Second Stop and Sbottom Searches with a Stealth Stop

The top squarks (stops) may be the most wanted particles after the Higgs boson discovery. The searches for the lightest stop have put strong constraints on its mass. However, there is still a search gap in the low mass region if the spectrum of the stop and the lightest neutralino is compressed. In that case, it may be easier to look for the second stop since naturalness requires both stops to be close to the weak scale. The current experimental searches for the second stop are based on the simplified model approach with the decay modes $\tilde{t}_2 \to \tilde{t}_1 Z$ and $\tilde{t}_2 \to \tilde{t}_1 h$. However, in a realistic supersymmetric spectrum there is always a sbottom lighter than the second stop, hence the decay patterns are usually more complicated than the simplified model assumptions. In particular, there are often large branching ratios of the decays $\tilde{t}_2 \to \tilde{b}_1 W$ and $\tilde{b}_1 \to \tilde{t}_1 W$ as long as they are open. The decay chains can be even more complex if there are intermediate states of additional charginos and neutralinos in the decays. By studying several MSSM benchmark models at the 14 TeV LHC, we point out the importance of the multi-$W$ final states in the second stop and the sbottom searches, such as the same-sign dilepton and multilepton signals, aside from the traditional search modes. The observed same-sign dilepton excesses at LHC Run 1 and Run 2 may be explained by some of our benchmark models. We also suggest that the vector boson tagging and a new kinematic variable may help to suppress the backgrounds and increase the signal significance for some search channels. Due to the complex decay patterns and lack of the dominant decay channels, the best reaches likely require a combination of various search channels at the LHC for the second stop and the lightest sbottom.

contributions from the stop loops [1][2][3]. This could happen if the stops are heavy and/or the trilinear A t term of the stop sector is large [4][5][6][7][8]. To keep the fine-tuning minimal, it is preferable to have a large A t term so that the stops masses can be lowered to ∼ 1 TeV or below. A large A t term implies large off-diagonal masses of the stop mass matrix so there will be a substantial mixing between the left-handed and the right-handed stops in the mass eigenstates. As a consequence, there will also be a sizable mass difference between the two stop mass eigenstates.
As a key to the naturalness problem, the stops have been extensively searched for at the LHC. Assuming that the lightest neutralinoχ 0 is the lightest supersymmetric particle (LSP) and is stable, the search limit for thet → tχ 0 decay (assuming 100% branch fraction) has reached ∼ 860 GeV for mχ0 250 GeV at the current Run 2 of LHC [9][10][11][12][13][14]. it is difficult to be distinguished from the SM top pair production background and it is still possible to have a relatively lightt 1 . There have been studies trying to identify useful variables to probe this compressed region but the reach is limited [15][16][17][18]. A more promising strategy is to consider the stop pair production with a hard initial state radiation (ISR) jet, then the neutralinos are boosted in the opposite direction to the ISR jet, giving rise to some MET. It may have a discovery reach up to mt 1 ∼ 400 − 500 GeV at LHC 13 TeV with 300 fb −1 [19][20][21][22].
Since naturalness needs both stops to be not too heavy, ift 1 happens to lie in the compressed region, it may be easier to search fort 2 even though it is heavier. Indeed, both ATLAS and CMS have performed searches for the heavier stop for mt 1 ≈ m t + mχ0 1 so thatt 1 escapes the detection [23][24][25][26]. These searches adopted the simplified model approach, assuming that the heavier stop decays to the lighter stop plus a Z or a Higgs boson (t 2 →t 1 + Z/h) with a 100% branching fraction. The exclusion limit for thet 2 mass goes up to ∼ 730 GeV for thet 2 → Zt 1 decay mode with 13 TeV Run 2 [26] and about 600 GeV for thet 2 → ht 1 decay mode with the 8 TeV Run 1 data [23] (the corresponding Run 2 analysis has not appeared yet).
The simplified model approach is a good strategy if there is a dominant decay channel or the search limit is dictated by a certain decay channel. One can easily recast the search result to a wide range of models which have similar decay processes and final states, by rescaling the cross sections and branching ratios. However, it is seldom a good approximation for the system of the two stops. Because the left-handed stop and sbottom belong to an SU (2) W doublet, there is always a sbottom with mass within the vicinity of the two stops. The presence of the sbottom will give additional decay modes oft 2 . In addition, there could be other charginos and neutralinos lighter thant 2 besidesχ 0 . If this is the case, there will also be decay chains going through them as intermediate states. As a result, thet 2 decays often have many decay channels without a dominant one [27,28]. Different decay channels have different final states and hence require different signal selection criteria. It makes thẽ t 2 search in the compressed region in a realistic scenario more complicated than simply rescaling the simplified model analysis.
An alternative approach to the simplified model is the pMSSM [29][30][31], which parametrizes MSSM with some modest assumptions. The assumptions include R parity conservation with the lightest neutralino being the LSP, minimal flavor violation at the TeV scale with no CP violation in the SUSY sector, and degenerate sfermion masses for the first two generations.
It contains 19 phenomenological parameters and a scan over these parameters generates a large model samples for phenomenological studies. However, if one is interested in the stop system, scanning over the full 19-parameter space may be an overkill. 1 To study the possible interesting decay patterns of the stop and sbottom system and their experimental signals one should focus on the most relevant parameters. This is the approach taken in this paper.
We divide the models into two scenarios. In the first scenario all neutralinos and charginos except the LSP are heavier thant 2 so they decouple. The only relevant particles aret 1 , t 2 ,b 1 , andχ 0 1 , whose masses and interactions are only governed by a few parameters. We scan through them and find model points with different characteristic decay patterns. We identify categories of signal channels which may be sensitive to various final states of the decay chains and study the signal significance over the backgrounds. The real search reach may require division and combination of many different channels. In the second scenario we consider additional charginos and neutralinos below the mass oft 2 , which can introduce even more complicated decay patterns. We perform the similar study as in the first scenario for the model points where the additional charginos and neutralinos play important roles in the decay chains. This paper is organized as follows. In Sec. II, we discuss the spectrum of the third generation squarks in MSSM based on the naturalness and the Higgs boson mass. We also summarize the current experimental constraints. In Sec. III, we consider some benchmark points for the stop and sbottom spectrum wheret 1 is hidden in the compressed region. The benchmark models are divided into two types, depending on whether there are additional neutralinos and charginos which can appear in the decay chains oft 2 andb 1 , We present the branching ratios of various decay chains and the fractions of final states of these benchmark points. In Sec. IV, we discuss categories of signals for thet 2 andb 1 searches whent 1 is hidden. We perform simplified collider studies for the benchmark models to explore the future sensitivities at the 14 TeV LHC. The conclusions are drawn in Sec. V. The compatibilities of our benchmark points with the current experimental constraints are examined in the Appendix. 1 A light stop study in the pMSSM approach can be found in Ref. [32].

