Theory Predictions
The decomposition <Decomposition> of the input model as a sum of elements <element> (simplified models) is the first step for confronting the model with the experimental limits. The next step consists of computing the relevant signal cross sections (or theory predictions) for comparison with the experimental limits. Below we describe the procedure for the computation of the theory predictions after the model has been decomposed.
As discussed in Database Definitions <DatabaseDefinitions>, the SModelS database allows for two types of experimental constraints: Upper Limit constraints (see UL-type results <ULtype>) and Efficiency Map constraints (see EM-type results <EMtype>). Each of them requires different theoretical predictions to be compared against experimental data.
UL-type results <ULtype> constrains the weight of one element <element> or sum of elements <element>. The element <element> weight is defined as σ × BR × ℱ, where σ is the total production cross-section, BR is the product of all branching ratios for the decays appearing in the element and ℱ is the product of all the lifetime reweighting factors (ℱlong and ℱprompt), as discussed in the SLHA decomposition <slhaDecomp> (for the LHE-type decomposition <lheDecomp> ℱ = 1).
Therefore, in order to apply the experimental constraints, SModelS must first compute the theoretical value of σ × BR × ℱ summing only over the elements <element> appearing in the respective constraint <ULconstraint>. This is done applying a 1 (zero) efficiency (ϵ) for the elements which appear (do not appear) in the constraint <ULconstraint>. Then the final theoretical prediction is the sum over all elements <element> with a non-zero value of ∑σ × BR × ℱ × ϵ. This value can then be compared with the respective 95% C.L. upper limit extracted from the UL map (see UL-type results <ULtype>).
On the other hand, EM-type results <EMtype> constrain the total signal (∑σ × BR × ℱ × ϵ) in a given signal region (data set<DataSet>). Consequently, in this case SModelS must compute σ × BR × ℱ × ϵ for each element <element>, using the efficiency maps for the corresponding data set<DataSet>. The final theoretical prediction is the sum over all elements <element> with a non-zero value of σ × BR × ℱ × ϵ. This value can then be compared with the signal upper limit for the respective signal region (data set<DataSet>).
For experimental results for which the covariance matrix is provided, it is possible to combine all the signal regions (see Combination of Signal Regions <combineSRs>). In this case the final theory prediction corresponds to the sum of σ × BR × ℱ × ϵ over all signal regions (and all elements) and the upper limit is computed for this sum.
Although the details of the theoretical prediction computation differ depending on the type of Experimental Result<ExpResult> (UL-type results <ULtype> or EM-type results <EMtype>), the overall procedure is common for both type of results. Below we schematically show the main steps of the theory prediction calculation:
As shown above the procedure can always be divided in two main steps: Element Selection and Element Clustering. Once the elements <element> have been selected and clustered, the theory prediction for each data set<DataSet> is given by the sum of all the element <element> weights (σ × BR × ℱ × ϵ) belonging to the same cluster:
theory prediction = ∑cluster(element weight) = ∑cluster(σ × BR × ℱ × ϵ)
Below we describe in detail the element selection and element clustering methods for computing the theory predictions for each type of Experimental Result<ExpResult> separately.
- Theory predictions are computed using the theoryPredictionsFor method
Computation of the signal cross sections for a given UL-type result <ULtype> takes place in two steps. First selection of the elements <element> generated by the model decomposition <Decomposition> and then clustering of the selected elements according to their masses. These two steps are described below.
An UL-type result <ULtype> holds upper limits for the cross sections of an element <element> or sum of elements <element>. Consequently, the first step for computing the theory predictions for the corresponding experimental result is to select the elements <element> that appear in the UL result constraint <ULconstraint>. This is conveniently done attributing to each element <element> an efficiency equal to 1 (0) if the element <element> appears (does not appear) in the constraint <ULconstraint>. After all the elements <element> weights (σ × BR) have been rescaled by these ''trivial'' efficiencies, only the ones with non-zero weights are relevant for the signal cross section. The element <element> selection is then trivially achieved by selecting all the elements <element> with non-zero weights.
The procedure described above is illustrated graphically in the figure below for the simple example where the constraint <ULconstraint> is [[[e+]], [[e−]]] + [[[μ+]], [[μ−]]].
