The pMSSM10 after LHC Run 1
We present a frequentist analysis of the parameter space of the pMSSM10, in which the following 10 soft SUSY-breaking parameters are specified independently at the mean scalar top mass scale : the gaugino masses , the first-and second-generation squark masses , the third-generation squark mass , a common slepton mass and a common trilinear mixing parameter , as well as the Higgs mixing parameter , the pseudoscalar Higgs mass and , the ratio of the two Higgs vacuum expectation values. We use the MultiNest sampling algorithm with points to sample the pMSSM10 parameter space. A dedicated study shows that the sensitivities to strongly-interacting sparticle masses of ATLAS and CMS searches for jets, leptons + signals depend only weakly on many of the other pMSSM10 parameters. With the aid of the Atom and Scorpion codes, we also implement the LHC searches for electroweakly-interacting sparticles and light stops, so as to confront the pMSSM10 parameter space with all relevant SUSY searches. In addition, our analysis includes Higgs mass and rate measurements using the HiggsSignals code, SUSY Higgs exclusion bounds, the measurements of by LHCb and CMS, other -physics observables, electroweak precision observables, the cold dark matter density and the XENON100 and LUX searches for spin-independent dark matter scattering, assuming that the cold dark matter is mainly provided by the lightest neutralino . We show that the pMSSM10 is able to provide a supersymmetric interpretation of , unlike the CMSSM, NUHM1 and NUHM2. As a result, we find (omitting Higgs rates) that the minimum with 18 degrees of freedom (d.o.f.) in the pMSSM10, corresponding to a probability of %, to be compared with in the CMSSM (NUHM1) (NUHM2). We display the one-dimensional likelihood functions for sparticle masses, and show that they may be significantly lighter in the pMSSM10 than in the other models, e.g., the gluino may be as light as at the 68% CL, and squarks, stops, electroweak gauginos and sleptons may be much lighter than in the CMSSM, NUHM1 and NUHM2. We discuss the discovery potential of future LHC runs, colliders and direct detection experiments.
KCL-PH-TH/2015-15, LCTS/2015-07, CERN-PH-TH/2015-066,
DESY 15-046, FTPI-MINN-15/13, UMN-TH-3427/15, SLAC-PUB-16245, FERMILAB-PUB-15-100-CMS
- 1 Introduction
- 2.1 Parameter Ranges
- 2.2 MasterCode Framework
- 2.3 Electroweak, Flavour, Cosmological and Dark Matter Constraints
- 2.4 Higgs Constraints
- 2.5 LHC Constraints on Sparticle Masses
- 3.1 Mass Planes
- 3.2 The Best-Fit Point
- 3.3 Sparticle Masses
- 3.4 Benchmark pMSSM10 Models
- 3.5 The Anomalous Magnetic Dipole Moment of the Muon
- 3.6 Interplay of the , and Dark Matter Constraints
- 3.7 Higgs Physics
- 3.8 Decay
- 3.9 Direct Dark Matter Detection
- 4 Extrapolation to High Scales
- 5 Prospects for Sparticle Detection in Future LHC Runs
- 6 Prospects for Sparticle Detection at a Future Collider
- 7 Conclusions
The quest for supersymmetry (SUSY) has been among the principal objectives of the ATLAS and CMS experiments during Run 1 of the Large Hadron Collider (LHC). However, despite searches in many production and decay channels, no significant signals have been observed [1, 2]. These negative results impose strong constraints on -conserving SUSY models, in particular, which are also constrained by measurements of the mass and other properties of the Higgs boson , by precision measurements of rare decays such as [4, 5, 6, 7] and other measurements. Overall, these constraints tend to reduce the capacity of SUSY models to alleviate the hierarchy problem. However, their impact on a possible resolution of the discrepancy between the experimental measurement of and theoretical calculations in the Standard Model (SM) depends on further assumptions as will be discussed below.
