Prospects for Electroweakino Discovery at a 100 TeV Hadron Collider

We investigate the prospects of discovering split Supersymmetry at a future 100 TeV proton-proton collider through the direct production of electroweakino next-to-lightest-supersymmetric-particles (NLSPs). We focus on signatures with multi-lepton and missing energy: $3\ell$, opposite-sign dileptons and same-sign dileptons. We perform a comprehensive study of different electroweakino spectra. A 100 TeV collider with 3000/fb data is expected to exclude Higgsino thermal dark matter candidates with $m_{\rm{LSP}}\sim 1 $ TeV if Wino NLSPs are lighter than about 3.2 TeV. The $3\ell$ search usually offers the highest mass reach, which varies in the range of (2-4) TeV depending on scenarios. In particular, scenarios with light Higgsinos have generically simplified parameter dependences. We also demonstrate that, at a 100 TeV collider, lepton collimation becomes a crucial issue for NLSPs heavier than about 2.5 TeV. We finally compare our results with the discovery prospects of gluino pair productions and deduce which SUSY breaking model can be discovered first by electroweakino searches.

The lack of discovery of Supersymmetry (SUSY) at the first run of the LHC started to place some tension on natural SUSY. Even though it is premature to abandon the idea of a natural spectrum, an attractive scenario is split SUSY [1][2][3][4]. We refer to, e.g., Refs. [5][6][7][8][9][10][11] for developments along this line.
In these models, gauginos and higgsinos are the lightest SUSY particles and provide most important collider search channels as SUSY scalars are much heavier. Collider searches of gluino pair production usually lead to the easiest discovery if gluinos are not much heavier than other gauginos and higgsinos. According to Refs. [12,13], gluinos up to about 11 TeV can be discovered at a 100 TeV proton-proton collider with 3 ab −1 data. If gluinos are heavier, electroweakinos 1 can be the best channel to discover split SUSY. In any case, electroweakino studies are essential for precision measurements of the superpartner mass spectrum. Gluino-focused studies are not enough in this regard as M ef f from gluino pair decays is not very sensitive to the electroweakino mass spectrum [13].
Electroweakino searches can also probe the WIMP (weakly interacting massive particle) nature of neutralino lightest superparticles (LSPs). Either Higgsinos, Winos or welltempered neutralinos can serve as thermal relic cold dark matter (DM) candidates with full relic abundance as needed to satisfy cosmological data [14]. 1 TeV and 3.1 TeV are the masses of potential Higgsino and Wino full thermal DM [15]. Testing the split parameter space up to these masses is both an important mission and a useful goal of a future collider.
Direct LSP collider searches are the most model independent tests of the scenario. According to dedicated studies in Refs. [16,17], Wino DM can plausibly be probed at a 100 TeV collider, but probing Higgsino DM through those searches will be unlikely.
In this paper, we study the 100 TeV proton-proton collider prospects of NLSP electroweakino searches in multi-lepton final states. In particular, we will discuss the potential of probing Higgsino dark matter from the pair production of Wino NLSPs and Wino dark matter from the pair production of Higgsinos NLSPs. Finally, we will compare the search capabilities of these channels to those based on direct gluino production and decay.
Meanwhile, this parameter space of SUSY, with relatively light (at most few TeV) electroweakinos and much heavier scalar superpartners, must be studied in qualitatively different ways in several aspects, compared to the previous studies of O(100) GeV SUSY at the LHC.
As is well known, boosted phenomena and electroweak radiation phenomena become central issues at a 100 TeV proton collider; see, e.g. Ref. [18]. Moreover, more analytic approaches are possible for this higher energy environment with only electroweakinos accessible, as a smaller number of particles and parameters are relevant to the final signatures. The very high energy of the collisions with relatively light electroweakinos create, in fact, an environment where the Goldstone equivalence theorem generically applies. Therefore, the various electroweakino decay branching ratios (BRs) are inherently related. Interestingly, the NLSP BRs involving Higgsinos (either as decaying mother particles or daughter particles) are greatly simplified in this parameter space. All the underlying dependences from tan β and from the signs of gaugino and higgsino masses essentially vanish as a result of (1) summing the effects of two indistinguishably degenerate neutral Higgsinos to calculate what we actually observe at the collider [19] and (2) the Higgs alignment limit dictated by Higgs signal strength data [20]. We emphasize that these relations did not hold previously in general, especially when the electroweakinos under consideration are light. At the same time, they become very good approximations for TeV scale electroweakinos. Various other relations are also revealed in a similar way and analytic understanding of BRs are aided [19,21].
Throughout this paper, we present results obtained by full numerical computation of BRs. As already mentioned, we have model independent BRs in the scenarios with Higgsino-NLSP or -LSP. In the case of heavier Higgsinos, µ > M 2 , M 1 , the results will be more model dependent. For this reason, we will consider several choices of parameters with heavier Higgsinos and provide analytic discussion.
The paper is organized as follows. We first introduce multi-lepton searches and our collider analysis strategy in Sec. II. Sec. III contains our main results: we provide discovery and exclusion prospects for several scenarios containing different types of NLSPs and LSPs.
We also compare our results with the discovery prospects of split SUSY via gluino pair productions. We further discuss potential issues regarding detector and object measurements at a future 100 TeV hadron collider in Sec. IV. We finally reserve Sec. V to our conclusions. We estimate several uncertainties involved in our analysis in the Appendix.