A. Argument of Naturalness
In MSSM, after minimizing the Higgs potential, the Z boson mass is given by [33] where m 2 Hu and m 2 H d are the soft SUSY breaking masses of the H u and H d doublets at an UV cutoff scale Λ, δ Hu and δ H d represent the radiative contributions to the soft SUSY breaking masses below the cutoff, and µ is the SUSY-preserving Higgs mass parameter which is also the approximate Higgsino mass. To avoid fine-tuning on the Z mass, there should not be a large cancellation among various terms in the above equation. In particular, the radiative correction δ Hu receives the largest contribution from the stop loops: where m Q 3 , m u 3 are soft-breaking mass terms for the left-handed top-bottom doublet and the right-handed top squarks, A t is the trilinear soft-SUSY breaking of the corresponding Yukawa interaction, and µ IR is taken to be the geometric average of the stop masses mt = √ mt 1 mt 2 .
The tuning of m 2 Z due to the stop mass contribution Hu | is already at the level of 1% for m Q 3 , m u 3 ∼ 1 TeV and Λ ∼ 100 TeV [34]. Therefore, naturalness argument would prefer both stops to have masses below or around 1 TeV.

B. Higgs boson mass
At the tree level, MSSM also predicts the light Higgs boson mass to be less than the Z mass: where mt = √ mt 1 mt 2 as defined previously and X t = A t − µ cot β is the stop mass mixing parameter. From the formula one can see that without the X t term, the stop masses need to be raised to very high values in order to generate a Higgs massif 125 GeV. This would be in severe conflict with naturalness. To minimize the fine-tuning, the second term in the bracket should be large and the one-loop correction is maximized when X t = √ 6mt. Such a large X t implies a large mixing between the left-handed and right-handed stops, which has interesting phenomenological implications. 2

C. Stop and sbottom masses
The mass matrices for the stops,t L ,t R , and sbottoms,b L andb R are given by where X b = A b − µ tan β is the term related tob L andb R mixing from trilinear couplings, represent the D-term contributions.
Notice that in the limit where there is no mixing, botht L andb L masses are controlled by the soft breaking mass m Q 3 and they are expected to be nearly degenerate if m Q 3 m t , m Z , with mt L mb L . However, with a large X t , there is a significant mixing betweent L andt R and the mass spectrum of the stops will be modified. The two mass eigenstates are repelled from one another by the mixing term and the mass gap between them further increases.
Consequently, there is at least one sbottom lighter than the heavier stop. The spectrum of 2 If there are additional contributions to the Higgs quartic coupling, such as in the Next-to-Minimal Supersymmetric Standard Model (NMSSM) [38], it is easier to have a 125 GeV Higgs boson with light stops compatible with naturalness [4,39]. The stop mixing does not need to be large in that case and the stop spectrum could be more compact. Nevertheless, the results in this paper also applies to a large region of parameter space in NMSSM. the stop and sbottom sector has important implications for their decay patterns and collider searches as we will see.
To scan the MSSM parameter space we specify the parameters at the cutoff scale Λ which is taken to be 100 TeV. Since we are focusing on the spectrum of the third generation squark (and neutralino/chargino for the decay patterns), we decouple the first two generation of sfermions and third generation sleptons by setting their soft SUSY-breaking masses to 3 TeV. We also set the gluino mass to 2.5 TeV, beyond the current and near future reaches. 3 In order to generate stop masses which are potentially within the LHC Run 2 reach, the diagonal stop soft breaking masses m Q 3 and m u 3 are varied from 250 GeV to 1.4 TeV. The We use FeynHiggs [40] to generate the SUSY spectrum and to calculate the Higgs boson mass. Given the uncertainties in different approaches in the Higgs mass calculation and higher order corrections, we require the resulting Higgs boson mass to be bigger than 122 GeV as our selection criterion. 4 The masses of the two stop mass eigenstates and the corresponding mixing term X t which can satisfy the Higgs boson mass requirement are shown in Fig. 1 We can see that the lightest stop mass can be as low as 250 GeV in extreme cases, while the second stop can be as low as around 600 GeV. When it comes to how the sbottom and the second stop decay, an interesting question is the mass differences between the stops and the sbottom. In Fig. 2 we show mt 1 vs. mt 2 for allowed points. There is always a significant gap between the two stop masses due to the large mass mixing term. Most points have a mass difference greater than 300 GeV, which means thatt 2 →t 1 + Z(h) decays are always open. Fig. 3 shows the mass difference betweenb 1 andt 1 vs. the mass difference between 3 A lighter gluino does not affect the direct stop and sbottom productions. However, it may give the first SUSY signals at the LHC. Its decays through stops and sbottoms will mix into the signals for direct stop and sbottom productions, so we choose a heavier gluino to avoid this complication. 4 The effective field theory approach generally gives a lower Higgs boson mass [41] so we do not impose an upper limit on the Higgs mass. GeV.
For all points we have either mb 1 − mt 1 > m W or mt 2 − mb 1 > m W , and there is also a significant fraction of points where both inequalities hold. For mt 2 − mb 1 > m W , the decay channelt 2 →b 1 + W is open, which has not been considered in currentt 2 searches. Similarlỹ [49,50]. These decay channels should be included in searches fort 2 andb 1 since they occur naturally in MSSM. There will be even more possible decay channels if some charginos and neutralinos have masses between these stop and sbottom states. We will perform some benchmark point studies in the rest of the paper to point out the final states and channels that are relevant for the second stop and sbottom searches.

