Fastlim: a fast LHC limit calculator
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Abstract
Fastlim is a tool to calculate conservative limits on extensions of the Standard Model from direct LHC searches without performing any Monte Carlo event generation. The program reconstructs the visible cross sections (cross sections after event selection cuts) from precalculated efficiency tables and cross section tables for simplified event topologies. As a proof of concept of the approach, we have implemented searches relevant for supersymmetric models with Rparity conservation. Fastlim takes the spectrum and coupling information of a given model point and provides, for each signal region of the implemented analyses, the visible cross sections normalised to the corresponding upper limit, reported by the experiments, as well as the \(\mathrm{CL}_\mathrm{s}\) value. To demonstrate the utility of the program we study the sensitivity of the recent ATLAS missing energy searches to the parameter space of natural SUSY models. The program structure allows the straightforward inclusion of external efficiency tables and can be generalised to Rparity violating scenarios and nonSUSY models. This paper serves as a selfcontained user guide and indicates the conventions and approximations used.
Keywords
Signal Region Decay Chain Gluino Mass SUSY Particle SUSY Model1 Introduction
1.1 Motivation
In the 3 years of LHC operation, ATLAS and CMS have conducted many direct new physics searches. These searches have put significant constraints on the parameter space of new physics models. The experimental collaborations have so far interpreted their results in simplified scenarios of full models like the Constrained MSSM (CMSSM) or various simplified models, which are defined by effective Lagrangians with a small number of new physics particles and couplings; see e.g. [1, 2, 3, 4]. On the other hand, many models have not been covered and most of the parameter space of the studied models (e.g. the MSSM with \({\sim }20\) phenomenological parameters) has been left unexplored, except for a few very computationally intensive efforts in the MSSM [5, 6, 7, 8, 9, 10, 11, 12].
An important question is how sensitive current analyses are to models that have so far been ignored by ATLAS and CMS and if there are holes in the coverage in the models that have been studied. Existing experimental analyses are often sensitive to alternate models, so there is not necessarily any additional effort required for the experiments in the limit setting process – it is only a matter of reinterpreting existing results. While the experimental collaborations can do this, it is often not a good use of their computing resources and the effort required in reinterpreting results could be spent in performing new analyses.
Recently, various groups have started to recast direct LHC searches to extract limits on new physics scenarios; see e.g. [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. However, this usually requires a tedious task for which requires a chain of Monte Carlo (MC) simulations is needed: event generation, detector simulation and efficiency estimation – taking often in total a few hours to test a single model point and a large computing cluster for days to perform parameter scans. Tuning the MC simulations and validating the efficiency estimation for each analysis can also be cumbersome, especially when several analyses are considered.
On the other hand, for models like the MSSM, the idea of Simplified Models provides the basis to decouple the (slow) MC event generation and simulation steps necessary to estimate the efficiencies, from the (much faster) limit setting steps. It is therefore desirable to develop a tool which is simple in use and can calculate a conservative limit in less than a minute per model point by using this principle. We present such a tool (Fastlim) in this paper. We have developed the first version of Fastlim specialising on Rparity conserving supersymmetric models but the approach can be generalised to any new physics model.
A novel feature of the program is that it does not perform any MC simulation to calculate visible cross sections. Instead, the program reconstructs the visible cross sections from the contributions of the relevant simplified event topologies. The visible cross section for each event topology^{1} and signal region^{2} is obtained by interpolating the precalculated efficiency tables and the cross section tables, which are provided together with the program. In this approach, the reconstructed visible cross section may only be underestimated because only the available simplified topologies and searches are considered. In other words, the limits obtained by Fastlim are always conservative. Including additional topologies may strengthen the bounds.^{3} The first version of Fastlim contains a set of event topologies which can cover the natural SUSY model parameter space. The input of the program are the masses and decay branching ratios of SUSY particles which must be given in the Supersymmetry Les Houches Accord (SLHA) [35, 36] format. The running time is between a couple of seconds and about a half minute depending on the model point and the CPU speed. For a short guide of the installation and a quick start of Fastlim, see Appendices and .
The paper is organised as follows: the next section describes the method and the calculation procedure of the program. In Sect. 3, the definition of the event topologies and our nomenclature for their identification are given. Section 4 explains the output files, in which the users can find the constraints set by the direct SUSY searches on the input model. Several useful approximations are introduced in Sect. 6, which can be used to enhance the performance of the program when there is a mass degeneracy in the spectrum. Section 7 provides the detailed information on version 1.0. In Sect. 8, we study the direct SUSY search constraints on the natural SUSY models using Fastlim 1.0. Section 9 is dedicated to a summary and future developments.
