# Analysing parameter space correlations of recent 13 TeV gluino and squark searches in the pMSSM

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## Abstract

This paper examines unexplored correlations in the parameter spaces probed by recent ATLAS analyses for gluinos and squarks, addressing various shortcomings in the literature. Six 13 TeV ATLAS analyses based on 3.2 fb\(^{-1}\) of integrated luminosity are interpreted in the 19-parameter R-parity conserving phenomenological minimal supersymmetric extension to the Standard Model (pMSSM). The distinct regions covered by each search are independent of prior, and we reveal particularly striking complementarity between the 2–6 jets and Multi-b searches. In the leptonic searches, we identify better sensitivity to models than those used for analysis optimisation, notably a squark–slepton–wino scenario for the SS/3L search. Further, we show how collider searches for coloured states probe the structure of the pMSSM dark sector more extensively than the Monojet analysis alone, with sensitivity to parameter spaces that are challenging for direct detection experiments.

## 1 Introduction

The ATLAS and CMS collaborations pursue a rich programme of supersymmetry (SUSY) searches, but statistically significant signals remain absent [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. While conventional theoretical expectations for weak scale SUSY are challenged [22, 23, 24, 25, 26, 27, 28, 29, 30], this equally motivates assessment of present experimental search strategies. Simplified models introduce a small number of kinematically accessible superpartner particles (sparticles), which are typically used to design, optimise and interpret collider searches [31, 32, 33, 34].

A broader framework for assessing the robustness of such strategies is the phenomenological minimal supersymmetric extension to the Standard Model (pMSSM) [35, 36, 37, 38, 39, 40, 41, 42, 43, 44], which also facilitates dark matter (DM) interpretations [45, 46, 47, 48, 49, 50, 51, 52]. Notably, the ATLAS collaboration examined the sensitivity of 22 Run 1 analyses within the context of a 19-parameter pMSSM [53], while CMS undertook a similar survey using different assumptions [54].

- (a)
*How distinct are the regions of parameter space being probed by individual analyses?*Interpretations using the pMSSM often present combined constraints from multiple searches as fractions of models excluded [39, 48, 53, 54, 56]. Overlap matrices were recently used in the literature [53, 55] to quantify the complementarity of these searches, namely the fractional exclusion of the same subset of points by two analyses. However, this marginalisation not only obscures which analyses had greatest sensitivity to different pMSSM subspaces, but also depends on the prior distribution from the parameter scan and non-LHC constraints. - (b)
*To what extent are analyses over-optimising to a set of simplified models, which may preclude sensitivity to a wider class of scenarios?*The pMSSM offers a greater variety of decay chains, such as those with suppressed branching fractions and placing those with more intermediate sparticles on-shell, which alter kinematics. How well these simplified model oriented searches are capturing the wider classes of signatures remains relatively unexplored. - (c)
*What neutralino DM scenarios can be probed competitively by 13 TeV collider searches for coloured sparticles?*Simplified dark matter models are stimulating the collider frontier for DM searches [57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67]. ATLAS uses these to perform an explicit DM interpretation for the Monojet search [6]. However, its sensitivity is greatly altered in richer dark sectors of the pMSSM, shaped by the composition of the neutralino \(\tilde{\chi }^0_1\) lightest SUSY particle (LSP) [68, 69, 70, 71, 72] being dominantly bino, Higgsino or wino (defined in Table 2 of Appendix A.1). Furthermore, studies of neutralino DM often combine exclusions of multiple analyses [48, 53] or only focus on the electroweakino sector [52, 73, 74, 75], omitting potentially important coannihilation roles of other sparticles.

Section 2.3.2 addresses question (b) by considering the analyses that select leptonic events as a case study. We examine to what extent the simplified models used by the 1-lepton analysis [3] and the same-sign or 3-lepton (SS/3L) search [5] map onto pMSSM points, and identify scenarios beyond those considered for analysis optimisation.

Section 3 addresses question (c) by ascribing DM interpretations to each 13 TeV search considered, allowing for comparisons to the Monojet search. Striking correlations are exhibited, and we discuss how distinct decay cascades are influenced by the bino, Higgsino or wino content of the LSP, while coloured sparticles may act as early-universe coannihilators [76]. Further, we identify the collider parameter space affected by recent limits from Xenon-target direct detection experiments [77, 78].

List of the ATLAS 13 TeV (3.2 fb\(^{-1}\)) analyses used to constrain the 181.8k model points that survived Run 1. The column \(N_\mathrm{Lowest\, CLs}/N^\mathrm{Excluded}_{\mathrm{13\, TeV\, 3.2 \,fb}^{-1}}\) denotes the fraction of the 28.5k excluded models for which the indicated analysis was the most sensitive, i.e. had the lowest CLs value \(N_\mathrm{Lowest\,\, CLs}\) as discussed in Sect. 2.1; these figures may not sum to 100% due to rounding. The column \(N^\mathrm{Excluded}_{\mathrm{13\, TeV\, 3.2\, fb}^{-1}}/{N}^\mathrm{Survived}_\mathrm{ATLAS\, Run\, 1}\) is the ‘fractional exclusion’ displaying the total percentage of points excluded by each of the analyses \(N^\mathrm{Excluded}_{\mathrm{13\, TeV\, 3.2\, fb}^{-1}}\) out of the points that survived ATLAS Run 1 \(N^\mathrm{Survived}_\mathrm{ATLAS \,Run\, 1}\). The right-most column quantifies the subset of points that are excluded uniquely by the corresponding analysis and not by any of the other five considered \(N^\mathrm{Excluded \,Uniquely}_{\mathrm{13\, TeV\, 3.2\, fb}^{-1}}\). For all these figures, care must be taken with interpretation as they are prior dependent. Models with long-lived gluinos, squarks and sleptons are not considered