D. Current experimental bounds
The third generation squarks are extensively searched at the LHC experiments. Here we give a brief summary of the experimental constraints most relevant to our discussion.
For mt 1 ≈ m t + mχ0 1 , the current bound on thet 1 mass is rather weak. Both ATLAS and CMS performed searches for thet 2 states with the decays to Zt 1 , ht 1 . The bound on mt 2 from the most recent ATLAS Run 2 analysis is up to 730 GeV for 100% decay to Zt 1 [26].
The analysis for the ht 1 mode with Run 2 data has not come out yet. The limit from Run 1 data is about 600 GeV [23,25] and similar limits were obtained ift 2 decays to a mixture of the Zt 1 , ht 1 , and tχ 0 1 final states.

III. BENCHMARK POINTS FOR CASE STUDIES
We are interested in searches for the heavier stop and sbottom in the case when the lightest stop is hidden. Therefore we choose the benchmark points of our study to satisfy GeV where the bound is weakest. Of course, for a heaviert 1 the difference can be bigger from current bounds. For mt 1 being much closer to m 0 χ 1 , there have been several studies focusing on these scenarios [39,65,66]. We also assume thatχ 0 1 is Bino-like so that there is no nearby chargino state. Otherwise thet 1 → bχ ± 1 search would provide a strong constraint. Moreover, ifb 1 is close to or even lighter thant 1 , then the sbottom search inb 1 → bχ 0 1 decay will provide the strongest constraint. As we see from the summary in the last section, the constraint on b-jets + MET is very strong and the exclusion limit has reached ∼ 1 TeV for these decay modes. Thus, we will focus on points whereb 1 is somewhat heavier where other decay modes such asb 1 → Wt 1 are open, so that the traditional search based onb 1 → bχ 0 1 is not effective. We assume that the gluino is heavy enough so that the stops and sbottoms are dominated by the direct pair production.
Otherwise the gluino cascade decays would be the strongest probe.
Given the spectrum m 0 χ 1 < mt 1 < mb 1 < mt 2 , we can find two main classes of model points depending on whether there are other neutralinos and charginos lying between them.
We denote Type A models to have the lightest charginos and the second lightest neutralinos heavier than our second stop, so that they decouple from our discussion, given that their direct pair production rate is much smaller than that of squarks of the same mass. In Type B models, the second neutralino and accompanying charginos are lighter thant 2 so that they may appear in the cascade decays oft 2 . These neutralinos and charginos may be either Wino-like or Higgsino-like, or both. From the naturalness point of view, it is preferable to have the Higgsinos not too heavy. In addition, Winos couple to the squarks via the SU (2) gauge coupling instead of the large top Yukawa coupling. The branching fractions of stop and sbootom decays through the Winos are often small. Then the decay patterns are mostly similar to the Type A models. Therefore for Type B models we focus on the cases where the relevant charginos and neutralinos are mostly Higgsino-like. The benchmark points are selected to be compatible with the current experimental constraints as examined in the Appendix.

