Measurement of the differential cross-section of highly boosted top quarks as a function of their transverse momentum ins=8  TeVproton-proton collisions using the ATLAS detector

The differential cross-section for pair production of top quarks with high transverse momentum is measured in 20.3 fb$^{-1}$ of proton-proton collisions at a center-of-mass energy of 8 TeV. The measurement is performed for $t\bar{t}$ events in the lepton+jets channel. The cross-section is reported as a function of the hadronically decaying top quark transverse momentum for values above 300 GeV. The hadronically decaying top quark is reconstructed as an anti-$k_t$ jet with radius parameter $R=1.0$ and identified with jet substructure techniques. The observed yield is corrected for detector effects to obtain a cross-section at particle level in a fiducial region close to the event selection. A parton-level cross-section extrapolated to the full phase space is also reported for top quarks with transverse momentum above 300 GeV. The predictions of a majority of next-to-leading-order and leading-order matrix-element Monte Carlo generators are found to agree with the measured cross-sections.


Introduction
The large number of top-antitop quark (tt) pairs produced at the LHC provide a unique opportunity to improve our understanding of tt production and test the Standard Model (SM) at the TeV scale. New phenomena beyond the Standard Model may distort the top quark transverse momentum (p T ) spectrum, in particular at high p T (see, e.g., Refs. [1, 2]), and could thus be revealed by a precise measurement. Moreover, due to their high cross-section at the LHC and rich experimental signature, tt events constitute a dominant background to a wide range of searches for new massive particles. A better understanding of the production of high-momentum top quarks, including a more precise determination of the parton distribution functions (PDF) of the proton, would be of great benefit to the broader LHC program.
The initial measurements of tt production at the LHC have focused on a determination of the inclusive production cross-section. Now that the experimental uncertainties on these measurements (see, e.g., Refs. [3][4][5]) are comparable to or lower than the uncertainties on the next-to-next-to-leading-order plus next-to-next-to-leading-logarithmic order (NNLO+NLLL) theory prediction [6][7][8][9][10][11], the interest in differential top quark cross-section measurements has gained traction. Measurements of the differential cross-section as a function of the kinematics of the top quark, or the top-antitop quark pair, have been performed by the ATLAS [12][13][14] and CMS collaborations [15,16], where the highest measured top quark p T range is 350-800 GeV [13].
In this paper a measurement using techniques specifically designed to deal with the collimated decay topology of highly boosted top quarks is presented. In particular, the hadronic top quark decay is reconstructed as a single large-radius (large-R) jet. The selection and reconstruction are based on an algorithm developed [17] and used in tt resonance searches [18][19][20][21] that increases the tt selection efficiency at high top quark p T and extends the kinematic reach into the TeV range. This analysis utilizes the lepton+jets channel where one W boson decays hadronically and the other leptonically to an electron or a muon, assuming each top quark decays to a W boson and a b-quark. The cross-section is measured as a function of the hadronically decaying top quark p T . A particle-level cross-section is measured in a kinematic region close to the detector-level selection, referred to in the following as fiducial region. A parton-level differential cross-section is also reported as a function of the hadronically decaying top quark p T , by further extrapolating to the full kinematic phase space except for a lower limit on top quark p T of 300 GeV. The measured cross-sections are compared to the predictions of several MC generators and PDF sets.
The object definition, event selection, and background determination used in this analysis follow closely the ones used in the search for tt resonances [20]. More details of these aspects of the measurement can be found in the corresponding reference.

The ATLAS detector
ATLAS is a multipurpose detector [22] that provides nearly full solid angle 1 coverage around the interaction point. Charged-particle trajectories are reconstructed by the inner detector, which covers pseudorapidity |η| < 2.5 and is composed of a silicon pixel detector, a silicon microstrip detector, and a transition radiation tracker (TRT). The inner detector is surrounded by a solenoid that provides a 2 T magnetic field. Sampling calorimeters with several different designs span the pseudorapidity range up to |η| = 4.9. High-granularity liquid-argon (LAr) electromagnetic (EM) calorimeters are used up to |η| = 3.2. Hadronic calorimetry based on scintillator-tile active material covers |η| < 1.7 while LAr technology is utilized for hadronic calorimetry from |η| = 1.5 to |η| = 4.9. The calorimeters are surrounded by a muon spectrometer. A magnetic field in the spectrometer is provided by air-core toroid magnets. Three layers of precision gas chambers track muons up to |η| = 2.7 and muon trigger chambers cover |η| < 2.4.

