The discriminant is a likelihood which takes four variables as input, and is trained to discriminate t¯t from W+jets based on MC simulation. The training of the likelihood discriminant and its evaluation is done using TMVA [128], which trains for signal divided by signal over background, S/S+B. The shape of W+jets is taken as representative of all of the background processes for the training process. It is expected to be the dominant source of background events, as can be seen in Table 7.1. For t¯t and W+jets the MC is split in half, where one half is used as a reference sample to train the likelihood and the other half is used in evaluation, that is, to create the template for the process to be used in the fit. The discriminant is evaluated for all other background processes and data as well.
The W+jets sample includes all of the various flavor contributions, scaled as discussed in Section 5.1. The analysis is flavor-sensitive due to the inclusion of continuous b-tagging as an input to the discriminant likelihood. The variables which are used as input are described here, and the discriminant is shown before the fit.
Lepton Pseudorapidity
Lepton pseudorapidity (η) is a simple physical quantity which is quite useful in discriminating t¯tfromW+jets, due to the differing production mechanisms of the leptonically decayingW. In
µ+jets Channel 1 Jet 2 Jet 3 Jet 4 Jet ≥5 jet
tt¯ 20±9 85±22 165±23 156±18 109±27
W+jets 18000±2100 4300±1000 980±410 220±140 59±38 Fakes (QCD multijet) 316±95 180±54 79±24 19±6 11±3
Single Top 58±10 64±11 31±7 11±4 4±2
Z+jets 700±140 210±50 58±26 14±10 5±4
Diboson (W W,W Z,ZZ) 67±10 56±9 16±4 3±2 0.6±0.8 Total Predicted 19200±2380 4900±1140 1320±500 420±180 187±74
Data Observed 20076 5039 1289 436 190
e+jets Channel 1 Jet 2 Jet 3 Jet 4 Jet ≥5 jet
tt¯ 15±5 63±12 117±17 109±15 76±19
W+jets 8500±1100 2160±500 520±220 124±77 35±23 Fakes (QCD,γ+jets) 410±200 160±81 64±32 12±6 8±4
Single Top 36±7 42±8 21±5 7±3 3±2
Z+jets 166±38 147±43 60±28 21±15 8±6.0
Diboson (W W,W Z,ZZ) 36±6 29±6 9±3 2±1.5 0.4±0.6 Total Predicted 9140±1350 2600±650 800±300 275±117 129±55
Data Observed 9849 2568 755 261 123
Table 7.1: Selected events in both data and from expectations in theµ+jets (top) ande+jets (bottom) channels. The uncertainties shown are those which affect rate the most, namely limited MC statistics, theoretical cross sections, JES, luminosity. For t¯t, ISR/FSR are included as well.
Observed event counts agree with predictions within uncertainty.
a W+jets event the lepton is produced rather homogeneously in η in the detector frame, while it tends to be central in at¯tevent.
For a muon,η of the object is taken: ηℓ =η. For electrons it is not quite so simple due to the discontinuity of acceptance caused by the calorimeter crack region. In the electron case, the η of the calorimeter cluster associated to the electron is used, yielding a clean distribution with precisely no events in the gap region (using the reconstructed electron’s η, which is actually the η of the track associated with the electron, would give a small tail into the vetoed region caused by events whose track are within theη range of the calorimeter crack but whose cluster is actually outside of the gap). In order to make this distribution continuous, η is transformed in the electron channel as:
ηℓ=
η for |η|<1.37 η−0.15 for 1.52< η <2.47 η+ 0.15 for −2.47< η <−1.52.
Normalized Transverse Energy, HT,3p
It is a rough but true statement that tt¯events are on average more energetic than W+jets events, due to the fact that the threshold for production of tt¯is higher than that of W+jets (2mtop ≫ MW ), giving the Ws (and their decay constituents) less transverse momentum on average in aW+jets event than in at¯tevent. This combined with the rapidly increasing strong
7.2 The Input Distribution
coupling strength at low energies means that the additional jets due to QCD jet production in association with the W will tend to be soft, indeed far softer on average than the jets in a tt¯ event.