A. Type A
From the parameter space scan described earlier, we select several benchmark points of Type A spectrum, listed in Table I. 5 The range oft 2 andb 1 masses is chosen from ∼ 700 GeV up to ∼ 1.2 TeV. A1 and A2 have relatively light spectra which are not far from the current bounds. They may soon be discovered or excluded at LHC Run 2 with more luminosities.
Benchmarks A4-A6 have heavyt 2 andb 1 which are close to the reach limits of the 14 TeV LHC. As expected, to obtain a Higgs boson mass close to 125 GeV, these benchmark points all have large mixing |X t /mt| between the left-and right-handed stops.  Due to the large mixing term, it is typical to have a large mass gap betweent 1 andt 2 , leaving a relatively large phase space fort 2 →t 1 + Z/h decays. In the scanned parameter 5 In the first version of the paper, the A1 point has been ruled out after the release of new data at ICHEP 2016, so it is removed and the original A2 is moved to A1. We add two new benchmark points A2 and A3 which have similar decay patterns as the original A1, but with heavier spectra. The original A3-A5 are shifted to A4-A6.
space it is also common to have the mass difference mt 2 − mb 1 ≥ m W , which opens up the decay modet 2 →b 1 + W . This decay mode has not been considered by the experimental analysis oft 2 searches. In Table II we list the branching ratios of various decay modes oft 2 andb 1 for the benchmark points. The branching ratios are calculated by SDECAY [67].  As it can be seen, in Type A models, thet 2 →t 1 Z decay branching fraction is always significant, and it is dominant for A1 and A5. Thet 2 →t 1 h branching ratio is smaller than thet 1 Z branching ratio. It is controlled by the difference between two diagonal soft breaking masses. More specifically, the ht 2t1 coupling is proportional to cos 2θ t where θ t is the mixing angle diagonalizing the stop mass matrix. If the two diagonal soft breaking masses are exactly equal, m u 3 = m Q 3 , then θ t = π 4 and the mass eigenstatest 1 ,t 2 will be equal mixtures betweent L andt R . In this case the ht 2t1 coupling cancels between the contributions coming from the left-handed and the right-handed stops, and thet 2 →t 1 + h decay will be highly suppressed. In models wheret 2 →b 1 +W decay is kinematically allowed, the branching ratio of this channel increases rapidly as the allowed phase space expands, and easily becomes comparable to thet 1 + Z decay channel, as in models A2-A4. The model A6 is chosen such that thet 2 →b 1 + W decay is closed while thet 2 →t 1 h branching ratio is significant.
On the other hand, in Type A spectrum,b 1 decays predominately tot 1 W if the phase space allows. This is not covered by the current experimental searches for direct pairproduction ofb 1    in the future directb 1 pair production search analysis. In these benchmark models, the branching ratios of the direct decays oft 2 andb 1 to an LSP plus t or b are small, so the searches using these decays will not be effective.
In Table III we list the fractions of the final states in terms of t, b, W, Z, h (without their subsequent decays) of thet 2t2 andb 1b1 pair productions, aside from a pair ofχ 0 1 's which are implicitly understood. The recent advances in jet substructure techniques to tag hadronically decayed top quarks may help to identify final states which contain them [73,74], so in the list we keep the top quark instead of its decay product in the final states. As a result, all W bosons and b jets listed in Table III are coming from SUSY particle decays instead of top decays.
It is pretty common for Type A models the heavy stop pairs produce an excess in W bosons. Since each top quark itself gives another W boson in its decay, the chances for thet 2 pairs in our model points to give a final state with 4 or more W 's varies up to ∼ 70%. The high multiplicity of W bosons can lead to excesses in same-sign dilepton and multiple lepton events. As a result, these types of signals are also useful fort 2 searches in our benchmark models. Also, top pair associated with additional ZZ, Zh or hh constitutes considerable fractions of final states for some model points. They are the basis of the existing experimental searches fort 2t2 direction production. Sbottom pairs in our benchmark points predominately decay into 4W + 2b final state, which also leads to an excess in same-sign dilepton and multiple lepton signals.  The spectra of Type B benchmark points are listed in Table IV. The mass gap betweeñ t 2 ,b 1 andχ ± ,χ 0 2 are chosen to be big enough to allow sufficient phase space for decays through the additional neutralinos or charginos, such ast 2 → tχ 0 2 or bχ ± . Therefore these benchmark points have relatively heavyt 2 , and mt 2 ranges from ∼ 900 GeV to 1.3 TeV. For a more compressed spectrum, the decays oft 2 andb 1 into second neutralino and charginos are suppressed by the phase space, then the decay patterns will be similar to those of Type A benchmark points. Ifχ 0 2,3 ,χ ± are lighter thanb 1 ,b 1 can also decay to bχ 2,3 or tχ ± (if kinematically open), in addition thet 2 → bχ ± , tχ 0 2,3 decays. The branching ratios of SUSY particle decays for the benchmark points are listed in Table V.  From the Table V we see thatt 2 → tχ 0 2,3 decay branching ratios are usually small, due to the phase space suppression. Thet 2 → bχ ± branching ratio, on the other hand, can get up to about 1/4 for model B4. Similarly, theb 1 → tχ ± branching ratio is typically small but b 1 → bχ 0 2,3 branching ratios can be quite big. In model B2, the sum of the branching ratios of b 1 → bχ 0 2 and bχ 0 3 is close to 70%. The heavy neutralinos tends to decay toχ 0 1 + h/Z, unless the decay channel to tt 1 is open (as in model B4), in which case it becomes the dominant decay channel. The chargino decays to bt 1 or Wχ 0 1 . Which branching ratio is larger depends on the model point.
Similar to the Type A case, we can list the fractions of the final states of thet 2t2 andb 1b1 pair productions, in terms of t, b, W, Z, h aside from theχ 0 1 . Because there are many more possible decay channels for Type B models, we only list the final states of a single decay chain in Table VI. The complete final states can be obtained by simply squaring the Table. In addition to the final states which have been present in Type A models, Type B models can produce final states with a large number (up to 6) of t or b quarks. Therefore, search channels for multiple tops or bottoms could be interesting and important for some Type B spectra (e.g., B4).