Data and Monte Carlo samples
The cross-section is measured using data from the 2012 LHC pp run at √ s = 8 TeV, which corresponds to an integrated luminosity of 20.3 ± 0.6 fb −1 . The luminosity was measured using techniques similar to those described in Ref.
[23] with a calibration of the luminosity scale derived from beam-overlap scans performed in November 2012. The average number of pp interactions per bunch crossing (pileup) in 2012 was around 21. The sample was collected using the logical OR of two single-electron triggers with transverse momentum thresholds of 60 GeV, lowered to 24 GeV in the case of isolated electrons, and two single-muon triggers with transverse momentum thresholds of 36 GeV, lowered to 24 GeV in the case of isolated muons.
Samples of Monte Carlo (MC) simulated events are used to characterize the detector response and efficiency to reconstruct tt events, estimate systematic uncertainties, predict the background contributions from various physics processes, and to compare the theoretical predictions with the measurement. The simulated events are weighted such that the distribution of the average number of pp interactions per bunch crossing agrees with data. The samples were processed through the GEANT4 [24] simulation of the ATLAS detector [25]. For the evaluation of some systematic uncertainties, generated samples are passed to a fast simulation using a parameterization of the performance of the ATLAS electromagnetic and hadronic calorimeters [26]. Simulated events are reconstructed using the same algorithms that are applied to the data.
The nominal signal tt sample is generated using the Powheg (Powheg-hvq patch4) [27] method, as implemented in the Powheg-Box generator [28], which is based on next-to-leading-order (NLO) QCD matrix elements. The h damp parameter, which effectively regulates the high-p T radiation in Powheg, is set to the top quark mass. The CT10 [29] PDF are employed and the top quark mass is set to m top = 172.5 GeV. Parton showering and hadronization are simulated with Pythia v6.425 [30] using the Perugia 2011 C set of tuned parameters (tune) [31] and the corresponding leading-order (LO) CTEQ6L1 [32] PDF set. Unless otherwise noted, electroweak corrections extracted with Hathor 2.1alpha [33], implementing the theoretical calculations of Refs. [34][35][36], are applied as weights to the events of this sample. The prediction of Powheg is compared to that obtained with other generators such as MC@NLO v4.01 [37] with CT10 for the PDF set, interfaced to Herwig v6.520 [38] for parton showering and hadronization, Jimmy v4.31 [39] for the modeling of multiple parton scattering. In Herwig and Jimmy the CT10 PDF is used and the ATLAS AUET2 tune [40] is employed for the parton shower and hadronization settings. In addition, the LO multileg generator Alpgen v2.13 [41] interfaced to Herwig is used where up to four additional partons in the matrix element are produced; the MLM [42] matching scheme is employed to avoid double counting of configurations generated by both the parton shower and the matrix-element calculation; the CTEQ6L1 [32] PDF set is employed; heavy-flavor quarks are included in the matrix-element calculations to produce the tt + bb and tt + cc processes; the overlap between the heavy-flavor quarks produced from the matrix-element calculations and from the parton shower is removed. For the evaluation of systematic uncertainties due to the parton showering and hadronization models, a Powheg+Herwig sample is compared to a Powheg+Pythia sample. The uncertainties due to QCD initial-and final-state radiation (ISR and FSR) modeling are estimated with samples generated with AcerMC v3.8 [43], interfaced to Pythia for which the parton shower parameters are varied according to a measurement of the additional jet activity in tt events [44]. The tunes for samples used to describe tt production show a reasonable agreement over a broad range of observables and kinematic regions in tt events [45][46][47]. The electroweak corrections that are applied to the nominal Powheg+Pythia sample are not applied to the other samples. The tt samples are normalized to the NNLO+NNLL cross-section 2 [6-11]: σ tt = 253 +13 −15 pb. Leptonic decays of vector bosons produced in association with several high-p T jets, referred to as W+jets and Z+jets, constitute the largest background in this analysis. Samples of simulated W/Z+jets events with up to five additional partons in the LO matrix elements are produced with the Alpgen generator interfaced to Pythia for parton showering using the MLM matching scheme. Heavy-flavor quarks are included in the matrix-element calculations to produce the Wbb, Wcc, Wc, Zbb, and Zcc processes. The overlap between the heavy-flavor quarks produced by the matrix element and by parton showering is removed. W+jets samples are normalized to the inclusive W boson NNLO cross-section [49,50] and corrected by applying additional scale factors derived from data, as described in Sec. 5.
Single top quark production in the t-channel is simulated using the AcerMC generator, while production in the s-channel and the production of a top quark in association with a W boson are modeled with Powheg [51][52][53][54]. Both generators are interfaced with Pythia using the CTEQ6L1 PDF set and the Perugia 2011 tune for parton shower modeling. The cross-sections multiplied by the branching ratios for the leptonic W decay employed for these processes are 28.4 pb (t-channel) [55], 22.4 pb (Wt production) [56], and 1.8 pb (s-channel) [57], as obtained from NLO+NNLL calculations.
Diboson production is modeled using Sherpa [58] with the CT10 PDF set and the yields are normalized to the NLO cross-sections [59].

