Implications of a Stop Sector Signal at the LHC

Naturalness arguments suggest that the stop sector is within reach of the Large Hadron Collider (LHC). We investigate how the observation of a third generation squark signal could predict masses and discovery modes of other supersymmetric particles, or potentially test the Higgs boson mass relation and the validity of the Minimal Supersymmetric Standard Model (MSSM) at the high luminosity LHC. We illustrate these ideas in three distinct scenarios: discovery of a light stop, a sbottom signal in multileptons, and a signal of the second (heavier) stop in boosted dibosons.


I. INTRODUCTION AND MOTIVATION
If the electroweak scale is natural, third generation squarks should be among the first supersymmetric particles to be discovered at the Large Hadron Collider (LHC). The latest results from Run II of the LHC place strong limits on their masses under various assumptions about the mass spectrum and decay channels [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Nevertheless, windows for light ( ∼ < TeV) third generation squarks still exist, and there have even been recent hints of signals of such light states (e.g. [17,18]). The discovery of a third generation squark at the LHC in the next few years remains an exciting possibility.
Such a discovery carries important implications, both theoretical and observational. One of the primary appeals of a stop sector discovery is that it is intricately tied to the mass of the Higgs boson [19][20][21]. Given the measurements of the mass and properties of the Higgs in recent years, this connection provides strong constraints on the possible values of stop masses and mixing, which, in turn, determine their decay branching ratios (see e.g. [22]). Furthermore, the left-handed stop is part of a doublet that also contains the left-handed sbottom, hence their masses are related: in particular, after mixing in the stop sector, the left-handed sbottom mass lies between the two stop mass eigenstates provided sbottom mixing is not too large.
Such correlations imply that an initial signal can enable predictions of subsequent signals at the LHC. Establishing discrepancies between the observed Higgs mass and that predicted from third generation sparticle measurements could rule out the Minimal Supersymmetric Standard Model (MSSM) as the underlying theory behind these signals, establishing the need for a nonminimal version of supersymmetry, such as the Next-to-Minimal Supersymmetric Standard Model (NMSSM). Such predictions and consistency checks can remain largely insensitive to the remainder of the supersymmetric mass spectrum.
In this paper, we study such theoretical and observational implications of a stop sector signal at the LHC within a few specific scenarios. These are not intended to provide comprehensive coverage of all possibilities, but rather offer qualitative illustrations of the various ways in which progress can be made once a signal is observed. 1 In the event of a relevant discovery of the type described here, it would be of interest to carry out the corresponding theory calculations with higher precision (i.e. higher loop level) and examine the collider aspects (i.e. event and 1 Earlier ideas using measurements to constrain parameters in the stop sector include [23,24]. background simulations) with additional care.
Finally, we elaborate on the philosophy behind the structure of this paper. MSSM parameter space studies generally scan over all parameters in the theory over some range, calculate the Higgs mass at two or three loops, and include all relevant constraints from flavor, dark matter, and other relevant aspects. While we also scan over stop parameters in this paper, our setup is manifestly different. Our studies are driven by hypothetical observations: in particular, we are interested in scenarios where stop parameters are known with some uncertainty due to observed signals, but other parameters in the underlying theory, such as the gluino mass, are not known at all. Then it becomes impossible to calculate the Higgs mass at higher order, and we instead allow the Higgs mass at one loop within a reasonable window that includes all potentially consistent regions of parameter space (see Sec. II for details). Likewise, given the lack of information on other parameters, we also do not include any constraints from flavor, dark matter, or other similar considerations that rely on additional assumptions or parameters not relevant to our study of the stop sector. A proper inclusion of such constraints or the calculation of the Higgs mass with greater precision would eliminate a subset of the parameter space points we consider in various sections in this paper; however, this would not falsify any of the statements or conclusions in these sections, but only make them sharper and stronger.
The paper is structured as follows. The basic theoretical framework and relevant observational constraints are reviewed in Sec. II. In the next three sections, we study distinct scenarios where the Higgs mass relation can be used to perform consistency checks of the MSSM frame- this section is also supplemented with a benchmark case study.

II. THEORETICAL FRAMEWORK
We denote the lighter and heavier stop mass eigenstates ast 1 andt 2 respectively. We denote the stop mixing angle as θ t , witht 1 = cos θ ttL + sin θ ttR , wheret L ,t R are the stop gauge eigenstates, so that θ = 0 corresponds to the scenario where the lighter stop is left-handed. In terms of these parameters, the Higgs boson mass at one-loop in the MSSM is [25] where s t (c t ) = sin θ t (cos θ t ) and y t is the top Yukawa coupling. For tanβ sufficiently large that cos 2 (2β) ≈ 1, but not so large that (s)bottom loops are significant, the Higgs boson mass at one-loop is therefore determined by the three parameters mt 1 , mt 2 , and θ t .