IV. COLLIDER STUDIES FOR LHC 14 TEV
Given the complex decay chains and many possible final state combinations of the second stop and the sbottom, we expect that there exist many possible signal channels. The best experimental reach may come from a combination of different signals. In this section we perform a rudimentary collider study for the benchmark points discussed in the previous section for the 14 TeV LHC and point out the interesting channels. Even though some of the channels were not considered in thet 2 simplified model analyses by ATLAS and CMS, most of them have been used in other SUSY or new physics searches. A purpose of this study is to point out their relevance for thet 2 andb 1 searches. These existing analyses could readily be adapted to the current case, maybe with some optimizations of cuts for the current scenario.
We adopt the backgrounds generated by the Snowmass 2013 Energy Frontier Simulation group [81,82]. No-pileup effects are included for either the signals or the backgrounds. For this analysis the dominant background is the top-pair production, and top pair plus an extra boson (W , Z or h). We also include other backgrounds such as the single/multiple boson(s) with jets, and the single top production.

A. Basic cuts and tagging for backgrounds and signals
The lepton and b jet tagging efficiencies are taken to be the same as those of the Snowmass 2013 Energy Frontier Simulations. Additionally, we drop all leptons having a ∆R 0.4 with any jets. For both e and µ, we require them to have p T > 15 GeV and η < 2.5. We will refer to these isolated leptons simply as leptons for the rest of our discussion.
Each jet candidate is required to have η < 2.5 and p T > 25 GeV, which is rather loose since our analysis is not very sensitive to non-b-tagged jets. The maximum tagging efficiency for b jets is ≈ 70%. The b-jet mis-tagging rate from light flavors is 0.1%. A b jet candidate must have p T > 30 GeV. Further selection rules for leptons and jets will be described in detail later.
Since all SUSY signals we discussed have two LSP which could lead to significant MET, an event with a higher MET is preferred, especially for those with no or only one lepton in the final state. On the other hand, SM background events with more leptons could get a large MET from the associated neutrinos. Therefore, for a higher lepton multiplicity we would have a lower the MET cut in order to include more signal events. As a result, we set up a preliminary criteria based on MET: for all-hadronic or one-leptonic channels, all signal events are required to have a MET greater than 200 GeV. On the other hand, events with two/more than two leptons would be vetoed if their MET is less than 150/100 GeV.
To further suppress the backgrounds, scalar H T could provide a good discrimination between the signal and backgrounds, since we are interested in the pair production of heavy particles. We require that each event should have scalar H T ≥ 800 GeV as a preliminary selection rule.
A pair of opposite-sign leptons of the same flavor can come from Z decays. The Z boson produced int 2 decay could be somewhat boosted due to the large mt 2 − mt 1 . To define the Z candidate, we require that the opposite-sign same-flavor lepton pair lepton pair to have | (Σp ) 2 − m Z | 10 GeV and ∆R < 1.5.
Opposite-sign lepton pairs also often arise from SM background such as tt. For our signals,t 2 pair decays usually produce extra W/Z bosons. If they are boosted and decay hadronically, they may be tagged to help discriminating signals and backgrounds. We follow the method presented in Ref. [83]: A Cambridge-Archen jet algorithm [84] is adopted to identify fat jets from vector boson decays, with ∆R = 0.8 and p T > 200 GeV. Any fat jet constructed this way with a invariant mass between 60 and 110 GeV would be our vector jet candidate. Furthermore, we require N -subjettiness [85] τ 21 = τ 2 τ 1 < 0.5, which means that its substructure is more likely to have two subjets rather than coming from QCD backgrounds. with an h or Z decay to a pair of b quarks andt 1t1 cascade to a pair of tops can give us 4 or more b jets. Thus, requiring more than 3 b jets in each event would be a good way to handle SM backgrounds [86]. On the other hand, signal events with two or less b jets would be overwhelmed by SM backgrounds such as tt production. Therefore, for events with Based on these considerations we have tried various search channels by N and N b . The set of useful signal channels that we found are classified as: 1. No lepton and no less than 3 b jets with a large MET and no less than 6 jets in total (0 3b).
2. 1 lepton and no less than 3 b jets and no less than 6 jets in total (1 3b).
3. 2 opposite-sign (OS) leptons forming a Z candidate with no less than 2 b jets, and 5 or more jets in total (Z2b).
4. 2 same-sign (SS) leptons with at least one b jet, also no less than 5 jets in total (SS + nb).
5. 3 or more leptons and at least one b jets, the number of jets in total 2 (Multi-).
Different channels are classified as such that for zero or one lepton channels, more b jets and total number of jets are required to optimize the sensitivity. As the lepton multiplicity increases, we loosen up the requirements on N b , N j and MET in order to keep more signal events. Some of the channels are essentially the same as those which are already used in experimentalt 2 searches. Others are also close to some other SUSY or new physics searches but have not been applied tot 2 searches. We also identified some additional requirements which may enhance the signal and background discrimination in some channels. For each signal channel, it would be beneficial to further divide into signal regions based on various energy distributions to utilize the possible different distributions between signals and backgrounds. This requires more sophisticated event simulations to produce accurate event distributions. For simplicity, here we will treat each signal channel as a whole and leave more detailed analyses for the experimental collaborations. In this channel, we require no isolated lepton in the final states, N b 3, and the total number of jets 6. For such fully-hadronic events to be triggered, we require that the leading jet has p T 250 GeV and two subleading jets have p T 90 GeV. For b jets we also require at least two b jets to have p T 90 GeV and the rest of b jets to have p T > 30 GeV.
A large / E T cut is imposed in this channel to greatly suppress the contribution from QCD background, allowing us to utilize the Snowmass 2013 Energy Frontier Simulation results.
The / E T distributions for some benchmark signals and the background are shown in Fig. 5. We can see that bothb 1b1 andt 2t2 events on average have higher / E T than background events. We impose / E T > 280 GeV for event selections in this channel.  The number of signal events of each benchmark model and the background events passing the selection in this channel for a 300 fb −1 integrated luminosity is listed in Table VII. We can see that the 0 3b channel can be useful for botht 2 andb 1 searches. Naïvely one might expect that this channel is more useful for thet 2 search than theb 1 search because the Z or h bosons fromt 2 decays give the extra b jets. This is true for some (e.g.,t 2 has a large branching ratio in decaying to h/Z in A1, A5, A6, which produces more b jets on average) but not all of the benchmark models. For example, in some Type B models (in particular B2), Z or h bosons can also arise from the cascade decays ofb 1 through heavier neutralinos, sob 1 can provide comparable number of events ast 2 . Even though in Type A modelsb 1b1 pair production can only give two b jets at the parton level, additional b-tagged jets can arise from QCD radiations and mis-tagged light jets. In addition,b 1 has a larger production cross section thant 2 because it is lighter. As a result, the number of events fromb 1 and fromt 2 can be comparable in many Type A models, and this channel can also be sensitive tob 1 pair production. Among the heavier benchmarks A4-A6, this channel is most sensitive to A6 and its contribution mainly comes fromt 2 because of the larget 2 → ht 1 branching ratio.
For Type B models, B3 has fewer signal events due to the more compressed spectrum which results in softer final state particles and hence a lower signal efficiency. This is also true for other signal channels discussed later. For B4, even though the production cross sections oft 2 andb 1 pairs are small, the final state particles are harder due to the large mass splittings. The signal efficiency is better. In addition, there are significant branching ratios oft 2 andb 1 decaying to tbb and ttb which also give a higher number of b jets.
2. One lepton with three or more b jets (1 3b) In this channel, the isolated lepton is required to have p T 25 GeV. We also require more  integrated luminosity for each model point.
This is one of the main channels for current experimentalt 2 searches [23,25]. From Table VIII we see thatb 1 could also give nearly comparable contributions for many benchmark points (except B2 where W 's are not produced inb 1 decays), due to the larger cross section for being lighter.