A. Search Channels
In split SUSY, all the scalars are much heavier than the electroweakinos and therefore elec- , the p T ratio of the second and leading leptons p T ( 2 )/p T ( 1 ), and the jet energy fraction H T (jets)/M ef f . Here, M ef f (H T (jets)) is the scalar sum of p T 's of reconstructed jets, leptons and MET (jets only). No explicit jet veto is applied. We will discuss in Sec. II C, however, that the upper cut on the jet energy fraction is analogous to a jet veto. The SM W Z/γ 2 production is the dominant background for most cases, while tribosons and ttV backgrounds can be relevant when the W h becomes the dominant contribution to the signal.
The latest 3 LHC8 searches can be found in Refs. [22,24]. Wino NLSPs up to ∼ 350 GeV are excluded for massless Bino LSPs in the simplified model, which assumes a 100% branching ratio for C ± 1 N 2 → W ± ZN 1 N 1 .
• Opposite-sign di-leptons (OSDL): The W + W − channel is the dominant signal contribution.
In our analysis, we require exactly two opposite-sign leptons of any flavor and veto any events with reconstructed jets (p T > 30 GeV, |η| < 2.5). The observables we optimize are M ef f , missing energy fraction E miss T /M ef f , p T ( 2 )/p T ( 1 ) and the transverse mass obtained from the two leptons and the missing energy, M T (E miss T , ). With our analysis, the SM W + W − is the dominant background and the SM W Z is also non-negligible.
The latest OSDL LHC8 searches can be found in Refs. [24,26]. Wino NLSPs up to ∼ 200 GeV are excluded for massless LSPs in the simplified model, which only includes the process C ± 1 C ∓ 1 → W ± W ∓ N 1 N 1 with assumed 100% BR.
• Same-sign di-leptons (SSDL): The W ± W ± channel is the dominant signal contribution, but it is absent in some NLSP-LSP configurations, for which W Z becomes the only channel contributing to this signature, if one of the three leptons is missed. Standard ATLAS and CMS searches show that, typically, this channel is the best search mode for electroweakino spectra with small mass gap between the NLSP and the LSP, for which one of the leptons coming from the NLSP decay might be too soft to be included in the 3 analysis.
In our analysis, we require exactly two same-sign leptons of any flavor and veto any events with reconstructed jets (p T > 30 GeV, |η| < 2.5). The observables we optimize ). The SM W Z is the dominant background while fake and mis-identified backgrounds can similar or larger [24], and the double parton scattering (DPS) production of W ± W ± is smaller 3 . Muons are perhaps cleaner against the fake backgrounds [29], but we include both e and µ with equal efficiencies 4 . The latest SSDL LHC8 searches can be found in Ref. [24]. Wino NLSPs up to ∼ 130 GeV are excluded for massless LSPs in the simplified model which only includes the process N 2 C ± 1 → W hN 1 N 1 with assumed 100% BR.
• 4 : Exactly four leptons of any flavor can be searched for. The ZZ channel is the dominant contribution. Due to the small branching ratio of the Z to leptons and to the smaller production cross section of neutral NLSPs, if compared to the associated production of a neutral and a charged NLSP, this channel is typically not a leading discovery channel. For this reason, we do not consider this signature further.
The latest 4 LHC8 searches can be found in Refs. [24,25]. Higgsinos up to ∼ 150 GeV are excluded in the context of GMSB models with the gravitino LSP. 3 The DPS W ± W ± produces softer leptons than the W + W − background [28], which make it less important for high mass searches. If it had the same kinematic distributions as the SM W + W − , it would contribute to the SSDL search by only ∼ 20% of the main SM W Z background. 4 We learn from Maurizio Pierini that the resolution of high-p T muon measurements can be quite worse than that of high-p T electrons depending on the performances of calorimeters and magnet strengths.

