6. Event Selection and Reconstruction 73
6.2. Event Reconstruction
6.2.1. Kinematic Likelihood Fitting
The selected events are reconstructed using a kinematic likelihood package (KLFitter) [148] based on theBayesian Analysis Toolkit (BAT) [149]. The KLFitter uses the t¯t decay topology as an input model with the mt and mW mass constraints on composite objects built from the input lepton, ETmiss, and jets to map the input objects to leading order partons and lepton from thet¯tdecay.
Since the detector has a limited energy/momentum resolution, the energy/momentum of the input objects is allowed to vary within the corresponding detector resolutions, while the coordinate information of these objects is assumed to be measured precisely.
This information is incorporated by the so-calledtransfer functions (TF), which describe the probability of detecting a final state object with energy Emeas originating from LO parton/lepton with true energy E. Separate TFs are derived for electrons, muons, light jets, b-jets, and ETmiss in different η ranges. The final two and three-body masses are evaluated with Breit-Wigner distributions using top quark andW boson masses fixed to mt = 172.5 GeV andmW= 80.2 GeV. The likelihood is defined as:
L =BW(mq1q2q3|mtΓt)·BW(mq1q2|mWΓW)·BW(mq4`ν|mtΓt)·BW(m`ν|mWΓW) Y4
i=1
Wjet(Eimeas|Ei)·W`(Emeas` |E`)·Wmiss(Exmiss|pνx)·Wmiss(Eymiss|pνy), (6.1) where Wi(Exmeas|Ei) are the transfer functions, Exmeas is the measured energy of a re-constructed object x, Ei is the ’true’ energy of the corresponding parton i, and the BW(mij(k)|mYΓY) are the Breit-Wigner functions used to evaluate the mass of com-posite reconstructed particles with respect to a set mass and width of particle Y. The in-depth discussion about the construction and use of the transfer functions is follows.
Permuting the jets in an event through all positions in the model hypothesis yields dif-ferent likelihood values for each permutation. To increase the reconstruction efficiency1of theKLFitter, the likelihood of a given permutation is extended to anevent probability by adding additional information such as b-tagging and kinematic differences between types of light jets. This extension is discussed in details in Section 6.2.2. After the calculation of the likelihood (and/or event probability) of each permutation, the permu-tation with the highest event probability defines the reconstructed event and chosen for
1The ratio of the correctly reconstructed events (all four jets match their corresponding partons) to all selected events
6.2. Event Reconstruction
measuring the angles to extract theW boson helicity fractions.
Reconstructed distributions from the leading permutation after applying the log like-lihood cut of >-48 are shown in Figs 6.6–6.9, where good agreement between data and prediction is observed in all channels.
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Events / 13 GeV
reco. top mass [GeV]
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Data/Pred. 0.6
Number of b-tags (MV1 @ 70%)
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Data/Pred. 0.60.811.21.4 75− 70− 65− 60− 55− 50− 45−
Events / 2
Figure 6.6.: Plots showing selected top quark kinematics, the log likelihood, and the event probability distributions of the leading permutation (ranked by event probability) in the electron channel with 1b-tag. All plots except for the log likelihood are shown after the log LH>−48cut. The displayed uncertainties represent the Monte Carlo statistical uncertainty as well as the background normalisation uncertainties.
Transfer Functions
The transfer functions are obtained fromt¯tevents simulated withMC@NLO[150,151], to map the energies and momenta of the final state objects at the detector level to the parton level energy at LO MC simulation. The final state objects at parton/lepton
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Events / 13 GeV
reco. top mass [GeV]
100 200 300 400 500
Data/Pred. 0.60.811.21.4
Number of b-tags (MV1 @ 70%)
0 1 2 3
Figure 6.7.: Plots showing selected top quark kinematics, the log likelihood, and the event probability distributions of the leading permutation (ranked by event probability) in the electron channel with ≥ 2 b-tags. All plots except for the log likelihood are shown after the log LH > −48 cut. The displayed uncertainties represent the Monte Carlo statistical uncertainty as well as the background normalisation uncertainties.