2 Methodology
2.1 The traditional “recasting” approach
This method is generic and applicable to any model. However, one has to tune the detector simulation and define the reconstructed objects (often on a per analysis basis), mockup the analyses and validate the codes in some way. This task becomes increasingly difficult as the analyses become more elaborate and their number and the number of signal regions increases. One of the solutions to this problem would be to develop a program that automatically evaluates efficiencies taking detector effects into account, in which well validated analyses are already implemented together with the appropriate detector setups. Along these lines, ATOM [55] has been developed and already applied to some studies [56, 57].^{4} ATOM also plays a crucial role in developing Fastlim version 1.0 as we will see in Sect. 7. Another issue is the computation time. Even if the efficiencies were automatically calculated, the whole process, including event generation and efficiency evaluation, can easily take tens of minutes to an hour per model point. This becomes a crucial problem when a parameter scan is performed, requiring large computing facilities. To overcome this problem, leveraging on the idea of simplified topologies, we take a different approach, which is described in the next subsection.
2.2 The method
If the decay chains in the topology \(i\) are sufficiency short, the \(\epsilon ^{(a)}_i\) may depend only on two or three mass parameters. For such topologies, one can precalculate the \(\epsilon ^{(a)}_i (\mathbf{m}_i)\) for every grid point in the parameter space, \(\mathbf{m}_i = \{m_i^{(1)}, m_i^{(2)},\ldots \} \), and tabulate its values. Once such tables are available, one can obtain the \(\epsilon ^{(a)}_i\) by interpolation and then reconstruct the visible cross section according to Eq. (4) without the need of carrying out a MC simulation again. In practice, due to the “curse of dimensionality”, it is computationally feasible to generate the efficiency tables currently only for topologies with two or three different SUSY particles.^{5} Therefore, some of the topologies may be neglected from the formula (4) and in this case the reconstructed visible cross section is underestimated. This means the derived limit is conservative. The detailed information on the currently available efficiency tables is given in Sects. 5 and 7. Additional tables are currently being produced and once available can be downloaded from the Fastlim website (http://cern.ch/fastlim).
Similarly to the precalculated \(\epsilon _i^{(a)}\), the program contains cross section tables for the various production modes. The cross section is obtained by interpolating the tables during the reconstruction of the visible cross sections. More details of the cross section calculation is given in Sect. 5.
2.3 The calculation procedure

The program first goes through all the decay chains starting with the SUSY particles specified in the main program file, fastlim.py, by following the decay modes listed in the input SLHA file. The program collects the branching fraction of each decay mode and calculates the total branching ratios for possible decay chains. In this process, PySLHA [61] is used to extract the masses and branching ratios from the SLHA file.

The production cross sections are then extracted for a given production mode by interpolating the cross section tables. It then computes the cross sections of the event topologies, \(\sigma _i\), by multiplying the production cross sections by the pairs of decay branching ratios. The set of \(\sigma _i\) contains interesting information on the model point. The list of the cross sections for the relevant event topologies (sorted from largest to smallest) is therefore given in the output file.

A loop through all the event topologies is then performed, where the program checks for the presence of the efficiency tables for the event topology under consideration. If the corresponding efficiency tables are found, the efficiencies for all the signal regions are obtained by interpolating the tables.^{6} The visible cross section for the topology, \(\sigma _i^{(a)}\), is then calculated by multiplying the cross section and the efficiency. A sum over all the topologies is performed to compute the total visible cross section, \(\sigma _\mathrm{vis}^{(a)}\), for the signal region \(a\) (the topologies whose efficiency tables are not available are ignored in this sum). The lists of \(\sigma _\mathrm{vis}^{(a)}\) and \(\sigma _i^{(a)}\) can also be found in the output file.