Object definition and event selection
Jets are reconstructed using the anti-k t algorithm [60] implemented in the FastJet package [61] with radius parameter R = 0.4 or R = 1.0, respectively called small-R and large-R jets in the following, using as input calibrated topological clusters [62][63][64]. These clusters are assumed to be massless when computing the jet four-vectors and substructure variables. Large-R jets containing hadronically decaying top quarks are selected by applying jet substructure requirements, which exploit the fact that they contain several high-p T objects and have a high mass, unlike most jets originating from the fragmentation of other quarks or gluons. The trimming algorithm [65] with parameters R sub = 0.3 and f cut = 0.05 is applied to large-R jets to mitigate the impact of initial-state radiation, underlying-event activity, and pileup. A correction for the number of additional pp interactions per bunch crossing is applied to small-R jets [66-69]. The p T of small-R jets and large-R trimmed jets and the large-R jet mass, obtained from the four-momentum sum of all jet constituents, are calibrated using energyand η-dependent correction factors. After this calibration, the p T and mass of the jets in simulated events correspond on average to the ones of the corresponding particle-level jets, which are built from the stable particles produced by the MC event generator [70,71]. Differences between the small-R jet response in data and MC simulation are evaluated from control samples and corresponding corrections are applied to data. Small-R jets are required to be in the fiducial region |η| < 2.5 and must have p T > 25 GeV. The jet vertex fraction (JVF) is a measure of the fraction of the jet's track momenta that originate from the primary vertex. It is computed as the summed p T of all tracks matched to the jet and the primary vertex, divided by the summed p T of all tracks matched to the jet. Small-R jets with p T < 50 GeV and |η| < 2.4 are rejected when JVF < 0.5, to reduce the contribution of jets generated by pileup interactions. 3 Trimmed large-R jets are considered for the analysis if |η| < 2.0 and p T > 300 GeV. More details on the reconstruction and performance of highly boosted top quarks in ATLAS can be found in Ref. [71,72].
Small-R jets containing a b-hadron are tagged using a neural-network-based algorithm (MV1) [73] that combines information from the track impact parameters, secondary vertex location, and decay topology inside the jets. The operating point corresponds to an overall 70% b-tagging efficiency in tt events, and to a probability to mistag light-flavor jets of approximately 1%.
Electron candidates are reconstructed as charged-particle tracks in the inner detector associated with energy deposits in the EM calorimeter. They must satisfy identification criteria based on the shower shape in the EM calorimeter, on track quality, and on the transition radiation observed in the TRT detector [74]. Electrons are required to be in the pseudorapidity region |η| < 2.47, excluding the transition region between the barrel and the endcap calorimeters (1.37 < |η| < 1.52). The EM clusters must have a transverse energy E T > 25 GeV. The associated track must have a longitudinal impact parameter |z 0 | < 2 mm with respect to the primary vertex, which is the vertex with the highest p 2 T of the associated tracks in the event.
Muon candidates are defined by matching track segments in the muon spectrometer with tracks in the inner detector. The track p T is determined through a global fit of the track that takes into account the energy loss in the calorimeters [75]. The track is required to have a longitudinal impact parameter |z 0 | < 2 mm, and a transverse impact parameter significance |d 0 /σ(d 0 )| < 3, indicating the track is consistent with originating from the hard-scattering vertex. Muons are required to have p T > 25 GeV and be in the fiducial region |η| < 2.5.
Lepton candidates are required to be isolated to suppress background leptons originating from jets. The variable "mini-isolation" [76] is used. It is defined as I mini = tracks p track T /p T , where p T is the lepton transverse momentum and the sum is over all good-quality tracks (excluding the lepton track) that have p T > 0.4 GeV and a distance from the lepton ∆R = (∆η) 2 + (∆φ) 2 < K T /p T . The parameter K T is set to 10 GeV and the isolation requirement I mini < 0.05 is applied for both the electrons and muons. An isolation cone that decreases in size with increasing p T improves the selection efficiency of the decay of high-p T top quarks.
Since leptons deposit energy in the calorimeters, an overlap removal procedure is applied in order to avoid double counting of leptons and small-R jets. In order to improve the reconstruction efficiency in the highly boosted topology, the same overlap removal procedure as used in Ref.
[20] has been adopted. First, jets close to electrons, with ∆R(e, jet R=0.4 ) < 0.4 are corrected by subtracting the electron four-vector from the jet four-vector and the JVF is recalculated after removing the electron track. The new e-subtracted jet is retained if it satisfies the jet selection criteria listed above, otherwise it is rejected. After this procedure, electrons that lie within ∆R(e, jet R=0.4 ) = 0.2 from a small-R jet are removed and their four-momentum added back to that of the jet. The muon-jet overlap removal procedure removes muons that fall inside a cone of size ∆R(µ, jet R=0.4 ) < 0.04 + 10 GeV/p T,µ around a small-R jet axis.
The missing transverse momentum E miss T is the magnitude of the vector sum of the transverse energy of all calorimeter cells [77]. Their energy is corrected on the basis of the associated physics object. The contribution of muons is added using their transverse momentum obtained from the tracking system and the muon spectrometer.
The event selection proceeds as follows. Each event must have a reconstructed primary vertex with five or more associated tracks with p T > 0.4 GeV. The events are required to contain exactly one reconstructed lepton candidate with p T > 25 GeV. The transverse mass of the lepton and E miss T is defined as m W T = 2p T E miss T (1 − cos ∆φ), where ∆φ is the azimuthal angle between the lepton and E miss T . Events are retained if E miss T > 20 GeV and E miss T + m W T > 60 GeV to suppress QCD multijet events.
The selection exploits the fact that the highly boosted top quark decay products tend to be collimated. Therefore events are selected by requiring the presence of at least one small-R jet close to the lepton (∆R( , jet R=0.4 ) < 1.5) and the existence of a reconstructed large-R trimmed jet with mass m jet > 100 GeV. To improve the rejection of background jets, originating from light quarks or gluons, a cut on the k t splitting scale [68, 69] of the large-R jets is made. The k t splitting scale is calculated by reclustering the large-R jet with the k t -clustering algorithm, and taking the k t distance between the two subjets of the final clustering step to be √ d 12 = min(p T1 , p T2 )∆R 12 , where p T1 and p T2 are the transverse momenta of the two subjets and ∆R 12 is the distance between them. It is expected to have large values for jets containing two hard subjets, as expected in the decay of massive objects. Events are selected if the large-R jet has √ d 12 > 40 GeV. The large-R jet must be well separated from the lepton (∆φ( , jet R=1.0 ) > 2.3) and from the small-R jet associated with the lepton (∆R(jet R=1.0 , jet R=0.4 ) > 1.5). The leading-p T trimmed large-R jet satisfying these requirements is referred to as the top-jet candidate. Finally, at least one of the two top quark candidates must be b-tagged. This implies that either the highest-p T small-R jet close to the lepton (∆R( , jet R=0.4 ) < 1.5) or at least one small-R jet close to the large-R jet (∆R(jet R=1.0 , jet R=0.
The event selection is summarized in Table 1. After these requirements the data sample contains 4145 and 3603 events in the electron channel and muon channel, respectively, of which ≈ 85% are expected to be semileptonic tt events.

Background estimation
After the event selection the background is composed primarily, in order of importance, of W+jets, tt dilepton, single top, and QCD multijet events. The W+jets background is obtained from MC simulation with normalization and heavy-flavor content adjusted in data control regions. The tt dilepton background is determined as a fraction of the full tt sample predicted by MC simulation. QCD multijet events are estimated with a fully data-driven method. Single top production as well as minor backgrounds (Z+jets and diboson) are determined from MC simulation normalized to the best available theoretical calculation of their cross-sections.
The W+jets background estimate uses as a starting point the Alpgen+Pythia samples normalized to the inclusive W boson NNLO cross-section. The normalization and heavy-flavor fraction of the W+jets background have large theoretical uncertainties, and are then determined from data. The overall W+jets normalization is obtained by exploiting the expected charge asymmetry in the production of W + and W − bosons at a pp collider [12,78]. This asymmetry is predicted precisely by theory, and other processes in the tt sample are symmetric in charge except for a small contamination from single top and WZ events, which is corrected by MC simulation. The total number of W+jets events in the sample can thus be estimated with the following equation: where r MC is the ratio of the number of events with positive leptons to the number with negative leptons in the MC simulation, and D + and D − are the number of events with positive and negative leptons in the data, respectively. The signal sample has too few events to apply At least one anti-k t R = 0.4 jet with ∆R( , jet R=0.4 ) < 1.5