For degenerate stops, the logarithmic stop correction in the first term in the square parenthesis is dominant. In this degenerate scenario, increasingly heavy stops masses can be made consistent with the measured Higgs mass m h = 125 GeV by appropriately decreasing the treelevel contribution to match the increasing loop contribution. However, as the mass splitting between the two stops increases, the remaining two terms in the loop correction grow stronger.
A key observation is that the final term switches sign and becomes negative for mt 2 ∼ > 2.7mt 1 .
For non-vanishing stop mixing and mt 2 mt 1 , this negative term can dominate. Therefore, for nonzero mixing, there exists an upper limit on mt 2 (as a function oft 1 and θ t ), beyond which it is impossible to accommodate m h = 125 GeV in the MSSM. 2 In other words, a measurement of mt 1 and some knowledge of θ t allows for an upper limit on mt 2 . Ruling out mt 2 in this window rules out the MSSM. This statement is independent of the rest of the supersymmetric spectrum at one-loop (see related discussion below). In this case, one can conclude that the supersymmetric theory must include additional corrections to the Higgs mass, as, for instance, in the Next-to-Minimal Supersymmetric Standard Model (NMSSM). Thus, even partial information 2 This is simply an alternate formulation (in terms of the physical stop masses) of the more familiar statement that the Higgs mass cannot be raised arbitrarily by increasing the stop trilinear term A t ; beyond a certain value, further increasing A t lowers the Higgs mass. Note that this statement is only valid for nonzero mixing, and remains applicable at low tan β. on the three parameters mt 1 , mt 2 , and θ t can suffice to make meaningful statements about the underlying supersymmetric model.
In this paper, we will make use of the analytic one-loop formula in Eq. (1) to calculate the Higgs mass. While a crude approximation, it is sufficient to illustrate our ideas. We also take the 120 ≤ m h ≤ 130 GeV mass window as potentially compatible with the measured mass of the Higgs boson; we allow this perhaps surprisingly generous 10 GeV window to account for several corrections not captured by this simple formula, which are known to amount to a few GeV. For example, the Higgs mass is sensitive to both the uncertainty in and the running of the top Yukawa (we use m t = 173 GeV); these are known to affect the Higgs mass by a few GeV [27][28][29]. Likewise, at higher loop order the Higgs mass is sensitive to the gluino mass, particularly for large stop mixing. Assuming the gluino is not too heavy (remains ∼ < 4 TeV), we find that the associated uncertainly in the Higgs mass remains a few GeV. We illustrate this dependence for a specific choice of stop masses and mixing angle in Fig. 1 Thus the lighter sbottom mass is fixed by the same three parameters that fix the Higgs mass, providing another constraint in the system. The upper limit discussed above for mt 2 (imposed by the Higgs boson mass) can also be translated to an upper limit on mb 1 . In this paper, we work with the most minimal possible spectrum, decoupling all particles other thant 1 ,t 2 ,b 1 , and a bino-like neutralino χ 0 , which we take to be the lightest supersymmetric particle (LSP). 4

A. Indirect Constraints on Light Stops
To appreciate the range of possible LHC signals, it is useful to first discuss indirect constraints on light third generation squarks. In particular, when stop mixing is significant, as might be suggested by the Higgs boson mass if the stops are light, there can be significant contributions to the ρ parameter or a modification of the Higgs boson production rate.
The one-loop contribution to the ρ parameter is [36][37][38][39] 3 We have verified with a scan with SUSY-HIT that all compatible points fall within this 10 GeV window on the Higgs mass. 4 Higgsinos are motivated to be light from naturalness considerations. The presence of both light charginos and neutralinos would lead to additional collider signatures. In this paper, for simplicity, we decouple the Higgsinos and keep only a light bino to demonstrate that progress is possible even with this minimal scenario.
For investigations of scenarios where Higgsinos are light and part of the phenomenology, see e.g. [32][33][34][35]. We also remain agnostic about whether the LSP can account for some or all of dark matter. We take the constraint from Ref. [40], ∆ρ = (4.2±2.7)×10 −4 . We demand consistency with this number to 2σ. In general, ∆ρ can increase for larger mass splitting or mixing angle. However, as discussed earlier, a large mass splitting with large mixing suppresses the Higgs boson mass on account of the large negative term in the loop contribution. Indeed, for mt 1 < 1 TeV, we find that points with the correct Higgs mass are correlated with small values of ∆ρ. These features are shown in Fig. 2.