B. Analysis Details
We model signals and backgrounds using MadGraph5 [30], interfaced with Pythia 6.4 [31], for parton showering. We allow up to one additional parton in the final state, and adopt the MLM matching scheme [32] with xqcut=40 GeV. The generated SM background processes are di-boson (W W , W Z and ZZ), tribosons and ttV . We also check tt and W h backgrounds for some cases. The backgrounds are generated in successively smaller phase spaces sectored by the scalar p T sum of intermediate dibosons [12,13] as done for the Snowmass studies [33].
To this end, we modify the MadGraph cuts.f code. Corresponding Pythia matched rates are used for background normalizations. Summing all sectored backgrounds yields a rate similar to the next-to-leading order results predicted from MCFM [34]. As for signal rates, we multiply the leading order MadGraph results by the assumed NLO K-factor, K = 1.2.
Leptons are required to have p T > 15 GeV and |η| < 2.5. To reconstruct jets, we use the anti-k T algorithm [35] with R = 0.4 implemented in FastJet-2.4.3 [36] on remaining particles. Jets are required to have p T > 30 GeV and |η| < 2.5. If a reconstructed lepton is found within ∆R = 0.4 of a reconstructed jet, the lepton is merged into the jet. We require leptons (both muons and electrons) to be separated by more than ∆R = 0.05. We cluster photons nearby a lepton within a cone of ∆R < 0.05, to reconstruct the lepton by taking into account QED radiation effects. Hard QED radiation is infrequent but resulting photons can carry away a non-negligible fraction of energy momentum. As shown in [37], this is especially important to reconstruct Z peaks more properly. Lepton identification efficiencies are adapted from the current ATLAS efficiencies [38,39]. Typically, as our leptons are energetic, more than 95% of the leptons are identified. We refer to Sec. IV for related discussions. We do not take into account any detector effects such as finite cell sizes and momentum smearing.
Our baseline selection requires no additional leptons: exactly 3 , OSDL and SSDL for corresponding searches. A jet veto is applied for the OSDL and SSDL. Any SFOS dileptons are required to have invariant mass, mSFOS > 12 GeV. In addition, the mSFOS closest to m Z , denoted by mSFOS(Z), should be within (outside of) 30 GeV of m Z if W Z → 3 (W h → 3 , W ± W ∓ → 2 , W ± W ± → 2 ) channels are searched for. For the OSDL, we additionally apply p T ( ) > 30 GeV, where p T ( ) is the transverse momentum of the lepton pair. Finally, we require either MET> 100 GeV or p T (l 1 ) > 100GeV. No specific trigger is applied but we expect that devising a working trigger will not be problematic, thanks to the high energy cuts used in our analysis.  Table I are applied except for those on p T ( 2 )/p T ( 1 ) and H T (jets)/M ef f .

C. The Variables Used in Analysis
The search for the high-mass and large-gap parameter space is based on high visible and invisible energy. Signal processes easily produce such high energy particles from decays of heavy mother particles. The SM backgrounds, instead, can reach high visible and invisible energies only by hard radiations. This implies certain alignment between the boost direction and the final state particle momentum so that certain particles are more efficiently boosted.  Fig. 1). The peak of the jet energy fraction distribution, H T (jets)/M ef f , is sharp and located at around 0.5 in the background, which means that all the remaining leptons and MET are recoiling against the radiated jets, in such a way that the total energy is balanced. On the other hand, the jet energy fraction is small for signal events, for which the leptons are already boosted thanks to the large mass splitting between NLSP and LSP. This is an interesting feature that generally appears in high-mass searches. Typically, the jet veto has been designed to suppress backgrounds with jets coming from the several particle decays, but now the jet veto can be very useful even to suppress backgrounds as W Z, in which jets only come from radiation. In our analysis we require the jet energy fraction, H T (jets)/M ef f , to be small to suppress the W Z background. 5 We do not employ a different and more dedicated strategy for the benchmarks contributing to the 3 signature mainly through the W h channel. hierarchical as can be seen in the right panel of Fig. 1. In our analysis, we will impose a lower cut on the lepton p T ratio, p T ( 2 )/p T ( 1 ).
The lepton p T ratio and the variable M ef f turn out to be more optimal for our analysis than the mere M ef f . This is partly because the information of p T ( 1 ) is used only by the lepton p T ratio but not by M ef f , which makes the two variables more independent and a better optimization possible. With these additional variables, the exclusion mass reach is improved by 400-800 GeV compared to the result based solely on M ef f , MET and M T (W ).
These additional variables are also useful to make sure that additional backgrounds from triboson and ttV remain small. The ttV background produces many jets from the top and vector boson decays, thus it has significant jet energy fraction as shown in the left panel of Fig. 2. Therefore, the upper cut on the jet energy fraction can suppress efficiently the ttV backgrounds. In contrast, the triboson background, W W W , does not produce any jets from the vector boson decays and needs hard radiation to reach a high M ef f , similarly to the W Z background. Thus, they similarly have a sharp peak in the distribution of the jet energy fraction at 0.5 as shown in Fig. 2. Nevertheless, W W W has a sharper peak at 0.5 and a smaller tail at small jet energy fraction. The accumulation of W Z events at small jet energy fraction is due to the M T (W ) cut preferring a boosted leading lepton and/or neutrino. The  Table II are applied. W W signal (blue) is from 2 TeV NLSP and massless LSP, and the background (red) is the SM W W process.
energy fraction can suppress triboson backgrounds even more efficiently than the diboson background.
Finally, in the right panel of Fig. 2, we also check that the distributions for p T ( 2 )/p T ( 1 ) are very similar for all backgrounds. This demonstrates the goodness of a lower cut on We first require a large M ef f . The missing energy fraction, E miss T /M ef f , is then a useful discriminator between signal and background. As signal events contain more missing particles, they tend to have a larger E miss T /M ef f (see left panel of Fig. 3). Once a lower cut on the missing energy fraction is applied, a neutrino from the W W background is likely aligned with the boost direction of its mother W so that large MET is obtained. Consequently, the charged lepton from the same side stays soft, and the p T of the two charged leptons tend to be hierarchical as shown on the right panel of Fig. 3. As will be shown in Table II, numerically optimized cuts on these ratio variables make the W + W − and W Z backgrounds similar in size.
Meanwhile, transverse mass variables are often used in OSDL searches in analogy of the M T (W ) in 3 searches. For example, the CMS h → W W * → 2 2ν analysis uses the transverse mass between MET and the lepton pair [40]. We find that using MET and this transverse mass is not any better in rejecting the W + W − background. The jet veto implies a simple momentum conservation among the two charged leptons and the missing particles,  Table III except for the ones drawn here are applied.
: in this respect, the transverse mass is redundant. Nevertheless, the transverse mass is useful in rejecting non-negligible W Z backgrounds. We thus apply a lower cut on this transverse mass. Finally, we comment that the tt background has similar distributions of these variables as those of the W + W − background because tt events are essentially reduced to W + W − events after jet veto. We have checked that the tt background is subdominant with our cuts.
With these additional variables, we can improve the mass reach by about 200-500 GeV compared to a simple MET and M ef f analysis.