 Finally the information as regards the signal region \(a\) necessary to set a limit is retrieved. Such information has been previously extracted from the experimental papers and it includes the 95 % CL upper limit on the visible cross section (reported by the experimental collaborations using the full likelihood), \(\sigma _\mathrm{UL}^{(a)}\), the contribution of the SM background, \(N_\mathrm{BG}^{(a)}\), together with its uncertainty, the observed data, \(N_\mathrm{obs}^{(a)}\), and the luminosity used for the analysis. A convenient measure for the exclusion is the ratio between the visible cross section and its 95 % CL upper limitThe model point is excluded at the 95 % CL if \(R^{(a)} > 1\). The program may also calculate an approximate \(CL_s^{(a)}\) variable by comparing \(N_\mathrm{obs}^{(a)}\) and \(N_\mathrm{BG}^{(a)} + N_\mathrm{SUSY}^{(a)}\) taking their uncertainties into account using an approximated likelihood \(\mathcal {L} = \mathrm{poiss}(N_\mathrm{obs}^{(a)}  N_\mathrm{SUSY}^{(a)} + {\bar{b}}) \cdot \mathrm{gauss}(N_\mathrm{BG}^{(a)},\delta N_\mathrm{BG}^{(a)} \,\, {\bar{b}}) \). The \(CL_s^{(a)}\) variable provides a conservative exclusion criterion [62] since it corrects for underfluctuations of the background. A model point is excluded if \(CL_s^{(a)} < 0.05\). We do not combine multiple signal regions between different analyses, since it requires detailed knowledge on the correlations of both systematical and statistical uncertainties. The program outputs \(R^{(a)}\) for all the signal regions and provides an approximate \(CL_s^{(a)}\) if specified. An interface to RooStats [63] is currently in testing and will be included in a future version.$$\begin{aligned} R^{(a)} \equiv \frac{\sigma _\mathrm{vis}^{(a)}}{\sigma ^{(a)}_\mathrm{UL}}. \end{aligned}$$
3 Nomenclature of the event topologies

the event topology should be defined such that the efficiency for the topology depends only on the masses of the onshell SUSY particles appearing in the event topology when the effect of the polarisation and the spin correlation is neglected;

the definition and classification should be as minimal as possible, otherwise the number of event topologies becomes unreasonably large, requiring unnecessary efficiency tables and slowing down the computation speed;

the name assigned to the event topology should be as simple and intuitive as possible and must be able to identify the event topology uniquely. It is desirable that the name of event topologies can be directly used as a directory or file name.
The names for the Reven (top) and Rodd (bottom) particles
Particle  \(g\)  \(\gamma \)  \(Z\)  \(h\)  \(H\)  \(A\)  \(W^{\pm }\)  \(H^{\pm }\)  \(q\)  \(t\)  \(b\)  \(e\)  \(\mu \)  \(\tau \)  \(\nu \) 

Name  g  gam  z  h  h2  h3  w  hp  q  t  b  e  m  ta  n 
Particle  \({\tilde{g}}\)  \({\tilde{\chi }_{1}^{0}}\cdots {\tilde{\chi }_{4}^{0}}\)  \({\tilde{\chi }_{1}^{\pm }}, {\tilde{\chi }_{2}^{\pm }}\)  \({\tilde{q}}\)  \({\tilde{t}}_1, {\tilde{t}}_2\)  \({\tilde{b}}_1, {\tilde{b}}_2\)  \({\tilde{e}}\)  \({\tilde{\mu }}\)  \({\tilde{\tau }}_1, {\tilde{\tau }}_2\)  \({\tilde{\nu }}, {\tilde{\nu }}_{\tau }\)  

Name  G  \(\mathtt{{N1}} \cdots \mathtt{{N4}}\)  C1, C2  Q  T1, T2  B1, B2  E  M  TAU1, TAU2  NU, NUT 
According to our wish list, in order to reduce the length of the decay chains, we do not specify the decay of the SM particles because the decay branching ratios for the SM particles are fixed and independent of the SUSY parameters.^{8} Similarly, we do not specify charges nor do we distinguish particles and antiparticles. This specification is not necessary for our purpose as long as CP is conserved, since the branching ratio is then the same for a process and its CP conjugate. The production cross sections are, on the other hand, different among those processes because the initial \(pp\) state at the LHC is not CP invariant. The ratio of the cross sections is, however, fixed once the masses of the produced SUSY particles are given. Consider, for example, \(pp \rightarrow {\tilde{d}} {\tilde{u}}^*\) and \(pp \rightarrow {\tilde{d}}^* {\tilde{u}}\). The productions are governed by QCD and the cross sections are fully determined by the masses of \({\tilde{u}}\) and \({\tilde{d}}\). The ratio \(\sigma ({\tilde{d}} {\tilde{u}}^*)/\sigma ({\tilde{d}}^* {\tilde{u}})\) is therefore fixed if the masses are specified. This means that for each grid point of the efficiency table the ratio between a process and its CP conjugation process is correctly taken into account and is independent of the other parameters. Therefore, the charge of the particle does not need to be specified in the event topology for our purpose. Finally, we also do not yet distinguish between light (s)quark flavours, although the full squark flavour implementation is in principle straightforward. For the effect of large mass splitting between the first two generations, see [57].
Fastlim 1.0 (and the discussion in this section) concentrate on the SUSY models with an (approximate) Rparity symmetry. However, the program is applicable also for nonSUSY models as long as the topology names and the corresponding efficiency tables are provided. Also in that case, the three points in the guideline above provide a useful way to determine the topology names.
