relic DM density with the other experimental measurements. However, it should be noted that in this context an LSP with M 1 ∼ mχ0 1 ∼ M Z /2 or ∼ M h /2 would yield a detectable cross-section e + e − →χ 0 1χ 0 1 γ in any future high-energy e + e − collider. Furthermore, depending on the values of M 2 and µ, this scenario likely yields other clearly detectable EW-SUSY production cross-sections at future e + e − colliders. We leave this possibility for future studies.
On the other hand, the possibility of A-pole annihilation was briefly discussed for all three scenarios. While it appears a rather remote possibility, it cannot be fully excluded by our analysis. However, even in the "worst" case ofl ± -coannihilation Case-L an upper limit on mχ0 1 of ∼ 250 GeV can be set. While not as low as in the case of Z or h-pole annihilation, this would still offer good prospects for future e + e − colliders. We leave also this possibility for future studies.

Conclusions
The electroweak (EW) sector of the MSSM, consisting of charginos, neutralinos and scalar leptons can account for a variety of experimental data: the CDM relic abundance with the lightest neutralino,χ 0 1 as LSP, the bounds from DD experiments as well as from direct searches at the LHC. Most importantly, the EW sector of the MSSM can account for the long-standing discrepancy of (g − 2) µ . The new result for the Run 1 data of the "MUON G-2" experiment confirmed the deviation from the SM prediction found previously.
Under the assumption that the previous experimental result on (g − 2) µ is uncorrelated with the new "MUON G-2" result we combined the data and obtained a new deviation from the SM prediction of ∆a µ = (25.1 ± 5.9) × 10 −10 , corresponding to a 4.2 σ discrepancy. We used this limit as a cut at the ±2 σ level.
In this paper, under the assumption that theχ 0 1 provides the full DM relic abundance we analyzed which mass ranges of neutralinos, charginos and sleptons are in agreement with all relevant experimental data: the new limit for (g − 2) µ , the relic density bounds, the DD experimental bounds, as well as the LHC searches for EW SUSY particles. These results present an update of Ref. [1], where the previous (g − 2) µ result had been used (as well as a hypothetical "MUON G-2" result).
We analyzed three scenarios, depending on the mechanism that brings the relic density in agreement with the experimental data:χ ± 1 -coannihilation,l ± -coannihilation with the mass of the "left-handed" ("right-handed") slepton close to mχ0 1 , Case-L (Case-R). We find in all three cases a clear upper limit on mχ0 1 . We find that the upper limits on the LSP mass are decreased to about 600 GeV forχ ± 1 -coannihilation, 540 GeV forl ± -coannihilation Case-L and 520 GeV in Case-R, confirming the collider targets w.r.t. the old (g − 2) µ . Similarly, the upper limits on the NLSP masses are confirmed to about 600 GeV, 600 GeV and 530 GeV in the three cases that we have explored, again compatible with the previous (g − 2) µ result.
For the HL-LHC we have briefly discussed the prospects to cover the parameter regions that are preferred by the new (g − 2) µ result. In particular thel ± -coannihilation Case-R can be conclusively tested at the HL-LHC, while the other scenarios are only partially covered. Concerning future high(er) energy e + e − colliders, ILC and CLIC, one can conclude that the new accuracy on (g − 2) µ confirms the upper limits on EW SUSY, and it can be expected that at least some particles can be discovered at the higher-energy stages of the ILC and/or CLIC. This holds in particular for the LSP and the NLSP. Therefore, the new (g − 2) µ constraint, confirming the deviation of a exp µ from the SM prediction, strongly motivates the need of future e + e − colliders. and in part by the AEI through the grant IFT Centro de Excelencia Severo Ochoa SEV-2016-0597. The work of M.C. is supported by the project AstroCeNT: Particle Astrophysics Science and Technology Centre, carried out within the International Research Agendas programme of the Foundation for Polish Science financed by the European Union under the European Regional Development Fund.