Hadronic top
The leading-p T trimmed anti-k t R = 1.0 jet has: At least one of: 1) the leading-p T anti-k t R = 0.4 jet with ∆R( , jet R=0.4 ) < 1.5 is b-tagged 2) at least one anti-k t R = 0.4 jet with ∆R(jet R=1.0 , jet R=0.4 ) < 1.0 is b-tagged Table 1: Summary of event selections for detector-level and MC-generated particle-level events described in Secs. 4 and 8.2, respectively. a sample enhanced in W+jets events is obtained by removing the b-tagging, ∆φ(jet R=1.0 , ), jet mass, and √ d 12 requirements. The heavy-flavor fraction scale factors correct for potential mismodeling in the generator of the fractions of W production associated with different flavor components (W + bb, W + cc, W + c). They are estimated in a sample with the same lepton and E miss T selections as the signal selection, but with only two small-R jets and no b-tagging requirements. The b-jet multiplicity, in conjunction with knowledge of the b-tagging and mistag efficiency, is used to extract the heavy-flavor fraction in this sample. A common scale factor is used for the W + bb and W + cc components. This information is extrapolated to the signal region using the MC simulation, assuming constant relative rates for the signal and control regions. The overall normalization and heavy-flavor scale factors are extracted iteratively because the various flavor components have different charge asymmetries. After correction the W+jets events are expected to make up approximately 5% of the total events in the signal region.
QCD multijet events can mimic the lepton+jets signature. This background is estimated directly from data by using the matrix-method technique [79]. A sample enhanced in fake leptons, i.e., nonprompt leptons or jets misidentified as prompt leptons, is obtained by loosening the lepton identification requirements. The number of events with fake leptons in the signal region can be predicted as: where and f are the efficiencies for leptons that passed the loose selections to also pass the tight (signal) selections, for real and fake leptons respectively, N T is the number of events with a tight lepton, and N L is the number of events with a loose lepton that failed the tight cuts. The efficiency f is measured using data in fake-lepton-enhanced control regions and is extracted from MC simulation and validated in data. QCD multijet events contribute to the total event yield at approximately the percent level.
Top quark pair events with both the top and antitop quarks decaying leptonically (including decays to τ) can sometimes pass the event selection, contributing approximately 5% of the total event yield, and are treated as background in the analysis. The fraction of dileptonic tt events in each p T bin is estimated using the same MC sample used to model the signal.

Systematic uncertainties
Systematic uncertainties, which arise from object reconstruction and calibration, MC generator modeling, and background estimation, are described below. The propagation of systematic uncertainties through the unfolding procedure is described in Sec. 8.4.

Detector modeling
The uncertainty on the large-R jet energy scale (JES), jet mass scale (JMS), and k t splitting scale is obtained using two different data-driven methods. For p T > 800 GeV for JES, and for all p T for the JMS and k t splitting scale, the ratio of the large-R jets kinematic variables reconstructed from the calorimeter clusters to those from inner-detector tracks is compared between data and MC simulation. For p T < 800 GeV for JES, the p T of large-R jets are compared to the well-calibrated p T of photons in a large sample of photon+jets events. An additional MC-based uncertainty, referred to as large-R JES topology uncertainty, is included to reflect the fact that the jets in these calibration samples have a different response (gluon or light-quark jets) than those in tt events (top-jets). The full difference between the response to these two types of jets is conservatively assigned as the corresponding systematic uncertainty. The uncertainty on the large-R jet energy resolution (JER) is determined by smearing the jet energy such that the resolution is degraded by 20% [80, 81] and evaluating the effect on the final result. The same smearing procedure is applied to determine the uncertainty due to the large-R jet mass resolution (JMR). The uncertainties on the large-R jets JES are the dominant contribution to the total uncertainty of this measurement, in particular the topology and photon+jet calibration uncertainties.
The small-R jet energy scale uncertainty is derived using a combination of simulations, test beam data, and in situ measurements [63,70,82]. Additional contributions from the jet flavor composition, calorimeter response to different jet flavors, and pileup are taken into account. Uncertainties in the jet energy resolution are obtained with an in situ measurement of the jet p T balance in dijet events [83].
The efficiency to tag b-jets and mistag light jets is corrected in Monte Carlo events by applying btagging scale factors, extracted in tt and dijet samples, that compensate for the residual difference between data and simulation. The associated systematic uncertainty is computed by varying the scale factors within their uncertainty [84][85][86]. The b-jet calibration is performed for jets with p T up to 300 GeV; for larger transverse momenta an additional MC-based extrapolation uncertainty is applied, which ranges from approximately 10% to 30%, increasing with b-jet p T from 300 GeV to 1200 GeV.
The lepton reconstruction efficiency in simulation is corrected by scale factors derived from measurements of these efficiencies in data using Z → + − enriched control regions. The lepton trigger and reconstruction efficiency scale factors, energy scale, and energy resolution are varied within their uncertainties [75,87].
The uncertainty associated with E miss T is calculated by propagating the energy scale and resolution systematic uncertainties on all physics objects to the E miss T calculation. Additional E miss T uncertainties arising from energy deposits not associated with any reconstructed objects are also included [77].
The uncertainty on the integrated luminosity is ± 2.8% and is derived following a methodology similar to that defined in Ref. [23].