Analysis | References | \(N_\mathrm{Lowest\, CLs}/N^\mathrm{Excluded}_{13\, \mathrm{TeV}\, 3.2\, \mathrm{fb}^{-1}} \) (%) | \(N^\mathrm{Excluded}_{13\, \mathrm{TeV}\, 3.2\, \mathrm{fb}^{-1}}/N ^\mathrm{Survived}_ \mathrm{ATLAS\, Run\, 1}\) (%) | \(N^\mathrm{Excluded\, Uniquely}_{13\, \mathrm{TeV}\, 3.2\, \mathrm{fb}^{-1}}/N^\mathrm{Survived}_\mathrm{ATLAS\, Run\, 1}\) (%) |
---|---|---|---|---|

2–6 jets | [1] | 72 | 12.6 | 11 |

7–10 jets | [2] | 0.3 | 0.6 | 0.02 |

1-lepton | [3] | 1.5 | 1.0 | 0.2 |

Multi-b | [4] | 23 | 4.2 | 3.5 |

SS/3L | [5] | 2.7 | 0.5 | 0.4 |

Monojet | [6] | 1.1 | 3.3 | 0.01 |

All analyses | – | 100 | 15.7 | 15.1 |

The pMSSM points we investigate were produced for the ATLAS study [53]. The 19 parameters of the R-parity conserving MSSM, where all flavour and CP violation resides in the CKM matrix, were scanned with flat priors with sparticle mass scales capped at 4 TeV [53]. However, the distinct regions of sensitivity we identify in our results are independent of sampling prior. Constraints were imposed from LEP searches [79], precision electroweak measurements [80, 81, 82, 83, 84, 85, 86, 87, 88, 89], heavy flavour physics [90, 91, 92, 93, 94, 95, 96, 97], DM direct detection [98, 99, 100] and the Planck relic density [101] upper bound.^{1} Further details of the theoretical assumptions and experimental constraints used by ATLAS [53] may be found in Appendix A.1. We consider the 181.8k points that survive Run 1 constraints from Ref. [53], after having removed models with long-lived (\(c\tau > 1\) mm as defined in Ref. [53]) gluinos, squarks and sleptons as these require dedicated Monte-Carlo simulation. The methodology we employed to interpret the six 13 TeV searches using fast detector simulation is detailed in Appendix A.2.

## 2 Complementarity between early 13 TeV searches

For the six 13 TeV (3.2 fb\(^{-1}\)) ATLAS searches considered in Table 1, we present the regions of sensitivity between these analyses. In Appendix A.3, we present results using existing practices in the literature, based on fractions of models excluded (marginalised distributions) and overlap matrices. This section addresses various shortcomings of these approaches as discussed in Sect. 1. Section 2.1 defines and quantifies the ‘most sensitive analysis’ used to exclude the points in our interpretation. Following this, Sect. 2.2 projects this information into two-dimensional subspaces of the pMSSM involving gluinos, squarks and the LSP, and discusses prior features in these planes. Finally, we examine the complementary sensitivity of each analysis to distinct regions of pMSSM parameter space in Sect. 2.3, partitioning our discussion between analyses that veto events with leptons with those that select on them.

### 2.1 Most sensitive analyses used for exclusion

In this study, a point is deemed excluded at 95% confidence level if at least one analysis returned a CLs value less than 0.05, using the CLs prescription [102]. Of the 181.8k points that survived Run 1 [53], a total of \({N^\mathrm{Excluded}_{\mathrm{13\, TeV } \,3.2\, \mathrm{fb}^{-1} }}= 28.5\mathrm{k}\) were excluded by our interpretation of the six 13 TeV analyses. We take the analysis with the smallest CLs value as the ‘most sensitive analysis’ used to exclude the point. In Table 1, we normalise the number of points satisfying this for each analysis \(N_\mathrm{Lowest\,\, CLs}\) to the total excluded \({N^\mathrm{Excluded}_\mathrm{13\, TeV\, 3.2\, fb^{-1}}}\). For example, the Multi-b search was the analysis with the lowest CLs value for 23% of the 28.5k excluded points. Indeed, almost 95% of the excluded points have either the 2–6 jets or Multi-b searches being the most sensitive. However, care must be taken when interpreting these fractions, as they are prior dependent and correlated with non-LHC constraints. The fractions indicate the relative number of points in forthcoming figures.

Less than 4% of the excluded points have two or more analyses associated with the same smallest CLs value. In such cases, the ‘most sensitive analysis’ is randomly chosen from this subset of analyses with smallest CLs value to minimise systematic selection bias. In the vast majority of these situations, this is done because the analyses share a CLs value of 0.0.

### 2.2 Features in mass plane projections

The distinct regions of sensitivity for each of the six searches become unambiguous when we project into various two-dimensional subspaces of the pMSSM. Figures 1 and 2 display each excluded point styled and coloured according to the analysis that returned the lowest CLs value, projected into the mass planes of gluino vs. LSP, gluino vs. squark and squark vs. LSP, respectively. Due to their large numbers, the 2–6 jets and Multi-b analyses are allocated smaller markers to improve clarity of other points.

*distributions*(but not the

*regions*to be discussed in Sects. 2.3 and 3.1) of excluded points do depend on the flat prior and importance sampling ATLAS used to scan the parameter space [53], in addition to non-LHC constraints. This is especially apparent in the mass planes involving the LSP (Figs. 1a, 2), and we comment on several features common to these figures to aid interpretation:

Within around 100 GeV of the grey hatched regions where 100% of models in the mass bins were excluded by Run 1 ATLAS searches, very few points are present. The number of models close to this boundary is often less than ten.

^{2}Typically, such points marginally survived Run 1 constraints and many have since been excluded by the six 13 TeV searches.There is a visible break in points at \(m(\tilde{\chi }_1^0) \sim 100\) GeV. Below this mass, there are few models with Higgsino- or wino-like LSP as these generally have a near-degenerate chargino, which are excluded by LEP direct searches. For the definition of the three LSP classifications, see Table 2 in Appendix A.1.

