Jets

Partons from the hard-scattering process or radiated quarks hadronize according to the QCD theory, since free partons cannot exist due to the color confinement. However, the top quark has a lifetime (τt≈ (0.5 × 10⁻²⁴) s [11]) smaller than the typical QCD hadronization timescale.

Due to their relatively short lifetime, top quarks decay before they hadronize into bound states of subatomic particles. In this analysis, the dynamics of parton hadronization are modeled with thePYTHIA [63] generator based on the Lund string model. As a result of the hadronization process, many stable particles like charged and neutral hadrons are produced. The mapping of multiple stable particles, which result from the hadronization process, to the original parton is subject to jet-clustering algorithms. Jet-clustering can be understood as a matching of exper-imental observations to theory predictions that are formulated on parton level. Typically, the direction and energy of reconstructed jets is related to the original (fragmenting) parton. How-ever, non-linear responses of the calorimeters, additional interactions, and noise effects require a jet-calibration of reconstructed jets.

Approximately 65% of the jet energy is carried by charged hadrons, 25% by photons, and 10% by neutral hadrons [160, 161]. Since the PF-event reconstruction is able to distinguish be-tween these types of particles, an improved jet reconstruction is possible. The charged-hadron momentum resolution is greatly improved when combining HCAL calorimeter deposits with the tracker information. Photons can be separated from charged-hadron-energy deposits such that also jet-energy reconstruction profits from the high granularity and energy resolution of the ECAL. Thus, about 90% of the typical jet-energy deposits are reconstructed with improved resolution when using the PF algorithm, leading to smaller particle-level-correction coefficients and jet-energy-scale uncertainties than conventional jet-reconstruction algorithms.

Jet-clustering algorithm The anti-kt algorithm [170] with a distance parameter of R = 0.5 is used as the jet-clustering algorithm. The algorithm runs on the list of stable, elementary particles which is obtained by the PF algorithm.

The following definition of the anti-kt algorithm is given in ref. [170]. Constituents which have the smallest distance di,j among each other are consecutively recombined. Those con-stituents i and j include particles as well as pseudo-jets, and jet-clustering of a particular jet continues until di,B is the smallest distance, whereas di,B is the distance between reconstructed jet i and the beam B. If a jet is found, particles related to that jet are removed from the list and the clustering algorithm continues. The distance measure di,jscales the transverse momentum relative to the geometrical distance and is defined as

di,j = min(k_t,i^2p, k_t,j^2p)∆²_i,j

R² (3.12)

and di,B = k^2p_t,i, (3.13)

in which ∆²_i,j = (y_i− yj)²+ (φ_i− φj)²with rapidity yi, azimuth φi, and transverse momentum k^2p_t,i of particle i. For the anti-ktalgorithm, p =−1 is chosen as the distance parameter.

The anti-kt-clustering algorithm provides infrared- and collinear-safe jets [170], which means that the number of hard jets is insensitive w.r.t. the addition of new soft particles or collinear

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splittings [171]. Moreover, the resulting jet boundaries are robust against soft-radiation effects, i.e. jets reconstructed with the anti-kt algorithm usually have a regular shape with a circular cone with radius R if they are not too soft [170].

A potential ambiguity in the event reconstruction arises if electrons or muons from subse-quent hadron decays are used once within the jet clustering and once again reconstructed as charged leptons. This ambiguity is resolved by using a relIso criterion during the PF-event reconstruction. Well-isolated charged leptons are excluded from the input-particle list of the jet-clustering algorithm and reconstructed as prompt, charged leptons, and vice versa. Recon-structed τ -lepton candidates and photons are clustered always into jets in this analysis.

Furthermore, charged hadrons are subtracted from the input-particle list of the jet-clustering algorithm if their tracks are not compatible with the primary vertex of the event. The vertex-compatibility criterion aims at suppressing deposits from additional interactions (Pile-Up) that cannot be related to partons from the hard-scattering process. Remaining energy deposits from neutral hadrons are subtracted during jet-energy calibration.