Signal and background modeling
The tt parton shower and hadronization uncertainty is computed by comparing the results obtained with Powheg+Pythia (without electroweak corrections applied) and Powheg+Herwig. The tt generator uncertainty is evaluated by taking the difference between the results obtained with Powheg+Herwig and MC@NLO+Herwig. Both uncertainties are symmetrized. The procedure to compute the PDF uncertainty on the signal is based on the PDF4LHC recommendations [88] using the MC@NLO+Herwig sample with three different PDF sets (CT10 [29], MSTW [89] and NNPDF [90]). An intra-PDF uncertainty is obtained for each PDF set by following its respective prescription while an inter-PDF uncertainty is computed as the envelope of the three intra-PDF uncertainties. The modeling of ISR and FSR is evaluated separately using dedicated AcerMC + Pythia samples with variation of the Pythia parameters for QCD radiation.
The W+jets shape uncertainty is extracted by varying the renormalization and matching scales in Alpgen. The W+jets MC statistical uncertainty is also computed and its contribution to the crosssection uncertainty increases with the top-jet candidate p T from approximately 1% to 6%. A new set of W+jets normalization and heavy-flavor scale factors is extracted for each variation of the most important detector modeling uncertainties, allowing their correlated effect on the W+jets background, tt signal and background, and other MC-based background processes to be assessed.
The uncertainty on the fake-lepton background is determined by varying the definition of loose leptons, changing the selection used to form the fake-enhanced control region, and propagating the statistical uncertainty of parameterizations of the efficiency and the fake rate.
The single-top background is assigned an uncertainty associated with the theory calculations used for its normalization [55][56][57]. A generator uncertainty is included for the Wt channel, which provides the largest single-top contribution, by taking the difference between the yields predicted by Powheg and MC@NLO. An uncertainty on the interference between the tt and Wt processes is also included. A conservative uncertainty of 50% is applied to the normalization of the subdominant Z+jets and diboson backgrounds. Table 2 gives the number of observed and expected events for each process, where the systematic uncertainties on the background estimates, objects' energy scales and reconstruction efficiencies, and MC statistics are taken into account. The prediction is generally found to overestimate the data by approximately one standard deviation.  Agreement of the data with the prediction is further tested by studying the distributions of several variables of interest in Fig. 1. The systematic uncertainties on the objects' energy scales and reconstruction efficiencies, on the background estimates, luminosity and MC statistics are shown. While the prediction generally overestimates the data, as already seen in Table 2, the simulation reproduces the observed shapes in most cases. Exceptions include the tails of some kinematic variables such as the top-jet candidate p T . The distribution of the top-jet candidate p T constitutes the input to the unfolding procedure and is studied in more detail in the following sections.

Differential cross-section determination
Differential cross-sections are measured as a function of the p T of the top-jet candidate at particle level and the p T of the top quark at parton level. The electron and muon channels are first combined into a +jets sample at the detector level. The detector-level p T spectrum is corrected for detector inefficiencies and finite resolution to obtain particle-and parton-level differential cross-sections. The particle-level measurement is performed in a specific fiducial region of phase space close to the event selection. The systematic and statistical uncertainties are propagated through the unfolding procedure. Finally a covariance matrix is computed to perform a quantitative comparison of the measured crosssections with MC predictions.   Figure 1: Distributions of (a) transverse momentum p T of the lepton candidates, (b) p T of selected small-R jets, (c) missing transverse momentum E miss T , (d) and pseudorapidity η, (e) mass and (f) p T of the leading selected anti-k t R=1.0 jets for the +jets channel. The tt prediction is obtained using the nominal Powheg+Pythia sample. The ratio of the MC prediction to the data is shown in the insets below the histograms. The hashed area includes all the object-related uncertainties (on the jet, lepton, and E miss T ), and the uncertainties from the background estimation, luminosity and MC statistics. The vertical lines indicate the data statistical uncertainty.

Combination of decay channels
The e+jets and µ+jets selections are combined into a +jets sample at the detector level. The combined +jets signal and background samples take into account the efficiencies of the two selections. This procedure is well motivated given that the relative yields of the two channels agree well between data and MC simulation, as shown in Table 2. The combination method is cross-checked by performing the unfolding in each channel individually to the +jets phase space described in Sec. 8.2 and comparing these alternative cross-section estimates with the one based on the combined data. The final results are found to be consistent.

Particle-and parton-levels fiducial region definitions
Particle-level corrections to the data are derived from leptons and jets in simulated tt events that are constructed using stable particles, with a mean lifetime greater than 0.3 × 10 −10 seconds, which result directly from the hard-scattering pp interaction or from subsequent decays of particles with a shorter lifetime.
All leptons (e, µ, ν e ,ν µ , ν τ ) not from hadron decays are considered as prompt isolated leptons. The leptons from τ decays are accepted only if the parent τ is not a hadron decay product itself. The four-momenta of photons within a cone of ∆R = 0.1 around the electron or muon direction are added to those of the leptons (dressed leptons). Both the small-R and large-R jets are reconstructed using all stable particles except for the selected dressed leptons. The trimming procedure applied to detectorlevel jets is also applied to particle-level jets. A small-R jet with p T > 25 GeV and |η| < 2.5 is considered to be "b-tagged" if there exists at least one b-hadron with p T > 5 GeV clustered in the jet. 5 The missing transverse momentum E miss T is the magnitude of the vector sum of the momenta of neutrinos not resulting from hadron decays.
To minimize the theoretical input to the measurement, the fiducial region is chosen to follow the detector-level event selections closely, including the kinematic requirements on the objects and the requirements on the event topology. In contrast to the detector-level selection, no overlap removal procedure is applied to the leptons and jets, and no isolation requirement is imposed on the leptons. Using the particle-level objects defined above, the fiducial region is defined by requiring: • Exactly one lepton (electron or muon) with p T > 25 GeV, |η| <2.5.
• E miss T > 20 GeV and E miss T +m W T > 60 GeV. • At least one small-R jet with p T > 25 GeV, |η| < 2.5, and a distance ∆R <1.5 from the lepton. If there is more than one such jet, the one with the largest p T is considered to be the leptonic b-jet candidate (the b-jet associated to the leptonic top quark decay).
• At least one trimmed large-R jet with p T > 300 GeV, mass > 100 GeV, √ d 12 > 40 GeV, and |η| < 2, well separated from both the lepton (∆φ > 2.3) and the leptonic b-jet candidate (∆R > 1.5). The jet mass is reconstructed from the four-vector sum of the particles constituting the jet.
If more than one large-R jet satisfies these criteria, the one with largest p T is chosen. The jet passing this selection is referred to as the particle-level top-jet candidate.
• At least one b-tagged small-R jet such that ∆R(jet R=1.0 , jet R=0.4 ) < 1 and/or the leptonic b-jet candidate is b-tagged.
The particle-level event selection is summarized in Table 1. Fiducial particle-level corrections are determined by using only simulated tt events in which exactly one of the W bosons, resulting from the decay of the tt pair, decays to an electron or a muon either directly or through a τ lepton decay. All other tt events are not used. The cross-section is then determined as a function of the particle-level top-jet candidate transverse momentum, p T,ptcl .
For the parton level, the top quark that decays to a hadronically decaying W boson is considered just before the decay and after QCD radiation, selecting events in which the momentum of such a top quark, p T,parton , is larger than 300 GeV. Parton-level corrections are determined by using only simulated tt events in which exactly one of the W boson decays to an electron or a muon or a τ lepton (including hadronic τ decays). The correction to the full parton-level phase space defined above is obtained by accounting for the branching ratio of tt pairs to the +jets channel.

