Boosting Stop Searches with a 100 TeV Proton Collider

A proton-proton collider with center of mass energy around 100 TeV is the energy frontier machine that is likely to succeed the LHC. One of the primary physics goals will be the continued exploration of weak scale naturalness. Here we focus on the pair-production of stops that decay to a top and a neutralino. Most of the heavy stop parameter space results in highly boosted tops, populating kinematic regimes inaccessible at the LHC. New strategies for boosted top-tagging are needed and a simple, detector-independent tagger can be constructed by requiring a muon inside a jet. Assuming 20% systematic uncertainties, this future collider can discover (exclude) stops with masses up to 6.5 (8) TeV with 3000 fb^-1 of integrated luminosity. Studying how the exclusion limits scale with luminosity motivates going beyond this benchmark in order to saturate the discovery potential of the machine.


I. Introduction
Exploring the nature of our Universe at the smallest possible scales is the primary goal of the particle physics community. This pursuit will require extending the energy frontier program beyond the 14 TeV LHC.
Recently the idea of building a 100 TeV circular proton-proton collider has gained momentum, starting with an endorsement in the Snowmass Energy Frontier report [1], and importantly followed by the creation of two parallel initiatives: one at CERN [2] and one in China [3]. For some recent studies of the capabilities of a 100 TeV collider see [4][5][6][7][8][9][10][11].
Regardless of what is discovered during the upcoming run of the LHC, data from the 100 TeV machine will still be utilized to push new particle searches to higher mass scales. The existence (or absence) of these states could have a dramatic impact on the way we think about fundamental questions. Of particular interest to this work is the question of weak scale naturalness, and specifically the possibility of TeV-scale top partners. Any discoveries of such particles at the LHC would likely require further study at a higher-energy machine.
However, even in the event that the LHC does not find any top partners, this program will continue to be of central importance by pushing the fine-tuning of the Higgs mass into qualitatively new regimes.
Here we focus on the stop in supersymmetric (SUSY) extensions of the Standard Model (for a SUSY status update after Run I of the LHC, see [12]). Naturalness considerations imply that the stops should be light [13,14] in order to regulate the Higgs mass, while the masses of first and second generation squarks are less constrained. Explicit models that realize the so-called "natural SUSY" spectrum have been constructed [15][16][17][18][19][20][21], and often the dominant collider signatures can be reduced to a set of now-standard Simplified Models [22] involving only the third generation squarks, a neutralino, and a gluino [23][24][25]. A discovery of stops, or at least an understanding of the allowed parameter space of these models, has direct implications for weak-scale naturalness.
We study the stop-neutralino Simplified Model, in which the stops are pair-produced, and each stop decays to a top and a stable neutralino 1 . This signature is well suited to compare the physics implications of different machine parameters such as √ s and total integrated luminosity.
Searches for direct stop production have been carried out at both ATLAS and CMS, providing limits on the stop mass of ≈ 800 GeV with 20 fb −1 at √ s = 8 TeV [28][29][30][31][32]. The high-luminosity upgrade of the LHC (HL-LHC) is expected to deliver 3000 fb −1 of data at √ s = 14 TeV, allowing for a discovery reach of ≈ 800 GeV stops and an exclusion reach of ≈ 1.5 TeV [33,34].
Beyond naturalness considerations, this study is motivated by the exploration of new kinematic regimes in top physics. In 100 TeV collisions, the tops from stop decays are so highly boosted that current LHC analysis strategies, usually based on resolving the individual decay products of the top, become ineffective. This work demonstrates that an analysis that relies on a muon inside a jet can be used to discover (exclude) stop masses up to ≈ 6.5 (8) TeV.
One issue in the specifications of the 100 TeV collider that has not yet been addressed is the integrated luminosity needed to fulfill its physics potential. The baseline integrated luminosity is taken to be 3000 fb −1 , but we also consider scenarios yielding 300 fb −1 and 30000 fb −1 . We find that 3000 fb −1 may be insufficient to saturate the physics reach of a high-energy machine.
The rest of this paper is organized as follows. Section II studies generic properties of heavy new physics decaying to boosted tops and compares the sensitivity of jet substructure techniques and muon-in-jet requirements. Section III presents a cut flow optimized for heavy stops that is based on the presence of a muon inside a jet and shows its sensitivity in the stop-neutralino mass plane; an additional analysis is presented which optimizes the reach for compressed spectra. Section IV summarizes the implications of the analysis on future accelerator and detector design, and discusses the implications of the mass sensitivity for fine-tuning.
The results presented here rely on events generated at parton level with MadGraph5 [35], showered with Pythia6 [36], and processed using Delphes [37] and the Snowmass combined detector card [38]. The Snowmass background samples [39] were used, augmented by a high statistics H T -binned QCD sample generated for this study.