The points below \(m(\tilde{\chi }_1^0) \sim 100\) GeV are therefore predominantly bino-like LSP models. These LSPs are close to half the mass of the \(Z^0\) or Higgs boson \(h^0\) so they can undergo resonant annihilation in the early universe in so-called ‘funnel regions’ to satisfy relic abundance constraints.

A large density of points is visible along the diagonals where the mass splitting between the LSP and the gluino or squark become small (\(\lesssim \)50 GeV). The enhanced density of points in this coannihilation region has physical origin: bino-like LSPs tend to oversaturate the observed relic abundance [103] unless there is a near mass-degenerate sparticle, such as a squark or gluino, to act as an early-universe coannihilator. This effect is made more apparent by using importance sampling to ensure the number of bino-like LSP models is of the same order as those of Higgsino- and wino-like LSP; see Ref. [53] for details.

### 2.3 Mass plane correlations of searches for squarks and gluinos

This subsection discusses the distinct regions of sensitivity correlated with each analysis considered. Importantly, the identified regions in the two-dimensional planes are independent of the prior distribution of points.

#### 2.3.1 Searches with a lepton veto

First, we investigate the excluded points where each of the 2–6 jets, Multi-b, 7–10 jets, and Monojet searches are most sensitive, in that order. These analyses all veto on leptons, and exhibit the most unambiguous correlations in the mass planes of the gluino, squark and LSP (Figs. 1, 2). Generally, these analyses select events with transverse momentum imbalance of magnitude \(E_\mathrm {T}^\mathrm{miss}\), varying jet multiplicity, together with discriminants dependent on mass scale or flavour. For full details, refer to the ATLAS references in Table 1.

*2–6 jets* Figure 1a reveals the extensive distribution in the gluino–LSP plane where the 2–6 jets is the most sensitive analysis (blue points). The 2–6 jets search uses the effective mass discriminant, with varying degrees of minimum jet multiplicity to target models where a squark or gluino directly decays to the LSP. Meanwhile, the larger jet multiplicity regions target scenarios where a chargino mediates the decay of the gluino to the LSP.

This analysis has almost exclusive sensitivity to points with gluino \(\tilde{g}\) masses below 1 TeV, where its mass splittings with the LSP in the range \(25 \lesssim m(\tilde{g}) - m(\tilde{\chi }^0_1) \lesssim 500\) GeV. Larger mass splitting scenarios for sub-TeV mass gluinos are excluded by Run 1 searches (hatched grey mass regions). For regions with gluinos above 1 TeV with large gluino–LSP mass splittings, we find the 2–6 jets has reduced sensitivity compared with other analyses, especially for LSP masses \(m(\tilde{\chi }^0_1) \lesssim 500\) GeV. For gluino masses \(m(\tilde{g}) \gtrsim 2\) TeV, we expect reduced sensitivity to gluinos, but points in this region are correlated with excluded points involving low-mass squarks. This is confirmed by comparing with the gluino–squark plane (Fig. 1b). Indeed, the 2–6 jets search is also predominantly the most sensitive analysis for light squarks in regions not far beyond Run 1 sensitivity. High mass gluinos in such scenarios can nevertheless contribute to production cross sections of the squarks as a mediator via *t*-channel diagrams.

The gluino–squark plane (Fig. 1b) also reveals a vertical strip with a lower density of points around gluino mass between about 1 and 1.2 TeV. This corresponds to a region where a lower fraction of models is excluded per mass interval (see also Figs. 11c, 13c in Appendix A). This reduced sensitivity, and lack of any other dedicated analysis probing this region, is due to gluino–LSP mass being moderately small (between around 25 and 200 GeV). Such scenarios are challenging for traditional ‘missing energy plus jets’ searches, but represent the greatest potential for high luminosity where novel techniques are being developed to target such regions [104].

Further interpretations for the 2–6 jets analysis apply in the squark–LSP plane (Fig. 2). The search has sensitivity to a wide variety of squark mass scenarios, gradually reducing above \(m(\tilde{q})\sim 1\) TeV. The strip of blue points for \(m(\tilde{q})\gtrsim 1.5\) TeV and \(600 \lesssim m (\tilde{\chi }^0_1)\lesssim 900\) GeV where we would expect reduced squark sensitivity are correlated with low-mass gluinos. These points largely have sub-TeV gluinos, an interpretation, again confirmed in the gluino–squark plane (Fig. 1b).

*Multi-b*Where the Multi-b analysis has most sensitivity (red points) is localised to regions complementary to the 2–6 jets analysis. In the gluino–LSP plane (Fig. 1a), the region is enclosed by gluino masses of \(1.2 \lesssim m(\tilde{g}) \lesssim 1.7\) TeV and LSP masses of \(100 \lesssim m (\tilde{\chi }^0_1)\lesssim 500\) GeV. This sensitivity also extends to smaller gluino–LSP mass splittings, though to a lesser extent.

The distinctiveness of this region of sensitivity remains in the gluino–squark plane (Fig. 1b). Here, points where the Multi-b analysis is most sensitive are highly correlated with squark masses above 1.5 TeV, largely untouched by other analyses, being strikingly separated from 2–6 jets and to some extent the 7–10 jets analysis. Figure 2 also confirms such correlations of the Multi-b in the squark–LSP plane and again sensitivity to squarks above 1.5 TeV is largely from gluino rather than squark production.

These correlations arise from the Multi-b search targeting \(\tilde{g}\rightarrow b\bar{b}\tilde{\chi }^0_1\) and \(\tilde{g}\rightarrow t\bar{t} \tilde{\chi }^0_1\) models with heavy gluino production decaying via off-shell stops \(\tilde{t}\) and sbottoms \(\tilde{b}\). This analysis selects events with large \(E_\mathrm {T}^\mathrm{miss}\) and at least three jets originating from bottom quarks. A subset of the signal regions selects loosely on boosted top quarks decaying from gluinos, including events enriched with lepton presence.