Unfolding to particle and parton levels
The procedure to unfold the distribution of p T,reco , the p T of the detector-level leading-p T trimmed large-R jet, to obtain the differential cross-section as a function of p T,ptcl is composed of several steps, outlined in: where N j reco is the number of observed events in bin j of p T,reco with the detector-level selection applied, N i ptcl is the total number of events in bin i of p T,ptcl that meet the fiducial region selection, ∆p i T,ptcl is the size of bin i of p T,ptcl , and L is the integrated luminosity of the data sample. The corrections that are applied to p T,reco are all extracted from the nominal Powheg+Pythia tt sample.
First, the non-tt background contamination, N j reco,bgnd , is subtracted from the observed number of events in each p T,reco bin. The contribution from non-+ jets tt events is taken into account by the multiplicative correction f tt, +jets , which represents the fraction of +jets tt events extracted from the nominal Powheg+Pythia tt sample.
In a second step the correction factor f j reco!ptcl , also referred to as acceptance correction, corrects the p T,reco spectrum for the tt events that pass the detector-level selection but fail the particle-level selection. For each p T,reco bin j, f j reco!ptcl is defined as the ratio of the number of events that meet both the detector-level and particle-level selections to the number of events that satisfy the detector-level selection. The distribution of the acceptance correction f j reco!ptcl is shown in Fig. 2(a) for various MC generators. It represents the ratio of the number of events that meet both the parton-level and particle-level to the number of events that satisfy only the particle-level selection requirements.
The third step corrects for detector resolution effects. A migration matrix is constructed to correlate the p T,reco -binned distribution to the p T,ptcl distribution. The matrix M i j represents the probability for an event with p T,ptcl in bin i to have a p T,reco in bin j. This matrix is shown in Fig. 3(a). It shows that approximately 50% to 85% of events have values of p T,ptcl and of p T,reco that fall in the same bin.
The inversion of the migration matrix to correct p T,reco to the particle level is carried out by an unfolding scheme based on Tikhonov regularization which is implemented through the singular value decomposition (SVD) of the matrix [91]. This scheme is chosen to reduce sizable statistical fluctuations that are introduced by instabilities in the inversion procedure. The unfolding regularization parameter, which characterizes the size of the expansion of the solution to the inversion problem, is optimized according to the procedure described in Ref. [91]. In parallel the bin size for the p T,ptcl (and p T,reco ) distribution is optimized such that systematic uncertainties are larger than statistical uncertainties in each bin, and such that the width of each bin corresponds to at least one and a half times the expected resolution in that bin. The former requirement is introduced to minimize statistical fluctuations when estimating systematic uncertainties. The typical expected fractional resolution for p T,reco in tt simulated events ranges from 7% to 3% for p T,reco values between 250 GeV and 1.2 TeV. Finally, the optimization requires the unfolding to be unbiased, i.e., that a given input p T,ptcl spectrum is recovered on average by the unfolding procedure. After rounding to the nearest 50 GeV, this procedure results in bin widths of 50 GeV between 300 GeV and 550 GeV, 100 GeV between 550 GeV and 750 GeV, while the last bin spans 750 GeV to 1200 GeV. Just one event with reconstructed p T = 1535 GeV falls outside this region in the µ + jets sample, and none in the e + jets sample.
The fourth step is to apply a bin-by-bin correction factor f i ptcl!reco , also referred to as efficiency correction, which restores the contribution of tt events that fulfill the particle-level selection but not the detector-level selection. This factor is defined as the ratio of the number of events that satisfy both the particle-level and detector-level selections to the number that meet the selection at particle level only. The distribution of the efficiency correction f i ptcl!reco is shown in Fig. 4(a).  Figure 3: (a) Migration matrix between the particle-level p T,ptcl and reconstructed detector-level p T,reco . (b) Migration matrix between the generated p T,parton and the particle-level p T,ptcl . The unit of the matrix elements is the probability (expressed in percentage) for an event generated at a given value to be reconstructed at another value (each row adds up to 100%).  Figure 4: (a) Distribution of the correction factor f ptcl!reco as a function of p T,ptcl . It represents the ratio of events that meet both the particle-level and detector-level to those that satisfy the particle-level selection requirements. (b) Distribution of the correction factor f parton!ptcl as a function of p T,parton . It represents the ratio of events that meet both the parton-level and particle-level to those that satisfy the parton-level selection requirements.
The ability of the full correction procedure to recover a distribution that is significantly different from the nominal tt sample is tested. Simulated tt events are reweighted such that the p T,reco distribution matches the data. The corresponding p T,ptcl spectrum of the distorted p T,reco input spectrum is recovered with subpercent accuracy after unfolding.
The differential cross-section as a function of p T,parton is then derived according to: Similarly to Eq. (2), N j ptcl is the total number of events in bin j of p T,ptcl that enter the particle-level fiducial region described in Sec. 8.2, N k parton is the number of events in bin k of p T,parton in the full phase space, ∆p k T,parton is the size of bin k of the parton-level p T,parton (and of p T,ptcl ), L is the total integrated luminosity of the data sample, and B=0.438 [92] is the branching ratio for tt events with exactly one of the W bosons, from the decay of the tt pair, decaying to an electron or a muon or a τ lepton.
The corrections that are applied to the p T,ptcl variable are derived following steps similar to the ones described to derive dσ tt /d p T,ptcl . They are also extracted from the nominal Powheg+Pythia tt sample. First, the factor f j ptcl!parton corrects the p T,ptcl spectrum for the tt events that pass the particle-level selection but fail the parton-level selection, shown in Fig. 2(b). Effects relating p T,parton to p T,ptcl are corrected with the same matrix unfolding procedure used for detector effects. This migration matrix M jk is shown in Fig. 3(b). A final correction factor f k parton!ptcl is applied in bins of p T,parton to correct the result from the particle level to the partonic phase space, shown in Fig. 4(b).
To test the two-step derivation, the cross-section is also obtained by directly correcting the reconstructed distribution to parton level in a single step. The results are found to be consistent.