There have been many analyses that combine these constraints in global statistical fits within specific SUSY models based on the minimal supersymmetric extension of the Standard Model (MSSM) . Many of these analyses assume that the low-energy soft SUSY-breaking parameters of the MSSM may be extrapolated using the renormalization-group equations (RGEs) up to some grand unified theory (GUT) scale, where they are postulated to satisfy some universality conditions. Examples of such models include the constrained MSSM (CMSSM) [9, 10, 11], in which the soft SUSY-breaking mass parameters and are assumed to be universal at the GUT scale, as are the trilinear parameters . Other examples include models that relax the universality assumptions for the soft SUSY-breaking contributions to the Higgs masses, the NUHM1  and NUHM2  (see also, e.g., Ref. ), but retain universality for the slepton, squark and gaugino masses. Such models are particularly severely constrained by the LHC searches for colored sparticles, the squarks and gluino, which also place indirect limits on the masses of sleptons and electroweak gauginos and higgsinos via the GUT scale constraints, while the direct search limits on these particles have much less impact.
An alternative approach is to make no assumption about the RGE extrapolation to very high energies, but take a purely phenomenological approach in which the soft SUSY-breaking parameters are specified at low energies, and are not required to be universal at any input scale, a class of models referred to as the phenomenological MSSM with free parameters (pMSSM) . This is the framework explored in this paper. Favoured mass patterns in a pMSSM analysis might then give hints for (alternative) GUT-scale scenarios.
In the absence of any assumptions, the pMSSM has so many parameters that a thorough analysis of its multi-dimensional parameter space is computationally prohibitive. Here we restrict our attention to a ten-dimensional version, the pMSSM10, in which the following assumptions are made. Motivated by the absence of significant flavor-changing neutral interactions (FCNI) beyond those in the Standard Model (SM), we assume that the soft SUSY-breaking contributions to the masses of the squarks of the first two generations are equal, which we also assume for the three generations of sleptons. The FCNI argument does not motivate any relation between the soft SUSY-breaking contributions to the masses of left- and right-handed sfermions, but here we assume for simplicity that they are equal. As a result, we consider the following 10 parameters in our analysis (where “mass” is here used as a synonym for a soft SUSY-breaking parameter, and the gaugino masses and trilinear couplings are taken to be real):
All of these parameters are specified at a low renormalisation scale, the mean scalar top mass scale, , close to that of electroweak symmetry breaking.
In any pMSSM scenario such as this, the disconnect between the different gaugino masses allows, for example, the U(1) and SU(2) gauginos to be much lighter than is possible in GUT-universal models, where their masses are related to the gluino mass and hence constrained by gluino searches at the LHC. Likewise, the disconnect between the different squark masses opens up more possibilities for light stops, and the disconnect between squark and slepton masses largely frees the latter from LHC constraints.
An important feature of our global analysis is that the possibilities for light electroweak gauginos and sleptons reopen an opportunity for an significant SUSY contribution to in the pMSSM, a possibility that is precluded in simple GUT-universal models such as the CMSSM, NUHM1 and NUHM2 by the LHC searches for strongly-interacting sparticles. As we discuss in detail in this paper, the pMSSM10’s flexibility removes the tension between LHC constraints and the measured value of , with the result that the best fit in the pMSSM10 has a global probability that is considerably better than in the CMSSM, NUHM1, NUHM2 or SM.
The main challenges for a global fit of the pMSSM10 are the efficient sampling of the ten-dimensional parameter space and the accurate implementation of the various SUSY searches by ATLAS and CMS. As in , here we use the sampling algorithm MultiNest  to scan efficiently the pMSSM10 parameter space. To achieve sufficient coverage of the relevant parameter space, approximately pMSSM10 points were sampled. However, confronting all these sample points individually with all relevant collider searches is computationally impossible. In order to overcome this problem and still to apply the SUSY searches in a consistent and precise manner, we split the LHC searches into three categories. In the first category we consider inclusive SUSY searches that mainly constrain the production of coloured sparticles, namely the gluino and squarks. To apply these searches to the pMSSM10 parameter space, we follow closely an approach proposed in , which uses a variety of inclusive SUSY searches covering different final states to establish a simple but accurate look-up table that depends only on the gluino, squark and LSP masses. Then, in order to implement the other two categories of LHC constraints on the SUSY electroweak sector and compressed stop spectra, we treat the LHC searches for electroweakly-interacting sparticles via trileptons and dileptons, and for light stops, separately using dedicated algorithms validated using the Atom  and Scorpion  codes. In all cases we consider the latest SUSY searches from ATLAS and CMS that are based on the full Run 1 data set, as detailed later in the paper. We perform extensive validations of the applications of these searches to the pMSSM10, so as to ensure that we make an accurate and comprehensive set of implementations of the experimental constraints on the model.