Figure 3 takes the points where the Multi-b is most sensitive and illuminates the mass distributions of various pertinent sparticles. The light flavour squarks \(\tilde{q}\) are centred around 2 TeV with gluinos around 1.4 TeV, where Fig. 1b indicates that squarks are predominantly heavier than gluinos. This fact suppresses gluino decays to light flavour quarks, which proceed via the three body \(\tilde{g} \rightarrow q \bar{q} \tilde{\chi }^0_1\) process. By contrast, the mass distribution of sbottoms \(\tilde{b}_1\) peaks around 800 GeV in Fig. 3 and have a preference to be lighter than the gluinos, allowing on-shell \(\tilde{g}\rightarrow \tilde{b}b\) decays to be favoured. Indeed, this demonstrates favourable sensitivity to the on-shell counterpart of the simplified models considered for optimisation by ATLAS. Meanwhile, the distribution of stop masses is relatively uniform for \(m(\tilde{t}_1)\gtrsim 900\) GeV. The requirement of three or more jets originating from bottom quarks therefore favours such scenarios. We furthermore note that out of the models excluded where the Multi-b is most sensitive, 56% models have a Higgsinos-like LSP while 22% are wino-like, which have light charginos consistent with the corresponding mass distribution in Fig. 3. This preference of the Multi-b analysis for Higgsino-like LSP models can be understood by the higher Yukawa couplings to heavy flavour quarks, which also enhance decays of the gluino to bottom and/or top quarks.

From the region in Fig. 1 where the density of red points is greatest, we show a representative point with model number 148229034 (Fig. 4a). This contains an LSP with a bino–Higgsino mixture at a mass of 175 GeV, a relatively low-mass 1.2 TeV gluino and a 1.3 TeV sbottom enabling \(\tilde{g} \rightarrow b \bar{b} \tilde{\chi }^0_1\) branching ratios to be preferred.

*7–10 jets*The concentration of magenta triangles in Fig. 1 illustrates that the 7–10 jets search is the most sensitive analysis for models having light LSPs with large mass splitting from the gluino. Most of these LSPs have masses below 100 GeV, which are bino-like as discussed in Sect. 2.2, and undergo early-universe annihilation through a \(Z^0\) or a \(h^0\) boson in so-called ‘funnel’ regions. Since this annihilation mechanism requires Higgsino or wino admixtures to proceed, such scenarios typically have light charginos \(\tilde{\chi }^\pm _1\) and next-to-lightest neutralinos \(\tilde{\chi }^0_2\), with relatively small mass splittings between one other.

These types of decays are consistent with the simplified models containing long decay chains that this analysis optimises for. The observed missing transverse energy \(E_\mathrm {T}^\mathrm{miss}\) therefore tends to be smaller than that required by the 2–6 jets or Multi-b searches. Indeed no explicit requirement on \(E_\mathrm {T}^\mathrm{miss}\) is made by the search (instead the main discriminant is a ratio \(E_\mathrm {T}^\mathrm{miss}/\sqrt{\sum p_\mathrm {T}^\mathrm{jet}}\) involving the missing energy and scalar sum of transverse jet momentum). The 7–10 jet search also has a looser requirement on jets originating from bottom quarks compared with the Multi-b analysis. Together, this allows the 7–10 jets analysis to maintain a unique coverage of models. In the gluino–squark plane (Fig. 1b), the 7–10 jets points occupy a similar space as Multi-b, but the longer cascade chains mean the gluino mass reach is lower. The remainder of the spectrum can be relatively decoupled. Figure 4b displays model number 227558023, which is representative of the models where the 7–10 jets analysis is most sensitive, where we selected the one with lowest gluino mass which had an LSP below 100 GeV.

*Monojet* The dedicated Monojet analysis selects events with an energetic jet from initial-state radiation recoiling off a system of large missing transverse momentum and up to three additional jets. In terms of SUSY models, this is optimised for scenarios where the mass of the squark is almost equal to that of the LSP, so-called ‘compressed scenarios’.

In Figs. 1a and 2, the excluded points (cyan rings) where this analysis is most sensitive involve very small mass splittings between the coloured sparticle and LSP. Though there is significant overlap in regions of sensitivity for the Monojet and 2–6 jets analyses, the former is exclusively the most sensitive analysis involve squark–LSP splittings below 30 GeV. We note that the 2–6 jets search includes a similar signal region but requiring a minimum of two jets called ‘2jm’. The different jet multiplicity requirements ensure the dedicated Monojet maintains a unique sensitivity to the smallest squark–LSP mass splittings, again demonstrating the complementarity of searches. We find a small number (fewer than 10) of scenarios involving small mass splittings between the lightest third generation squark and the LSP in the models for which the Monojet is most sensitive.

#### 2.3.2 Searches selecting one or more leptons

Both the 1-lepton and SS/3L searches also target production gluinos and squarks, but require one or more leptons in selected events. In Figs. 1 and 2 involving the gluino, squark and LSP masses, the correlations between where the 1-lepton (orange plus) or SS/3L (green cross) analyses were most sensitive are less obvious. There points tend to cluster below squark masses of \(m(\tilde{q})\lesssim 1.5\) TeV, while few points are present for gluino masses \(m(\tilde{g})\lesssim 1.2\) TeV where other analyses dominate. Further investigation reveals other correlations driven by the light flavour slepton and gaugino masses not apparent in these figures, which we discuss in the following. We also identify scenarios where these searches had most sensitivity, which are beyond what ATLAS optimised for.