The existence of a light stop can also modify the Higgs-gluon-gluon coupling, which is constrained to be somewhat close to its Standard Model (SM) value. The stop contribution to this coupling is [41,42] We include the contributions from both stop mass eigenstates, though the contribution from the lighter eigenstate tends to dominate due to the 1/m 2 factor (the sbottom contribution, even when it is as light ast 1 , is generally negligible). LHC data constrain r gg to within ∼ 25% of the SM value [43]. The LHC is expected to probe this quantity to within 12 − 16% (6 − 10%) of the SM value with 300 fb −1 (3000 fb −1 ) of data, whereas the ILC and TLEP can probe it to percent level precision [44]. Such constraints on r gg can therefore result in strong bounds on the stop mixing angle as a function of the two stop masses.
We illustrate the potential power of such constraints in Fig. 3; the light (dark) green shades denote regions that would be compatible with future LHC runs with 300 (3000) fb −1 data.
Note that the corresponding constraint on the mixing angle becomes stronger ast 1 becomes lighter ort 2 becomes heavier. Notably, we see that the allowed regions of parameter space can cleanly separate into two distinct bands corresponding to small and large mixing angles, i .e. a mostly left-handed or right-handedt 1 . We also show regions incompatible with ∆ρ constraints in yellow, which become stronger as the mass splitting between the stop mass eigenstates increases, as seen earlier in Fig. 2.
In the next three sections, we demonstrate how the above ideas can be implemented in three distinct scenarios at the LHC, corresponding to qualitatively very different signals fromt 1 ,b 1 , andt 2 . In all cases, we demand compatibility with both ∆ ρ and r gg .

III. IMPLICATIONS OF A LIGHT STOP SIGNAL
The latest LHC results impose increasingly stringent constraints on light stops: limits exist for mt 1 ∼ m t + m χ [11,12], 3-body decay into bW χ 0 [10,11], 4-body decay into bf f χ 0 [10], as well as flavor violating decays into cχ 0 [16]. Together, these bounds essentially rule out stop masses below mt 1 ∼ < 450 GeV. In this section, we therefore focus on the mass window 450 ≤ mt 1 ≤ 600 GeV, where a light stop is potentially compatible with existing constraints, and discuss the implications of its discovery at the LHC. As described in the previous section, the existence of a light stop invites non-trivial constraints from r gg . For this section, we assume the optimistic reach with 3000 fb −1 of data at the LHC as reported in Ref. [44], which will constrain 0.94 ≤ r gg ≤ 1.06.
The constraint on r gg can be mapped onto the physical parameters mb 1 and mt 2 . This is plotted in the top row of Fig. 4, obtained by performing a scan over parameter space, demanding consistency with both 0.94 ≤ r gg ≤ 1.06 and ∆ρ < 9.6 × 10 −4 . We also find it instructive to look at the mass splittings ∆mb 1t1 ≡ mb 1 − mt 1 and ∆mt 2b1 ≡ mt 2 − mb 1 , which are plotted in the bottom row; the horizontal blue line denotes mass splitting equal to m W . Points that satisfy 120 ≤ m h ≤ 130 GeV in the MSSM are plotted in green (red) for primarily left-handed (right-handed)t 1 , while points outside this mass window are shown in gray. We see that while arbitrary mb 1 and mt 2 can be realized for the mixing angles allowed by r gg and ∆ρ, interesting patterns emerge with the additional requirement of reproducing the Higgs mass. As discussed in the previous section, this imposes an upper limit on mt 2 and mb 1 . It is convenient to separate the discussion into cases where the light stopt 1 is mostly left-handed (θ t ≤ π/4, green points) or mostly right-handed (θ t ≥ π/4, red points).
For mostly left-handed 5t 1 , this limit is not very meaningful for mt 2 , which can be at several TeV (the constraints do impose a lower bound on mt 2 ). However, it is sharp for mb 1 , constraining ∆mb 1t1 ∼ < 200 GeV in thet 1 mass window of interest, as seen in the left panels of Fig. 4. In addition to revealing the existence of a light sbottom, these correlations also reveal information about its decay channels: below (above) the m W line,b 1 decays primarily to bχ 0 (t 1 W ). 6 For 5 While additional observations are required to determine whether a stop is left-or right-handed, theoretical considerations may prefer one over the other -for instance, in gauge mediation, the lighter stop is generally left-handed [45]. 6 We also see points with mb 1 < mt 1 ; we do not address them further in this paper. mt 1 ∼ < 500 GeV, the relevant splitting is constrained to be smaller than m W , and the sbottom decays asb 1 → bχ 0 . This decay is strongly constrained by the LHC, with the latest bounds [46] ruling out mb 1 ∼ < 1 TeV for m χ 0 ∼ < 500 GeV, effectively eliminating this region (mt 1 ∼ < 500 GeV) of MSSM parameter space. For mt 1 ∼ > 500 GeV, the sbottom can decay primarily as which requires the stops to be split due to large mixing. As argued in the previous section, such large mixings, in turn, enforce strong upper limits on mt 2 for compatibility with the MSSM Higgs mass. We find that the ∆mb 1t1 > m W region is correlated with mt 2 ∼ < 1.2 TeV. Such masses are potentially within reach of the 14 TeV LHC, although discovery will be challenging and will require dedicated searches. For a discussion of possible detection strategies in various scenarios, see e.g. Ref. [47].