More information about the scan of the pMSSM10 parameter space using the MultiNest technique, as well as details about our implementations of the LHC searches, are provided in Section 2. Section 3 discusses the results of the pMSSM10 analysis, including the best-fit point and other benchmark points with low sparticle masses that could serve to focus analyses at Run 2 of the LHC. Section 4 discusses the extent to which the preferred ranges of pMSSM10 parameters permit renormalization-group extrapolation to GUT scales. Section 5 analyses the prospects for discovering SUSY in future runs of the LHC, Section 6 analyses the prospects for discovering SUSY at possible future colliders, and our conclusions are summarised in Section 7.
We describe in this Section how we perform a global fit of the pMSSM10 taking into account constraints from direct searches for SUSY particles, the Higgs boson mass and rate measurements, SUSY Higgs exclusion bounds, precision electroweak observables, -physics observables, and astrophysical and cosmological constraints on cold dark matter. We describe the scanned parameters and their ranges, the framework that we use to calculate the observables, and the treatment of the various constraints.
2.1 Parameter Ranges
As described above we consider a ten-dimension subset (pMSSM10) of the full pMSSM parameter space. The selected SUSY parameters were listed in Eq. (1), and the ranges of these parameters that we sample are shown in Table 1. We also indicate in the right column of this Table how we divide the ranges of most of these parameters into segments, as we did previously for our analyses of the CMSSM, NUHM1 and NUHM2 [21, 16].
The combinations of these segments constitute boxes, in which we sample the parameter space using the MultiNest package . For each box, we choose a prior for which 80% of the sample has a flat distribution within the nominal range, and 20% of the sample is outside the box in normally-distributed tails in each variable. In this way, our total sample exhibits a smooth overlap between boxes, eliminating features associated with box boundaries. An initial scan over all mass parameters with absolute values showed that non-trivial behaviour of the global likelihood function was restricted to and . In order to achieve high resolution efficiently, we restricted the ranges of these parameters to and in the full scan.
|(-1 , 1 )||2|
|( 0 , 4 )||2|
|(-4 , 4 )||4|
|( 0 , 4 )||2|
|( 0 , 4 )||2|
|( 0 , 2 )||1|
|( 0 , 4 )||2|
|(-5 , 5 )||1|
|(-5 , 5 )||1|
|( 1 , 60)||1|
|Total number of boxes||128|
2.2 MasterCode Framework
We calculate the observables that go into the likelihood using the MasterCode framework [22, 23, 24, 21, 16, 25], which interfaces various public and private codes: SoftSusy 3.3.9  for the spectrum, FeynWZ  for the electroweak precision observables, FeynHiggs 2.10.0 [28, 29] for the Higgs sector and , SuFla , SuperIso  for the -physics observables, Micromegas 3.2  for the dark matter relic density, SSARD  for the spin-independent cross-section , SDECAY 1.3b  for calculating sparticle branching ratios, and HiggsSignals 1.3.0  and HiggsBounds 4.2.0  for calculating constraints on the Higgs sector. The codes are linked using the SUSY Les Houches Accord (SLHA) .
2.3 Electroweak, Flavour, Cosmological and Dark Matter Constraints
For many of these constraints, we follow very closely our previous implementations, which were summarized recently in Table 1 in . Specifically, we treat all electroweak precision observables, all -physics observables (except for ), , and the relic density as Gaussian constraints. The contribution from , combined here in the quantity , is calculated using the combination of CMS  and LHCb  results described in . We incorporate the current world average of the branching ratio for BR() from  combined with the theoretical estimate in the SM from , and the recent measurement of the branching ratio for BR() by the Belle Collaboration  combined with the SM estimate from . We use the upper limit on the spin-independent cross section as a function of the lightest neutralino mass from LUX , which is slightly stronger than that from XENON100 , taking into account the theoretical uncertainty on as described in .
2.4 Higgs Constraints
We use the recent combination of ATLAS and CMS measurements of the mass of the Higgs boson: , which we combine with a one- uncertainty of in the FeynHiggs calculation of in the MSSM.