II. Boosted Tops at 100 TeV
Signal events in the stop-neutralino Simplified Model include pair-produced stops ( t) that decay promptly into a top quark and a stable neutralino ( χ 0 1 ). Under the assumption that the stops are produced at rest, the boost of the top quark is given by and the resulting top jet has a typical size of ∆R ∼ 1/γ t ∼ m t /p t T . The left panel of of Fig. 1 shows the p T distribution of the leading top quark for three different stop masses (assuming a massless neutralino). For stops with a mass of a few TeV or higher, the tops from the stop decay are highly boosted with p T m t . The right panel of Fig. 1 shows the mean distance between the W boson and the b from the decay of the top as a function of m t and m χ 0 1 . Given that the jet radius chosen for this study is ∆R = 0.5, the top will on average be contained within a single jet. Stop searches at a 100 TeV collider will therefore have to probe a kinematic regime not accessible to the 14 TeV LHC, where the top p T relevant for most searches is less than a TeV. One possible tool for separating signal from background is to tag these highly boosted tops. Note that top taggers constructed for LHC energies are optimized for large radius jets with ∆R ≈ 1.0 − 1.5 (for a review, see [40]). It is therefore interesting to understand if existing algorithms are suitable for events at 100 TeV. If the top tagger depends on an intrinsic angular scale, for example the Johns Hopkins top tagger [41], then the choices appropriate for tagging boosted tops at the LHC will need to be reconsidered. In contrast, the HEP top tagger [42] does not make any assumption about the angular separation of the top decay products.
Given the magnitude of the boost being considered, separating the individual constituents of the top decay requires detector granularities higher than presently available in hadronic calorimeters. For example, a 5 TeV top jet falls within a cone size of roughly ∆R ≈ 0.07, while the typical size of a calorimeter cell at ATLAS is ∆η × ∆φ ∼ O 0.1 × 0.1 [43]. In order to understand this effect quantitatively, we generated a sample of t t and QCD events at √ s = 100 TeV, with a minimum generator-level p T cut on the leading top at 500 GeV or 5000 GeV. The hadron-level events were passed through a FASTJET [44,45] based code. This framework was validated against the results in [42] using a sample of 14 TeV events. In order to investigate the impact of finite calorimeter resolution, a simple pixelation was applied by summing particle energies within square cells whose widths were allowed to vary from 0 to 0.1. The events were then clustered using the Cambridge/Aachen algorithm [46,47], where ∆R = 1.5 (0.5) was taken for p T > 500 (5000) GeV. The HEP top tagger was applied to the leading two jets in order to determine the efficiency for tagging a top jet and the probability of mis-tagging a QCD jet. The results are shown in Fig. 2, where top-tagging is found to be insensitive to the detector granularity for 500 GeV top jets, but with a cell width 0.02 is significantly degraded for 5 TeV top jets.
The jet radius changes approximately as the inverse of the top p T , ∆R ∼ m t /p t T , so in most of the parameter space of interest for this simplified model, this toy study demonstrates that a much finer calorimeter segmentation than that used for LHC detectors will be needed to exploit substructure techniques at higher-energy colliders. On the other hand, tracking systems have much finer granularity than is needed by the HEP top tagger, so it would be interesting to explore a Particle Flow or a purely track-based approach. We leave this for future studies. Instead, we consider an alternative strategy with less sensitivity to the detector response.
When a highly-boosted top decays leptonically, or when the resulting b (or even c) quark decay yields a lepton, it is very likely that the lepton(s) will be collinear with the top jet.
Requiring a hard lepton inside a jet can therefore be used to tag boosted tops much like leptonic b-tagging techniques implemented at the Tevatron [48][49][50][51][52][53] and at the LHC [54][55][56][57]. By definition these leptons will not be isolated from nearby tracks or calorimeter activity, removing a common handle for rejecting fake leptons. For simplicity we therefore only consider the case where a muon is collinear with a jet, and assume that a layered detector design similar to that employed by LHC experiments will provide adequate rejection of fake muons. Rejection of fake electrons without the use of an isolation requirement is more detector-dependent, and is not considered here. The signal efficiency for this requirement is roughly 15%, compared to t t + W/Z efficiencies of 3%, t t efficiencies of about 2% and QCD efficiencies around 0.4%.
For the t t background, the top quarks constitute only ∼ 60% to the total jet p T in the highly boosted regime, indicating a significant contribution from additional QCD radiation.
This leads to a lower efficiency for t t than in signal events, where more of the total p T is carried by top jets.
Our results in this section ignore the impact of any additional p p interactions (pile-up) in the event. However, we expect pile-up would only degrade the performances of hadronic taggers compared to the muon-in-jet requirement. Furthermore, it has been shown [58,59] 2 Due to the structure of the Snowmass detector card, we are using generator level muons when computing the muon-in-jet requirement. This procedure was validated against a dedicated sample that was produced with no lepton isolation requirements imposed, thereby giving detector level muons inside jets. that minimum bias events do not change dramatically going from ∼ 10 TeV to ∼ 100 TeV. A larger difference between the LHC and this machine will arise from changes in instantaneous luminosity and/or bunch spacing. If they are within a factor of a few of those at the LHC, it is not inconceivable that a Particle Flow based subtraction scheme could make the performances of substructure techniques (almost) pile-up independent at a 100 TeV collider.