For a mostly right-handedt 1 , both theb 1 and mt 2 are heavy ( ∼ > 700 GeV; top panels, red points), and searching for their signals is challenging. In this case, the absence of such signals at the LHC does not lead to any meaningful conclusions about the MSSM. In contrast, it is the lower bounds on these masses that are relevant. Should ab 1 ort 2 be discovered with mass lighter than what is shown in the figure, this would be inconsistent with the MSSM, pointing to physics -and contributions to the Higgs mass -beyond the MSSM. 7 For a mostly right-handedt 1 , the sbottom mass is more closely aligned with the heavier stop mass, and we find ∆mt 2b1 ∼ < 200 GeV (bottom right panel, red points). Moreover, for mt 1 ∼ < 500 GeV, ∆mt 2b1 < m W , which impliest 2 →b 1 W is not allowed in this window, motivatingt 2 searches in thet 1 Z (and possiblyt 1 h) channels (we will explore such signals in Sec. V). Likewise, increasing ∆mt 2b1 > m W , which appears possible for mt 1 ∼ > 500 GeV, again requires mixing in the stop sector, resulting in mt 2 ∼ < 1.2 TeV for compatibility with the Higgs mass in the MSSM, which represents an attractive target for the LHC. In Section IV, we perform a detailed study of a scenario where bothb 1 →t 1 W andt 2 →b 1 W are open, leading to a multitude of leptonic signals at the LHC.
With the above considerations in mind, we divide our discussion of the interpretations of ã t 1 signal into two distinct mass windows.
• In the MSSM, ift 1 is mostly left-handed, this discovery implies a light sbottom (mb 1 ∼ < 580 GeV) that decays asb 1 → bχ 0 , which is already ruled out by the latest LHC bounds [46, [48][49][50]. Discovering such a light stop would therefore imply that either the stop is mostly right-handed (red points) or the underlying theory is not the MSSM (gray points with ∆mb 1t1 > m W , but with the wrong Higgs boson mass).
• Ift 1 is mostly right-handed, ∆mt 2b1 < m W in the MSSM, thus this tells us thatt 2 decays primarily tot 1 Z,t 1 h. However, botht 2 andb 1 could be extremely heavy (several TeV) and escape detection. TeV and might have better detection prospects.
• For a mostly right-handedt 1 , we can conclude that eithert 2 →b 1 W is not allowed (in this case botht 2 andb 1 can be very heavy), or mt 2 ∼ < 1. search strategies involving leptons as applied at CMS: same-sign dileptons [5] (recently updated in [51]), and a multilepton search strategy [52] (recently updated in [53]), which searches for an excess in ≥ 3 l+jets+/ E T at the 13 TeV LHC, and explore whether a sbottom signal can be uncovered with these approaches in the future. 8 Before exploring the reach of these search strategies, we first discuss the implications of observing such a signal. A multilepton excess interpreted as ab 1 →t 1 W,t 1 → tχ 0 signal (further corroboration, such as an independent measurements oft 1 and the presence of b tags in the excess, will help solidify this interpretation) implies that botht 1 andb 1 are somewhat left handed (necessary forb 1 →t 1 W decays), and ∆mb 1t1 must be sufficiently large (for the leptons to be hard enough to be observed). Taken together, these imply appreciable mixing in the stop sector, as a purely right-handedt 1 precludes this decay channel altogether, while a purely left-handedt 1 does not result in a sufficiently large mass splitting withb 1 .
In the MSSM, these observations have consequences for the second (heavier) stop. We find mt 2 is correlated with the ∆mb 1t1 mass splitting, as shown in Fig. 5 (for mt 1 < 1 TeV The colored regions (red, black, green, and blue) correspond to different sbottom masses (mb 1 > 1000, 750 < mb 1 < 1000, 500 < mb 1 < 750, and mb 1 < 500 GeV respectively). At small ∆mb 1t1 ( ∼ < 150 GeV -more difficult to probe via multilepton searches), stop mixing is small,t 1 is mostly left-handed, and the desired Higgs mass can be obtained with a heavy (several TeV) t 2 . The splitting ∆mb 1t1 can be made larger by increasing the stop mixing angle (this correlation is plotted in Fig. 6); in this case, as discussed in Section II, consistency with the Higgs mass enforces an upper limit on mt 2 , which is indeed clearly visible in Fig. 5. Alternatively, the splitting can be raised without significant stop mixing by makingt 1 mostly right-handed; however, in this case,t 2 becomes approximately degenerate withb 1 and thus again faces an upper mass limit. For large ∆mb 1t1 , there is therefore an upper limit on mt 2 , which grows stronger for lighterb 1 , as seen in the various colored bands in Fig. 5. This becomes particularly sharp for a sub-TeV sbottom, which forces mt 2 into a narrow wedge-shaped region -for instance, for Thus, the observation of a multilepton+/ E T signal associated with a sub-TeV sbottom and large ∆mb 1t1 leads to a robust upper limit on mt 2 in the MSSM. These stops may well be within reach of the LHC. Detailed analysis of the multilepton excess can shed further light on the properties of mt 2 : inferring ∆mb 1t1 (from, e.g., the lepton p T distribution) and the sbottom mass (from, e.g., the signal rate) not only narrows the allowed range of mt 2 (Fig. 5) but also constrains the stop mixing angle (Fig. 6). It is therefore possible to not only predict a relatively narrow mass window fort 2 , but also get a profile of its decay channels. Ruling out such at 2 is sufficient to rule out the MSSM.