In addition, we refine substantially our treatment of the Higgs boson constraints, as compared with previous analyses in the MasterCode framework. In order to include the observed Higgs signal rates we have incorporated HiggsSignals , which evaluates the contribution of 77 channels from the Higgs boson searches at the LHC and the Tevatron (see Ref.  for a complete list of references). A discussion of the effective number of contributing channels is given in Sect. 3.2 below.
We also take into account the relevant searches for heavy neutral MSSM Higgs bosons via the channels [45, 46]. We evaluate the corresponding contribution using the code HiggsBounds , which includes the latest CMS results  based on of data 111The corresponding ATLAS results  have similar sensitivity, but are documented less completely.. These results include a combination of the two possible production modes, and , which is consistently evaluated depending on the MSSM parameters. Their implementation in HiggsBounds has been tested against the published CMS data, and very good qualitative and quantitative agreement had been found . Other Higgs boson searches are not taken into account, as they turn out to be weaker in the pMSSM10 that we study.
2.5 LHC Constraints on Sparticle Masses
A comprehensive and accurate application of the SUSY searches with the full Run 1 data of the LHC to the pMSSM10 parameter space is a central part of this paper. As most of these searches have been interpreted by ATLAS and CMS only in simplified model frameworks, we have introduced supplementary procedures in order to apply these searches to the complicated sparticle spectrum content of a full SUSY model such as the pMSSM10. For this we consider three separate categories of particle mass constraints that arise from the LHC searches: a) generic constraints on coloured sparticles (gluinos and squarks), b) dedicated constraints on electroweakly-interacting gauginos, Higgsinos and sleptons, c) dedicated constraints on stop production in scenarios with compressed spectra. We refer to the combination of all these constraints from direct SUSY searches as the LHC8 constraint, with sectors labelled as , , and , respectively. In the following subsections we provide further details about our implementations of these individual constraints, discussing in detail the validations of our procedures and the corresponding uncertainties.
We use two dedicated software frameworks for recasting the LHC analyses used in this paper. Both frameworks implement the full list of cuts of a given experimental search to obtain yields in the respective signal regions of the search. These signal yields are then confronted with the SM background yields and observations in data, as reported by the experimental searches. Based on these comparisons we construct the standard statistical estimator , which is also used by the experiments to determine the compatibility of their data with a given signal hypothesis. In this way it is possible to interpret the various LHC searches in any given SUSY model, such as those explored in our pMSSM10 scans.
To recast the ATLAS searches considered in this paper we use Atom , which is a Rivet  based framework. Atom models the resolutions of LHC detectors by mapping from the truth-level particles found for example in PYTHIA 6  event samples to the reconstructed objects, such as -jets and isolated leptons, according to the reported detector performances. In particular, the efficiencies of object reconstruction and the parameters associated with the momentum smearing are implemented in the form of analytical functions or numerical grids. The program has already been used in several studies , and the validation of the code can be found in .
For the CMS searches we use a private code called Scorpion  that was already used in . Scorpion obtains signal yields for a number of CMS searches based on events generated with PYTHIA 6  that are passed through the DELPHES 3  detector simulation package using an appropriate data card to emulate the response of the CMS detector. A significant effort was made to validate the modelling of these analyses by comparing the results obtained with the published results of the experimental collaboration. For further information on the validation of the CMS searches see .
The signal yields from Atom and Scorpion are confronted with the background yields and observations obtained from the individual ATLAS and CMS searches, and the corresponding is calculated using the LandS package . We convert the calculated value for a generic spectrum in the MSSM into a contribution by interpreting it as a p-value for the signal hypothesis assuming one degree of freedom.
2.5.1 LHC constraints on coloured sparticles
In the cases of the CMSSM, NUHM1 and NUHM2, we showed in [21, 16] that it was sufficient to extrapolate to other parameter values the exclusion contour in the CMSSM plane from the ATLAS search for jets+  that was given for specified values of and . We showed that the ATLAS exclusion is, to good approximation, independent of and [23, 16, 55] and, for the applications to the NUHM1 and the NUHM2, we checked that these limits in the plane were independent of the degrees of non-universality of the soft SUSY-breaking contributions to the Higgs masses, within the intrinsic sampling uncertainties.