SSDL Search:
Let us compare the signal arising from the same-sign W pair channel, W ± W ± , with the main SM background W Z. The most important variable distinguishing between them is the transverse mass variable, M T (E miss T , ). As shown in the right panel of Fig. 4, the background distribution is peaked at smaller values. The difference is more pronounced with heavier NLSPs. As, in the background, the second lepton is hierarchically softer than the leading lepton (see left panel of Fig. 4), the transverse mass is approximately , and thus has a strong drop at M W . We apply a lower bound on this variable.

D. Cut Optimization
For each benchmark that we simulate, we optimize the cuts on the variables discussed in the previous subsections to maximize the statistical significance, are the number of signal (background) events with the assumed luminosity (3000 fb −1 ). To this end, O(10) n -size arrays of cut efficiencies are generated for each process, where    the n is the number of variables that we optimize. We always require at least 5 signal events after all cuts. For the 3 and OSDL searches, we do not assume any systematic uncertainties and do not vary background normalization. For the SSDL search, instead, reducible backgrounds can be especially large and difficult to estimate. According to the current SSDL searches with jet vetos [24], the reducible backgrounds may be of similar size as the ones from diboson production. Thus, to take them into account, we conservatively multiply the simulated backgrounds by 2 in the SSDL analysis -this is denoted by N = 2 in the notation of Eq. (A1)-and S/ √ N B is maximized in our optimization. In Tables I-IV, we present our optimal cuts and results for the all multilepton searches using benchmark scenarios that will be discussed in Sec. III 6 . After all cuts, diboson backgrounds are generally dominant, while ttV and tribosons can only be important for W hdominated 3 searches, as shown in Table IV. As discussed, we checked that the SM tt background is smaller than W W and W Z backgrounds in the OSDL search in Table II   in Tab. IV. In particular, in the last one or two steps of each cut flow table, we also show the results of applying cuts on the new ratio variables introduced in the previous subsection.
In the next section, we will present discovery and exclusion prospects based on the strategies and cuts described in this section.

III. PROSPECTS OF A 100 TEV COLLIDER
We present results for the following cases: The heaviest electroweakino mass is always fixed to 5 TeV. We do not study the cases of (mainly) Bino NLSPs because of their small production rates. We do not use simplified models to present results. We rather take into account all the relevant branching ratios to

