In this section, we discuss the technical details of the main building blocks of SModelS: the decomposition procedure, the analysis database, and the matching of theoretical and experimental results.
Decomposition procedure
As explained in the previous section, under the SMS assumption, all the complexity of the BSM model can be replaced by the knowledge of the signal topologies and their weights, together with the relevant BSM masses.Footnote 4 The topologies can then be classified according to the number of vertices in each branch and the SM states appearing in each vertex. In order to properly classify the signal topologies, we introduce a formal labeling scheme, which is (i) model independent, (ii) general enough to describe any topology, (iii) sufficiently concise and (iv) allows us to easily combine topologies according to the assumptions and conditions in the experimental analyses.
We choose to use a labeling system based on nested brackets, which corresponds to a textual representation of the topology. As discussed in Sect. 2, we assume that all signal topologies respect a \(\mathbb {Z}_2\)-symmetry, hence BSM states are produced in pairs and cascade decay to a single BSM state and SM particles. Therefore a signal topology always contains two branches (one for each of the initially pair produced BSM states), which we describe by \([B_1,B_2]\). In each branch there is a series of vertices, which represent the cascade decays. From each vertex there is one outgoing BSM state and a number of outgoing SM particles. Thus, inside each branch bracket, we add an inner bracket for each vertex, containing the lists of outgoing SM particles. Note that there is no mention to the intermediate BSM particles (\(\tilde{\chi }_1\), \(\tilde{l} \),...), which makes our method explicitly model independent. The only information kept from the BSM states are their masses. To illustrate the labeling scheme just defined, we show in Fig. 4 a signal topology containing a 2-step cascade decay in one branch and a one step decay in the other. Following the above prescription, this topology is described by \([B_1,B_2]\), with \(B_1 {=}\, \hbox {[[particles in vertex 1]}, \hbox {[particles in vertex 2]]}\) and \(B_2\,{=}\,\hbox {[[particles in vertex 1]]}\), as seen in Fig. 4. Note that the brackets inside each branch are ordered according to the vertex number. For the specific final SM particles assumed in the figure, we finally obtain: topology \(= [[[l^+],[\nu ]], [[l^+,l^-]]]\). The corresponding mass vector, shown in parenthesis in Fig. 4, is given by \([[M_1,M_2,M_3],[m_1,m_2]]\), where once again the masses are ordered according to the vertices. Note that we allow for the possibility of distinct final state masses in each branch (\(M_3 \ne m_2\)). Finally, adding the topology weight (\({\sigma \times \mathcal {B}} \)), we have a full description of the signal topology.
Given a BSM model—described by its spectrum, branching ratios and production cross sections—we need to obtain all the possible signal topologies and compute their weights (\({\sigma \times \mathcal {B}} \)). The procedure of computing these corresponds to the decomposition of the full model in terms of simplified model topologies. There are two ways to actually perform the SMS decomposition. The first requires the generation of parton-level Monte Carlo (MC) events, followed by the mapping of each event into the corresponding SMS topology. The second method is purely based on the SLHA [20] spectrum file and decay table, supplemented by theoretically computed cross sections. Below we describe both approaches in more detail.
Monte Carlo based decomposition
The decomposition based on parton level MC events is the most general one, since it can decompose any type of BSM model, as long as it is possible to simulate MC events for it. In this case an event file in the LHE format is used as input and each event is mapped to a simplified model topology. The mapping is obviously not one-to-one, since more than one event can generate the same exact SMS topology. Then the sum of MC weights for all events contributing to the same topology directly gives \({\sigma \times \mathcal {B}} \) for the corresponding topology. The disadvantage of this method is the introduction of MC uncertainties in the decomposition result; this can however be easily solved by increasing the MC statistics. We also note that the recent advances on NLO MC generators will allow to produce decomposition results at NLO even in the MC based decomposition.
SLHA-based decomposition
In the SUSY case, there are a number of public codes for the computation of the mass spectrum and decay branching ratios with output in the SLHA [20] format. The production cross sections can then be computed at leading order (LO) through a Monte Carlo generator, or at next-to-leading order (NLO) with Prospino [21]. For gluinos and squarks, cross sections at next-to-leading log (NLL) precision can be computed using NLL-Fast [22]. These cross sections can be included in the SLHA file [23], which then holds all the required information for the SMS decomposition.