4 The output
First, the cross section for each production mode is given. Secondly, the list of cross sections (or production cross section times branching ratios) for the relevant event topologies is provided. This list is sorted from the largest cross section to the smallest one. The rate (“Rate”) with which this process contributes to the total cross section and the accumulated rate (“Accum”) up to the topology looked at are also shown. If the efficiency table for a certain event topology is implemented, the tag “\(<\) == Implemented” appears.
5 The numerical tables
The efficiency and cross section tables are provided in the form of a standard text file so that new tables can be added straightforwardly. In this section, we explain the conventions for the efficiency and cross section tables.
5.1 The efficiency tables
The information as regards the grids can be directly found in the efficiency table files. Although the experimental collaborations have not provided their results of the signal efficiencies for the 2013 SUSY searches, we will include them in our program whenever they will become publicly available. The efficiency tables installed in Fastlim 1.0 are generated by us using MadGraph 5 and ATOM. More detailed information is given in Sect. 7.
5.2 The cross section tables
6 The approximations
6.1 Treatment of soft decays
Several SUSY models predict partially degenerate SUSY mass spectra. For example, in anomaly mediation, the wino often becomes the lightest SUSY state. Since the wino is SU(2) triplet, it leads to almost degenerate \({\tilde{\chi }_{1}^{\pm }}\) and \({\tilde{\chi }_{1}^{0}}\). Another example is the higgsino LSP scenario. In this case, two higgsino doublets have similar masses, leading to almost degenerate \({\tilde{\chi }_{1}^{\pm }}\), \({\tilde{\chi }_{2}^{0}}\) and \({\tilde{\chi }_{1}^{0}}\).
If one SUSY particle decays to another which has a similar mass, the SM particles produced in the decay will tend to be very soft. Such SM particles may not be observed in the detector because of the low detector acceptance and the reconstruction efficiencies. Even if such objects are reconstructed, they hardly affect the signal region efficiency because the high\(p_T\) cuts employed in the SUSY searches are likely to ignore such objects. Therefore, barring the case of dedicated analyses looking for such soft objects or having low \(p_T\) jet vetos, if there is an event topology containing a decay associated with two nearly degenerate SUSY particles, it may be useful to truncate the decay from the topology and redefine it as a shorter effective event topology.
where procs_8 contains the information of all the relevant topologies together with their 8 TeV cross sections (as a Python dictionary). The above command replaces the string C1qqN1 by N1 in all topologies stored in procs_8. If the event topology name generated after this truncation already exists, the contributing cross sections are summed: for the above example the cross section of GbbC1qqN1_GbbC1qqN1 is added to the cross section of GbbN1_GbbN1 and the topology GbbC1qqN1_GbbC1qq N1 is removed from procs_8. In the current version of the program such possibility is implemented by default for N1, N2 and C1 if their mass splitting is smaller than 10 GeV. The extension of such checks to other cases, via a userdefined input file is planned for the next release of Fastlim.
Note that this replacement may introduce topologies in which the electric charge appears not to be conserved.^{9} For example, truncating C1qqN1 in GbbN1_GbtC1qqN1 introduces GbbN1_GbtN1. As will be discussed in Sect. 7.3, the program contains many such event topologies to increase the applicability to concrete models.
6.2 Topologies with similar decay structure
There are several event topologies among which the same efficiency table can be used. An obvious example is T1tN1_ T1tN1 and T2tN1_T2tN1. In general \(\tilde{t}_{2}\) and \(\tilde{t}_{1}\) decay kinematics depend on their \(\tilde{t}_{L,R}\) admixture. The top quarks coming from stop decays may be polarised depending on the \(\tilde{t}_{L,R}\) admixture of the stop. This is also known to affect the efficiencies of certain analyses to some level [66, 67]. While including top polarisation is a straightforward addition to Fastlim code (which will be included in later versions), at the moment we provide efficiencies for unpolarized tops only. This allows us to present an example of another simplification feature of the Fastlim code.
Because the polarisation effect is ignored in our calculation, the efficiencies of the two topologies are identical apart from the stop mass. As will be discussed in Sect. 7.3, we provide the efficiency tables only for T1tN1_T1tN1 but use them both for T1tN1_T1tN1 and T2tN1_T2tN1. The same efficiency tables can also be used for B1tN1_B1tN1 and B2tN1_B2tN1, which may arise after truncating the soft chargino decays in B1tC1qqN1_B1tC1qqN1 and B2tC1qqN1_B2tC1qqN1, respectively.