A. Higgsino LSP
When the Higgsino is the LSP, the production of Wino NLSPs can be used to probe the model. Multi-lepton signals arise from the following processes: • The 3 arises mainly from N 3 C 2 → W Z + N 1,2 N 1,2 , W Z + C 1 C 1 and C 2 C 2 → W Z + N 1,2 C 1 .
• The SSDL arises mainly from N 3 C 2 → W ± W ± + N 1,2 C 1 . 7 There are interesting exceptions from models with weakly interacting LSPs such as axinos or gravitinos due to slow decays of heavier Higgsinos [41]. In Table V, we decompose the multi-lepton signal rates into each diboson channel contribution for a benchmark with a 1 TeV Wino NLSP and a massless Higgsino LSP. As mentioned, the 3 , OSDL and SSDL channels get dominant contributions from the W Z, W + W − and W ± W ± diboson channels, respectively. In spite of the fact that BR(N NLSP → N LSP h) ∼ 0.25, the W h channel contributions are subdominant in all final states because the Higgs's leptonic branching ratio, h → W W * (ZZ * ) → ν ν, is small. Their contribution to the discovery reach is subdominant.
The corresponding reach is presented in Fig. 5. We do not specify our choice of additional parameters (t β and the sign of gaugino and Higgsino masses), since the branching ratios of the NLSP are model independent in this Higgsino LSP case. As expected, the 3 signature can probe the highest NLSP mass while the SSDL signature can be useful in the region with a smaller mass difference between the NLSP and the LSP.
It is important to note that a 100 TeV collider with 3000/fb data will be able to exclude Higgsino dark matter (m LSP ∼ 1 TeV) for Winos lighter than about 3.2 TeV and not too close in mass to the Higgsino. Achieving the significance needed for discovery of a 1 TeV Higgsino, however, is expected to be rather difficult (see left panel of Fig. 5). Ref. [16] shows that monojet and disappearing charged track searches at a 100 TeV collider also can have difficulties in probing 1 TeV Higgsino dark matter. In addition, Higgsino dark matter is a very challenging scenario to discover from the astrophysical side, since current astrophysical photon line/continuum searches lack sensitivity to 1 TeV Higgsinos as well [15].

B. Wino LSP
When the Wino is the LSP, Higgsino NLSPs can be used to probe the model. Multi-lepton signals arise from the following processes: • The 3 arises mainly from N 2,3 C 2 → W Z +N 1 N 1 , W Z +C 1 C 1 and C 2 C 2 → W Z +N 1 C 1 and N 2 N 3 → W Z + N 1 C 1 .
In Table VI, we decompose multi-lepton signal rates into each diboson contribution. The qualitative discussion in this table is the same as for Table V. The reach is presented in Fig. 6. As we already discussed, the branching ratios of the NLSP pairs to W Z, W + W − , W ± W ± and W h are again independent of the choice of parameters, as Higgsinos are involved in the decay. The 3 signature can probe the highest NLSP mass, while the SSDL signature can be useful in the region with smaller mass difference. Compared to the Wino NLSP results shown in Fig. 5, the reach here is worse, mainly because Higgsino NLSP production cross sections are smaller than the Wino ones.

C. Bino LSP with Higgsino NLSP
When the Bino is the LSP, either Higgsino or Wino NLSPs can be used to probe the model. In this subsection, we first consider the Higgsino NLSP case since it is simpler to discuss. Multi-lepton signals arise from the following processes: • The 3 arises mainly from N 2,3 C 1 → W Z + N 1 N 1 .
• The SSDL arises mainly from the W Z channel by accidentally loosing one lepton: The W ± W ± channel is not produced as shown in Table VII.
Multi-lepton signal rates are decomposed into each diboson contribution in Table VII.
The reach is presented in Fig. 7 and is independent of the particular choice of additional parameters. The 3 channel is by far the best. The SSDL signature gives now a much weaker bound than the OSDL signature because SSDL arises only from W Z by accidentally loosing