III. Search Strategy and Results
In the previous section we discussed some general aspects of searches with boosted tops at 100 TeV. Here we propose a detailed analysis strategy that utilizes the muon-in-jet requirement, and we show the expected discovery reach and 95% C.L. exclusion sensitivity.
In addition, we provide an alternative cut-flow that relies on isolated leptons in order to increase sensitivity in the compressed region where m t − m χ 0 1 ≈ m t .

A. Heavy Stops and Light Neutralinos
We make the following requirements: 1. At least two anti-k T jets [60] with cone parameter ∆R = 0.5 and kinematic cuts: |η| < 2.5 and p T > 1000 GeV.
2. At least one muon with p T µ > 200 GeV contained within a ∆R = 0.5 cone centered around one of the leading two jets.
3. Events with at least one isolated lepton with p T > 35 GeV and |η| < 2.5 are rejected. The isolation criterion demands the total p T of all particles within a ∆R < 0.5 cone around the lepton to be less than 10% of its p T .
4. ∆φ / E T J > 1.0, where ∆φ / E T J is the smallest |∆φ| between / E T and any jet with p T > 200 GeV and |η| < 2.5. 5. / E T > 3, 3.5 or 4 TeV. Out of the three choices, the cut is chosen for each mass point by optimizing the expected exclusion.
After imposing a muon-in-jet requirement on the background, the selected sample is composed mainly of boosted heavy quarks. The neutrinos and muons resulting from their decays will be highly collimated and the total / E T will tend to be aligned with the jet momenta. Therefore it is useful to impose an angular ∆φ cut between the / E T and all the jets. For q q, the maximum angle between each neutrino and the final q jet will be of order m q /p T . After a stringent ∆φ cut, the remaining background is then boosted t t + X events.
In particular, t t + W/Z is the dominant background in the signal region. The / E T and ∆φ / E T J distributions after all other cuts are applied are shown in Fig. 4. The low / E T region is mostly dominated by QCD, whereas the high / E T tail is dominated by The results of the cut-flow with / E T > 4 TeV for the background and three signal mass points are shown in Table I without uncertainties. We note that corrections for electroweak radiation of W or Z bosons within high-p T jets (e.g. in QCD dijets) could lead to muon-injet signatures, and at a high-energy machine these corrections can be large [61]. We expect that the ∆φ requirement will highly suppress such contributions as it already does in events where the W or Z is produced in the matrix element, but this should be verified in future studies.
As a baseline, we choose the relative background and signal uncertainty to be 20%, and an integrated luminosity of 3000 fb −1 . However, it is useful to explore the reach of this future collider for different choices of systematics and integrated luminosities, especially to study the impact of potential accelerator and detector designs. In Sec. III D we therefore show results for a range of integrated luminosities (with appropriate adjustments to the final / E T cut for optimization). In Sec. III E, we present discovery and exclusion reaches for different choices of systematic uncertainties.