Opposite-sign dilepton (Z) channels
Events with an OS lepton pair and b jets would be dominated by SM tt production.
Here we define the OS lepton channels by having two OS leptons with p T > 15 GeV, with / E T > 150 GeV. Also each event is required to have 2 b-jets with p T > 30 GeV and 5 jets of p T > 25 GeV in total. A rough estimation for this channel predicts ∼ 800 background events and 10 signal events for those heavy benchmark points such as A4-A6 or B2-B4, yielding a significance too low to be useful. Therefore, additional requirement is needed to suppress the background and we focus on the case where the lepton pair come from the Z decay, since there is always a significant branching ratio of producing Z's int 2 decays. In addition, Z can also arise from heavier neutralino decays in Type B models. To be identified as a Z, the OS leptons of the same flavor is required to have ∆R < 1.5 and an invariant mass |m − m Z | < 10 GeV.
The Z2b is also a standard search channel for thet 2 pair production [23][24][25]. There is also an attempt to explain the ATLAS Z+jets+MET excess [87] by the two mixed stop system [88]. Here we notice that most signal events contain additional W or Z bosons. Some of them may be boosted if they come from the decay of a heavy particle. We therefore explore the possibility that additional vector tags may be helpful with the signal and background discrimination. We divide the events passing the above requirements into exclusive channels with no vector tag (Z2b) and with at least one vector tag (Z2bV ). They are listed in Table IX. We see that the channel with a vector tag in general does better than without the vector tag, especially for model points with larger mt 2 − mt 1 and higher BR(t 2 → Zt 1 ) (e.g., A1, A6 and B4). Since the vector-tagging technique we used here is rather crude, further improvements with advanced vector-tagging techniques are possible.
These channels are in general more useful for thet 2 search, because in most benchmark models, Z is rarely produced inb 1 decays. The exceptions are B2 and B3, where Z can be produced from the neutralino decays. However, the signal efficiency for B3 is small due to its compressed spectrum. Only for B2b 1 can give a comparable contribution tot 2 .
The significance of this channel may be further improved if one can suppress the fake Z  With multiple leptons it is defined as In the SM fake Z events, since we require ∆R 1.5, the MET given by two neutrinos from W decay would tend to be in the same direction as the fake Z direction. On the contrary, there is much less such correlation for SUSY events, since Z produced byt 2 decay can have a different direction from the MET. As it can be seen in both plots in Fig. 7, most of the background events have a small L and a suitable cut on L can increase both the significance and S/B ratio effectively. To enhance the signal significance, we put an additional L > 40 GeV cut for both Z2bV and Z2b channels besides the cuts aforementioned. The numbers of events after the cut are listed in Table X. We can see that it significantly reduces the remaining background events while retaining most of the signal events. is the same-sign dilepton which is relatively rare in the SM. For this study we define the same-sign dilepton channel to have two leptons of the same charge and p T > 15 GeV, with / E T > 190 GeV for each event. Also, we require at least one b-jet with p T > 30 GeV and the total number of jets of p T > 25 GeV is required to be more than 4. We further divide these events into two channels: SS1b for events with exactly one b-tagged jet and SS2b for events with two or more b-tagged jets.
Ignoring the contribution of misidentified leptons, only a few SM processes can generate a pair of same-sign dilepton, such as multiple vector boson or tt+ boson production, etc. In our case, the dominant background is ttW/ttZ/tth production. However, these processes'  contribution suffer from small production cross sections (∼ 3 pb in total). The distributions of the signal and background events for the SS1b and SS2b channels in / E T are shown in Fig. 8. We see that our / E T cut can further reduce the SM backgrounds.
One can see from Table XI that the numbers of signal events are comparable for SS1b and SS2b channels. However, SS1b has much more background events due to mis-tagged b jets, so SS2b is expected to have better reaches. In general, for benchmark points wherẽ b 1 has a large branching ratio of decaying tot 1 W (i.e., other than B2 and B3), more signal events come fromb 1 due to the larger production cross section. Nevertheless,t 2 can also give a significant contribution. Ast 2 pair decays may produce more than 4 W 's in the final states, the probability of getting same-sign dileptons could be helped by the combinatorial factor. Also, in models where the mass difference betweenb 1 andt 1 is small, the leptons coming fromb 1 are softer and hence have a lower signal efficiency. For example, A1 and A2 have similar production rate forb 1 pairs, but the signal efficiency of the latter is almost halved because of its smallb 1 -t 1 mass difference. In this case, the contributions fromt 2 and b 1 can even be comparable.
The same-sign dilepton signals, although not used int 2 search yet, have been applied to many other new physics searches. Interestingly, excesses in the same-sign dilepton channel with b-jets and MET are found in both Run 1 and Run 2 of LHC in many separate analyses by the ATLAS and CMS collaborations. These include CMS SUSY Search [70], ATLAS SUSY Search [71], CMS tth Search [89], ATLAS Exotica Search [90], and ATLAS tth Search [91] in Run 1 and ATLAS tth Search [92], CMS tth Search [93] in Run 2. On the other hand, there are no significant excesses in Run 2 SUSY searches [64,94]. The SUSY analyses were based on the simplified model of sbottom decayb 1 → t + (χ ± 1 → W ± +χ 0 1 ). However, the branching ratio of such a decay chain is never close to 100% due to the presence of other decays (b 1 → b +χ 0 2 orχ 0 1 ), and these other decays generally give stronger constraints as we saw in the previous section. Ref. [95] proposed to explain the excesses from the right-handed stop production with the decayt R → t + (B → W ± +W ∓ ) whereW ∓ is closely degenerate with the neutral Wino which is assumed to be the LSP, and hence its decay products are too soft to be seen. With a suitable arrangement of the spectrum, the branching ratio of this decay chain can be close to 100%, and the excesses could be explained by at R of ∼ 550 GeV.
The spectrum in our study provides an alternative way to explain the excesses. From Table III, we can see that the sbottom pair decays to the 2t + 2W + / E T final state which gives the desired signals are close to 100% in Type A models. In addition, a substantial fraction oft 2 pairs also give ttW W + / E T + X final states which contribute to the signal.
We expect that our benchmark models with light spectra may produce same-sign dilepton both channels are shown in Fig. 9. We see that there are still substantial backgrounds for both channels. The numbers of background and signal events for the benchmark points with a 300 fb −1 integrated luminosity are listed in Table XII. We note that Z bosons are mostly produced int 2 decay so Zb signal ist 2 specific. As a result the Zb channel has less total signal events than the 3 b channel. Anyway, the large number of background events make either channel less effective for our benchmark models than the previous channels.

