8. Statistical Analysis and Results 109
8.2. Systematic Uncertainties
8.2.1. Detector Systematics
The luminosity estimate has an uncertainty of 2.8%, determined using beam-separation scans, also known as van der Meer scans, where the absolute luminosity can be inferred from direct measurements of the beam parameters [199, 200]. This systematic uncertainty is applied to all contributions obtained from MC simulation.
Uncertainties on Physics Objects
Leptons Uncertainties associated with the lepton selection arise from the trigger, reconstruc-tion, identificareconstruc-tion, isolation and lepton momentum scale and resolution. In total, uncertainties associated with electrons (muons) include 5 (6) components.
The uncertainty on the reconstruction, identification and trigger efficiency of electrons and muons are derived by applying the tag-and-probe techniques onZ →`+`− (`=e, µ) events, as already introduced in Section 4.2 and 4.3.
The accuracy of the lepton momentum scale and resolution in simulation is checked using reconstructed distributions of the Z →`+`− and J/ψ→`+`− masses. In the case of electrons, E/p studies, using the combined measurement of track momentum in the inner detector p and the energy in the calorimeterE, are also used, profiting from the large statisticsW →eνsample.
In the case of muons, uncertainties on both the momentum scale and resolutions in the muon spectrometer and the tracking system are considered, and varied separately.
Jet Energy Scale The jet energy scale (JES) uncertainty is split into 22 uncorrelated sources which model the pT and η dependencies of the JES uncertainty [126] and are treated indepen-dently:
• In situ calibration techniques: the residual in situ corrections outlined in Section 4.4.1 exploit the transverse momentum balance between the jet and a well-measured reference object, and is defined as:
R(pjet
T , η) = (pjet
T /pref
T )data (pjet
T /pref
T )MC
. (8.9)
Jets in the central region are calibrated using photons or Z-bosons (Z →e+e−) as refer-ence objects up to a transverse momentum of 800 GeV. For higher pT jets, a calibration using a system of low-pT jets recoiling against a high-pT jet is used. The corresponding uncertainties on the jet response ratio R for each in situ technique in the central region as a function of the jet pT can be seen in Figure 8.2, showing the jet pT range that each technique covers. The different components of this uncertainty are split depending on their source and correlated into the categories: detector description (Det), statistics/method (Stat), physics modelling (Model), and mixed detector and modelling (Mixed). From the initial 54 in situ components, a reduced set of 12 is obtained while retaining the infor-mation on the correlations with a “diagonalisation and reduction” method [126]. In each category, the components are ordered by their effect, beginning with the largest.
The calibration of jets in the forwardη region of the detector relative to jets in the central η region, denoted as η-intercalibration, exploits the transverse momentum balance in dijet events with a well calibrated jet in the central region and a jet in the forward region.
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8.2. Systematic Uncertainties
In this way, the dependence of the detector response to jets within 0.8 ≤ |η| < 4.5 is removed, by equalising it with the one for jets within |η| < 0.8. The uncertainties are divided into a statistical component and a MC modelling component, the latter being the dominant one in the forwardη region.
• Single Particle Response: the uncertainty in the calorimeter response to jets can be obtained from the response uncertainty in the individual particles constituting the jet.
In situ measurements of the single hadron response in pp collisions and test-beam data significantly reduce the uncertainty due to a limited knowledge of the detector geometry, such as the presence of dead material, and the modelling of the interaction of particles with the detector [201].
• Pile-up correction: uncertainties on the pile-up correction to the jet energy are obtained from the comparison of in situ measurements of the slopesα=∂pT/∂NP V andβ =∂pT/∂µ with the corresponding simulation and the results based on a MC simulation only approach, whereNP V is the number of reconstructed primary vertices, a measure of the in-time pile-up, and µis the average number of pile-up interactions per bunch crossing, a measure of the out-of-time pile-up. Hence, the uncertainty covers possible systematic biases due to mismodelling of the effect of pile-up on simulated jets.
• Flavour related: the calorimeter response is different for different jet flavour types:
gluon, or light jets. This can be attributed to the difference in the fragmentation and showering properties of the jet origin, or flavour. The derived uncertainties on the flavour composition and flavour response are dependent on the topology.
• Jets with heavy-flavour content: the main observable, rtrk, used to study jets from b-quarks, is defined as:
rtrk = |P−→ptrackT |
pjetT , (8.10)
where−→ptrackT is the sum of the transverse momentum vectors from all tracks in the jet cone, andpjetT is the calorimeter jet transverse momentum. Comparisons between data and MC simulations show agreement within systematic uncertainties of approximately 3%, with a weak dependence on the jet pT [126].
Figure 8.3 shows the fractional contribution of each component to the total JES uncertainty from the 2012 dataset. As expected, the uncertainty on the pile-up correction dominates at low jet pT, whereas it is negligible at higher jet pT. At high jet pT, the uncertainty is driven by the in situ JES uncertainty. The uncertainty in the forward η region is dominated by the contribution of the in situη-intercalibration, while in the central region, the uncertainty on the flavour response of jets originating from quarks or gluons dominates.
Jet Vertex Fraction As mentioned in Section 4.4.1, the JVF uncertainty is evaluated and propagated to the analysis by varying the nominal JVF cut value up and down to cover the discrepancies in JVF efficiency between data and simulation. Figure 8.4 shows the relative variations of the JVF uncertainty, ±1σ, with respect to the nominal JVF cut, for the Z+jets (top) andt¯t(bottom) background processes in the 2`OSZ and 2`OSZveto fit regions, respectively.