In the SLHA-based decomposition the cross sections for pair production of BSM states and the corresponding branching ratios are used to generate all possible signal topologies. The only theoretical uncertainties in this case come from the cross-section uncertainty, hence being much smaller than the uncertainties in the MC-based method, which depends on the MC statistics. In order to avoid dealing with a large number of irrelevant signal topologies, only the ones with \({\sigma \times \mathcal {B}} \) above a minimal cut value are kept. For the results presented below we take this cut value to be \(0.03\) fb.Footnote 5
Compression of topologies
Although the decomposition described here (both for the MC and SLHA based methods) is fairly straightforward, the model independent language introduced above can be used to perform non-trivial operations on the signal topologies. One possibility is the compression of topologies containing a series of invisible decays at the end of the cascade decay chain, as illustrated by Fig. 5 (left). In this case the effective final state BSM momentum is given by the sum of the neutrinos and the final BSM state momenta. Therefore, for the experimental analyses, this topology is equivalent to a compressed one, where the effective BSM final state includes the neutrino emissions, as shown in Fig. 5 (right). Using the notation introduced previously, this compression simply corresponds to \([[X',[\nu ]],[X,[\nu ,\nu ],[\nu ]]] \rightarrow [[X'],[X]]\), where \(X,X'\) represent the other cascade decay vertices. When performing this compression we must also remove the corresponding BSM masses from the mass vector, so the effective final state BSM mass becomes heavier. The compression procedure allows us to constrain the original topology using experimental constraints for \([[X],[X']]\). More importantly, such “invisible compression” can be automatically performed in a model-independent way using the nested bracket notation.
Another case of interest corresponds to topologies which contain spectra with very small mass splittings. In this case one can again approximately map the original topology into a smaller one, where decays between quasi-degenerate states are omitted. This “mass compression” approximation is only reliable when the energy carried away by the SM particles emitted in the decay of quasi-degenerate states is negligible for all experimental purposes. If this is the case, the original topology is equivalent to a compressed one, as shown in Fig. 6. For the results presented in Sect. 5 we perform the mass compression for quasi-degenerate states if their mass difference is below 5 GeV. Once again, this compression procedure allows us to constrain the original topology using experimental constraints for shorter cascade decays. We also point out that, whenever (invisible and/or mass) compressions are performed, care must be taken to avoid double counting topologies. If both the original and the compressed topologies are kept after the decomposition, they should not be combined later, since this would result in a double counting of the topology weight.
Analysis database
Anatomy of an SMS result
The interpretation of the BSM search results in the context of simplified models has become the de-facto standard for the experimental collaborations. The ATLAS and CMS collaborations typically produce two types of SMS results: for each simplified model, values for the product of the experimental acceptance and efficiency (\(A \times \epsilon \)) are determined to translate a number of signal events after cuts into a signal cross section. From this information, a 95 % confidence level upper limit (UL) on the product of the cross section and branching fraction (\(\sigma \times \mathcal {B}\)) is derived as a function of the BSM masses appearing in the SMS. Finally, assuming a theoretical “reference” cross section for each mass combination, an exclusion curve in the plane of two masses is produced.
Figure 7 gives two examples. The plot on the left is from the CMS analysis of chargino–neutralino production with \(\tilde{\chi }^{\pm }_1 \rightarrow \tilde{\tau }^{\pm } \nu \) and \(\tilde{\chi }^0_2 \rightarrow \tilde{L}^{\pm } L^{\mp }\), with \(\tilde{L}^{\pm } \rightarrow L^{\pm }\tilde{\chi }_1^0\), where \(L = e,\mu ,\tau \). Shown are the 95 % CL UL on \(\sigma \times \mathcal {B}\) together with the expected and observed mass limit curves. The plot on the right is an ATLAS result for the case of slepton pair production and direct decay to the LSP, \(\tilde{l} \rightarrow l \tilde{\chi }^0_1\). Our approach builds upon the binned UL on \(\sigma \times \mathcal {B}\), which is the information collected in our analysis database. Neither the efficiency plots nor the exclusion lines are used in our procedure.