Other potential channels
As we discussed, the decay chaint 2 → Wb 1 → W Wt 1 → W W tχ 0 1 is common for a typical spectrum and has a significant branching ratio in several benchmark models (e.g., A2-A4).
In this case there can be up to 6 W 's in the final state, which could give rise to same-sign trilepton events. The SM background would be extremely low. The signal event rate is low so one does not expect it to be the discovery channel. However, it may provide useful information about the spectrum after the discovery.
For type B models, the long decay chains may even produce multiple t's in the final states. This occurs in the benchmark B4, where thet 2 decay can give 3 t's and theb 1 decay can produce ttb. With 6 tops it can also produce same-sign trilepton events. In addition, specialized searches for multiple top final states such as Ref. [96] may be sensitive to this type of spectrum, although it is not expected to be the first discovery channel either.
The systematic uncertainties in general depend on signal channels. Without knowing the exact numbers we will calculate the significances with the assumption of a 10% uncertainty in background normalization for each channel and compare them with the significances obtained without systematic errors to see their effects. The result for Type A and Type B models by combining signal events from botht 2 andb 1 are listed in Table XIII and      b 1 , they can appear in the decay chains and make the decay pattern even more complex.
In this paper, we perform a study of collider searches oft 2 andb 1 at 14 TeV LHC assuming thatt 1 is hidden in the compressed region. The study is based on general MSSM but focuses on the stop and sbottom sector. The spectra are divided into two types, depending whether there are additional charginos and neutralinos besides the LSP below thet 2 mass. We derive the branching ratios of various decay modes and obtain the fractions of possible final states.
From there we can identify potentially useful signal channels for thet 2 andb 1 searches. In additional to the standard signals based on multi-b-jets and leptonic decaying Z's of current experimental searches, we find that same-sign dilepton and multi-lepton signals are also important fort 2 andb 1 searches, because many benchmark models produce large fractions of multi-W final states. The same-sign dilepton excesses observed in the LHC Run 1 and Run 2 data may be explained by some of our benchmark models if they turn out to be real. For the standard leptonic decaying Z, we find that additional vector-tags and a new kinematic variable which we called "leverage" can further help to increase signal significance.
Due to the complex decay patterns, which signal channels are most useful fort 2 and b 1 searches depend on the models and spectra. More often than not there is no dominant decay mode and a combination of many different signal channels is needed to obtain the best reach. Also, most signal channels receive contributions from botht 2 andb 1 . Some signals with a reconstructed Z may be thought as moret 2 specific. However, in Type B models, Z can also appear in theb 1 decay chain fromχ 0 2 orχ 0 3 decay if they are lighter thanb 1 . It is therefore difficult to perform independent searches fort 2 andb 1 in a realistic scenario.
Observation of signals in multiple channels and their kinematic distributions will be needed to help disentangling the ultimate underlying theory and its spectrum. In this Appendix, we check our benchmark models against the experimental constraints from the most recent LHC 13 TeV Run 2 results. For our analyses, the most relevant search channels from current LHC public results are multiple-b jets with 0 or 1 lepton and samesign dilepton with one or more b-jets. Other channels are either less significant such as multi-lepton channels, or not available yet such as the Z2b channel.