D. Bino LSP with Wino NLSP
Wino NLSPs can also be used to probe the Bino LSP scenario. The multi-lepton signals arise from the following processes: • The 3 arises mainly from N 2 C 1 → W Z + N 1 N 1 .
• The SSDL arises mainly from the W Z channel by accidentally loosing one lepton: The W ± W ± channel is not produced as shown in Table IX.
The branching ratios of Winos depend now sensitively on the choice of parameters: 8 t β , sign(µM 2 ), and sign(M 2 M 1 ). (1) In this section, we fix |µ| = 5 TeV. In Fig. 8, we collect our results for the 3 channel using the six sets of parameters listed in Table VIII. The remaining two possible choices are not much qualitatively different from these choices. The OSDL and SSDL results are presented in Fig. 9. Here we consider only one benchmark (Case 5) since the reach of the OSDL channel is rather model independent and the 8 Only two of the sign(M 1 M * 2 ), sign(µM * 2 ) and sign(M 1 µ * ) are physical. The former two are most convenient choices to understand our numerical results. Without loss of generality, we assume that mass parameters are real and M 1 ≥ 0.   reach of the SSDL channel is weak. The OSDL channel receives the main contribution from chargino pair production and chargino pairs always lead to the W + W − channel. It has the highest reach in this Wino-Bino model among all models we have investigated (note that this model has the highest rate for W + W − , see Table IX).
The OSDL can exclude up to about 3 TeV NLSPs. The 3 can exclude higher or lower masses depending on the parameters. At best, it can exclude ∼4.3 TeV NLSPs (Case 4 and 6) while only ∼1.3 TeV at worst (Case 3).
We now discuss various features of the 3 results in Fig. 8. We first collect them here, and explain them analytically below.
1. Flatness of the reach curves: For Case 1 and 2, the reach curves are relatively flat, whereas wider regions of mild-or small-gap can be probed for Case 5 and 6. 5σ discovery reach (solid) and 95% CL exclusion limit (dashed) are shown. We show individual results from W Z (blue) and W h (red) channels in separate colors, but we also show combined results (black) when both channels contribute similarly. Parameters in each case are also tabulated in Table VIII. Case 2 and 5, only W Z (blue curve in the figure) is the dominant channel. Differently, in Case 1, 4 and 6, there is a transition from W h to W Z dominance: the W h channel is best at small NLSP masses but the W Z takes over in the high mass region.
In the parameter space with well-separated electroweakino masses, the relative branching ratios into the Z and Higgs bosons can be approximated using the Goldstone equivalence theorem [19] valid in the approximation |M 1 | M 2 and where r ≡ m B /m W M 1 /M 2 can either be positive or negative depending on the relative sign of parameters. The mixing angles N ij are approximated in the heavy Higgsino limit by [43]    where N 13 (N 14 ) are the Bino-like mass eigenstateH 0 d (H 0 u ) components, and N 23 (N 24 ) are the Wino-like mass eigenstateH 0 d (H 0 u ) components. By plugging Eq. (3) into Eq. (2) and taking the limit M 1 → 0, we arrive at where we used |µ| > |M 2 | in the second approximation. This relation keeps all the leading dependences on relative signs between µ and M 2 that can lead to important cancellations.
The approximation is valid up to O(M 2 2 /µ 2 ) terms. If we further assume that 2|µ|s 2β |M 2 |, the ratio gets the familiar form In this limit, it is evident that the Wino dominantly decays to Binos via Higgs bosons rather than Z bosons [27,43]. The statement is further supported by the observation that the Wino-Bino-Higgs coupling needs only one small mixing insertion while the Wino-Bino-Z coupling needs two. This statement is generally true if Higgsinos are very heavy. However, in a large part of the parameter space with mildly heavy Higgsinos, the condition 2|µ|s 2β |M 2 | is not satisfied, and the Goldstone equivalence theorem inherently relates the Wino-Bino-Z process with the Wino-Bino-Goldstone process which needs only one mixing insertion [19]. This is especially true when the µ and M 2 have opposite signs and lead to a partial cancellation in the denominator of Eq. (4). They can lead to the dominance of Wino decays to Z bosons.
If we restore the leading dependence on M 1 , Eq. (4) becomes (still in the limit of |µ| This expression keeps all the leading dependences on the relative signs of mass parameters. The approximation is valid up to O(M 2 1 /µ 2 , M 2 2 /µ 2 , M 2 1 /M 2 2 ). All the features discussed above in the 3 reach can be understood from these analytic approximate expressions. We also show BRs of NLSP Winos in Fig. 10 to help understanding the results.
• Let us consider the M 1 → 0 limit. Only the relative sign(M 2 µ) and t β are relevant (see Eq. (4)). Case 2 and 5 differ only by the sign(M 2 M 1 ), and thus they have the same mass reach along the massless LSP line (M 1 = 0). Likewise, Case 4 and 6 have the same reach with massless LSPs.
• The flatness of the reach curve is dictated by the sign(M 2 M 1 ). From Eq. (6) we see that the sign determines how the branching ratio changes with the mass-gap. As M 1 approaches |M 2 |, the Z mode branching ratio becomes larger if sign(M 2 M 1 ) < 0; thus, the reach curve extends towards the small-gap region covering a wider parameter space. Otherwise, the reach curve tends to be flatter. Case 1, 2 vs. 5 as well as case 4 vs. 6 can be compared to observe this behavior.
• The shape of the reach curve at the high mass end is also explained by the sign(M 2 M 1 ).
As shown in Eq. 6, if sign(M 2 M 1 ) < 0, the branching ratio to W Z becomes larger as we raise the LSP mass, resulting in better reach. Of course, this effect is limited if the mass gap is too small. As usual, this compressed region suffers from low efficiencies and therefore worse sensitivities.
• The mode N 2 → N 1 h is dominant at small t β as most clearly shown in Eq. (5). It is especially dominant when sign(µM 2 ) > 0, where no cancellation in the Higgs partial width is possible, as shown in Eq. (4). If the sign(µM 2 ) < 0, even a small value of tan β does not guarantee the dominance of the h mode. This behavior can be seen by comparing Case 3 with sign(µM 2 ) > 0 to Case 4 and 6 with sign(µM 2 ) < 0.
• The transition of W h channel dominance to W Z channel dominance is generically dictated by the suppression factor M 2 2 /µ 2 in Eq. (5). As M 2 grows, the W Z signature becomes relatively more important. The behavior generally appears in the high mass region M 2 3 TeV, which is not far from the value we chose for the µ parameter: What if Higgsinos are much heavier than 5 TeV, as assumed in our Figs. 8, 9? If Higgsinos are heavy enough to satisfy Eq. (5) reasonably well, Higgs channels always dominate and the 3 reach becomes weaker. The reach will be rather low, similar to that of Case 3. On the other hand, the OSDL searches are not affected by the exact choice of the µ parameter, as long as |µ| |M 2 |, so that the chargino is mainly Wino-like, because the relevant BR, BR(C 1 C 1 → N 1 N 1 W W ), is always close to 100%. For this reason, the OSDL channel can become the leading discovery channel and a hint for a spectrum with very heavy Higgsinos.