In the case of the pMSSM10, however, the implementation of the direct searches for coloured sparticles is less straightforward. It is computationally impossible to apply all the LHC search constraints individually to each of the parameter choices in our sample. For example, PYTHIA 6 and DELPHES 3 take several minutes for the generation of 10,000 events followed by detector simulation, which is required to determine the signal acceptance and of each point sampled in the parameter space. Instead, we follow an approach outlined in , which constructs universal mass limits on coloured sparticles by combining an inclusive set of jets + + searches, as we now describe.
As was shown in , it is possible to establish lower limits on the gluino mass, , and the third-generation squark mass, , that are independent of the details of the underlying spectrum, within the intrinsic sampling uncertainties, by combining a suitable set of inclusive SUSY searches. In this approach the limits only depend on , and the mass of the lightest sparticle . The essence of the idea is that strongly-interacting sparticles decay through a variety of different cascade channels, whose relative probabilities depend on other model parameters. However, if one combines a sufficiently complete set of channels of the form jets + + , one will capture essentially all the relevant decay channels.
In order to apply this idea to the pMSSM10 parameter space, we have to extend this approach to include also the generic first- and second- generation squark mass, , as a free parameter. We then construct a ‘universal’ function that depends only on , , , and , as detailed below. This function defines our implementation of this constraint. There are two caveats to this approach. One is that the region of parameter space where is small, which is the object of dedicated searches, requires special attention. The other is that searches for electroweakly-produced sparticles (sleptons, neutralinos and charginos) fall outside the scope of the constraint. We have developed dedicated approaches to establish accurate LHC limits for the special cases of electroweakly-produced sparticles and the compressed-stop scenario with , as described in Sections 2.5.2 and 2.5.3, respectively.
In order to construct as a function of , , , and , we first generate a sample of points on a dimensional grid, which we use for linear interpolation. We construct this grid starting from values of . For each of these values of , we select the following values of and : , whereas takes values , where the dots indicate steps of 100, so that the total number of points in the grid is 25,564. The choice for this grid is motivated by the need for a fine granularity at low masses, while also capturing the parameter behaviours at higher masses.
We associate a SUSY spectrum to each point on the grid, by setting the first- and second-generation squark masses equal to , and the third-generation squark masses equal to . For each SUSY spectrum we generate coloured sparticle production events using PYTHIA 6  and pass them through the DELPHES 3  detector simulation code using a detector card that emulates the CMS detector response. We then pass the resulting events through Scorpion , which emulates the monojet, MT2, single-lepton, same- and opposite-sign dilepton (SS and OS) and 3-lepton CMS searches [56, 57], to estimate the numbers of signal events in each of the signal regions. After this we calculate the using the LandS package , by combining all signal regions from these searches. If searches have overlapping signal regions, we take the strongest expected limit, as is the case for the CMS monojet and single-lepton searches.
In Fig. 1 we show a three-dimensional overview and a pair of two-dimensional slices through this grid. The top panel shows the full three-dimensional grid for and illustrates the fine and coarse granularity of the grid at low and high values of , and , respectively. The lower left panel shows the two-dimensional slice for the same neutralino mass and , highlighting that there is only a small, though non-negligible, dependence of the function on for values of . The lower right panel shows the function as a function of and , for fixed and , illustrating that for different values of different grids are defined in , , and .
In order to apply the constraint to a generic pMSSM10 spectrum, we calculate () as the cross-section-weighted average of the first- and second- (third-)generation squark masses, to ensure that the constraint reflects the actual production cross-sections. This is especially relevant for the third-generation squark masses, as they generally have large splittings. The contribution for is obtained by linear interpolation of the values on the -dimensional grid. There is one special case when : here the standard searches listed above are less sensitive, and the universality of the limits is expected to break down. In this case, we calculate assuming zero cross-section for the lighter stop, and consider separately the impacts of dedicated stop searches in this region, as described in Section 2.5.3.