A TeV scalet 2 can be probed in several ways. If it decays primarily viat 2 →t 1 Z, leptonic decays of boosted Z-bosons offer a promising search strategy [54]. Fort 2 →b 1 W , the cascadẽ t 2 →b 1 W ,b 1 →t 1 W ,t 1 → tχ 0 can produce several high p T leptons; in this case, there could be excesses in both ≥ 3 and ≥ 4 lepton searches. Likewise,t 2 →t 1 h can be probed by reconstructing the h with boosted b-jets (see e.g. [55]). Note that even whent 2 →t 1 h is large, t 2 maintains the decayt 2 →t 1 Z, so the search for leptonic Z, boosted hadronic Zs, or a mix of Z and h might be possible.
As an aside, it is important to verify that the original multilepton excess originates primarily fromb 1 decays rather than fromt 2 decays. It is possible to be initially fooled: boosted leptons fromt 2 decays in the blue region of Fig. 5 might mimic ab 1 →t 1 W signal, leading to the erroneous interpretation that ∆mb 1t1 is large. However, in such cases, thet 2 search strategies discussed in the previous paragraph can reveal the heavier stop, and help clarify the extent to which it might give rise to a multilepton excess.

A. Benchmark Case Study
We choose an MSSM benchmark that produces the correct Higgs boson mass and has third generation squarks below the TeV scale. The masses and branching ratios of the relevant particles, generated with SUSY-HIT [26], are listed in Table I

Same-sign dileptons search
We first investigate the same-sign dilepton (SSDL) search as discussed in the CMS paper 9 We change the b-tagging efficiency to 0.7 to match the CMS analysis we rely on for cuts and background, but otherwise use default parameters from our implementation of Madgraph5 and Delphes. To ensure that the default implementations are similar to the CMS analysis that we mirror, we simulate the ttZ/h background, which is one of the dominant background for our analyses, and verify that the efficiency for this background contribution matches that from the CMS analysis. [5] (recently updated in [51]), also advocated by recent phenomenological studies [22,47] as a promising search strategy for heavier superpartners. We begin by mirroring the analysis in this CMS paper, imposing the following cuts: • Require same sign dileptons with p T ≥ 25 GeV.
• Impose a Z veto: reject events with opposite sign, same-flavor dilepton pairs with an invariant mass between 76 and 106 GeV.
• Require two or more b-jets, N b−jets ≥ 2.
• Require 300 ≤ H T ≤ 1125 GeV, where H T denotes the sum of transverse momenta of all the jets in the event.
• Require 5 or more jets in the event, N jets ≥ 5.
The resulting number of signal and background events with 3000 fb −1 of data at the 13 TeV LHC are shown in Table II. The signal primarily results fromb 1 decays. The expected background is taken from the CMS analysis [5], scaled up to 3000 fb −1 of data. We also list the significance of the signal, calculated as S ≡ S / B + σ 2 bg B 2 ; the second term in the denominator denotes systematic uncertainties, which are currently around 30% [52], but should improve with future studies and additional data. We calculate the significance for σ bg = 0, 0.1, and 0.3 to span the range of possibilities. Our results show that close to a 3σ signal is possible  with an improvement to σ bg = 0.1, while further improvements would push the significance towards a 5σ discovery.

Multileptons search
Next, we mirror the CMS search for a signal in multileptons (Signal Region (SR) 14, "off-Z" analysis as defined in [52]) by imposing the following set of requirements on the generated event sample (henceforth referred to as "≥ 3loffZ"): • Require three or more electrons or muons with p T ≥ 20, 15, 10 GeV.
• Require two or more jets, N jets ≥ 2.
• Impose a Z veto: reject events with opposite sign, same-flavor dilepton pairs with an invariant mass between 76 and 106 GeV.