One can construct a class of variables known as HT which are a measure the transverse momentum of the event as a whole. It has been found by testing that the transverse momentum of third and fourth jets yield a particularly high separation power; their sum therefore forms the numerator of the variable used. In the 3-jet bin it is only the third jet, and in the lower jet bins the variable is not defined. Higher jets were left out in order to control pileup and large radiation effects, because the 5-jet bin is inclusive and could have many more jets present. The denominator is the sum of the absolute value of the longitudinal momentum, pz, of all objects in the event, including the neutrino, which is calculated using the event kinematics and solved for by taking the smaller neutrino solution. This variable, calledHT,3p, is defined as:
HT,3p =
PNjets≤4 i=3 |p2T,i| PNobjects
j=1 |pz,j|.
This definition as such yields a particularly clumped distribution, so it is transformed as HT,3p →exp(−4×HT,3p) to obtain a smoother input distribution for the likelihood discrimi-nant.
Aplanarity
The aplanarity A is a measure of the topography of the event. A is defined as 1.5 times the smallest eigenvalue of the momentum tensor, which is defined as:
Mij =
PNobjects
k=1 pikpjk PNobjects
k=1 p2k ,
where pik is the i-th momentum component of object k and pk is the absolute value of its momentum. All jets in the event and the charged lepton are considered in the definition. The result is a clumped distribution which is smoothed by transformingA→exp(−8×A).
Mean of Two Highest JetProb Weights
The output of the JetProb algorithm, discussed in Section 4.4, is used as an input to the likelihood. Based on the knowledge that a t¯t event should contain two b-jets, a variable is constructed which is the mean of the b-tagging algorithm’s value for two most b-like jets. This variable, wJP, shows a strong discriminating power due to the fact that W+jets events are dominated by light jet production. Nonetheless, the uncertainty on fWHF is large enough that this becomes a sizable uncertainty in the analysis. Furthermore the uncertainty in the calibration of the algorithm, a process described in Section 4.4, becomes a large systematic uncertainty.
After the calibration procedure described in Section 4.4 was applied, a remnant discrepancy in the background dominated events in the light part of the spectrum – below the lowest calibrated working point – was observed. While unfortunate this is not particularly surprising, as the algorithm is only very roughly calibrated in this regime. This is handled in two ways in the analysis. In the distribution, all points below the lowest working point are lumped together into a single bin, averaging out the coarse calibration. The problem is solved in the bins which have a high signal to background ratio (4 and 5 jets). In the background dominated 3-jet bin it is thought that the mismodeling is too dramatic to be trustworthy. Accordingly the variablewJP
is not used in the 3-jet bin but is still used in the others.
The Input Likelihood
The four variables (three in the 3-jet bin, withoutwJP) are combined into a likelihood discrim-inant for each of the six physics channels: e+jets and µ+jets, which are further split into 3-jet exclusive, 4-jet exclusive, and 5-jet inclusive events. These six physics channels are treated on equal footing in the fit, with 20 bins in each histogram, yielding a total 120-bin fit. In addition to being necessary following the theoretical considerations in the introduction, for similar reasons this serves to mitigate systematic uncertainties by absorbing signal events which migrate away from the 4-jet bin. Furthermore, the 3-jet bins serve to constrain the background contribution.
The expected separation between W+jets and tt¯can be seen in each channel in Figure 7.1.
One can see that the 3-jet bin in both channels suffers from a significantly lower discriminating power due to omitting wJP in the likelihood discriminant. A clean separation between signal and background is achieved in particular in the 4-jet bins and the µ+5 jet bin. This is not the case ine+5 jets due to the significantly reduced background fraction in the channel, which comes from the harsher cuts to control the fake rate, reducing an already small background in the 5-jet bin to be even smaller in the electron channel. The sum of all prediction contributions is compared to the data before the fit in Figure 7.2.
Validation
The variables have been chosen based on expected discriminant power, with potential sensitivity to systematic uncertainties taken into account in the decision as well. It is essential to validate the choices to ensure that the variables are well-modeled, in particular in the background dominated regions. The set of variables used in the analysis is shown both in the two-jet control region and in the signal region for both channels. There is generally a good agreement between MC predictions and measured data. The data-MC agreement is shown for the variables of interest in Figures 7.7-7.14 at the end of this chapter. No discrepancies beyond uncertainties are observed.