In order to validate the constraint and to gauge quantitatively its uncertainty, we have performed a number of studies and tests. First, we randomly selected 1000 model points from our sample where at least one of the sparticle masses is low enough to have been within the reach of LHC Run 1 ( and either , or ) and relative to the global minimum. For these points we compare the values interpolated from the look-up table () with the obtained by running the full chain of event generation, detector simulation and analyses (). The left panel of Fig. 2 shows a histogram of the differences for the 1000 randomly-selected points. As indicated in the legend of this figure, the standard deviation on this distribution is .
The right panel of Fig. 2 shows a scatter plot in the plane of the values obtained from the two approaches. They would agree perfectly along the diagonal where , and the lighter- and darker-shaded blue strips are the and bands around this diagonal. The vertical and horizontal dashed lines in this plot correspond to the 95% in each approach. For the majority of points, the interpolation and the full analysis agree whether the point is excluded at the 95% , or not, and most of the remaining points lie within .
We then assess how the uncertainty in our implementation of the constraint translates into uncertainties in sparticle mass limits: see the upper left panel of Fig. 3 222The other panels of Fig. 3 show the corresponding uncertainties in our treatments of the and constraints, which are discussed later.. For this estimate, we bin the 1000 points of the first test, and calculate the standard deviation, , for points with , and . We then apply the constraint in three ways: with the nominal implementation, and shifting the penalty up and down according to these binned standard deviations. The results are shown in the upper left panel of Fig. 3 as solid and dotted red (blue) contours in the plane corresponding to the nominal and up- and down-shifted cases for the 68 (95)% CL, respectively 333Here and in subsequent analogous parameter planes, we treat the and 5.99 contours as proxies for the 68% and 95% CL contours.. A dedicated study of points within the 68% and 95% CL regions confirms that our implementation of the constraint is valid within these uncertainties, and our estimate of at the best-fit point differs from the Scorpion evaluation by less than one 444Our and analyses described later also differ by less than one from the corresponding Scorpion/Atom evaluations. This is also true for the benchmark points introduced later..
We conclude that the uncertainty in our estimate is generally reliable, and translates into an uncertainty of in the limits on the gluino and squark masses, which is fully sufficient for the purpose of our studies.
2.5.2 LHC constraints on electroweak gauginos, Higgsinos and sleptons
Unlike the searches for coloured sparticles, where we were able to construct a computationally-efficient, approximately universal limit, the LHC constraints on electroweakly-produced sparticles vary strongly in sensitivity, depending on the mass hierarchy of sparticles and their corresponding decay modes and final states. For example, searches in the three-lepton plus missing energy channel constrain the chargino and neutralino masses up to GeV for GeV, if and decay exclusively into on-shell sleptons [58, 59], whereas a much weaker limit, GeV for GeV, was found in an analysis of the two-lepton plus missing energy channel [59, 60], assuming that the and decay exclusively into the in association with and , respectively, and not taking into account the decay [61, 62]. The same two-lepton analyses constrain slepton pair production, leading to the limits (200) GeV for (50) GeV [59, 60]. Therefore, the universal limit approach that we use to combine and characterise searches for coloured sparticles is inapplicable to searches for electroweakly-produced sparticles, and we use an alternative method.
For model points where the production of electroweakly-produced sparticles provides a non-trivial constraint, they must be much lighter than the coloured sparticles, since otherwise the much higher rates of production of coloured sparticles would already exclude the model points. Therefore, in the region of interest, there can be only a few particles lighter than the electroweakly-produced sparticles, implying that one can use a combination of a few simplified models (SMS) to approximate the sensitivities of the LHC searches for the production of these sparticles. Depending on the decay mode and final state, we select ATLAS and/or CMS limits derived from relevant simplified models to calculate the contributions of these searches to our global function. For the LHC searches that constrain electroweakly-produced gauginos, Higgsinos and sleptons, to a good approximation all relevant contributions can be extracted from simplified chargino-neutralino and simplified smuon and selectron models.
For each simplified model limit we construct a function that depends on the two relevant masses: () for the simplified chargino-neutralino model and () for the simplified slepton model. We assume that in the bulk of the region excluded in the simplified model, and that this penalty vanishes exponentially when crossing the boundary to the allowed region, with the general form
where the subscripts indicate the simplified model exclusion contour to the left and right (in the horizontal direction, i.e., or ) of the point on the contour with the largest value of , is the branching ratio of the decay in question (as calculated with SDECAY ), is the closest distance in to the contour, and and control the precise fall-off of the function, so as to mimic the experimental uncertainty bands, and are functions of . We note that if one sets then , so that the exclusion on the contour corresponds approximately to the 95% . Finally, to avoid an unphysically slow fall-off outside the 95% limit we set and adjust accordingly if and (and hence ).