These cuts are not optimized for the signal, but they allow a robust determination of the background as determined by the experiment (which includes a not insignificant contribution from tt + fake leptons, which is difficult to estimate via naive simulation). The expected number of background events is again taken from the CMS analysis [52] and scaled up to 3000 fb −1 of data; the paper also lists a detailed breakdown of individual background contributions (see SR14 "off-Z" entries in Table 3 in that paper).  We improve on this CMS search strategy by further imposing the following additional requirements: • Reject events with no b-jets. This is particularly effective in suppressing the significant W Z background (see SR14 entry, Table 3 in [52]).
To estimate the background suppression from the b-tag requirement, we look at how individual background contributions drop when this requirement is imposed in the 50 ≤ / E T ≤ 300 GeV, 60 ≤ H T ≤ 600 GeV region -this information is readily available in the CMS analysis [52] (see Table 3, SR 1-12; see Table 1 for their definitions). To estimate the effect of the stronger / E T cut, we simulate the SM ttZ/h background (one of the major backgrounds for this signal), observe how it falls for increasing / E T , and make the simplifying assumption that all SM backgrounds scale in the same manner (as noted above, other large backgrounds include tt + jets with a fake lepton, which are also expected to fall at large / E T [52].) The resulting efficiencies, number of events, and signal significance are presented in Table III. A 3σ significance appears possible with improvements in systematic uncertainty in background to σ bg = 0.1, and further reducing it could even enable a 5σ discovery. This search strategy is therefore slightly more efficient than the SSDL analysis in extracting the signal, although a larger fraction of the signal now comes from the heavier stop.
Heavier stop search As discussed earlier, a light sbottom discovery with a large ∆mb 1t1 in the MSSM allows us to predict a TeV scalet 2 . The large mass splitting can be established, for instance, by looking at the p T spectrum of the hardest lepton. To motivate that it is possible to draw such conclusions, in Fig. 7 we plot, in red, the (normalized) p T spectrum of the hardest lepton in the signal events in the ≥ 3loffZ(II) analysis above, which corresponds to ∆mb 1t1 ≈ 240 GeV for our benchmark point. For comparison, we also plot, in blue, the corresponding spectrum for ∆mb 1t1 ≈ 170 GeV. The red spectrum is broader and has a stronger tail (above ∼ 330 GeV). By making use of such features, it is plausible that the ∆mb 1t1 splitting can be determined to within 100 GeV. For our benchmark scenario, this would enable us to predict mt 2 ∼ < 1.2 TeV. Moreover, this also enables us to deduce that there is significant mixing between the stops (see Fig. 6), and thus Br(t 2 →t 1 Z) should be significant. The next step, therefore, is to devise a search strategy for such at 2 .
Again, we make use of the CMS analysis as above (≥ 3loffZ), except we now require a Zreconstruction rather than a Z-veto in order to search for Z bosons fromt 2 →t 1 Z decays (this is defined as SR14, "on-Z" analysis in the CMS paper [52]). The background is again taken from the CMS paper [52] (SR14, Table 4) and scaled up to 3000 fb −1 . We optimize the CMS search strategy by imposing the following additional requirements: • Reject events with no b-tagged jets. Again, this is particularly effective in suppressing the dominant W Z background (see SR14 entry, Table 4 in [52]). • Require / E T ≥ 200 GeV instead of 50 ≤ / E T ≤ 300 GeV. Sincet 2 is significantly heavier thanb 1 , thet 2 signal contains a higher / E T distribution, motivating an even higher / E T cut than that employed in the CMS analysis.
We estimate the modified background contribution in the same manner as for ≥ 3loffZ.
We use information from Table 4, SR 1-12 from [52] to extrapolate the effects of the b-tag requirement on background, and simulate ttZ/h to determine the effects of the increased / E T cut, taking it to be representative of all background. 10 A similar search proposal fort 2 →t 1 Z in [54] also employed a narrower cut on the Z-boson lepton pair invariant mass as a strategy to suppress background, particularly the combinatoric background from tt. In the CMS analysis, the tt background is claimed to be largely suppressed by the strong / E T and H T cuts, hence we do not pursue this strategy in our analysis but note this could provide a further handle.
We denote the above search as "≥ 3lonZ", and present the resulting efficiencies, number of events, and signal significance in Table IV. As with the sbottom search strategies, we see that a ∼ 3σ signal is possible with improvements to σ bg = 0.1, and a ∼ 5σ discovery is possible with further improvements. For comparison, we also list the signal significance for the sbottom contribution only, which makes it clear that sbottom pollution to this signal is minimal. 10 Note that / E T ≥ 300 GeV is a part of SR15 (/ E T ≥ 300 GeV, H T ≥ 600 GeV) and not SR14 in the CMS analysis [52]. We have appropriately scaled the background in SR15 using results of stronger H T cuts on our simulated tt Z/h sample to estimate the modified background contribution from this region.