6.3 Reduction of multidimensional topologies
Let us finally consider the case of GtT1tN1_GtT2tN1. This event topology involves four onshell SUSY particles: G, T2, T1, N1, and in principle requires fourdimensional efficiency tables. However, if e.g. the masses of T1 and T2 are close to each other, one may use the efficiency tables for GtT1tN1_GtT1tN1, which are three dimensional. By default, the efficiencies for GtT1tN1_GtT2tN1 are taken from those for GtT1tN1_GtT1tN1 if \((m_\mathtt{{T2}}  m_\mathtt{{T1}})/m_\mathrm{T2} < 0.1\). The average mass, \((m_\mathtt{{T2}} + m_\mathtt{{T1}})/2\), is used for the mass of the intermediate particle between G and N1 in the interpolation. This approximation can be performed automatically for particles sharing the same type of decay modes. The same procedure and condition are used for instance for GbB1bN1_GbB2bN1 and GbB1bN1_GbB1bN1. As in the case of soft decays, we plan to provide additional user control over this feature in the next Fastlim version by suitable input configuration files.
7 Fastlim version 1.0
7.1 Generation of efficiency tables
The simplified model efficiency tables for the 2013 SUSY searches have yet to be provided by the experimental collaborations. The tables included in Fastlim 1.0 have therefore been precalculated by us using ATOM. The calculation procedure we used is as follows: \(5 \cdot 10^4\) events are generated using MadGraph 5.12 [52] for each grid point in the respective SUSY mass plane (independent of the topology and the mass spectrum). The samples include up to one extra hard parton emission at the matrix element level, matched to the parton shower (carried out by Pythia 6.426 [50]) using the MLM merging scheme [68], where the merging scale is set to \(m_\mathrm{SUSY}/4\) with \(m_\mathrm{SUSY}\) being the mass of the heavier SUSY particles in the production.
The event files are then passed to ATOM [55], which evaluates the efficiencies for various signal regions taking detector effects into account. ATOM estimates the efficiencies for many implemented signal regions. We have validated the implementation of the analyses in ATOM using the cutflow tables provided by ATLAS. The validation results are given in Appendix and the Fastlim website (http://cern.ch/fastlim).
7.2 The available analyses
The analyses available in Fastlim version 1.0
Name  Short description  \(E_\mathrm{CM}\)  \(\mathcal{L}_\mathrm{int}\)  # SRs  Refs. 

ATLAS_CONF_2013_024  0 lepton \(+\) 6 (2 b)jets \(+\) MET [Heavy stop]  8  20.5  3  [69] 
ATLAS_CONF_2013_035  3 leptons \(+\) MET [EW production]  8  20.7  6  [70] 
ATLAS_CONF_2013_037  1 lepton \(+\) 4(1 b)jets \(+\) MET [Medium/heavy stop]  8  20.7  5  [71] 
ATLAS_CONF_2013_047  0 leptons \(+\) 2–6 jets \(+\) MET [squarks & gluinos]  8  20.3  10  [72] 
ATLAS_CONF_2013_048  2 leptons (\(+\)jets) \(+\) MET [Medium stop]  8  20.3  4  [73] 
ATLAS_CONF_2013_049  2 leptons \(+\) MET [EW production]  8  20.3  9  [74] 
ATLAS_CONF_2013_053  0 leptons \(+\) 2 bjets \(+\) MET [Sbottom/stop]  8  20.1  6  [75] 
ATLAS_CONF_2013_054  0 leptons \(+\) \(\ge \)7–10 jets \(+\) MET [squarks and gluinos]  8  20.3  19  [76] 
ATLAS_CONF_2013_061  0–1 leptons \(+\) \(\ge \)3 bjets \(+\) MET [3rd gen. squarks]  8  20.1  9  [77] 
ATLAS_CONF_2013_062  1–2 leptons \(+\) 3–6 jets \(+\) MET [squarks and gluinos]  8  20.3  13  [78] 
ATLAS_CONF_2013_093  1 lepton \(+\) bb(H) \(+\) Etmiss [EW production]  8  20.3  2  [79] 
7.3 The implemented event topologies
There are several event topologies in which the electric charge appears not to be not conserved. These topologies can arise after the soft decays are truncated as mentioned in Sect. 6.1. We also include the loop induced \({\mathtt{{G}} \rightarrow \mathtt{{gN1}}}\) decay, which can have a sizeable branching fraction if the twobody modes and GttN1 are kinematically forbidden. The decay rate is also enhanced if the stop and higgsino masses are small and the trilinear \(A_t\) coupling is large. These conditions can often be found in natural SUSY models.