- The element selection is implemented by the getElementsFrom method
Naively one would expect that after all the elements <element> appearing in the constraint <ULconstraint> have been selected, it is trivial to compute the theory prediction: one must simply sum up the weights (σ × BR × ℱ) of all the selected elements <element>. However, the selected elements <element> usually differ in their masses1 and the experimental limit (see Upper Limit constraint <ULconstraint>) assumes that all the elements <element> appearing in the constraint <ULconstraint> have the same mass (or mass array). As a result, the selected elements <element> must be grouped into clusters of equal masses. When grouping the elements <element>, however, one must allow for small mass differences, since the experimental efficiencies should not be strongly sensitive to small mass differences. For instance, assume two elements <element> contain identical mass arrays, except for the parent masses which differ by 1 MeV. In this case it is obvious that for all experimental purposes the two elements <element> have identical masses and should contribute to the same theory prediction (e.g. their weights should be added when computing the signal cross section). Unfortunately there is no way to unambiguously define ''similar masses'' and the definition should depend on the Experimental Result<ExpResult>, since different results will be more or less sensitive to mass differences. SModelS uses an UL map-dependent measure of the distance between two element <element> masses, as described in Mass Distance <massdist>.
If two of the selected elements <element> have a mass distance <massdist> smaller than a maximum value (defined by maxDist), they are gouped in the same mass cluster, as illustrated by the example below:
Once all the elements <element> have been clustered, their weights can finally be added together and compared against the experimental upper limit.
- The clustering of elements is implemented by the clusterElements method.
As mentioned above <ULcluster>, in order to cluster the elements <element> it is necessary to determine whether two elements <element> have similar masses (see element <element> and Bracket Notation <bracketnotation> for more details on element <element> mass). Since an absolute definition of ''similar masses'' is not possible and the sensitivity to mass differences depends on the experimental result, SModelS uses an ''upper limit map-dependent'' definition. For each element <element>'s mass array, the upper limit for the corresponding mass values is obtained from the UL map (see UL-type result <ULtype>). This way, each mass array is mapped to a single number (the cross section upper limit for the experimental result). Then the distance between the two element <element>'s masses is simply given by the relative difference between their respective upper limits. More explicitly:
where MA, MB (x, y) are the mass arrays (upper limits) for the elements <element> A and B, respectively. If the mass distance of two elements <element> is smaller than maxDist, the two masses are considered similar.
Notice that the above definition of mass distance quantifies the experimental analysis sensitivity to mass differences, which is the relevant parameter when clustering elements <ULcluster>. Also, a check is performed to ensure that masses with very distinct values but similar upper limits are not clustered together.
- The mass distance function is implemented by the distance method
In order to compute the signal cross sections for a given EM-type result <EMtype>, so it can be compared to the signal region limits, it is first necessary to apply the efficiencies (see EM-type result <EMtype>) to all the elements <element> generated by the model decomposition <Decomposition>. Notice that typically a single EM-type result <EMtype> contains several signal regions (data sets<DataSet>) and there will be a set of efficiencies (or efficiency maps) for each data set<DataSet>. As a result, several theory predictions (one for each data set<DataSet>) will be computed. This procedure is similar (in nature) to the Element Selection<ULselection> applied in the case of an UL-type result <ULtype>, except that now it must be repeated for several data sets<DataSet> (signal regions).
After the element <element>'s weights have being rescaled by the corresponding efficiencies for the given data set<DataSet> (signal region), all of them can be grouped together in a single cluster, which will provide a single theory prediction (signal cross section) for each data set<DataSet>. Hence the element clustering <EMcluster> discussed below is completely trivial. On the other hand the element selection <EMselection> is slightly more involved than in the UL-type result <ULtype> case and will be discussed in more detail.
The element selection for the case of a EM-type result <EMtype> consists of rescaling all the elements <element> weights by their efficiencies, according to the efficiency map of the corresponding data set<DataSet>. The efficiency for a given data set<DataSet> depends both on the element <element> mass and on its topology and particle content. In practice the efficiencies for most of the elements <element> will be extremely small (or zero), hence only a subset effectively contributes after the element selection2. In the figure below we illustrate the element selection for the case of a EM-type result <EMtype>/data set<DataSet>:
If, for instance, the analysis being considered vetoes jets and τ's in the final state, we will have ϵ2, ϵ4 ≃ 0 for the example in the figure above <EMselectionfig>. Nonetheless, the element selection for a data set<DataSet> is usually more inclusive than the one applied for the UL-type result <ULtype>, resulting in less conservative values for the theory prediction.
- The element selection is implemented by the getElementsFrom method
Unlike the clustering required in the case of UL-type result <ULtype> (see Element Clustering for an UL analysis <ULcluster>), after the efficiencies have been applied to the element's weights, there is no longer the necessity to group the elements <element> according to their masses, since the mass differences have already been accounted for by the different efficiencies. As a result, after the element selection <EMselection> all elements belong to a single cluster:
- The (trivial) clustering of elements is implemented by the clusterElements method.
As discussed in
Database Definitions <databaseDefs>,UL-type results <ULtype>have a singledata set<DataSet>.↩When refering to an
element <element>mass, we mean all theintermediate state <odd states>masses appearing in theelement <element>(or theelement <element>mass array). Twoelements <element>are considered to have identical masses if their mass arrays are identical (seeelement <element>andBracket Notation <bracketnotation>for more details).↩