Reconstructed Jets

Detector Level

-Pile-Up and Noise Correction

Simulation Truth Calibration

Relative Correction

Residual

Absolute Correction

Residual

Calibrated Jets

Particle Level

-Simulation Data

Figure 3.10: Sequence of jet-energy corrections that are applied to relate the energy of recon-structed detector jets to corresponding particle jets. First, jet-energy corrections are derived from simulated events. Second, residual corrections are derived from data and are applied on top of the previous corrections.

Jet-Energy-Calibration Strategy On detector level, jet reconstruction uses combined informa-tion of calorimeter and tracker with the PF algorithm (sec. 3.3.2). The jet-clustering algorithm is applied to the list of stable, elementary particles, and a detector-jet-four-momentum vector P_jet^rawis obtained. On particle level (or generator level), jets are clusters of stable particles which stem from the fragmentation process of a parton. Their four-momentum is referred to as P_jet^true in the following. Technically, the same jet algorithm is used for jet-clustering on both detector and particle level. However, the energy of a detector jet cannot be directly related to the jet en-ergy on particle level due to non-linear responses of the calorimeters, additional interactions, and noise effects. Jet-energy calibration is needed to translate, on average, the jet reconstructed on detector level to a particle-level jet. The jet-energy calibration is often also referred to as

“jet-energy scale”.

The CMS experiment uses a multiplicative approach to account for jet-energy calibration, P_jet^true = C(p^raw_T , η)· Pjet^raw, (3.14) in which C(p^raw_T , η) refers to the pT- and η-dependent corrections, which are applied to ev-ery component of the four-momentum [71]. The jet-energy corrections C itself are factorized further into four sub-components,

C(p^raw_T , η) = C_Offset(p^raw_T )· CSimulation truth(p⁰_T, η)· Crelative(η)· Cabsolute(p⁰⁰_T), (3.15) namely the Pile-Up-and-noise-offset correction, the simulation-truth calibration, the η-dependent residual relative correction, and the residual pT-dependent absolute correction [71].

While all four corrections are sequentially applied to reconstructed jet four-momenta in data, only the former two corrections are applied to reconstructed jets in simulation.

The offset correction removes energy from Pile-Up contributions, electronics noise, and un-derlying event. The offset due to deposits from additional interactions is corrected for with the jet-area method [71]. The offset linearly depends on the number of reconstructed vertices and typically is of the order of a few GeV/c [172]. The offset from electronics noise is approximately 250 MeV/c [71].

The simulation-truth calibration corrects the reconstructed detector-jets back to particle-level jets using information from simulated events. An average correction is derived in QCD-multijet events, which are simulated with thePYTHIA generator [71]. Here, detector-level jets are matched to particle-jets within a ∆R<0.25 cone, and the average response is expressed as a function of pTand η. The calibration factor can be quite large with up toO(20%) [71] in the tran-sition region between barrel and end-cap, while being much less elsewhere. The parton-flavor composition of low-pTjets from the QCD-multijet sample is dominated by jets from gluon frag-mentation. Jets from fragmenting quarks usually have higher transverse momenta. The energy response is expected to be dependent on the flavor of the fragmenting parton due to diversified particle-multiplicity patterns, as well as different energy spectra. A small flavor dependence is indeed observed at a level of O(3%) for jets with pT > 10 GeV/c within the barrel region (|η| < 1.3). The response significantly profits from the precise charged-particle-momentum resolution of the PF algorithm. Simulations withPYTHIA show that the response among dif-ferent jet-flavors is enveloped by the response of light jets and gluon jets [71, 172]. Jets from c-or b partons lie in between those two extremes. Furthermc-ore, a comparison between PYTHIA

and HERWIG++ simulations shows that the light-jet (u, d, s partons) and gluon-jet responses marginally depend on the fragmentation model. The difference in responses isO(1%) for a jet with pT> 30 GeV/c within the central region (|η| < 1.3) of the detector (cf. [71]).