Although the event topologies are chosen to cover natural SUSY models, many of the topologies appear also in other models. A large rate of the gluino pair production is relatively common in a wide range of the SUSY models because of the largest colour factor of the gluino among the MSSM particles. Many models tend to predict light stops, since the interaction between the Higgs and stops (with a large top Yukawa coupling) pulls the stop mass down at low energies through the renormalisation group evolution, leading to larger branching ratios for GtT1tN1 and GttN1. The set of the event topologies implemented in Fastlim 1.0 has a very good coverage also for split SUSY models if the wino or the bino is heavier than the gluino.
Additional topologies are currently being evaluated and it will be possible to download them from the Fastlim website (http://cern.ch/fastlim) as they will become available. Furthermore, any additional 3rdparty efficiency map for a topology not currently covered by Fastlim can be easily added by formatting a text file according to the criteria exposed in Sect. 5.1. This is particularly useful to incorporate the efficiency maps that will be available from [80].
8 The constraint on natural SUSY models
In this section, we study the direct SUSY search constraints on the natural SUSY models using Fastlim. Since this is a well studied region of the SUSY parameter space [33, 56, 81, 82, 83, 84, 85, 86, 87, 88]. it provides a good test case to illustrate the usage of the program.
Figure 8a shows the constraints from all the SUSY searches implemented in Fastlim 1.0 (see Table 2). In this plot (and the following ones of the same type) only the names of the analyses providing an exclusion are listed on the plot, using the same colour as the exclusion contour. The exclusion regions are plotted on top of each other. As can be seen, only ATLAS_CONF_2013_024 and ATLAS_CONF_2013_053 exclude the parameter region in the plot. ATLAS_CONF_2013_024 is designed to constrain the T1tN1_T1tN1 topology focusing on the hadronic top decays. Because T1tN1_T1tN1 is subdominant in this model, the constraint from this analysis is slightly weaker than the corresponding exclusion plot in Ref. [69] assuming \(Br( {\tilde{t}}_1 \rightarrow t {\tilde{\chi }_{1}^{0}}) = 1\). ATLAS_CONF_2013_053, on the other hand, has been originally designed for the B1bN1_B1bN1 topology. In this model, T1bN1_T1bN1 has the largest or the second largest rate among the possible topologies depending on the parameter region, and the constraint is quite strong. It roughly excludes \(M_{U_3} < 500\) GeV with \(\mu < 200\) GeV.
From Fig. 9a, one can see that ATLAS_CONF_2013_053 only constraints the left hand side of the blue dashed line. This can be understood because the analysis is tailored for the \({\mathtt{{T1bN1\_T1bN1}}}\) and \({\mathtt{{B1bN1\_B1bN1}}}\) topologies. On the other side of the blue dashed line, the \({\mathtt{{T1tN1\_T1tN1}}}\) and \({\mathtt{{B1tN1\_B1tN1}}}\) topologies dominate. In this region, ATLAS_CONF_2013_024 and ATLAS_CONF_2013_037 are particularly constraining because they are designed for the hadronic–hadronic and hadronic–leptonic top modes for the \({\mathtt{{T1tN1\_T1tN1}}}\) topology, respectively. ATLAS_CONF_2013_024 excludes \(M_{Q_3}\) values from \(\sim \)400 up to 750 GeV for \(\mu \lesssim 250\) GeV at the \(95~\%\) CL. Because of the transition between different dominant decay modes, there is a gap in the exclusion region near the blue dashed line. In this particular region, \(M_{Q_3} = 400\) GeV and \(\mu = 200\) GeV is still allowed by all the analyses implemented in Fastlim.
Nevertheless, one can see from Fig. 10a that many analyses provide exclusion regions in this parameter slice because of the large cross section of the gluino pair production. Among them, ATLAS_CONF_2013_024 and ATLAS_CONF_2013_061 yield the most stringent constraints. ATLAS_CONF_2013_024 mainly constrains T1tN1_T1tN1 and B1tN1_B1tN1 topologies, and the bound on the gluino mass gradually decreases as the stop and sbottom masses increase together with the higgsino mass. On the other hand, the limit from ATLAS_CONF_2013_061 is almost independent of the higgsino mass. This analysis looks for the events with 0–1 lepton plus \({\ge }3\) \(b\)jet, targeting the gluino pair production processes with gluino decaying to the third generation quarks either through an on and offshell \({\tilde{t}}_1\) and \({\tilde{b}}_1\). The analysis roughly excludes 1.2 TeV gluino regardless of the \(\mu \) parameter at the \(95~\%\) CL.