In order to illustrate Eq. (2), we display in Fig. 4 for the decay via sleptons. In the left panel is shown for a fixed value of where the green (blue) line corresponds to , (), whereas vertical dashed lines indicate the position of the contour. The right panel shows the same (in colour) as a function of and , and the 95% exclusion contour found in Fig. 7(a) of  (blue line). Note that we apply no constraint for , the highest value on the blue experimental contour.
In order to establish we tuned the and parameters for each simplified model to reproduce best the values that we obtained using Atom for a representative set of model points from our sample. Table 2 summarises the implementations of the simplified model exclusion limits that contribute to . Note that, as described above, the large value of for the limit from production and decay via is replaced by setting and adjusting accordingly when (and hence ). Also, we had to produce our own contour for the direct production of right- and left-handed sleptons (selectrons and smuons), corresponding to their production cross-sections. Note that this simplified model contour is also applied when left-handed sleptons decay via and .
|via||Fig. 7(a) in ||(-5, 5)||(-40, 40)|
|via||Fig. 7(b) in ||(-20, 20)||(-300, 300)|
|Generated using Atom||(-20, 10)||(-40, 30)|
In order to validate our method and to determine quantitatively its uncertainty, we compare the contributions to the global function calculated with this limit approach,
to results from a full recast of all the above-listed searches as implemented in Atom. In this recast the full analysis is simulated, so that it is possible to determine for any arbitrary SUSY spectrum the value (and hence the corresponding ) with which a given search penalizes the SUSY spectrum. We obtain a set of 1000 model points from our sample by binning the plane in bins, selecting one point randomly per bin, and then take a random subset of 1000 of these points. This procedure was employed to ensure a representative set of the decay modes in our sample.
Fig. 5 displays scatter plots in the plane of the contributions to the global function for these 1000 model points as calculated using the method () (left panel) and the Atom code (right panel), with the indicated colour code in each plot. The immediate visual impression is that the colours in the two scatter plots are generally quite similar, indicating that the two procedures deliver similar contributions overall. A closer inspection of the plots reveals similar bands of low- points with small in a chargino coannihilation strip region, while elsewhere we see similar disfavouring of points with low and larger . However, even within this band we see a sparse set of points with relatively low that appear similarly in both the analysis based on simplified models and the Atom implementation of the full searches. These are mainly due to the decay , thus weakening the stronger -based limit.
For a more quantitative comparison of our method and Atom we turn to Fig. 6. We see in the left panel that the difference between and is relatively small, with an r.m.s. difference . The correlation between and is visible in the scatter plot in the right panel of Fig. 6. We see that most points are either excluded with in both analyses, or allowed with in both cases. Last but not least, there are relatively few ‘off-diagonal’ points with large , which form the small non-Gaussian tail of the distribution seen in the left panel of Fig. 6.
To quantify the impact of this uncertainty on our analysis, we follow the same procedure as for our limits on coloured sparticles, and translate the (binned analogously) into a band for our 68% and 95% CL contours in the important and planes. As can be seen in the upper right and lower left panels of Fig. 3, the uncertainty associated with is in general small in the 68% CL region of our fit, although it is larger at the 95% CL level in the plane. The effects on the best-fit point of these upward and downward shifts in the treatment are shown in these panels as open green stars. The downward shift has very little effect, and is essentially invisible in the plane. The upward shift increases the best-fit values of and while reducing that of , though the variations are contained well within the 68% CL region, clearly indicating that the corresponding uncertainties do not impact the overall conclusions.
2.5.3 LHC constraints on compressed stop spectra
In their searches for stop production, ATLAS and CMS have placed special emphasis on compressed spectra, which pose particular challenges for LHC searches. Whilst limits on stop production in the region where are fully included in the limits described in Section 2.5.1, a dedicated treatment of the compressed-spectrum region is required in order to include properly all the relevant collider limits. In this region we calculate the contribution of stop searches to the global in a similar way as for the for electroweakly produced sparticles described in Section 2.5.2. We refer to this dedicated limit-setting procedure as .