However, small differences are found when comparing the simulated jet response with the jet response measured in data. Thus, empirical corrections, the so-called “residual jet-energy corrections”, are applied in addition to the jet-energy corrections obtained from simulation.

These corrections are applied to jets in data only. Residual jet-energy corrections (cf. [71]) are derived in measurements with data corresponding to an integrated luminosity of 4.7 fb⁻¹ at

√s = 7 TeV, and applied to reconstructed jets which are used in this analysis. These corrections include an η-dependent relative correction and a pT-dependent absolute correction.

The η-dependent correction is derived in dijet events. Here, the conservation of transverse momentum is used. The relative response of a (probe) jet at an arbitrary pseudo-rapidity η w.r.t.

a jet within the central region of the detector (|η| < 1.3) is measured in events in which both jets are back-to-back in azimuth φ [71]. The η-dependent correction mostly affects the transition region between barrel and end-cap with scale factors up toO(15%) [71].

The pT-dependent absolute correction is determined in γ-plus-jet events and Z-plus-jet events with leptonically decaying Z bosons, since their pTresponse is precisely known from the ECAL, tracker, or muon subdetectors [71]. Both processes provide complementary information since they cover varied pT ranges, use different subdetectors with diversified resolution pat-terns, and have different production cross sections, which means varied trigger requirements and data-taking periods. A pTbalancing between the jet and the γ or Z boson is used, in which a central (|η| < 1.3) jet is required. Isolation criteria are used to reduce effects due to initial- and final-state radiation. A good agreement between simulation and data is observed, resulting in a small pT-dependent absolute correction of approximately 1%.

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Jet-Energy-Scale Uncertainties The total uncertainties of the jet-energy calibration (fig. 3.11) typically are of O(3%) for PF jets with pT = 30 GeV/c, and O(1%) for PF jets with pT = 100 GeV/c [172]. The total uncertainties are much larger (up to 5%) in the barrel-endcap-transition region due to an observed instability of the derived jet-energy corrections with in-creasing run numbers (“time stability”). This instability is expected to be caused by radiation damage to the HF and transparency loss of the ECAL crystals that is not yet corrected for. The most important uncertainty at low pTis the uncertainty of the Pile-Up correction. At medium-jet-pT range, the most important uncertainty is due to an altered response of quark and gluon jets in different fragmentation models, which estimated by comparingPYTHIAand HERWIG++

simulations. At high jet-pT, data statistics are limited. An extrapolation of the single-particle response and fragmentation modeling is done by combining information from simulation and data [71]. The uncertainty due to this extrapolation is the most important contribution for jets with large transverse momenta. In conclusion, the uncertainties of the jet-energy calibration are rather small when PF particles are used as input to the jet-reconstruction algorithm.

Figure 3.11: Uncertainties on jet-energy calibration expressed as a function of the jet pTfor jets at η = 0 (left) and as a function of jet η for jets at pT = 100 GeV/c (right). Both figures are from ref. [172].

Correction of Jet-Transverse-Momentum Resolution The resolution of transverse momenta of jets is found to be lower in data than in simulated events. The conservation of transverse momenta is utilized in dijet and γ-plus-jet events [71] to derive scale factors that correct the jet-pTresolution in simulated events. The transverse-momentum resolution of each jet is corrected for by scaling the reconstructed jet pT with the difference between reconstructed jet pT and matched generator-jet pT, where the difference is multiplied with a certain correction factor.

The resolution correction depends on the pseudo-rapidity η of the jet, and is determined by the CMS collaboration (based on the methodology in ref. [71]) to

• 1.05 ± 0.06 for jets within |η| ≤ 0.5 ,

• 1.06 ± 0.06 for jets within 0.5 < |η| ≤ 1.1,

• 1.10 ± 0.06 for jets within 1.1 < |η| ≤ 1.7,

• 1.13 ± 0.10 for jets within 1.7 < |η| ≤ 2.3,

• and 1.29 ± 0.20 for jets within |η| > 2.3.