*1-lepton* Figure 5a shows the distribution of masses for various sparticles from the excluded points where the 1-lepton analysis had the lowest CLs value. The 1-lepton analysis requires events with exactly 1 electron or muon with various minimum jet multiplicities and large \(E_\mathrm {T}^\mathrm{miss}\). Though the analysis was optimised for gluino production, there is also a prevalence of light squarks, whose distribution peaks around \(m(\tilde{q})\sim 1\) TeV. The distributions of the chargino and to a less extent next-to-lightest neutralino \(\tilde{\chi }^0_2\) are skewed towards low masses, peaking around 300 GeV, with a tail that extends above 1 TeV. The tendancy for the next-to-lightest neutralino and chargino to be light is characteristic of Higgsino content in the LSP, or a bino-like LSP with wino-like pair near in mass to the LSP.

*x*of models where 1-lepton was most sensitive in

*x*, and indeed the majority of models in the histogram have \(0<x<1\).

*SS/3L* Though there are similarities with the 1-lepton analysis, many salient differences appear for points where the SS/3L search is the most sensitive analysis. This search selects events with at least two leptons, and if there are exactly two, they are required to have the same electric charge.

One prominent difference is that points where the SS/3L is most sensitive again have almost exclusively wino-like LSPs, and so a nearly mass-degenerate chargino. This is consistent with the mass distributions of these models in Fig. 5b: compared with the 1-lepton case (Fig. 5a), the SS/3L has a smaller tail of high chargino mass while the next-to-lightest neutralino \(\tilde{\chi }^0_2\) distribution is no longer skewed to lower masses. We note that a future Run 2 version of the ‘Disappearing Track’ analysis [16], not considered in this work, could be sensitive to these models, since the small chargino-LSP mass splitting in such models ensures that the chargino is typically long-lived on collider time scales.

For these points, the gluinos are all relatively heavy and not strongly correlated with a particular mass scale, being fairly uniformly distributed for \(m(\tilde{g}) \gtrsim 1.5\) TeV, in contrast to the 1-lepton discussion. This implicates that the SS/3L is not the most sensitive analysis to light gluinos, where other analyses such as the 2–6 jets are most sensitive. By contrast, squarks retain a peaked distribution centred around 1 TeV.

Figure 6 demonstrates, in a similar way to the 1-lepton analysis, that the slepton mass is almost always between the chargino and the lightest squark. We find a negligible number of models (a single entry) has \(x>1\), corresponding to a slepton mass being above that of the lightest squark. Thus, we find that points where the SS/3L has best sensitivity are strongly correlated with one common feature: a squark–slepton–chargino–LSP \(\tilde{q}\)–\(\tilde{\ell }\)–\(\tilde{\chi }^\pm _1\)–\(\tilde{\chi }^0_1\) ordered mass spectrum. The squark can undergo a three-body decay to a quark, lepton and a slepton \(\tilde{q} \rightarrow q \ell \tilde{\ell }\) if the intermediate neutralino is off-shell \(m(\tilde{\chi }^0_2) > m(\tilde{q})\). This hierarchical structure is displayed in model number 11733067 (Fig. 4d). Among the models where SS/3L is most sensitive, this has the lightest squark mass at 436 GeV.

Moreover, we find one signal region ‘SR0b3j’ was used to exclude 98% of these models where the analysis was most sensitive. This suggests other analyses had better sensitivity to models targeted by the other three signal regions, for example the Multi-b analysis is particularly sensitive to \(\tilde{g} \rightarrow t\bar{t} \tilde{\chi }^0_1\) scenarios. The ‘SR0b3j’ signal region requires at least three leptons and was optimised to capture a \(\tilde{g}\)–\(\tilde{\chi }^0_2\)–\(\tilde{\ell }\)–\(\tilde{\chi }^0_1\) decay chain, distinct from the \(\tilde{q}\)–\(\tilde{\ell }\)–\(\tilde{\chi }^\pm _1\)–\(\tilde{\chi }^0_1\) scenario we just identified.

Taken together, we draw two noteworthy conclusions from our findings for the SS/3L search. First, this analysis is the most sensitive analysis for a different scenario we identified in the pMSSM than all the simplified models those used for analysis optimisaton. Second, the exclusivity of signal region used to exclude this scenario indicates that the SS/3L lacked competitive sensitivity to points in the pMSSM corresponding to the simplified models considered.

## 3 Probing the dark sector with strong SUSY searches

^{3}

### 3.1 Impact of 13 TeV constraints on dark matter observables

#### 3.1.1 Relic density

The LSP relic abundance \(\varOmega _{\chi ^0_1}h^2\) is set by early-universe thermal freeze-out, the hallmark of the weakly interacting massive particle (WIMP) paradigm. We do not require the neutralino to be the sole constituent of dark matter, as other well-motivated candidates such as axions can contribute [69], making the points considered more general than Ref. [56]. Thus the Planck measurement of the cold dark matter (CDM) abundance [53] only serves as an upper bound.

The composition of the LSP strongly influences the early-universe annihilation mechanism of the LSP and the resulting cosmological relic density. Figure 8a illustrates the relic density against the mass of the LSP that survived the constraints from six 13 TeV analyses, coloured by the dominant composition to the LSP, as defined in Table 2. Notably, the Higgsino- and wino-like LSP models are concentrated along straight diagonal lines in the plot. This is because the thermally averaged cross section \(\langle \sigma v\rangle \propto m^{-2}_{\tilde{\chi }_1^0}\) [70], and the relic density \(\varOmega _{\tilde{\chi }^0_1}h^2\) is therefore nearly proportional to the LSP mass squared. Mass splittings between the LSP and coannihilating chargino are typically a few GeV (\(\sim \)100s MeV) for Higgsino- (wino-) like LSPs. There are also no Higgsino- or wino-like LSPs below around 100 GeV. This is because such models have near-degenerate charginos to the LSP, which are excluded by LEP lower bounds.