Propagation of statistical and systematic uncertainties
The propagation of statistical and systematic uncertainties is performed in the same way for both the particle-level and parton-level results. The impact of the data statistical uncertainty is evaluated by performing 1000 pseudoexperiments in which independent Poisson fluctuations in each p T,reco bin are assumed. The statistical uncertainty due to the limited size of the signal and background MC samples used to correct the data are estimated by performing 1000 pseudoexperiments using the bootstrap method [93], which builds 1000 statistically connected (co-varied) replicas of individual simulated signal or background spectra and derives the associated corrections.
For each systematic uncertainty arising from detector modeling, background modeling, and the electroweak correction factor, a varied p T,reco distribution is obtained and unfolded using corrections extracted from the nominal signal and background samples. The correlation between each systematic uncertainty's effect on the signal and background spectra is taken into account. For the tt generator,  Figure 5: Relative uncertainties on (a) the particle-level differential cross section dσ tt /d p i T,ptcl and (b) the parton-level differential cross section dσ tt /d p i T,parton . The total uncertainty (band) is shown along with the effect of the dominant uncertainties. The components "Large-R (JES) stat." and "Large-R (JES) data vs MC" are, respectively, the statistical uncertainty and the systematic uncertainty associated with the difference in jet response between data and MC simulation when balancing p T in photon+jet events. parton shower, and ISR/FSR uncertainties, a systematic uncertainty variation is defined as the difference between the generated and unfolded cross-section of a given generator, with unfolding corrections extracted with an alternative generator (or alternative generator setting). The PDF uncertainty is computed by unfolding the nominal sample with correction factors extracted by reweighting the nominal sample at the hard-process level for each variation of the PDF. Figure 5 shows the effect of the statistical and systematic uncertainties on dσ tt /d p T,ptcl and dσ tt /d p T,parton . The total uncertainty generally increases with the measured p T and ranges from 13% to 29% for the particle-level cross-section, and from 15% to 41% for the parton-level cross-section. The dominant uncertainty for the particle-level cross-section is the large-R jet energy scale, in particular its components due to the topology uncertainty at low p T and the uncertainty from p T balance in photon+jet events at high p T . The experimental uncertainties have a comparable size at parton level. However, the reported parton-level cross-section has significantly larger systematic uncertainties than the particle-level cross-section since it is affected by larger tt modeling uncertainties. The parton shower or generator uncertainties are dominant for nearly all p T bins of the parton-level cross-section, which illustrates the benefit of defining a particle-level cross-section in a fiducial region closely following the detector-level selection. A detailed breakdown of the systematic uncertainties is provided in Appendix A.
A covariance matrix including the effect of all uncertainties is calculated at particle level to make quantitative comparisons with theoretical predictions. This covariance matrix is obtained by summing two covariance matrices. The first covariance matrix incorporates uncertainties from detector and background modeling by performing 250,000 pseudoexperiments. In each pseudoexperiment, the data p T,reco distribution is varied following a Poisson distribution. Gaussian-distributed shifts are coherently added for each systematic uncertainty effect by scaling each Poisson-fluctuated bin with the relative variation from the associated systematic uncertainty effect. Differential cross-sections are obtained by unfolding each varied p T,reco distribution with the nominal corrections, and the results are used to compute a covariance matrix.
The second covariance matrix is obtained by summing four separate covariance matrices corresponding to the effects of tt generator, parton shower, ISR/FSR, and PDF uncertainties. The standard deviations of the covariance matrices are derived by scaling the measured cross-section with the appropriate relative systematic uncertainty. The bin-to-bin correlation value is set to unity for the generator, parton shower, and ISR/FSR matrices, while it is set to 0.5 for the PDF matrix. This value is motivated by the fraction of the bins in which a single PDF set dominates in the determination of the envelopes used for their respective estimates. The procedure for these signal modeling uncertainties is needed because these effects cannot be represented by a variation at the detector level, and so cannot be included in the pseudoexperiment formalism used to build the first covariance matrix.
The correlation matrix derived from the particle-level covariance matrix is shown in Table 3. Agreement between the measured differential cross-sections and various predictions is quantified by calculating χ 2 values employing the covariance matrix and by inferring corresponding p-values. The χ 2 are evaluated using: where V is the vector of differences between measured differential cross-section values and predictions, and Cov −1 is the inverse of the covariance matrix.  Table 3: Correlation matrix between the bins of the particle-level differential cross-section as a function of p T,ptcl .