E. Comparison with Nearby Gluino Reach
The gluino pair is usually a better discovery channel if gluinos are not too much heavier than electroweakinos. It is interesting to identify in which circumstances heavy gluinos are more difficult to search for than electroweakino NLSP pairs studied here.
Gluino pairs can be excluded at a 100 TeV collider with 3/ab when gluinos are lighter than about 14 TeV [12,13]. As long as gluinos are lighter than about 12-13 TeV, up to 4 TeV LSPs can be excluded regardless of gluino masses. Meanwhile, as we have shown in our paper, only up to 1-2 TeV LSPs can be excluded from multi-lepton NLSP searches. Thus, if the gluino is lighter than 12-13 TeV, it is generally an earlier discovery channel.
In the majority of SUSY models [44], gaugino masses are predicted to have order-one ratios of each other, which means that gluinos are typically not much heavier than the other gauginos. In such scenarios, if the gluino is out of the reach of a 100 TeV collider, > 13 TeV, can we still have prospects of discovering the lighter electroweakinos? As examples, we consider a couple of well known SUSY breaking models.
With the mSUGRA relation, M 1 : M 2 : M 3 1 : 2 : 6, the 13 TeV gluino implies a 2 TeV Bino and a 4.2 TeV Wino. If Higgsinos are LSPs, lighter than the 2 TeV Binos, no exclusion is expected from Bino NLSP production nor Wino NNLSP productions (see Fig. 5). No exclusion is also expected when the Higgsino is the NLSP with mass between 2 and 4.2 TeV (see Fig. 7).
The AMSB scenario is more interesting, as it predicts a larger gluino-wino mass splitting.
The relation, M 1 : M 2 : M 3 3 : 1 : 8 -renormalized at 2 TeV by including two-loop gauge coupling runnings and one-loop threshold corrections [13,45] -implies that Winos can be as light as 1.6 TeV (while the 5 TeV Bino is irrelevantly heavy) when the gluino is above 13 GeV. If Higgsinos are lighter than Winos, the 1.6 TeV Wino NLSPs can probe up to 1.2 TeV Higgsino LSPs (see Fig. 5). If Higgsinos are NLSPs, however, a 1.6 TeV Wino LSP is not expected to be excluded from Higgsino NLSP pair productions (see Fig. 6).
In all, there are chances that multi-lepton searches of NLSPs can lead to an earlier discovery of SUSY than direct gluino searches, for example, in the AMSB scenario.

IV. DETECTOR OPTIMIZATION ISSUES
In this section, we briefly discuss possible detector developments that can improve and optimize our multilepton searches. The pair of leptons coming from heavy electroweakino decays, NLSP → LSPZ, Z → , will be collimated at a 100 TeV collider, if the mass splitting between the NLSP and the LSP is sizable. In Fig. 11, we show distributions of minimum angular separation between any two leptons from the 3 and OSDL signal events. Typical angular separation between the pair is ∆R ∼ m Z /2m N LSP , which can be smaller than the lepton separation criteria we use in our analysis, ∆R > 0.05. In that circumstance the two leptons will be reconstructed as a single jet. This can degrade the performance of multi-lepton searches.
We illustrate this issue in the left panel of Fig. 12, where we show the dependence of the 3 results on the lepton separation criterion. In particular, we present the luminosity needed for the 95% CL exclusion with separation criterion varied between ∆R > 0.1 and 0.05. With the ∆R > 0.1 criterion, the degradation of the 3 reach compared to reach obtained with ∆R > 0.05 begins to appear at NLSP masses at around 2.5-3.0 TeV with about 1/ab of data. For example, the luminosity needed to probe a 3.5 TeV Wino would be almost doubled with the separation requirement ∆R > 0.1, compared to the one with ∆R > 0.05.
We also verify that leptons are usually well separated in the OSDL (and SSDL) channels,  Finally, a 100 TeV collider will be an environment full of hadronic activity. Lepton-jet isolation techniques can thus be important. As an example, if we relax the isolation criteria to allow soft jets nearby a lepton (specifically, if a nearby jet is softer than the lepton, they are separately and properly reconstructed), we can have up to 30% more signal samples.
Such intrinsic uncertainty may reside in our analysis of the future high-energy collider, and more careful assessment will be useful when detector performances become known.