V. HEAVIER STOP MULTIPLE DECAY CHANNELS IN BOOSTED DIBOSONS
If superpartners are discovered at the LHC, the high luminosity LHC will be able to follow up with measurements in multiple channels with significant statistics. A particularly illustrative example is the decay of the heavier stopt 2 , which can occur in multiple channelst 1 Z,t 1 h,b 1 W , and tχ 0 , with branching ratios determined by the stop masses and mixing angle.
In this section, we focus on the two decayst 2 →t 1 Z andt 2 →t 1 h, which give rise to boosted dibosons if the mass splitting between the two stop mass eigenstates is large. The tree-level decay widths for these two processes, in the decoupling limit in the Higgs sector are [58] Γ(t 2 →t 1 Z) = g 2 256π where the phase space factor is λ(a, b, c) ≡ a 2 + b 2 + c 2 − 2ab − 2ac − 2bc. The ratio of the two widths is: The phase space factors effectively cancel for mt 2 − mt 1 m h , m Z . We expect many experimental uncertainties to cancel in this ratio as well. If the two stop masses are known from other measurements, this ratio offers a clean dependence on the stop mixing angle, 11 enabling a check of the MSSM Higgs mass relation. It should be clarified that we are not advocating R hZ as the most precise measurement of the stop mixing angle, but rather as a measurement with a particularly simple dependence on an important parameter in the theory.
An important caveat is that the above expressions only hold at tree level and will be modified by loop corrections to both Γ(t 2 →t 1 h) and Γ(t 2 →t 1 Z). The loop corrections are particularly important where the tree level contributions vanish (θ → 0, π/2, for both Eq.  [59,60], we estimate that loop contributions can modify R hZ substantially for θ t < 0.1 and θ t > 1.5; we therefore exclude these regions in our analysis. 12

A. Benchmark Case Study
The masses and branching ratios for our chosen benchmark point are listed in Table V; here we computed m h , mb 1 , and thet 2 branching ratios analytically using formulae listed in the previous sections. We have chosen a point with a large combined branching fraction to Z and h in order to maximize our signal by renderingt 2 →b 1 W kinematically inaccessible. This spectrum results in a too-light Higgs in the MSSM, so a sufficiently precise measurement of such a spectrum would imply additional new physics beyond the MSSM. For this benchmark point, R hZ = 0.53.
12 Experimentally, one might be able to confirm that nature is away from these "loop sensitive" windows, either by measurements of r gg , see Fig. 3, or by the absence oft 2 → tχ 0 decays, which should be present if the branching ratios tot 1 Z andt 1 h are very small.
Measuring the ratio R hZ with reasonable precision requires high statistics, motivating searches for the boosted Z and h bosons in their dominant (hadronic) decay channels rather than the cleaner decays into leptons or photons. The prospect of probing such signals by reconstructing the boosted dibosons via fat jets was studied in Ref. [55], which found that a ∼ 4 − 5σ discovery of a TeV scale mt 2 was possible with 100 fb −1 of data at the 14 TeV LHC with a combined diboson signal fromt 2t2 →t 1t1 + (hZ, ZZ, and hh). For these channels, Ref. [55] estimates a total background cross section (after cuts) of 0.16 fb, dominated by events with two W bosons. With a relatively narrow invariant jet mass window for Higgs boson candidates as in [55], which can further be augmented by jet charge, we estimate that the probability of a "W -jet" faking a "Higgs jet" is very small, likely < 1% (see Fig. 2(c) in Ref. [61]). Scaling the backgrounds from [55] by this "mistag" probability, we expect the SM backgrounds to be negligible for the hZ and hh channels. Incidentally, we also expect the probability for a Z jet to fake a Higgs jet to be small. R hZ can thus be determined in an essentially background-free environment by considering events with at least one Higgs jet: 13 where n ab denotes the number of signal events where the two boosted dibosons are tagged as a and b. The error in the calculated ratio R i j = n i /n j is Since the associated backgrounds are negligible, we estimate ∆n i = √ n i .
Our benchmark point is similar those in Ref. [55] in terms of mass spectra and branching ratios into various decay channels. This allows for a straightforward extrapolation of the results of this study. We extract the signal efficiency from this paper and apply it to our benchmark point, applying a modest correction for the branching ratios. With the simplifying assumption that this analysis is equally efficient in extracting Z and h events (likely approximately true given the nearly identical branching ratios to fully hadronic final states), we estimate an overall 13 This strategy also avoids the possibility of contamination fromb 1 decays with mb 1 ∼ mt 2 , where the fat jets from W bosons fromb 1 →t 1 W can be misinterpreted as Z bosons fromt 2 →t 1 Z. Theb 1 contribution was not considered in [55].
The resulting number of events and the corresponding uncertainty on ∆R hZ for 3000 fb −1 of data at the 14 TeV LHC are: n hh = 47, n Zh = 176, ∆R hZ = .07 (12) For our benchmark point, R hZ can thus be measured to within ∼ 12%. Whether the fat jet analyses of the type employed here can remain effective in the high luminosity environment is a question for further study.