We now look at the constraint on the (\(m_{\tilde{g}}\), \(M_{U_3/Q_3}\)) plane, where we take \(M_{U_3} = M_{Q_3}\), \(\mu = 200\) GeV, \(\tan \beta = 10\), \(X_t = 0\). Figure 11b shows that the cross section coverage can become as small as 60 % at the vicinity of the \({\mathtt{{G}} \rightarrow \mathtt{{t T1}}}\) threshold line. In this region, again, the asymmetric gluino decays (e.g. GbB1bN1_GtT1tN1 in the region slightly above the \({\mathtt{{G}} \rightarrow \mathtt{{t T1}}}\) threshold line, and e.g. GbB1bN1_GttN1 slightly below the line) become sizeable. One can see from Fig. 11a that the exclusions on the gluino mass and the stop mass are roughly independent of each other. The gluino mass is excluded up to 1280 GeV, almost independently of the stop mass.^{10} The most stringent constraint comes from ATLAS_CONF_2013_061. Near the \({\mathtt{{G}} \rightarrow \mathtt{{t T1}}}\) threshold line the exclusion is degraded because Fastlim 1.0 does not include the topologies with asymmetric gluino decays, though the degradation is only \({\sim }100\) GeV on the gluino mass. The soft mass parameters for the third generation squarks are, on the other hand, constrained up to 750 GeV. ATLAS_CONF_2013_024 provides the strongest limit in the region where \(m_{\tilde{g}}> 1.2\) TeV, by excluding the stop production processes independently of the gluino mass.
As can be seen, ATLAS_CONF_2013_024 again places the most stringent limit on the soft mass for the third generation squarks for both the \(M_{U_3}/M_{Q_3} = 1\) and the \( = 2\) cases. The blue dashed curves show the \({\tilde{t}}_1\) mass contours. One can see that the exclusion limit on \((M^2_{U_3} + M^2_{Q_3})^{1/2}\) does not change much when \(A_t\) is varied, although the limit on the \({\tilde{t}}_1\) mass changes from 780 to 600 GeV as \(A_t\) changes from 0 to 2 TeV (for \((M^2_{U_3}+M^2_{Q_3})^{1/2} \simeq 1\) TeV) in the \(M_{U_3}/M_{Q_3} = 1\) scenario. Increasing \(A_t\) results in making the mass splitting between \({\tilde{t}}_1\) and \({\tilde{t}}_2\) larger. However, the changes in the cross section times efficiency from the \({\tilde{t}}_1 {\tilde{t}}_1^*\) and \({\tilde{t}}_2 {\tilde{t}}_2^*\) processes tend to cancel each other and the resulting visible cross sections are more or less stable against the variation of \(A_t\). For \(M_{U_3}/M_{Q_3} = 2\) scenario, \({\tilde{t}}_1\) is mostly composed of \({\tilde{t}}_L\) and the dependence of \(A_t\) on the \({\tilde{t}}_1\) mass itself is very mild.
The green curves represent the Higgs mass contours, where we allow 3 (dashed) and 2 (solid) GeV deviation from the central observed value, taking the theory uncertainties into account. We have calculated the Higgs mass using FeynHiggs 2.9.4 [91]. Most of the parameter space is constrained by the Higgs mass measurement in the \(M_{U_3}/M_{Q_3} = 1\) scenario, whereas in the \(M_{U_3}/M_{Q_3} = 2\) scenario the ATLAS_CONF_2013_024 analysis excludes (at \(95~\%\) CL) a significant part of the parameter space where the Higgs mass condition is satisfied.
9 Discussion and future developments
In this paper we presented a program (Fastlim) which calculates the constraints from direct SUSY collider searches starting from a given SLHA model input file. A novel feature of the program is that it does not run any MC simulation to calculate the visible cross section. The program instead reconstructs the visible cross section for each signal region by adding the contributions from various event topologies. The cross section and efficiencies for each event topology and each search signal region are obtained by interpolating the precalculated cross section and efficiency tables. Similar ideas have also been discussed in the literature [1, 3, 92, 93].
A similar but different approach has recently been taken and implemented in [94]. In this approach, one checks if the model contains the event topologies on which the cross section upper limit is reported by the experimental collaborations.^{12} If such event topologies are found, the program calculates the cross section time branching ratios for those topologies and if one of them exceeds the experimental upper limit, it declares the model to be excluded. This method provides generally weaker but more conservative limits compared to our approach (assuming the same analyses are tested) since there is no attempt made to reconstruct the full BSM contribution to each signal region.
To implement our visible cross section reconstruction method, we have introduced a minimal and intuitive naming scheme for the event topology, which can also be conveniently used as a directory or file name for the efficiency tables. We have also introduced useful approximations which are used to enhance the applicability and speed of the program. Such approximations include shortening the decay chains in presence of mass degeneracies in the spectrum, or recycling efficiency maps in presence of different SUSY particles sharing similar decay modes.