By contrast, bino-like LSP models have suppressed early-universe thermally averaged annihilation cross sections,

^{4}leading to larger relic abundance, as generally seen in Fig. 8a. Therefore to satisfy the Planck bound, bino-like LSPs must either have non-negligible mixing with winos and/or Higgsinos, or there must be a near-degenerate next-to-LSP to act as a coannihilator. In Ref. [53], the natures of the coannihilators were displayed in the planes involving relic density. No distinction was made between light flavour squarks and gluinos, yet they have different phenomenological roles as coannihilators. We therefore elucidate this in Fig. 9, differentiating between coannihilators being gluinos (light orange), light flavour squarks (dark blue) and third generation squarks (green). We observe that light gluino coannihilators are more stringently excluded, with few points \(m(\tilde{g}) \lesssim 800\) GeV than for squarks due to powerful Run 1 constraints.Focusing now on electroweak particles (‘Other’, light grey), for LSP masses \(m (\tilde{\chi }^0_1) \lesssim 250\) GeV, the coannihilation mechanism is predominantly via uncoloured sparticles due to stringent LHC constraints on squarks and gluinos. The two peaks centred around \(m (\tilde{\chi }^0_1 ) \approx 45\) and 63 GeV involve resonant annihilation through a \(Z^0\) or Higgs boson. Meanwhile, for \(90 \lesssim m (\tilde{\chi }^0_1 ) \lesssim 250\) GeV, the coannihilators are predominantly slepton or gauginos, which are bounded from below by the LEP limit.

Returning to the discussion of points excluded by individual analyses in the relic density plane (Fig. 7a), the strong SUSY searches considered are sensitive to models with a wide variety of \(\varOmega _{\tilde{\chi }^0_1} h^2\). As previously discussed in Sect. 2.3, the 7–10 jets search is particularly sensitive to models with \(m (\tilde{\chi }^0_1 ) \lesssim 100\) GeV, where bino-like LSPs are associated with the \(Z^0\) and \(h^0\) funnel region. Meanwhile, the SS/3L analysis is most sensitive to wino-like LSPs models while the Multi-b analysis had preferential sensitivity to Higgsino-like LSP scenarios, as indicated by the clustering of green crosses and red dots along the wino and Higgsino respective diagonal bands (compare with Fig. 8a).

The 1-lepton analysis has sensitivity away from the Higgsino and wino diagonal bands, for LSP masses below about 300 GeV, where slepton and gaugino coannihilators are prevalent. The extensive presence of blue dots and cyan rings away from these bands shows respectively that the 2–6 jets and Monojet searches are particularly sensitive to gluinos and squarks that have small mass splitting with the bino-like LSP. Crucially, these impact scenarios where such coloured sparticles are the coannihilators (compare with Fig. 9) and therefore indirectly probe LSP masses higher than those currently accessible by direct electroweakino searches at the LHC. Nevertheless, the 2–6 jets analysis also has sensitivity covering Higgsino and wino-like LSP points, with clusters along the Higgsino and wino bands.

#### 3.1.2 Direct detection

^{5}cross section \(\sigma ^\mathrm{SI}_{N\mathrm{-}\tilde{\chi }^0_1}\) of the LSP interacting with a Xenon nucleus, normalised per nucleon, and LSP mass. Again we discuss several features, with reference to other projections where appropriate, to aid interpretation:

- Direct detection experiments typically interpret results assuming the LSP fully saturates the cold dark matter (CDM) relic abundance measured by Planck. As the LSP in the pMSSM need not be the sole constituent of dark matter, we rescale the direct detection interaction cross sections \(\sigma ^\mathrm{SI}_{N\mathrm{-}\tilde{\chi }^0_1}\) by a factorThis accounts for the reduction in direct detection sensitivity due to a lower local density of neutralino LSPs.$$\begin{aligned} R \equiv \varOmega _{\tilde{\chi }^0_1}h^2/\varOmega _\mathrm{CDM}^\mathrm{Planck}h^2. \end{aligned}$$(5)
Recently, the PandaX-II [77] and LUX [78] collaborations presented results that extend sensitivity for WIMP masses \(m (\tilde{\chi }^0_1) \gtrsim 20\) GeV by a factor of 2 and 4, respectively, beyond the LUX 2015 result [109]. As a guide to the sensitivity of direct detection, the observed limits from LUX 2016 [78] and the projected sensitivity of LZ [107] are overlayed in Fig. 7b.

In defining the permitted pMSSM points, ATLAS conservatively increased by a factor of four the upper limit on the spin-independent cross section \(R\cdot \sigma ^\mathrm{SI}_{N\mathrm{-}\tilde{\chi }^0_1}\) from LUX 2013 [98] to account for uncertainties in nuclear form factors [53], before rejecting points in the pre-selection. This pre-selection constraint carves out the points at the highest \(R\cdot \sigma ^\mathrm{SI}_{N\mathrm{-}\tilde{\chi }^0_1}\) cross sections around \(10^{-44}\) cm\(^2\).

Figure 8b reveals points that survived constraints from the six 13 TeV searches considered in this study, in the plane of spin-independent cross section against LSP mass. The points are coloured according to the whether the dominant contribution to the LSP is the bino, Higgsino, or wino. The composition of the LSP has a significant effect on \(R\cdot \sigma ^\mathrm{SI}_{N\mathrm{-}\tilde{\chi }^0_1}\). This is due both to the couplings of the LSP to nucleons and the relic density suppression (Eq. 5 and Fig. 8a). Notably, most wino-like LSPs have suppressed direct detection cross sections given the small coupling to the Higgs boson.

The 2–6 jets analysis (blue points) is sensitive to a large class of models, particularly those with gluino or squark coannihilators. As this difference has important phenomenological consequences, we display the strong sector coannihilators in Fig. 9b for bino-like LSPs of points that survived Run 1 constraints. In this projection, it is evident that coannihilation points involving light flavour squarks have enhanced cross section \(R\cdot \sigma ^\mathrm{SI}_{N\mathrm{-}\tilde{\chi }^0_1}\) compared with gluinos. This is due to the *s*-channel diagram involving the quarks and LSP scattering via an intermediate squark, a point we will elaborate further in Sect. 3.2. Thus, the 2–6 jets and Monojet analyses share sensitivity to many squark coannihilator scenarios with LUX 2016.