Multiple b + 0/1
For multiple b + 0/1 channels, the strongest constraints come from the searches of gluinos decaying via stop or sbottom [98]. In order to test the viability of our benchmark points, we adopt a cut similar to the ATLAS study: All candidate jets should have p T > 30 GeV with |η| < 2.5. All candidate leptons should have p T > 20 GeV with |η| < 2.5. The 4 leading hardest jets are required to have a ∆φ > 0.4 from the MET. At least 3 candidate jets should be b-tagged. We define the inclusive effective invariant mass M eff to be the scalar sum of p T of all candidate jets and leptons and MET. The transverse mass of b jets M b T,min is defined to be Min( 2p T b (1 − cos ∆φ i )), i = 1, 2, 3, the minimum is taking among three leading b jets. Finally, we define the total jet mass variable M Σ J to be the mass sum of the 4 leading fat jets with p T > 100 GeV. Five signal regions used to search forg →tt are given by: Name Definition ATLAS 95% CL limit These are then applied upon our benchmark points. For A1, we expect 0.4 and 0.9 events for two 0L signal regions. For 1L + multi-b channels Gtt1L-A/B and C, A1 gives 0.1, 0.3 and 0.5 signal events with 14.8 fb −1 . All of them are an order of magnitude less than the 95% CL exclusion limit. The next lightest benchmark point A2 contributes 0.2, 0.7, 0.03, 0.08 and 0.2 to these 5 signal regions. Other Type A benchmark points yield smaller numbers.
For Type B models, B1 would contribute 0.2 and 0.5 events to two 0L signal regions, and 0.06, 0.2, 0.3 events for the 1L regions. The contributions from other type B benchmark points are even less. The smallness of the number of events is partly due to the hard cuts imposed in the analyses which are designed for the gluino search. Therefore, none of our benchmark points are excluded by the multi-b +0/1 lepton searches.

Same-sign dilepton +b jets
For this signal we compare to the ATLAS same-sign dilepton SUSY search [64]. The most relevant signal region is SR1b and SR3b. We require each candidate jet must have p T > 25 GeV and |η| < 2.5. Each candidate lepton need to have p T > 10 GeV and |η| < 2.5.
The signal requires a pair of same-sign lepton or more than 3 leptons. We find that at 13 TeV and the same luminosity, the lightest benchmark A1 gives 8.0 events for SR1b and 0.9 events for SR3b, thus is not excluded. Other benchmark points produce even less events.
For instance, A2 gives 4.9 and 0.4 for these two signal regions while B1 gives 2.5 and 0.2.
Note that with more data, A1 might be excluded soon, while testing other benchmark points may take longer time because their contributions are well below the upper limits.

CMS jets+MET data
The models points can also be constrained by SUSY searches with multi-jets and MET.
In particular, CMS showed a deficit in the aggregate search region SR11 in "Additional As a test of our benchmark models, the lightest A1 point gives 35 events for SR11, which roughly equals the 1σ uncertainty of the predicted background events, and A2 contributes about 22 events. They are in some tension with the observed data, but not much more than the SM itself. We note that among the 12 combined signal regions, SR11 is the only one showing a significant deficit. We hope that with more statistics and a cross-check with ATLAS will clarify this situation soon.