By construction, the JVF uncertainty is not expected to be symmetric, and shows significantly larger variations in the lowHThad region, corresponding to lowpT jets. The overall normalisation variation per fit region goes up to 8% in the high jet multiplicity regions, (≥5j,1b + 2b) and (≥5j,2b).
8. Statistical Analysis and Results
[GeV]
jet
pT
20 30 40 102 2×102 103
MC / Response DataResponse
0.9
Total in situ uncertainty Statistical component
Figure 8.2.: Jet response ratio of data to the MC simulation as a function ofpT for three in situ techniques, which are combined to determine the in situ energy scale correction:
Z+jet (squares), γ+jet (full triangles) and multijet (empty triangles). The error bars indicate the statistical and the total uncertainties. The results are shown for LCW+JES calibrated jets. The dark line shows the combination of the in situ techniques, with the total in situ uncertainty band (green) and the statistical uncertainty fraction band (orange) [202].
[GeV]
Flav. composition, inclusive jets Flav. response, inclusive jets Pileup, average 2012 conditions
η
pT Total uncertainty
JES
Flav. composition, inclusive jets Flav. response, inclusive jets Pileup, average 2012 conditions
Figure 8.3.: Fractional jet energy scale systematic uncertainty components as a function of (left): pT for jets at|η|= 0.0, and (right): η for jets withpT = 40 GeV, using the LCW+JES calibration scheme. The total uncertainty (all components summed in quadrature) is shown as a filled blue region topped by a solid black line. Average 2012 pile-up conditions were used, and topology dependent components (flavour response and composition) were taken from inclusive dijet samples [202].
Jet Energy Resolution The jet energy resolution is measured using the in situ technique called the bisector method [203]. The method is based on a transverse balance vector −→
PT, defined as the sum of the momenta of the two leading jets in dijet events,−→pT,1 and−→pT ,2. This vector is projected along an orthogonal coordinate system in the transverse plane, (Ψ, η), where η is chosen in the direction that bisects the angle formed by −→pT ,1 and −→pT ,2. Different sources can cause fluctuations from the perfectly balanced dijet event,−→
PT = 0, giving rise to a non-zero variance on its Ψ and η components.
Figure 8.5 shows the fractional jet energy resolution as a function of the average jet pT in the centralηregion for the bisector technique. The jet energy resolution in MC agrees in general with
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8.2. Systematic Uncertainties
[GeV]
had HT
100 200 300 400 500 600 700
Relative difference (%)
100 200 300 400 500 600 700
Relative difference (%)
Figure 8.4.: Relative differences of the JVF uncertainty on the Z+jets discriminants in the three fit regions in the 2`OSZ region, (top left): (3j,2b), (top middle): (4j,2b), and (top right): (≥ 5j,2b), and on the t¯t discriminants in the two fit regions using shape information in the 2`OSZveto region, (bottom left): (4j,1b + 2b) and (bottom right): (≥5j,1b + 2b).
the one measured in data, except for some small differences in somepT andηregions. Therefore, instead of applying a smearing to the nominal measurement, a systematic uncertainty is defined as the difference in quadrature between the jet energy resolutions for data and simulation. To estimate the impact of this systematic uncertainty on this analysis, the energy of each jet in the simulation is smeared by this residual difference, and the changes in the normalisation and shape of the final discriminant are compared to the default prediction. Since jets in the simulation cannot be “under-smeared”, the resulting uncertainty is one-sided by definition, i.e. only the
“up” variation exists, and then the effect of the systematic uncertainty on the discriminant is symmetrised.
Dijet Balance: Monte Carlo (PYTHIA) Dijet Balance: Data
Bisector: Monte Carlo (PYTHIA) Bisector: Data
Figure 8.5.: Fractional jet energy resolution as a function of the average jet transverse mo-menta measured with and bisector (circles) techniques using the LCW+JES cal-ibration in MC simulation and 2011 data. The bottom plot shows the relative difference between data results and MC simulation (green) [204].
8. Statistical Analysis and Results
Jet Reconstruction The jet reconstruction efficiency is found to be about 0.2% lower in the simulation than in data for jets below 30 GeV, and consistent between data and MC simulation for higher jet pT. To evaluate the effect of this inefficiency, 0.2% of the jets with pT below 30 GeV are removed randomly and all jet-related kinematic variables are recomputed, and the event selection is updated accordingly. The effect of this systematic uncertainty is also symmetrised as in the jet energy resolution.
Heavy- and Light-Flavour Tagging The uncertainties on the b-tagging efficiency of jets originating fromb,c and light quarks or gluons, which are measured for the different jet types, as explained in Section 4.4.1, are split into sub-components using theeigenvector method. This method uses a diagonalisation procedure, preserving the correlation among jetpT bins from the measured SFs and providing components that can be treated as uncorrelated. There are as many independent systematic uncertainty components aspT bins used in the efficiency measurement:
six components for eachb-jet andc-jet efficiency (for the six jetpT bins), and twelve components for the mistag rate, since the efficiency is parametrised in six bins in jetpT, and in twoη regions.
In each jet type category, the components are labelled with an index to order them according to their absolute eigenvalue obtained from the diagonalisation procedure, beginning with the smallest effect.
Missing Transverse Energy Given that an explicit cut on theEmissT distribution is applied in the 2`OSZveto region, systematic uncertainties on the calculation of the ETmiss are included in the fit, in particular uncertainties on the scale and resolution of the soft terms (soft-jets and cell-out), introduced in Section 4.5.