To demonstrate the utility of the program, we have studied the direct SUSY search constraints on natural SUSY models. Using the results of the 2013 ATLAS SUSY searches, we have found that the stop is excluded up to about \(700\) GeV with \(\mu \lesssim 200\) GeV, whereas the gluino mass is excluded up to about \(1.2\) TeV with \(\mu \lesssim 400\) GeV. When \(A_t\) is varied, we found that the direct SUSY search constraint can be more stringent compared to the Higgs mass constraint in some parameter region, which was not the case when the 7 TeV data was considered [56]. Running Fastlim to extract the limits on the 4,836 parameter points composing the twodimensional plots shown in this paper took 18.7 h (14 s per point on average) on a single computer (single core, 2.4 GHz clock speed).
Fastlim version 1.0 contains the set of event topologies shown in Fig. 7. These topologies cover the natural SUSY model parameter space very well but they can also cover other models such as split SUSY models with a decoupled wino or bino. More topologies and analyses will be implemented in future updates very soon, thus extending the range of applicability of the approach. The code structure is flexible and the efficiency tables provided from other collaborations can be included straightforwardly (the steps necessary to include a new efficiency table are given in Appendix ). We particularly hope that the experimental collaborations will directly provide their efficiencies in a table format so that the results can be included and thus reinterpreted in a wide range of the SUSY models. Recasting LHC analyses to extend the number of topologies covered is becoming a coordinated effort [80]. Once enough topologies will be available Fastlim can be used for computationally lean pMSSM studies, which may give new insights into interesting SUSY models based on the LHC data.
Footnotes
 1.
In this paper the term topology refers to the full decay chain and not the observable final state signature. Please see Sect. 3 for the exact definition.
 2.
The signal regions are the sets of selection cuts defined in the experimental analyses.
 3.
This approach works with most of the currently available searches which are “cutandcount”, but may fail with shapeanalysis based searches where adding additional contributions may result in signal shapes more difficult to disentangle from the backgrounds.
 4.
 5.
In certain cases, topologies with more than three SUSY particles may be approximated by two or threedimensional topologies, as described in Sect. 6.
 6.
We use a linear extrapolation for the \(\ln x\), where \(x\) is the cross section or the efficiency.
 7.
We do not consider the SUSY particle decays into three or more SUSY particles.
 8.
A possibility to account for deviations in Higgs branching ratios from the SM values may easily be accounted in future releases.
 9.
The names of the topologies after the truncation of soft decays are the only exception where “topology” does not mean the full decay chain anymore.
 10.Here (and more generally in the discussion of the plots in this section) the exclusion refers to the \(95~\%\) CL exclusion given by the analysis that is most sensitive in that region.
 11.
This leading logarithmic approximation is generically valid for low scale SUSY breaking mediation models, while corresponding resumed expressions for high scale models can be found.
 12.
To derive the exclusion, the signal topologies are mapped to the topologies constrained in the experimental analyses. This implies that in some cases (in which the analyses target one topology) the exclusion is made from a single event topology, while in other cases (where the analyses constrain a sum of topologies, e.g. a sum over final state lepton flavours) a few topologies are “combined” correspondingly. No recasting of topologies (see Sect. 2.1) which are not covered by the experiments is performed in this approach. The code has a function similar to the replace function in Fastlim to truncate the soft decays. The check if a model point is explored is done after the truncation and combination.
 13.
The compatibility of Fastlim with different versions has been tested in cases. Fastlim can be used also with Python version 2.6, but the current version of our code is incompatible with Python version 3. NumPy versions newer than 1.6.1 and SciPy versions newer than 0.10.0 should work.
 14.
On HepData the efficiency and acceptance are given separately which need to be multiplied to be able to use these as efficiency tables in Fastlim.
Notes
Acknowledgments
We want to thank S. Caron, T. Cohen, M. D’Onofrio, B. Fuks, E. Halkiadakis, S. Heinemeyer, A. Hoeker, K. Howe, M. Mangano, Z. Marshall, M. McGarrie, M. Pierini, S. Plätzer, S. Prestel, M. Tonini, J. Wacker, G. Weiglein, and F. Wuerthwein for helpful discussions. This work has partially been supported by the Collaborative Research Center SFB676 of the DFG, “Particles, Strings and the early Universe”. The work of K.S. was supported in part by the London Centre for Terauniverse Studies (LCTS), using funding from the European Research Council via the Advanced Investigator Grant 267352. K.S. thanks the CERN Theory Group for hospitality during part of this work. K.S. thanks T. Becker, L. Oppermann, V. Selk, M. Wulf for helpful discussions.
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