Next, we discuss how this measurement can shed light on the Higgs boson mass relation and the validity of the MSSM. This requires some knowledge of the two stop masses, hence we assume that mt 1 has been measured to lie in the range 486 ± 40 GeV from monojet or charmtagged events, while mt 2 is known to fall in the 994.2±50 GeV range from various measurements (such as by combining the knowledge of mt 1 with information on p T (Z) int 2 →t 1 Z events).
The MSSM Higgs mass can then be calculated as a function of θ t using Eq. (1), which can be converted to a function of the ratio R hZ using Eq. (9). We plot this dependence in Fig. 8  Recall that the Higgs mass is small for vanishing stop mixing θ t → 0, π/2, which corresponds to R hZ ∼ cos 2 2θ t approaching 1. On the other hand, achieving the correct Higgs mass with sub-TeV stops requires large stop mixing, which correlates with a smaller value of R hZ . The trend in Fig. 8 is consistent with these observations. Thus, an inferred value of R hZ above some cutoff value R 0 (≈ 0.45 in this case) is incompatible with the MSSM Higgs mass relation. Such an observation would rule out the MSSM, pointing to the need for additional contributions to the Higgs mass. In Fig. 8, in the golden band we show the uncertainty in the calculated value of R hZ for our benchmark point. Under our assumptions, exclusion of the MSSM region is borderline; however, the MSSM can be clearly excluded with either better measurements of the stop masses (darker red band) or with an improved analysis with better signal efficiency (recall that here we simply used the efficiency from the analysis in Ref. [55]). This benchmark study serves as a proof of concept that measurements of the two decay channelst 2 →t 1 Z and t 2 →t 1 h can be used as a consistency check of the Higgs mass and possibly rule out the MSSM.
We conclude this section with a few miscellaneous comments. With approximate knowledge of the two stop masses, requiring that r gg remain consistent with observations also constrains the stop mixing angle. For our benchmark point (with the narrower stop mass windows discussed above), we find that r gg measurements at the 3000 fb −1 LHC can constrain R hZ to 0.2 < R hZ < 0.6, therefore providing complementary handles on the Higgs mass relation. Likewise, if the sbottom has not already been discovered, the above measurements can also be used to predict the mass and decay channels of the sbottom, aiding in its discovery. For our benchmark scenario, the sbottom is degenerate witht 2 and decays almost exclusively tob 1 →t 1 W , with mt 1 decaying ast 1 → cχ 0 . Dedicated searches optimized towards accepting W jets instead of Z jets might prove fruitful in discovering such a sbottom.

VI. SUMMARY
In this paper, we investigated the implications of a third generation squark signal discovery at the LHC. It is possible to make use of the relation between the stop sector and the Higgs boson mass in the MSSM in a wide variety of scenarios to test the consistency of the MSSM and predict the masses and decay channels of other superpartners, therefore offering clear subsequent targets for the LHC. We elaborated these ideas with studies in three distinct scenarios: • For a light (450 ≤ mt 1 ≤ 600 GeV) stop, constraints on the Higgs-gluon-gluon coupling strongly limit the stop sector parameters, which can be translated into bounds on thẽ b 1 andt 2 masses, leading to interesting patterns in the MSSM. For instance, the MSSM Higgs mass relation forces ∆mb 1t1 < m W for mt 1 ∼ < 500 GeV ift 1 is left-handed, which is incompatible with current LHC constraints. Likewise, scenarios involvingb 1 →t 1 W or t 2 →b 1 W require significant stop mixing, and consistency with the Higgs mass in the MSSM leads to the prediction mt 2 ∼ < 1.2 TeV.
• In the event of a sbottom signal in same-sign dileptons or multileptons+jets+/ E T from b 1 →t 1 W , the correlation between ∆mb 1t1 and mt 2 in the MSSM can be used to predict thet 2 mass and decay channels, thereby aidingt 2 searches at the LHC.
• For fixed stop masses, the ratio oft 2 →t 1 Z andt 2 →t 1 h decay widths is determined by the stop mixing angle, and measuring this ratio with sufficient precision can test the MSSM Higgs mass relation and therefore check the validity of the MSSM. We illustrated the plausibility of this scenario with a case study of the reconstruction of the boosted dibosons from fat jets at the high luminosity LHC with 3/ab of data.
These examples do not cover the full range of signals or spectra that are possible in the MSSM, and it might also be interesting to perform similar studies focusing on signals involving, e.g., light Higgsinos or the gluino. Likewise, we employed several approximations in our studies.
Higher precision calculations and more careful simulations would be warranted should a relevant signal actually be discovered at the LHC. Nevertheless, the scenarios we studied here illustrate the power and applicability of the Higgs mass relation in unravelling the supersymmetric sector.