7 Analysis Strategy
7.2 Template Reweighting
significantly. Despite having only little sensitivity to the decay width, the advantages of the angular observables are a small sensitivity to jet-related uncertainties, the addition of information from the hadronic decay branch to the fit and the reduction of leading systematic uncertainties in the combination with m`b. Particularly∆Rmin(jb,jl) performs promisingly and does not suffer from modelling issues which is why this observable is used in the fit in addition tom`b. The observable choice is justified and explained in more detail by testing various possible fit configurations in Ch. 9.
7.2 Template Reweighting
This section addresses the derivation of decay width templates required for the fit. Due to the absence of MC simulated t¯t samples with Γt values different from the SM expectation of Γt = 1.33 GeV, the signalΓt templates were created by a reweighting procedure. On the assumption that the distributions follow a Breit-Wigner shape, verified by additional cross-checks, they were reweighted from the nominal t¯t sample withΓt=1.33 GeV.
For this purpose, the distributions of reconstructed events based on the nominal t¯t full simulation sample were weighted employing theoretical Breit-Wigner curves for 54 different top quark decay width values. The procedure is explained in the following. The decay width range from 0.1<Γt<
5.0 GeV is covered by 50 templates in steps of∆Γt=0.1 GeV, four additional templates followed from a reweighting toΓt =0.01, 6, 7, 8 GeV to also account for very small and very large decay width values. The Breit-Wigner function to rescale the distributions on an event-by-event basis is defined as:
BW(x) = 2p
2mtΓt
Æm2t(m2t+Γt2) πÇ
m2t +Æ
m2t(m2t +Γt2)·((x2−m2t)2+m2tΓt2)
. (7.1)
The top quark mass is set to mt = 172.5 GeV as the mass of the nominal t¯t MC sample. The truth top quark masses per event are denoted by x. The mass distributions at truth parton level perfectly agree with the shape of a Breit-Wigner curve whereas the distributions are smeared at reconstruction level because of resolution effects. Both top quarks enter the reweighting procedure.
For observables of the hadronic decay branch of thet¯t events, only the true MC top quark mass of the hadronic top quark is used to reweight the distributions and, similarly, for observables of the leptonic decay branch only the leptonic top quark mass is used for the reweighting. These truth masses are defined after ISR but before FSR, according to the applied generator definitions.
The weights obtained from the reweighting procedure as a function ofmtare given in Fig. 7.1. The absolute weights per decay width template depicted as histograms are shown in Fig. 7.2. Since the nominal template enters the fit as well, in total 55 templates are used in the fit to extract the best fitted value ofΓt. The template distributions for the decay widths in the range from 0.7 to 3.0 GeV are shown in Fig. 7.3 and Fig. 7.4 for three example observables at reconstruction level as they are defined in Sec. 7.1.
[GeV]
mt
150 155 160 165 170 175 180 185 190 195
Weights
150 155 160 165 170 175 180 185 190 195
Weights
Figure 7.1:MC weights used for the template reweighting as a function of the underlying top quark masses which are passed as arguments to the Breit-Wigner equation. Weights are binned in the plots to allow for a better visibility. (a) also contains the extreme decay width values in the available range, (b) illustrates weights for templates closer to the nominal value ofΓt=1.33GeV.
Weights
Figure 7.2:MC weights used for the template reweighting filled into histograms for some exam-ple top quark decay width values.
7 . 2 T E M P L AT E R E W E I G H T I N G
Figure 7.3:Reweighted templates at reconstruction level in the range0.7≤Γt≤3.0GeV in the electron+jets (left) and muon+jets (right) channel using events with at least twob-tagged jets in the region|η| ≤1. Compared are templates for the observablesD32,m`band∆Rmin(jb,jl). The lower panels illustrate the ratio of the reweighted templates with respect to the nominal sample generated atΓt=1.33GeV.
Events
Figure 7.4:Reweighted templates at reconstruction level in the range0.7≤Γt ≤3.0GeV in the electron+jets (left) and muon+jets (right) channel using events with at least twob-tagged jets in the region|η|>1. Compared are templates for the observablesD32,m`band∆Rmin(jb,jl). The lower panels illustrate the ratio of the reweighted templates with respect to the nominal sample generated atΓt=1.33GeV.
7 . 2 T E M P L AT E R E W E I G H T I N G
The ratio plots underline the sensitivity of the observables toΓt. D32offers the largest differences, but suffers from large systematic uncertainties. The sensitivity ofm`b toΓt is well suited for the desired measurement while∆Rmin(jb,jl)adds only little sensitivity to the decay width which is compensated by the other advantages of using this observable.