The cross section upper limits reported in this way are however subject to a number of assumptions made in the analysis, which also have to be described in the database. In the SModelS language described in Sect. 3.1, the topologies appearing in Fig. 7 (left) are \([[[L],[L]],[[\nu ],[\tau ]]]\), where \(L = e,\mu ,\tau \) and the relevant mass vectors are \((m_{\tilde{\chi }_1^{\pm }},\, m_{\tilde{L}},\, m_{\tilde{\chi }^0_1})\) and \((m_{\tilde{\chi }_2^0},\, m_{\tilde{L}},\, m_{\tilde{\chi }^0_1})\). The experimental analysis in our example assumes degenerate sleptons and \(m_{\tilde{\chi }_2^0} = m_{\tilde{\chi }_1^{\pm }}\), so there are only three independent masses. Furthermore this analysis constraints the sum over lepton flavors and charges for the topologies listed above under the assumption that each term contributes equally (flavor democratic decays). Moreover, the SMS result illustrated in Fig. 7 (left) is for the particular mass relation \(m_{\tilde{l}}= (m_{\tilde{\chi }^{\pm }_1} + m_{\tilde{\chi }_1^0})/2\) and can thus only be applied for (approximately) these \(m_{\tilde{l}}\) and \(m_{\tilde{\nu }}\) values. However, in Refs. [24, 25], results for distinct slepton mass values are also provided, which allows us to interpolate between them and apply these results to more generic models. All these constraints and conditions are described in our analysis database in the form of “metadata”. As an example we show the metadata for the CMS result in Fig. 7 in Table 1. The entry ‘constraint’ lists the (sum of) topologies being constrained by the analysis. If charges and/or flavors are not explicitly listed, a sum over charges and/or flavors is assumed. The additional analysis assumptions (each lepton flavor contributes equally to the total weight, \(\sigma \times \mathcal {B}\)) can actually be relaxed because the efficiency for \(\mu \)’s is higher than the efficiency for \(e\)’s, which is higher than the one for \(\tau \)’s. Therefore, instead of requiring an equal weight (\(\sigma \times \mathcal {B}\)) from each flavor, it suffices to impose:
$$\begin{aligned}&\sigma \times \mathcal {B}\left( \tilde{l} \tilde{l} \rightarrow \mu ^+\mu ^- \tilde{\chi }_1^0 \tilde{\chi }_1^0 \right) \ge \sigma \times \mathcal {B}\left( \tilde{l} \tilde{l} \rightarrow e^+ e^- \tilde{\chi }_1^0 \tilde{\chi }_1^0 \right) \\&\quad \ge \sigma \times \mathcal {B}\left( \tilde{l} \tilde{l} \rightarrow \tau ^+ \tau ^- \tilde{\chi }_1^0 \tilde{\chi }_1^0)\right) . \end{aligned}$$
These assumptions are included in the entry ‘fuzzycondition’ and the function \(Cgtr(x,y)\). This function uses the theoretical predictions for \(x\) and \(y\) to map the condition \(x > y\) into a number in the interval 0–1.Footnote 6 If \(Cgtr(x,y) = 0\), the condition is fully satisfied (\(x > y\)) and if \(Cgtr(x,y) = 1\), the condition is fully violated (\(x \ll y\)). If the conditions are strongly violated (\(Cgtr(x,y) > \) minimal value) by the input model, the corresponding analysis should not be used to constrain the model. In this way, the user can decide how strictly the conditions are enforced and ignore all constraints which have a too large value of \(Cgtr(x,y)\). In the following we ignore all constraints which have \(Cgtr(x,y) > 0.2\).
Table 1 Metadata describing the SMS result from CMS in Fig. 7 (left plot)
Finally, the entry ‘axes’ describes the available slices of the \((m_{\tilde{\chi }_1^{\pm }},\, m_{\tilde{L}},\,m_{\tilde{\chi }_1^0})\) parameter space, which in this example corresponds to \(m_{\tilde{l}}= 0.5m_{\tilde{\chi }^{\pm }_1} + 0.5m_{\tilde{\chi }_1^0}\), \(m_{\tilde{l}}= 0.95m_{\tilde{\chi }^{\pm }_1} + 0.05m_{\tilde{\chi }_1^0}\) and \(m_{\tilde{l}}= 0.05m_{\tilde{\chi }^{\pm }_1} + 0.95m_{\tilde{\chi }_1^0}\). This information is used to interpolate between the values in the experimental results. Since different experimental analyses adopt distinct slicing methods of the parameter space, we use a general interpolation procedure which works for any slicing choice. It is based on a tesselation of the mass vector space and a linear interpolation on each simplex.