As discussed in previous sections, the 7–10 jets is most sensitive to light mass LSPs with significant bino content. Many of the points excluded by this search are below the current LUX 2016 sensitivity. Meanwhile, the Multi-b analysis tends to favour scenarios Higgsino-like LSP scenarios where squark masses are above 2 TeV. There is particular sensitivity to a region centred around cross section from 10\(^{-46}\) cm\(^{2}\) to 10\(^{-49}\) cm\(^{2}\) and LSP mass of 300 to 700 GeV (Fig. 7b). Many Higgsino-like LSP models inhabit this region and are beginning to be probed by LUX 2016, but the majority of points where the Multi-b analysis is most sensitive are below the LUX limit.

Notably, ATLAS strong SUSY searches are sensitive to scenarios with direct detection cross section \(R\cdot \sigma ^\mathrm{SI}_{N\mathrm{-}\tilde{\chi }^0_1}\) well below even the projected sensitivity of LZ based on 1000 days of data taking [107]. The SS/3L reach into this regime is especially prominent, being most sensitive to wino-like LSP models with highly suppressed cross sections \(R\cdot \sigma ^\mathrm{SI}_{N\mathrm{-}\tilde{\chi }^0_1}\). Many gluino coannihilator scenarios occupy this region (Fig. 9b) and are dominantly probed by the 2–6 jets analysis. The projected LZ sensitivity is within an order of magnitude of the irreducible neutrino background ‘floor’, which is a challenging regime for Xenon-target direct detection experiments.

Concluding this subsection, we demonstrated the important complementarity of strong SUSY searches for probing models beyond both Monojet-like collider interpretations and the reach of direct detection experiments. This enables colliders to indirectly constrain bino-like LSPs with coloured coannihilators, in addition to Higgsino LSP scenarios for example before electroweak SUSY searches gain direct sensitivity. This motivates construction of simplified DM models based on such interpretations, but is beyond the scope of this work.

### 3.2 Impact of direct detection constraints on squarks and gluinos

Finally, turning the question around, we explore the impact of LUX 2016 constraint on the parameter space of squarks and gluinos relevant to LHC searches. Considering the set of points that survived ATLAS Run 1 constraints [53], we deem any point with scaled cross section \(R \cdot \sigma ^\mathrm{SI}_{N\mathrm{-}\tilde{\chi }^0_1}\) above the 90% confidence level upper limit observed by LUX 2016 to be excluded. We take the observed central value of the upper limit as reported by LUX 2016 [78], without rescaling to account for the nuclear form factor uncertainties^{6} (as was done in Ref. [53]). The upper limit derived by LUX 2016 excludes 30.3% of such pMSSM points constraints (with long-lived gluinos, squarks and sleptons removed). Figure 8b shows that LUX is particularly sensitive to pMSSM points with bino-like LSP, due in part to the relic density suppression of Higgsino and wino-like models.

Figure 10a projects the fractional exclusion^{7} by LUX 2016 into the gluino vs. LSP plane. There is a band of modest exclusion for LSP masses \(m(\tilde{\chi }^0_1)\lesssim 250\) GeV, which is relatively uncorrelated with gluino masses. This apparent enhancement of sensitivity is partly an artefact due to the systematic oversampling of coannihilators for bino-like LSPs in Ref. [53]. This region is rich in gaugino and slepton coannihilators due to their weak LHC constraints compared with strongly interacting sparticles. Due to smaller annihilation cross sections of this electroweak process in the early universe, the relic abundance \(\varOmega _{\tilde{\chi }^0_1}h^2\) for a given bino-like LSP mass is larger. These models are thus not scaled down as far by the \(R=\varOmega _{\tilde{\chi }^0_1}h^2/\varOmega _\mathrm{CDM}^\mathrm{Planck}h^2\) factor.

A prominent feature of the points excluded by LUX appears when projecting into the squark vs. LSP plane (Fig. 10b). The diagonal region where the mass splitting between the LSP and squark is small \(m(\tilde{q}) - m(\tilde{\chi }^0_1) \lesssim 50\) GeV shows a distinctly higher exclusion fraction; this was absent in the gluino vs. LSP plane. Furthermore, mass bins with higher exclusion fraction are correlated with lower squark masses. This is particular salient along the diagonal, where points are predominantly squark coannihilators. The band of modest exclusion fraction for LSPs with \(m(\tilde{\chi }^0_1)\lesssim 250\) GeV remains, as with the gluino vs. LSP plane.

*s*-channel squark exchange diagram in direct detection

*q*inside the nucleons. Resonant scattering occurs when the LSP and squarks are nearly mass degenerate, enhancing the sensitivity of LUX along the squark–LSP diagonal of Fig. 10b. Thus, direct detection experiments are particularly sensitive to bino-like LSP scenarios with a squark coannihilator. The cross section \(\sigma ^\mathrm{SI}_{N\mathrm{-}\tilde{\chi }^0_1}\) decreases with heavier squarks due to the \(\sim \)\(1/(m_{\tilde{q}}^2 - m_{\tilde{\chi }^0_1}^2)\) suppression in the propagator.

These observations are consistent with the LSP–nucleon cross section vs. LSP mass plane with coannihilation mechanism identified (Fig. 9b). Squarks of the first or second generation (dark blue) have particularly enhanced direct detection cross sections. The points with gluino coannihilators (light orange) feature \(R\cdot \sigma ^\mathrm{SI}_{N\mathrm{-}\tilde{\chi }^0_1}\) primarily below \(10^{-46}\) cm\(^2\) and will only begin to be probed at the direct detection frontier by future experiments such as LZ. We also note that coannihilation points involving third generation squarks (light blue) also tend to have suppressed direct detection cross sections compared with squarks, due to the negligible third generation content in nucleons.