The reweighting method was validated using MCt¯t samples generated with PO W H E Ginterfaced with PY T H I A6, similar to the nominal sample described in Sec. 5.2, but withhhdamp=∞, based onΓt =0.7 GeV andΓt=3.0 GeV. The top quark mass distributions of these alternative samples were compared to corresponding mass distributions obtained from the reweighting procedure at parton level. The good agreement between the reweighted distributions for the alternative Γt samples is reflected in Fig. 7.5 and no significant differences beyond the statistical uncertainties in the individual bins are visible. This is also confirmed by results of Kolmogorov-Smirnov (KS) and χ2tests which quantify the agreement between the two histograms.
(a) (b)
Figure 7.5:Results of a closure test to validate the reweighting method. The nominal sample based onΓt=1.33GeV (cyan, dashed) is given and the corresponding reweighted curve (black, dotted), which is compared to a sample with a decay width of (a)Γt=0.7GeV and (b)Γt=3.0GeV (blue in each case). The plots comprise all events with at least oneb-tagged jet after summing up the events in the electron+jets and muon+jets channel. The lower panels contain the ratio with respect to the reweighted sample. The shown uncertainty bands contain statistical uncertainties which cover the differences between the two compared samples.
The Kolmogorov-Smirnov test is designed to check the consistency of one-dimensional distributions and, broadly speaking, quantifies the difference between two probability distributions or histograms, as it is the case here. A resulting KS value of around one verifies a very good agreement between two compared distributions. A bad agreement corresponds to a KS value close to zero. The KS test is particularly suited to compare the shapes of distributions which is why they were initially
defined to compare continuous distributions.
Aχ2test marks another possibility of checking distances between two distributions or histograms.
χ2methods rely on bin-by-bin comparisons as such a method requires a discretisation in general.
As a consequence thereof, aχ2 test is more sensitive to statistical fluctuations while a KS test is presumed to yield better results than aχ2 test in case of histograms with a lack of statistics. The calculation in this thesis is based on the implementation within the ROOT framework[279], resting mainly on Ref.[285]for the KS test and Ref.[286–288]for theχ2test. A further discussion would go beyond the scope of this thesis.
This closure test was repeated at reconstruction level for both chosen observables. Reweighted distributions were compared to alternative decay width distributions obtained from the above mentioned MC t¯t samples form`b and∆Rmin(jb,jl). Fig. 7.6 presents example plots confirming the agreement at reconstruction level within the statistical uncertainties.
Distributions for each analysis region are composed of 20 bins for both observables. The associated ranges are chosen depending on the two η regions in order to minimize the number of empty bins or bins which possess very low statistics. The effect of the applied binning was estimated by performing the analysis with less or more than 20 bins in each analysis region for the observables, without modifying the parameter ranges. Halving the number of bins reduces the sensitivity to different decay width templates and thus leads to a larger statistical uncertainty by about 15-20%.
An increase in the number of bins above 20 per analysis region results in statistical fluctuations of the fit due to a growing number of bins with low statistics. In summary, a number of 20 constitutes a good compromise.
After reducing the number of bins per analysis region to ten, fluctuations in the bins with a low number of events diminish.
The impact of the chosen binning of the observable templates was evaluated quantitatively by merging the bins in the tail regions with only a very limited number of entries. For this setting, the last six bins were combined to three bins with an equal bin width and, depending on the pseudorapidity region, the first four or two bins were combined into two or only one bin, respectively.
Consequently, the resulting distributions per analysis channel contain 15 or 16 bins instead of 20.
The bins in the tails are thus twice as large as the bins in the central peak region of the distributions.
The fluctuations in the tail regions of the templates decrease, especially for decay width templates outside of the region of extreme decay width values close to 0 GeV. The fit yields a result that differs by 0.12 GeV from the final quoted result, and the statistical uncertainty reduced to 0.02 GeV. Hence, the statistical uncertainty covers the effect due to the binning well.
7 . 2 T E M P L AT E R E W E I G H T I N G
(a) (b)
(c) (d)
Figure 7.6:Results of a closure test to validate the reweighting method at reconstruction level.
The generated sample atΓt =1.33 GeV (cyan, long dashes) is shown and the corresponding reweighted curve (black, dotted), which is compared to a sample with a decay width ofΓt = 3.0GeV (blue, short dashes) for (a,b)m`b and (c,d)∆Rmin(jb,jl). The plots comprise all events with at least two b-tags in the region|η|> 1in the (a,c) electron+jets and (b,d) muon+jets channel. The lower panels contain the ratio with respect to the black reweighted sample. The shown uncertainty bands comprise statistical uncertainties that cover the differences between the two compared samples except for a few outliers.