Another illustrative example is the ATLAS dilepton search [25] for slepton pair production and decay shown in Fig. 7 (right): \(\tilde{l} ^+ + \tilde{l} ^- \rightarrow (l^+ \tilde{\chi }^0_1) + (l^- \tilde{\chi }^0_1)\), where \(\tilde{l} = \tilde{e}, \tilde{\mu }\). In this case selectrons and smuons are assumed to be mass degenerate and the experimental collaboration constrains the sum of lepton flavors: \(\sigma ([[[e^+]],[[e^-]]]) + \sigma ([[[\mu ^+]],[[\mu ^-]]])\). Once again, each lepton flavor is assumed to contribute equally. The metadata for this example is given in Table 2. As in the previous example, we assume the muon efficiency to be higher than the electron one, hence in ‘fuzzycondition’ we require \(\sigma ([[[e^+]],[[e^-]]]) \le \sigma ([[[\mu ^+]],[[\mu ^-]]])\) instead of equal flavor contributions.
Table 2 Metadata describing the SMS result from ATLAS in Fig. 7 (right plot)
List of analyses in database
The analyses that are currently implemented in the database are:
Gluino and squark searches
-
ATLAS: SUSY-2013-04 [26], ATLAS-CONF-2012-105 [27], ATLAS-CONF-2013-007 [28], ATLAS-CONF-2013-047 [29], ATLAS-CONF-2013-061 [30], ATLAS-CONF-2013-062 [31], ATLAS-CONF-2013-089 [32]
-
CMS: SUS-11-022 [33], SUS-11-024 [34], SUS-12-005 [35], SUS-12-011 [36], SUS-12-024 [37], SUS-12-026 [38], SUS-12-028 [39], SUS-13-002 [40], SUS-13-004 [41], SUS-13-007 [42], SUS-13-008 [43], SUS-13-012 [44], SUS-13-013 [45]
Electroweakino searches
-
ATLAS: ATLAS-CONF-2013-028 [46], ATLAS-CONF-2013-035 [47], ATLAS-CONF-2013-036 [48], ATLAS-CONF-2013-093 [49]
-
CMS: SUS-11-013 [50], SUS-12-006 [51], SUS-12-022 [52], SUS-13-006 [24], SUS-13-017 [53]
Direct slepton searches
3rd generation: stop and bottom searches
-
ATLAS: ATLAS-CONF-2012-166 [54], ATLAS-CONF-2013-001 [55], ATLAS-CONF-2013-007 [28], ATLAS-CONF-2013-024 [56], ATLAS-CONF-2013-025 [57], ATLAS-CONF-2013-037 [58], ATLAS-CONF-2013-047 [29], ATLAS-CONF-2013-048 [59], ATLAS-CONF-2013-053 [60], ATLAS-CONF-2013-062 [31], ATLAS-CONF-2013-065 [61], SUSY-2013-05 [62]
-
CMS: SUS-11-022 [33], SUS-12-028 [39], SUS-13-002 [40], SUS-13-004 [41], SUS-13-008 [43], SUS-13-011 [63], SUS-13-013 [45]
Of course the database is continuously being extended as new results become available.
Matching theoretical and experimental results
Once a BSM spectrum is decomposed according to the procedure described in Sect. 3.1, all the relevant information for confronting the model with the experimental results is encapsulated in the SMS topologies plus their mass vectors and weights. Any specific model dependent information can be dropped at this point. As discussed in Sect. 3.2, it is however often the case that the experimental result constrains a sum of topologies instead of a single one. Before a direct comparison with the experimental constraints, it is necessary to combine single SMS topologies (which means adding their weights) according to the experimental analysis’ assumptions (described in the metadata of the analysis). Furthermore, it is always implicitly assumed that all summed topologies have a common BSM mass vector (for the slepton pair production example this means \((m_{\tilde{e}},m_{\tilde{\chi }_{1}^{0}}) = (m_{\tilde{\mu }},m_{\tilde{\chi }_{1}^{0}})\)). Therefore, when combining signal topologies according to the analysis constraints, we must ensure that they have similar mass vectors.
