Discriminant sideband
4.3.3. Control Distributions
In order to verify the goodness of the data model components, the effects of the event prese-lection and MC corrections are visualised using control distributions of physical observables.
In this work, following the common prescriptions adopted in the ATLAS top quark working group, two levels of the pre-processing are considered: the PreTag and the Tag sample, corresponding to the selection levels before and after the b-tag requirement, respectively.
Exemplary plots of the reconstructed ETmiss and mWT distributions before and after the b-tag requirement selection in the three-jet bin are shown in Fig. 4.2 and 4.3.
Normalisation of MC Histograms and Treatment of Statistical Errors
In order for the Monte Carlo distributions to represent the data faithfully, the histograms of all the simulated processes are normalised to the total integrated luminosity Ldata of the data sample which they are compared to. In practice, the number of entries NMC,i of each distribution of each MC process is replaced by a normalised quantity:
NMC,inorm=NMC,iσMC,i·Ldata
NMC,isample . (4.8)
HereσMC,i is the Monte Carlo cross section associated to the i-th process (listed in Tab.2.3) andNMC,isample is the total number of events that was initially generated for the i-th MC sample.
Inserting in the original number of generated events has the effect of including the selection acceptance in the normalisation factor. The sum PNMC,inorm of all MC histograms is then
compared to the data histogram. In this sense, in each distribution the top of the simulation stack represents the sum of the MC histograms of all the considered processes. Together with the number of expected events, the uncertainties of each bin entries, ∆NMC,i, are scaled according to Eq. (4.8) for each process. The definition of the statistical error depends on the number of entries populating the single bin of each of the distributions. After the selection, if the number of bin entries is large enough, the uncertainty is the width of the Poisson distribution, ∆NMC = √
NMC. In this case the error definition adopted is symmetrical around the central value. On the contrary, when the number of entries in the bin is low, the uncertainties follow the definition given by the Feldman-Cousins method [FC98]. Using asymmetrical definitions for central values close to zero, the unphysical intervals which exceed the domain of the variable are avoided; in this work, 33 bin entries were chosen as a suitable threshold for the error regime transition. The total uncertainty of the stacked MC histograms in each bin then follows from simple error propagation:
∆NMCtotal =sX
i
(∆NMC,inorm)2 (4.9)
Throughout this study, the error ∆NMCtotal is drawn with a pink hatched area above the stacked histograms of all MC samples. The same treatment of the statistical uncertainties, depending on the number of entries populating each bin on the MC histograms and using also a threshold of 33 entries to define the error definition transition, is used in a study performed in parallel to this work, where the kinematic fit technique is used to identify single top events in the t-channel [Her14].
PreTag Control Distributions
ThePreTagcontrol distributions of ETmissandmWT are shown in Fig.4.2for both the electron and muon channels in the 3 jet bin. The distributions of the missing transverse energy and the transverse mass of the reconstructed W boson are used to verify the goodness of the W+jets scale factor estimation described in Sec. 4.3.2. The signal and background MC samples, including W+jets, are normalised to the theoretical cross sections, reported in Tab.2.3. The QCD multi-jet part is normalised using the factors extracted by the Jet-Electron fit for the PreTag selection. Since the scale factors for the W+jets samples can not be extracted for the PreTag selection level, it is here chosen to display the simulation data after a further scaling of the simulation stack with a common factor, to best fit the collision data. As can be seen from the figures, the W+jets background component dominates the PreTag level of the selection.
Tag Control Distributions
All the simulation samples after the Tag selection level are normalised to the data lumi-nosity, according to the cross sections shown in Tab. 2.3. At this stage, the QCD multi-jet background component obtained from the Jet-Electron method is normalised by means of the scale factors presented in Tab. 4.2. The W+jets components are normalised using the W+jets scale factors, given in Tab. 4.3, as they are found with the data-driven method.
Jet Bin Kbb/cc Kc Kll WN
e 3 1.153525 0.939875 0.981052 0.854447 4 1.145241 0.933125 0.974007 0.909405 µ 3 1.229149 0.974645 0.956930 0.898860 4 1.215753 0.964022 0.946500 1.004964
Table 4.3.: Jet-bin dependent W+jets normalisation factors for each flavour fraction used in the analysis. The figures in the table have been obtained from the tag counting method. The statistical uncertainty on Kbb/cc,Kc and Kll is Xbb/cc%,Xc%,and Xll% respectively. The scale factors KNs are used for the overall normalisation of the W+jets contribution. The remaining flavour factors, Kis, are used for the normalisation of the differently flavoured components of the W+jets spectrum. The values found by us are in agreement with the current W+jets scaling settings obtained independently by other groups involved in top physics analyses at ATLAS [A+12k].
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(a)Missing transverse energy. µ+3 jets, pre-tag.
(GeV)
(b)Transverse W mass. µ+3 jets, pre-tag.
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(c)Missing transverse energy, e+3 jets pre-tag.
(GeV)
(d)Transverse W mass. e+3 jets, pre-tag.
Figure 4.2.: Distribution of the missing transverse energy and the transverse W mass, requiring the presence of exactly one lepton and three jets without b-tagging requirements, for the muon ((a), (b)) and electron ((c), (d)) selection. The jet-electron model described in Sec. 4.3.1for modelling the multi-jet background contribution, using the normalisation factors of Tab.4.2. For the W+jets fraction the kinematic distribution are taken from the MC simulation, while the data-driven factors in Tab 4.3 extracted from the charge asymmetry event yield are used for the normalisation. The remaining samples are normalised according to the theory. The full MC stack plot is scaled by an overall 1.08 factor. After this final normalisation step a good shape agreement between the data and the MC is seen. The last bin contains the sum of the events in that bin or higher.
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Missing transverse energy,µ+3 jets selection.
(GeV)
Transverse W mass,µ+3 jets selection.
40 60 80 100 120 140 160 180 200
Missing transverse energy, e+3 jets selection.
(GeV)
Transverse W mass, e+3 jets selection.
Figure 4.3.: Distribution of the missing transverse energy and the transverse W mass, requiring the presence of exactly one lepton and three jets of which exactly one is b-tagged, for the muon ((a), (b)) and electron ((c),(d)) selection. The jet-electron model described in Sec.4.3.1for modelling the multi-jet background contribution, using the normalisation factors of Tab.4.2. For the W+jets fraction the kinematic distribution are taken from the MC simulation, while the data-driven factors in Tab 4.3 extracted from the charge asymmetry event yield are used for the normalisation. The remaining samples are normalised according to the theory. The last bin contains the sum of the events in that bin or higher.
In this chapter the basic principles of the kinematic fitting of high energy physics events are presented along with its implementation in the analysis framework. This serves as a manual for the application of the fitter to the reconstruction of events where a single top quark is produced in association with a W boson, described in Chap. 6.