Taken together, this highlights the important implications of LHC searches for squarks in the context of direct detection experiments, especially when the squark–LSP mass splitting is small. Yet these squark coannhilation scenarios are challenging for colliders, where direct detection experiments can provide a complementary probe.

## 4 Conclusion

In this work, we interpreted six published 13 TeV (3.2 fb\(^{-1}\)) ATLAS SUSY searches for gluinos and light flavour squarks in a 19-parameter pMSSM. The purpose of this study was to analyse previously unexamined correlations between the most sensitive analyses with distinct regions of pMSSM parameter space. Our study addressed various shortcomings in the literature, presented under three questions in Sect. 1 (Sect. 1), which we now summarise.

Firstly, we examined these correlations in collider parameter spaces, providing substantially richer information than overlap matrices used in the literature. For the two-dimensional projections into gluino, LSP and squark masses, the separation in regions probed by the 2–6 jets and Multi-b analyses were particularly distinct. The Multi-b was the most sensitive analysis for models with larger gluino–LSP mass splittings, where the 2–6 jets search began to lose sensitivity. The regions identified are independent of the priors in the pMSSM points. Further, while the Monojet and 2–6 jets share substantial overlapping sensitivity, the tighter jet requirements of the former is needed for the scenarios where the coloured sparticle and LSP are near mass-degenerate.

Secondly, we identified classes of models beyond those ATLAS used for optimisation. Arguably the most striking realisation of this was the SS/3L search. Despite optimising to four distinct simplified models, we found one signal region to be most sensitive to a different scenario in the pMSSM not considered by the ATLAS search. It involved a light flavour squark cascading to a slepton and wino-like LSP with a nearly mass-degenerate chargino, which could be used by the experimental collaborations to refine future searches. Meanwhile, though the 1-lepton search optimised for a single simplified model, we showed it was sensitive to scenarios that included squark production, as well as intermediate sleptons and gauginos.

Finally, while ATLAS performed an explicit DM interpretation for their Monojet search, our study manifested the prominent role other searches for coloured sparticles have when the dark sector is beyond non-minimal regimes as in the pMSSM. Bino-like LSPs may rely on coloured coannihilators to be consistent with the observed relic abundance, which are probed by the 2–6 jets and Monojet analyses being sensitive to small squark–LSP and gluino–LSP mass splittings. Light flavour squarks enhance LSP–nucleon scattering cross sections, and squark lower mass bounds can still be below 500 GeV in the pMSSM. In addition, the SS/3L analysis had preferential sensitivity to wino-like LSP scenarios, which are particularly challenging for direct detection experiments.

Using our findings to design novel search strategies or interpretations was beyond the scope of this work and is deferred to future studies. It would also be of interest to perform similar assessments for the third generation squark and electroweak sectors including long-lived sparticles, interpret searches based on simplified models of DM, as well as develop surveys for non-minimal SUSY scenarios. Previously unprobed regions of the pMSSM will be explored further as LHC luminosity continues to rise.

## Footnotes

- 1.
A recent independent study [56] considered ATLAS constraints with up to 14.8 fb\(^{-1}\), but on the subset of points whose neutralino relic abundance was within 10% of Planck. We instead take the Planck measurement as an upper bound, allowing for non-SUSY contributions to DM.

- 2.
The prior distributions for these plots are shown in Fig. 13 of the appendix.

- 3.
Such indirect detection constraints were examined in Ref. [48], which finds that current constraints from Fermi-LAT are not expected to constrain the pMSSM space. Nevertheless, it was found that the future Cerenkov Telescope Array (CTA) [106] is projected to have sensitivity to high mass LSPs scenarios not accessible to direct detection and collider searches.

- 4.Pure binos do not couple to any gauge or Higgs bosons. This is seen from the couplings \(g_{Z\tilde{\chi }_1^0\tilde{\chi }_1^0}\) (\(g_{h\tilde{\chi }_1\tilde{\chi }_1}\)) of the neutralino to the \(Z^0\) and (Higgs) bosons are given at tree-level by [108]$$\begin{aligned} g_{Z\tilde{\chi }_1^0\tilde{\chi }_1^0}&= \frac{g_2}{2\cos _W}(\left| N_{13}\right| ^2 - \left| N_{14}\right| ^2),\end{aligned}$$(3)Here we have the neutralino mixing matrix elements \(N_{ij}\) defined in Table 2, the SU(2) gauge coupling \(g_2\), Weinberg angle \(\theta _W\) and the Higgs mixing parameter \(\alpha \). Pure binos have \(N_{12} = N_{13} = N_{14} = 0\).$$\begin{aligned} g_{h\tilde{\chi }_1^0\tilde{\chi }_1^0}&= g_2\cos \alpha (N_{11} - N_{12}\tan \theta _W )\left( N_{13}\tan \alpha + N_{14}\right) . \end{aligned}$$(4)
- 5.
- 6.
Had we weakened this constraint by a factor of four [53], the key qualitative features in the discussion are unaffected; only the numerical fraction of models excluded is reduced to 18.0%.

- 7.
Although \(N_\mathrm{LUX\, 2016}^\mathrm{Excluded} / N_\mathrm{ATLAS\, Run\, 1}^\mathrm{Survived}\) is prior dependent, we will momentarily discuss this, while the forthcoming comparison of gluino and squark coannihilation regions has minimal prior dependence.

## Notes

### Acknowledgements

We are grateful to Moritz Backes, Fady Bishara, Claire Gwenlan, Will Fawcett, Will Kalderon, Olivier Lennon, John March-Russell and Mike Nelson for helpful discussions, together with the computing support of Dennis Liu and Kashif Mohammad. This research is supported by the Science and Technology Facilities Council (STFC). The authors would like to acknowledge the use of the University of Oxford Advanced Research Computing (ARC) facility in carrying out this work [110].

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