B. Compressed Spectra
As the neutralino mass approaches the stop mass, both the / E T and the top p T are reduced. By relaxing some of the cuts in the previous section and trading the muon-in-jet  requirement for an isolated lepton requirement, sensitivity to this region of parameter space can be improved. Our cut-flow targeting the compressed region is: 1. At least two anti-k T jets with cone parameter ∆R = 0.5. The kinematic requirements |η| < 2.5 and p T > 500 GeV are imposed.
2. Two isolated leptons (either electrons or muons) with p T l > 35 GeV. A lepton satisfies the isolation cut when the total p T of all particles in a cone of ∆R = 0.5 around the lepton is less than 10% of its p T .
4. ∆φ / E T J, l > 1.0, where ∆φ / E T J, l is the smallest |∆φ| between / E T and any jet with p T > 200 GeV and |η| < 2.5, and any isolated lepton with p T l > 35 GeV and |η| < 2.5 These requirements yield increased sensitivity for m t 3 TeV close to the diagonal of the (m t , m χ 0 1 ) plane. Table II gives the results of this cut flow for the background and three signal mass points. Note that the / E T > 2 TeV requirement implicitly relies on the presence of extra QCD radiation in association with the signal. This implies some uncertainty on initial-state radiation that we assume is covered by the systematic uncertainties applied on the signal samples. Note that this cut-flow is much more sensitive to detector and machine details than the previous one. We therefore present it only as a a proof of principle that going to higher energies does not necessarily imply sacrificing sensitivity to compressed, i.e.  [58,59]. Most of this energy can be subtracted using common pile-up suppression techniques, so it is reasonable to expect small modifications to jet physics given the p T thresholds relevant for the models considered here. The only possible exception is the ∆φ requirement, which would be affected by resolution effects. We verified that raising the jet p T threshold to 500 GeV does not considerably impact our reach, giving us confidence that the impact of pile-up on the jet requirements will remain small. However, lepton isolation may suffer more significantly, which would impact the results for the compressed scenarios. Studies of such effects would require detailed assumptions of the detector performance, and thus we leave them for future work.

Cuts
Signal m t , m χ 0   A signal point is rejected for CL s < 0.05. Alternatively, a signal is discovered when the CL s for the background only hypothesis is less than ∼ 3 × 10 −7 , corresponding to 5σ. The expected exclusion limits and ±1σ contours are computed using ROOSTATS [62].
Stops with masses up to ≈ 6.5 TeV can be discovered when the neutralino is massless, assuming 3000 fb −1 of integrated luminosity. The exclusion reach is ≈ 8 TeV, which corresponds to roughly 110 signal events before cuts. Note that this agrees with the estimate obtained by extrapolating the number of excluded signal events at √ s = 8 TeV [63].
The searches proposed here also have good discriminating power away from the massless neutralino limit. A 2 TeV stop could be discovered in the compressed region of parameter space. It is possible to exclude neutralino masses up to 3 TeV in most of the parameter space.
All of the results presented here have been obtained with very minimal cut-flows that do not rely on b-tagging or jet substructure techniques. Additional refinements should increase the search sensitivity, at the price of making assumptions on the future detector design. for 3000 fb −1 of total integrated luminosity. At each signal point, the significance is obtained by taking the smaller CL s between the heavy stop and compressed spectra search strategies, and converting CL s to number of σ's. The blue and black contours (dotted) are the expected (±1σ) exclusions/discovery contours using the heavy stop and compressed spectra searches.