We show in Fig. 7 a colour-coded scatter plot in the plane of the decay modes with branching ratios % for 1000 randomly-selected pMSSM10 points in the region of interest. We see that the mode (shown in light green) dominates for the majority of points, and that this decay can be important throughout the parameter region displayed. We also find that, when this is the dominant stop decay mode, in most cases the and are almost mass degenerate. To constrain the final states with this decay mode we implement the simplified model limit presented in Fig. 6 of the ATLAS di-bottom analysis , where GeV is assumed, applying this for the model points with GeV.
If , the 3-body mode can dominate stop decay. The points for which this mode is dominant are shown by purple dots in Fig. 7. For this decay mode we implement the simplified model limit presented for in Fig. 15 of the ATLAS single-lepton analysis .
In the region, the decays (red dots in Fig. 7) and (grey dots) can be the dominant stop decay modes. The mode (green dots) may also dominate stop decay in this region, as well as in the region, as can also be seen in Fig. 7.
Due to the variety of different stop decay modes that are relevant in this compressed region, we cannot use only the limits from simplified models provided by the experiments, as they do not cover all relevant decay chains and assume branching ratios of 100%. However, these missing, in part rather complex, decay chains can effectively be constrained by hadronic inclusive searches such as those we have already used for our limits. In particular, the CMS hadronic search  has rather high sensitivity for these decay chains, as the kinematic phase space covered by the search makes no special assumptions on the final state, other than it having a purely hadronic signature.
Based on these inclusive searches, we derive limits for simplified models for and decays. For the simplified model we assume GeV when creating the limit in the (, ) plane. We do not implement a simplified model limit for because this decay mode has negligible impact on our study, as can be seen in Fig. 7. Using these simplified model limits, we constrain the stop decay modes following a procedure very similar to what we used for , using an interpolating function of the form (2) to mimic the uncertainty (yellow) band in, e.g., Fig. 6c in . We summarise our implementation of the simplified model limits in Table 3. When establishing these limits we use values of the parameters and that depend on . Whenever multiple values of these parameters are given for different values of , the parameters for intermediate values of are obtained by linear interpolation, and taken as constants elsewhere.
|Fig. 6(c) in ||210||(10, 20)||(-50, 50)|
|300||(-250, 200)||(-200, 200)|
|Fig. 15 in ||100||(-20,50)||(-70, 50)|
|Generated using||-||(-50, 50)||(-20, 50)||Based on , assuming|
|Generated using||-||(-20, 20)||(-20, 20)||Based on |
As for our and limit implementations, it is also important to determine accurately the uncertainty in the dedicated limit procedure for the compressed stop region. Note that in the compressed region not only the constraints from but also those from play a role. Therefore we first assess the qualitative agreement between and the “true” as calculated using the Scorpion and Atom codes, for points with . Fig. 8 compares scatter plots in the plane of (left panel) and (right panel). The colour code used is indicated on the right-hand sides of the panels, and we see that the patterns of colours in the two scatter plots are qualitatively similar. This is remarkable, given the interplay of so many different decay chains.
More quantitative comparisons of the contributions to the global function calculated on the basis of the simplified model searches for stops and electroweakly produced sparticles () with results from Scorpion and Atom for these 1000 randomly-selected pMSSM10 points () are shown in Fig. 9. The left panel shows a histogram of the difference between and , showing that it is relatively small, with an r.m.s. difference . The right panel of Fig. 9 displays a scatter plot in the plane. We see that points that are (dis)favoured at the 95% level in the simplified approach are, in general, also (dis)favoured at the 95% level in the more sophisticated approach based on Scorpion and Atom.
To determine quantitatively the effect of the uncertainty in the procedure, we translate the impact of the above-mentioned uncertainty into the plane in the lower right panel of Fig. 3. This shows the impacts of variations on our 68% and 95% contours in this plane, which is rather small except for small values of and .
Based on this study, we conclude that the computationally-manageable simplified approach is sufficiently reliable for our physics purposes. Specifically, we note that there are points with low that survive the full LHC constraints with relatively low .