Results and interpretation
The unfolding procedure is applied to the observed top-jet candidate p T distribution. The crosssections are provided in Table 4 and Fig. 6 for the particle-level cross-section, and in Table 5 and Fig. 7 for the parton-level cross-section. The higher efficiency of reconstruction techniques for highly boosted top quarks allows measurement of the top quark p T spectrum up to 1200 GeV. The differential cross-section is measured over two orders of magnitude. The measured differential cross-sections are compared to the predictions from Alpgen+Herwig, MC@NLO+Herwig, Powheg+Herwig, and Powheg+Pythia tt samples normalized to the NNLO+NNLL inclusive cross-section. The electroweak corrections are not applied to the Powheg+Pythia prediction in these figures in order to compare it on an equal footing with the other generators. All generators produce a top quark p T spectrum that is harder than the one observed, with a difference that generally increases with p T . The MC prediction   to data ratio is approximately the same at both the particle and parton levels for Powheg+Pythia, which was used to extract the unfolding corrections. However, it changes significantly when going from particle level to parton level for the other MC generators, in particular for Powheg+Herwig, and Alpgen+Herwig, due to the different parton-level corrections in these MC generators. The level of agreement is better at parton level than at particle level because the parton level is affected by larger systematic uncertainties.
The χ 2 and p-values that quantify the level of agreement between the particle-level predictions and data are listed in Table 6. Within uncertainties, the differences are not significant for Powheg+Pythia, Powheg+Herwig and MC@NLO+Herwig, for which p-values of 0.11 (for Powheg+Pythia without electroweak corrections), 0.41, and 0.14 are obtained, respectively. Only the prediction of Alp-gen+Herwig is significantly disfavored by the data at the particle level with a p-value of 5.9 · 10 −5 .
The measured differential cross-sections are compared in Fig. 8 to the predictions of Powheg+Pythia with and without the electroweak corrections applied. The electroweak corrections lead to a slightly softer p T spectrum, increasing the particle-level p-value from 0.11 to 0.28 without and with the corrections, respectively. The measured differential cross-sections are also compared in Fig. 9 to Powheg+Pythia predictions using either the HERAPDF [94] or CT10 PDF sets, and two different values of the Powheg h damp parameter, the nominal value h damp = m top and one with h damp = ∞, which increases the amount of hard radiation and yields a lower p-value of 0.05. Better agreement with data is obtained when using the HERAPDF set instead of CT10, which reduces the difference between    data and MC simulation by up to about 20%. The Powheg+Pythia prediction that provides the best description of the data is the one that simultaneously employs the HERAPDF set and h damp = m top , corresponding to a p-value of 0.31 at particle level.
The measured parton-level cross-section is compared to the prediction of the parton-level NLO MCFM generator [95], which is interfaced with Applgrid [96] to convolve the perturbative coefficients with the strong coupling and the PDF. The inclusive cross-section computed by MCFM is used to normalize the prediction and no electroweak corrections are applied. Several PDF sets are compared: CT10, MSTW, NNPDF, and HERAPDF. The renormalization scale µ R and factorization scale µ F are dynamic: µ R = µ F = m 2 top +p 2 T,top , wherep T,top is the average p T of the two top quarks in the event. The uncertainties on the prediction include the PDF uncertainties estimated according to the prescription of each set and variations of the strong coupling constant, µ F , and µ R . The predictions are compared to the measured parton-level cross-section in Fig. 10. All predictions are in good agreement with the measured cross-section within the quoted uncertainties, which are dominated by systematic uncertainties correlated between p T bins.

Conclusions
The differential tt production cross-section in √ s = 8 TeV pp collisions has been measured as a function of the hadronically decaying top quark p T in a high-p T regime, using a dataset corresponding to   Figure 9: (a) Fiducial particle-level differential cross-section as a function of the hadronic top-jet candidate p T and (b) parton-level differential cross-section as a function of the hadronically decaying top quark p T , both compared to Powheg+Pythia predictions using either the HERAPDF or CT10 PDF sets, and the Powheg h damp parameter set to ∞ or m top . MC samples are normalized to the NNLO+NNLL inclusive cross-section σ tt = 253 pb. No electroweak corrections are applied to the predictions. The lower part of the figure shows the ratio of the MC prediction to the data. The shaded area includes the total statistical plus systematic uncertainties. The points of the various predictions are spaced along the horizontal axis for presentation only; they correspond to the same p T range. an integrated luminosity of 20.3 fb −1 collected by the ATLAS detector at the LHC. Boosted hadronically decaying top quarks with p T > 300 GeV are reconstructed within large-R jets and identified using jet substructure techniques. The measured p T spectrum is extended in this analysis relative to previous measurements. A particle-level cross-section is measured in a fiducial region that closely follows the event selection. The measurement uncertainty ranges from 13% to 29% and is generally dominated by the uncertainty on the jet energy scale of large-R jets. A parton-level cross-section is also reported, with larger systematic uncertainties due to its greater reliance on tt MC generators to correct the data. The measured cross-sections are compared to the predictions of several NLO and LO matrix-element generators normalized to NNLO+NNLL QCD calculations, and using various PDF sets. Previous measurements suggest that the top quark p T spectrum is well predicted at low p T by NLO and matrixelement MC generators, both in normalization and shape, but that their predictions exceed the data at high p T . The current analysis, focused on the boosted topology and extended to higher p T values, also observes such a trend. However, a statistical analysis shows that the measurements are compatible with the majority of MC generator predictions within the quoted uncertainties.

A. Detailed tables of systematic uncertainties
Tables 7 and 8 report the detailed breakdown of the systematic uncertainties as a percentage of the measured differential cross sections.       [12] ATLAS Collaboration, Measurements of top quark pair relative differential cross-sections with ATLAS in pp collisions at √ s = 7 TeV, Eur. Phys. J. C 73 (2013) 2261, arXiv:1207.5644 [hep-ex].