V. CONCLUSIONS
In this paper, we have studied the discovery prospects of multi-lepton searches of electroweakinos at a 100 TeV proton-proton collider. In particular, we have studied the 3 , opposite sign di-lepton (OSDL) and same sign di-lepton (SSDL) final states and considered various possible NLSP-LSP combinations in the MSSM. We summarize our results in  These results represent a great improvement from the expected discovery reach at the 14 TeV LHC [46,47]. Most notably, the whole parameter space of a Higgsino-like WIMP dark matter can be probed via Wino NLSPs if the Wino is lighter than about 3.2 TeV and not too close in mass to the Higgsino. Wino-like dark matter, on the other hand, is not fully probed in these searches as Wino DM is required to be quite heavy (∼ 3.1 TeV) and Higgsino NLSP production cross section is smaller.
We find that the 3 search, usually, has the highest signal reach. In this search, important parameter dependences may arise from tan β and the signs of gaugino and higgsino masses.
In the case of Higgsino LSPs or NLSPs, the results do not depend sensitively on them, as implied by the Goldstone equivalence theorem and the Higgs alignment limit [19]. As a result, the models with light Higginos (LSPs or NLSPs) can naturally serve as true simplified models with fixed BRs of NLSP neutralinos: BR(Z) = BR(h). On the other hand, if Higgsinos are heavier than Wino NLSPs and Bino LSPs, the parameter dependences introduce various features in the reach plot, as shown in Fig. 8 and discussed thereafter. The 3 reach is highest when the BR into the W Z channel is maximal.
The OSDL search has advantages in the sense that parameter dependences are weaker and the lepton collimation issue is almost absent. When the 3 reach is limited by these factors, e.g. in the scenario with very heavy Higgsinos in which the dominant W h channel only leads to a weak reach, the OSDL channel can still provide a complementary sensitivity. Furthermore, the SSDL signal is relatively good in the low-mass small-gap region, where the soft lepton identification becomes difficult. We comment on the small-gap region, for which we did not perform a careful study. Hard initial state radiations plus soft leptons plus correlated large MET would efficiently probe the small-gap region with m NLSP − m LSP 50 GeV [37,48]. This could also be studied with our kinematic variables, but we leave more dedicated assessments for future studies.
We have also studied when the direct electroweakino searches can offer an earlier discovery than the direct searches of gluino pairs. In the AMSB models, light Wino NLSPs decaying to lightest Higgsino LSPs can be discovered earlier than the gluino pairs. In other models, however, the gluino pair search is generally better.
Searching for new physics at multi-TeV scales also presents new challenges. Our study highlights a few of them. First of all, the decay products, in particular the Z boson, can be very boosted. Therefore, the two leptons from Z decays will be collimated and may fail the conventional lepton isolation cuts. Secondly, measuring the properties of a energetic lepton with p T > TeV, such as its flavor and charge, can be challenging. As we emphasize, both of these effect can significantly impact the reach. It will be important to optimize such performances in detector design and search strategies.
Note Added: As this work neared completion, Ref. [49] appeared, whose scope partially overlaps with ours. One notable difference of results is that our 3 reach is stronger due to our smaller lepton separation criteria. Furthermore, we have studied several scenarios in addition to just wino-higgsino, and introduced additional helpful kinematical variables and discussed their optimizations for various multilepton searches. In both panels, the luminosity needed for 95%CL limit from the 3 search is plotted using the Wino-Bino simplified model (see text for more details).
reach of those searches utilizing high-energy cuts much higher than the masses of particles because kinematic distributions at relevant high-energy regime are effectively independent of particle masses.
The scaled-up result is shown as the red-solid line in Fig. 14. In the following subsection, we compare this curve with the results we obtain varying several uncertainties; and we will see that they agree within reasonable uncertainties.

Uncertainties From Unaccounted Effects
In this section, we assess the impacts of potential systematic uncertainties, background normalization and the required minimum number of signal events after all cuts. We parameterize the first two sources of uncertainty in the signal significance σ as where S and B are the number of signal and background events after all cuts. The systematic uncertainties are multiplicatively parameterized with δ (δ = 0 means no systematic errors) and the background normalization is denoted by N . The background normalization (N > 1) may effectively account for subleading processes and reducible backgrounds that we did not simulate.
In the left panel of Fig. 14, we vary N and δ within N =1-1.5 and δ=0-0.3. The scaledup ATLAS 8 result mostly falls within this uncertainty band. In the right panel of Fig. 14, we also vary the condition of minimum S for the number of events that will be needed for the discovery. For S in the range 2-8, the search capacity is not significantly modified and ATLAS 8 results mostly fall within the band. Recall that we have chosen S > 5 throughout in this paper. We conclude that naively scaling up the LHC 8 ATLAS bound agrees reasonably well with our Simplified model results.