6.2. Event Reconstruction
Data/Pred. 0.60.811.21.4 100 150 200 250 300 350 400 450 500
Events / 13 GeV
reco. top mass [GeV]
100 200 300 400 500
Data/Pred. 0.60.811.21.4
Number of b-tags (MV1 @ 70%)
0 1 2 3
Figure 6.8.: Plots showing selected top quark kinematics, the log likelihood, and the event probability distributions of the leading permutation (ranked by event probability) in the muon channel with 1 b-tag. All plots except for the log likelihood are shown after the log LH>−48cut. The displayed uncertainties represent the Monte Carlo statistical uncertainty as well as the background normalisation uncertainties.
0 100 200 300 400 500 600
Data/Pred. 0.60.811.21.4 100 150 200 250 300 350 400 450 500
Events / 13 GeV
reco. top mass [GeV]
100 200 300 400 500
Data/Pred. 0.60.811.21.4
Number of b-tags (MV1 @ 70%)
0 1 2 3
Figure 6.9.: Plots showing selected top quark kinematics, the log likelihood, and the event probability distributions of the leading permutation (ranked by event probability) in the electron muon with≥2b-tags. All plots except for the log likelihood are shown after the log LH>−48cut. The displayed uncertainties represent the Monte Carlo statistical uncertainty as well as the background normalisation uncertainties.
6.2. Event Reconstruction
level are uniquely matched to the reconstructed objects to obtain a continuous function describing the relative energy (momentum in case of muons) difference between these two stages as a function of the parton-level (truth-level) energy (momentum). In order to derive the transfer functions, only the reconstructed objects are used that match one-to-one to partons from the hard process. An object is considered matched when the distance ∆R between the reconstructed and truth object is less than 0.3.
The energy difference is fitted using a double Gaussian function of the form:
W(∆E) = 1
As the detector response changes in different regions of the detector, individual parametri-sations are derived for different regions of |η|. Also, the detector response changes with respect to each particle species. Hence, individual transfer functions are derived for elec-trons, muons2, light jets, b-jets, and ETmiss. As an example, Figure 6.10a shows the TF set for light jets in the central region of the detector.
The parameters pi depend on the energy of the parton/lepton and are defined as:
pi =ai+bi·Etruth for i= 1,3,5, (6.3)
pi =ai/p
Etruth+bi for i= 2,4, (6.4)
whereai’s and bi’s are obtained from a global fit for each particle species and η region.
For muons, allPi’s are parametrised linearly. For jets and electrons,p2,4 in Equation6.4 represent the calorimeter resolution and are hence parametrised as3∼1/√
E, while these parameters are considered to be linear for muons4. A linear dependence is assumed for all other parameters as shown in Equation 6.3.
Since the resolution of the ETmiss depends on the scalar sum of the deposited energy in the calorimeters in the transverse plane (P
ET) [152], the width of the difference (Ex,ymiss−pνx,y) is parametrised as a function ofP
ET as:
σ(X
ET) =p0+ p1
1 +e−p2(PET−p3). (6.5)
2In the case of muons, the transfer function is given in terms of thepTof the reconstructed and truth object.
3The calorimeter resolution in higher energies behaves as σEE ∼ √1
4The muon momentum resolution decreases linearly as the muonpETincreases (σppT
T ∼pT).
Figure6.10b represents the transfer functions for neutrinos as a function ofP ET.
(a) (b)
Figure 6.10.:(a) TFs set for light jets in the central |η| region, and (b) TFs set for neutrinos/ETmiss parametrised as a function of P
ET [85]. The plots corre-spond to TFs obtained at 7 TeV (8 TeV TFs are used in this analysis).
For the reason of simplicity, the TFs model the energy resolution of the objects for a fixed (or narrow bin)Etruth via double Gaussian functions.