Since different analysis have different sensitivities to the mass vector, we do not use the simple mass distance in GeV in order to verify if two vectors are similar or not. Instead we consider the sensitivity of the analysis in question to the difference between the two vectors. In order to quantify this sensitivity we use the analysis upper limit for each individual mass vector. If both upper limits differ by less than a maximal amount (20 %), we render the mass vectors as similar with respect to this particular analysis. However, if the upper limits differ by more than 20 %, we consider the two vectors as distinct. Moreover, in order to avoid cases where two upper limits are coincidentally equal, but they correspond to completely different mass configurations, we also require the mass values not differ by more than 100 %.
Once again we illustrate this procedure using the slepton pair production constraint as an example. In the MSSM we usually have \(m_{\tilde{e}_L} {=}\,\, m_{\tilde{\mu }_L}\) and \(m_{\tilde{e}_R} {=}\,\, m_{\tilde{\mu }_R}\), but \(m_{\tilde{e}_L} \ne m_{\tilde{e}_R}\). The experimental constraint on slepton pair production requires to combine the topologies \([[[e^+]],[[e^-]]]\) and \([[[\mu ^+]],[[\mu ^-]]]\) with \((m_{\tilde{e}},m_{\tilde{\chi }_{1}^{0}}) \simeq (m_{\tilde{\mu }},m_{\tilde{\chi }_{1}^{0}})\). Since \(\tilde{e}_{L,R}^+\tilde{e}_{L,R}^-\) and \(\tilde{\mu }_{L,R}^+\tilde{\mu }_{L,R}^-\) contribute to \([[[e^+]],[[e^-]]]+[[[\mu ^+]],[[\mu ^-]]]\), we must first group the sleptons (\(\tilde{e}_{L}\), \(\tilde{e}_{R}\), \(\tilde{\mu }_{L}\), \(\tilde{\mu }_{R}\)) with similar masses before we can combine the topologies. In order to identify the similar mass vectors we first obtain, for the given analysis, the upper limit for each vector (\((m_{\tilde{e}_L},m_{\tilde{\chi }_{1}^{0}})\), \((m_{\tilde{\mu }_R},m_{\tilde{\chi }_{1}^{0}})\),...) and cluster together the masses with similar upper limit values. If the analysis is sensitive to the left-handed/right-handed slepton mass splitting, the upper limits will differ significantly and the grouped masses will correspond to (\(\tilde{e}_L\), \(\tilde{\mu }_L\)) and (\(\tilde{e}_R\), \(\tilde{\mu }_R\)). On the other hand, if the mass spliting is small and the analysis is not sensitive to it, all upper limits will be similar and all the sleptons will be grouped together.
After the topologies with similar mass vectors have been identified, we can combine them (add their \({\sigma \times \mathcal {B}} \)) according to the experimental constraint (\(\sigma ([[[e^+]],[[e^-]]])+\sigma ([[[\mu ^+]],[[\mu ^-]]])\) for the example above). However, as mentioned in Sect. 3.2, for most constraints involving a sum of single topologies, the experimental assumptions include conditions on each topology contributing to the sum, such as \(\sigma ([[[e^+]],[[e^-]]]) \simeq \sigma ([[[\mu ^+]],[[\mu ^-]]])\) for the slepton analysis. These conditions need to be taken into account when interpreting the experimental results, since each topology may have a different signal efficiency. Therefore we must also verify that these conditions are satisfied, otherwise the experimental upper limit can not be applied. Finally, if the experimental assumptions are satisfied, the resulting theoretical predictions (\(\sigma \times \mathcal {B}\)) obtained after combining the topologies can be directly compared to the corresponding experimental upper limit.
In Fig. 1 we summarize the main steps required to confront the BSM model predictions with the experimental constraints: the SMS decomposition, the combination of SMS topologies with identical or similar masses into the topology sums assumed by the analyses and finally the comparison with the experimental upper limits obtained from the database of LHC results described in Sect. 3.2.