D. Different Luminosities
An open question in the design for the 100 TeV proton-proton collider is the luminosity that is necessary to take full advantage of the high center of mass energy. As cross sections fall with increased center of mass energy, one should expect that higher energy colliders require more integrated luminosity to fulfill their potential. The necessary luminosity typically scales quadratically with the center of mass energy, meaning that one should expect that the 100 TeV proton-proton collider would need roughly 50 times the luminosity of the LHC at 14 TeV.
This section shows the scaling of our search strategy as a function of the number of collected events. As the luminosity changes, we re-optimize the / E T cut. For integrated luminosities of 300 fb −1 , a / E T cut of 3 TeV is chosen. For 30000 fb −1 , a / E T cut of 5 or 6 TeV is chosen, depending on the mass point. Table III lists the number of background events for the heavy stop search and these two choices of luminosity and / E T cut. Figure 6 (7) shows the expected CL s discovery and exclusion for 300 (30000) fb −1 of integrated luminosity. For 300 fb −1 , the discovery potential is limited, but we obtain a 3σ evidence in the bulk of the parameter space. Only a narrow range of 3 − 4.5 TeV stops may be discovered for a light LSP. With 30000 fb −1 , stops of 8.5 (10) TeV could be discovered (excluded), a clear improvement over the 3000 fb −1 result.
If we model the mass reach as a function of luminosity as then we find n 7 in the 300 fb −1 to 3000 fb −1 range of luminosities and n 10 in the 3000 fb −1 to 30000 fb −1 range. This indicates that the 100 TeV collider is still gaining significant reach at 3000 fb −1 and running out reach at 30000 fb −1 . Reaching a higher integrated luminosity implies running at higher instantaneous luminosity, with potential implications for detector performance that we do not consider here. However, we expect that improvements in detector design and pile-up mitigation strategies will minimize any loss of sensitivity from harsher running conditions.    Figure 8 shows the same for the discovery potential. The expected discovery changes modestly as the systematic uncertainty on the signal is increased. However, when the background systematic uncertainty is increased to 50%, discovery becomes impossible with 3000 fb −1 ; only 3σ evidence is possible in the bulk of the parameter space. A larger background systematic uncertainty implies that it is harder to reject the background hypothesis, so a precise understanding of the backgrounds will be crucial for discovery.

IV. Conclusions
In this paper we propose a robust search strategy targeting stops that decay to a top quark and a stable neutral particle at a 100 TeV proton-proton collider. A 6.5 (8) TeV stop could be discovered (excluded) at such a machine with 3000 fb −1 of integrated luminosity.
Such an exclusion would have a deep impact on our understanding of electroweak finetuning. In the Minimal SUSY SM (MSSM), the tuning of the electroweak scale, ∆ −1 , receives a large contribution from the SUSY breaking parameters in the stop sector. A rough estimate of the minimum contribution to the Higgs mass parameter yields [24]: where Λ is the SUSY breaking scale and tan β = v u /v d the ratio of the two Higgs doublets vacuum expectation values. A 100 TeV proton collider clearly has the potential to impact our understanding of electroweak naturalness to an unprecedented degree.
However stop masses approaching 10 TeV are above the typical range motivated by finetuning considerations. Nonetheless, this range of masses could be the consequence of the Higgs mass being so far above m Z . In the MSSM, the Higgs quartic coupling must receive sizable radiative corrections to raise the Higgs boson mass from m Z to the observed value.
The largest of these contributions arise from the top sector and comes in two forms. In the effective theory below the stop mass, the first is the contribution from the top quark and is logarithmically enhanced by the running from the mass of the top squark down to the top quark. The second contribution is given by the A-terms which at low energies can be viewed as finite threshold corrections. In order to have top squarks with masses in the range accessible by the LHC14, there need to be sizable A-terms. However, many calculable frameworks for coupling a SUSY breaking sector to the visible sector result in suppressed A-terms, e.g. gauge mediation [64], anomaly mediation [65,66], and gaugino mediation [67]. These classes of theories are the ones that have the best solutions for the SUSY flavor problem and hence are amongst the most favored. In the absence of sizable A-terms, only the logarithmically enhanced top quark contributions are left to raise the Higgs mass which results in top squarks with masses in the range 6 TeV m t 10 TeV at the 2σ level (for large tan β and small values of µ) [68]. These masses are outside the reach of the LHC14, but discoverable at a 100 TeV collider. Frequently the top squarks are amongst the lighter colored superpartners, meaning that it is possible that supersymmetry will be above the reach of the LHC14. This observation provides motivation for building another energy frontier machine, even in the case where no new physics is found at the LHC14.
Beyond the theory motivation, the lessons of this study can be generalized to a wide class Furthermore we have shown that it would be desirable to achieve higher integrated luminosity than presently used in 100 TeV studies. The current benchmark of 3000 fb −1 [4][5][6][7][8][9][10][11] does not saturate the physics reach of this machine. The ideal integrated luminosity would be 10000 − 30000 fb −1 .
Designing searches for heavy stops yields a concrete example of how a 100 TeV collider is qualitatively different from the LHC. The new energy regime that this machine will explore is so far above the electroweak scale as to render traditional search strategies ineffective. On the other hand, this makes the analyst's job easier since signals and backgrounds become more qualitatively different. This is exemplified by the sensitivity that can be derived using the simple cut-flows presented in this work.