6 Dataset, Event Selection and Reconstruction
6.3 Event Reconstruction
6.3.1 Kinematic Likelihood Fit
The reconstruction with the KLFitter framework is based on a certain input decay model,t¯tdecays in this analysis, and requires the usage of the Bayesian Analysis Toolkit (BAT)[282]. KLFitter maps the four partons of thet¯t decay in the lepton+jets decay channel to four reconstructed jets using constraints on both the top quark mass mt and theW boson mass mW. The four reconstructed jets with the highest pTvalues are used as input to KLFitter. Further possible configurations are compared in Sec. 6.3.4. For all resulting 4!=24 permutations, a likelihood Lis maximised, which is of the following form for each permutation:
L=BW(mq1q2q3|mt,Γt)·BW(mq1q2|mW,ΓW)·BW(mq4`ν|mt,Γt)·BW(m`ν|mW,ΓW)
·
4
Y
i=1
W(Eimeas|Ei)·W(E`meas|E`)·W(Emissx |pνx)·W(Emissy |pνy). (6.1) TheW(EmeasP |EP)are transfer functions, described in detail in Sec. 6.3.2, whereEmeasP is the mea-sured energy of the reconstructed objects P, EP is the energy of the corresponding original parton or leptonP, andpνx andpνy stand for the momentum components of the neutrinoνin the transverse plane. The energies EP and these momentum components are free parameters of the likelihood maximisation. The third neutrino momentum component pzνis initially computed with aW boson mass constraint of m2W = (pν+p`)2 with the four-momenta pν and p`. Then,pνz is another free parameter in the fit after this initial calculation. Transfer functions for electrons, muons (whereE is replaced by pT), b-jets, light jets (includingc-jets) andETmissare employed.
The BW(mi j(k)|mt/W,Γt/W) terms represent Breit-Wigner functions which characterise the proba-bility distribution of the reconstructedW boson or top quark mass given the assumed values for the masses mt/W and the decay widths Γt/W. Thus, the BW constraints serve to assign leptons, ETmiss and jets to the leading order partons/lepton from the hard t¯t decay, and fitted masses of composite reconstructed particles can be evaluated. The indicesq1-q4 refer to the four quarks that are mapped to the reconstructed jets.
The two mass parameters in the BW terms are set tomt=172.5 GeV andmW =80.4 GeV while
6 . 3 E V E N T R E C O N S T R U C T I O N
the parameters for the decay width are fixed toΓt=1.33 GeV andΓW =2.1 GeV. Since this analysis uses KLFitter merely to choose the best assignment of jets to partons and does not exploit the fitted parameters of reconstructed particles, as obtained from the kinematic fit, a variation ofΓt has no influence on the reconstructed distributions and the final result. This possible relationship between the input decay widthΓt and the resulting reconstructed observable distributions was studied and the results are shown in Sec. 6.3.4.
As described in Sec. 6.2, a cut on the logarithm of the KLFitter likelihood ln(L), as shown in Eq. (6.1), is applied. Around 56%-58% of events, depending on the analysis region, are removed by this requirement. Combinatorial background due to wrongly reconstructed events, which is mainly present at very small likelihood values, is suppressed to a large extent, as visible in Fig. 6.9.
Also, the purity of the selected sample increases as a larger fraction of background events than signal t¯t events is removed. The distributions of the logarithm of the KLFitter likelihood illustrate that events with ln(L) < −50 form a smaller second peak comprising mainly events without a correct match of all four reconstructed jets. The distributions of the reconstructed top quark mass lose their broader tails caused by those not correctly reconstructed events while the general shape of the distributions changes only very slightly when solely fully matched events are plotted.
The cut on ln(L) also affects the shapes of the distributions of the two observables, m`b and
∆Rmin(jb,jl). Especially the very large tail at higher mass values of m`b, which suffers notice-ably from the combinatorial background, is reduced after applying the cut. On the contrary, the peak region of them`b distributions hardly change. Later studies showed that the impact of this cut on the final result is covered by the statistical uncertainties, as described in Sec. 10.1.
Fig. 6.10 contains control plots of the logarithm of the likelihood for different analysis regions before the cut on this quantity is imposed. The fully matched events are shown separately. The fraction of events where all four partons are matched correctly increases from 13% to 23% and from 17% to 31% after applying the ln(L)cut for events with exactly one and at least twob-tagged jets, respectively.
This analysis does not rely on these efficiencies for matching all four jets correctly. The mass observablem`b, which provides most of the sensitivity toΓt, requires solely the reconstruction of theb-jet from the leptonically decaying top quark. The reconstruction efficiency for this jet amounts to 65% and 75% for events with one and at least two b-tagged jets, respectively. The fraction of events with a correctly matched b-jet from the leptonically decaying top quark is highlighted in the ln(L)distributions contained in Fig. 6.11.
The given figures demonstrate once more to which extent background events and combinatorial background is reduced in order to improve the entire sensitivity of the measurement.
The purely kinematic information in the KLFitter likelihood can be augmented by additional infor-mation as, for instance, b-tagging information. The likelihood definition in Eq. (6.1) is modified and converted into a so-called event probability based on such further event properties, outlined in Sec. 6.3.3.
(a) (b)
(c) (d)
Figure 6.9: Distributions of (a,b) the logarithm of the KLFitter likelihood and (c,d) the recon-structed top quark mass for events with at least one b-tag for (a,c) electron+jets and (b,d) muon+jets events. The distributions in (a) and (b) compare fully matched with not fully matched events while three different options are provided in the top quark mass plots: (1) KLFitter without cut, (2) KLFitter with the likelihood cut (“LL cut”) and (3) KLFitter with the likelihood cut for fully matched combinations.
KLFitter determines the likelihood and also the event probability for all permutations in the event.
Finally, the permutation with the highest event probability is regarded as the best estimate for the jet-to-particle association and used to reconstruct events entering the decay width measurement.
6 . 3 E V E N T R E C O N S T R U C T I O N
40000 Datatt, wrong match , correct match
Figure 6.10:Distributions of the logarithm of the likelihood obtained from the event reconstruc-tion algorithm for selected (a,b) electron+jets and (c,d) muon+jets events having (a,c) exactly one and (b,d) at least twob-tagged jets. The events withln(L)<−50form a secondary broader peak mainly composed of events that are not properly reconstructed, i.e. events for which not all four jets are correctly matched to partons (“wrong match”). Fully matched events are primarily existent in the largerln(L)regions. The hatched bands contain the normalisation uncertainty in the signal and background contributions as well as the signal model systematic uncertainties.
The first and last bins include underflow and overflow events, respectively.
−80 −70 −60 −50
40000 Datatt, wrong match , blep match
Figure 6.11:Distributions of the logarithm of the likelihood obtained from the event reconstruc-tion algorithm for selected (a,b) electron+jets and (c,d) muon+jets events having (a,c) exactly one and (b,d) at least twob-tagged jets. A large fraction of events possesses a leptonicbjet that is correctly matched to the corresponding parton, in particular in the region of largerln(L)values (“blep match”), compared to events without a correct match of this jet (“wrong match”). The hatched bands contain the normalisation uncertainty in the signal and background contributions as well as the signal model systematic uncertainties. The first and last bins include underflow and overflow events, respectively.
6 . 3 E V E N T R E C O N S T R U C T I O N