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PAPER-TopDiffXS8TeV.bib \addbibresourceATLAS.bib \AtlasTitleMeasurement of the differential cross-section of highly boosted top quarks as a function of their transverse momentum in = 8 \TeV proton–proton collisions using the ATLAS detector \AtlasRefCodeTOPQ-2014-15 \PreprintIdNumberCERN-PH-EP-2015-237 \AtlasJournalPhys. Rev. D. \AtlasAbstractThe differential cross-section for pair production of top quarks with high transverse momentum is measured in 20.3 fb of proton–proton collisions at a center-of-mass energy of 8 \TeV. The measurement is performed for 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- jet with radius parameter 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. \pdfstringdefDisableCommands\pdfstringdefDisableCommands

## 1 Introduction

The large number of top–antitop quark () pairs produced at the LHC provide a unique opportunity to improve our understanding of production and test the Standard Model (SM) at the \TeV scale. New phenomena beyond the Standard Model may distort the top quark transverse momentum () spectrum, in particular at high (see, e.g., Refs. [Atwood:1994vm, Englert:2012by]), and could thus be revealed by a precise measurement. Moreover, due to their high cross-section at the LHC and rich experimental signature, 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 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. [Aad:2014kva, Chatrchyan:2012bra, Chatrchyan:2013faa]) 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 [Cacciari:2011hy, Beneke:2011mq, Baernreuther:2012ws, Czakon:2012zr, Czakon:2012pz, Czakon:2013goa], 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 [Aad:2012hg, Aad:2014zka, Aad:2015eia] and CMS collaborations [Chatrchyan:2012saa, Khachatryan:2015oqa], where the highest measured top quark range is 350-800 \GeV [Aad:2014zka].

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-) jet. The selection and reconstruction are based on an algorithm developed [ATL-PHYS-PUB-2010-008] and used in resonance searches [Aad:2013nca, Aad:2012raa, Aad:2015fna, Chatrchyan:2012ku] that increases the selection efficiency at high top quark and extends the kinematic reach into the \TeV range. This analysis utilizes the lepton+jets channel where one boson decays hadronically and the other leptonically to an electron or a muon, assuming each top quark decays to a boson and a -quark. The cross-section is measured as a function of the hadronically decaying top quark . 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 , by further extrapolating to the full kinematic phase space except for a lower limit on top quark 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 resonances [Aad:2015fna]. More details of these aspects of the measurement can be found in the corresponding reference.

## 2 The ATLAS detector

ATLAS is a multipurpose detector [Aad:2008zzm] that provides nearly full solid angle111ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the center of the detector and the -axis along the beam pipe. The -axis points from the IP to the center of the LHC ring, and the -axis points upward. Cylindrical coordinates (,) are used in the transverse plane, being the azimuthal angle around the beam pipe. The pseudorapidity is defined in terms of the polar angle as . coverage around the interaction point. Charged-particle trajectories are reconstructed by the inner detector, which covers pseudorapidity 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 . High-granularity liquid-argon (LAr) electromagnetic (EM) calorimeters are used up to . Hadronic calorimetry based on scintillator-tile active material covers while LAr technology is utilized for hadronic calorimetry from to . 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 = 2.7 and muon trigger chambers cover .

## 3 Data and Monte Carlo samples

The cross-section is measured using data from the 2012 LHC run at = 8 \TeV, which corresponds to an integrated luminosity of

## 4 Object definition and event selection

Jets are reconstructed using the anti- algorithm [Cacciari:2008gp] implemented in the FastJet package [Cacciari:2011ma] with radius parameter or , respectively called small- and large- jets in the following, using as input calibrated topological clusters [topocluster, Aad:2011he, CONF-2015-017]. These clusters are assumed to be massless when computing the jet four-vectors and substructure variables. Large- jets containing hadronically decaying top quarks are selected by applying jet substructure requirements, which exploit the fact that they contain several high- objects and have a high mass, unlike most jets originating from the fragmentation of other quarks or gluons. The trimming algorithm [Krohn:2009th] with parameters and is applied to large- jets to mitigate the impact of initial-state radiation, underlying-event activity, and pileup. A correction for the number of additional interactions per bunch crossing is applied to small- jets [Cacciari:2008gn, Cacciari:2007fd, Ellis:1993tq, Catani:1993hr]. The of small- jets and large- trimmed jets and the large- jet mass, obtained from the four-momentum sum of all jet constituents, are calibrated using energy- and -dependent correction factors. After this calibration, the 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 [Aad:2014bia, Aad:2013gja]. Differences between the small- jet response in data and MC simulation are evaluated from control samples and corresponding corrections are applied to data. Small- jets are required to be in the fiducial region and must have \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 of all tracks matched to the jet and the primary vertex, divided by the summed of all tracks matched to the jet. Small- jets with \GeV and are rejected when , to reduce the contribution of jets generated by pileup interactions.222The jet is retained if no tracks are assigned to the jet. Trimmed large- jets are considered for the analysis if and  \GeV. More details on the reconstruction and performance of highly boosted top quarks in ATLAS can be found in Ref. [Aad:2013gja, ATLAS-CONF-2015-036].

Small- jets containing a -hadron are tagged using a neural-network-based algorithm (MV1) [CON-2014-046] 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% -tagging efficiency in 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 [Aad:2014fxa]. Electrons are required to be in the pseudorapidity region , excluding the transition region between the barrel and the endcap calorimeters (). The EM clusters must have a transverse energy \GeV. The associated track must have a longitudinal impact parameter  2 mm with respect to the primary vertex, which is the vertex with the highest 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 is determined through a global fit of the track that takes into account the energy loss in the calorimeters [Aad:2014rra]. The track is required to have a longitudinal impact parameter 2 mm, and a transverse impact parameter significance , indicating the track is consistent with originating from the hard-scattering vertex. Muons are required to have \GeV and be in the fiducial region .

Lepton candidates are required to be isolated to suppress background leptons originating from jets. The variable “mini-isolation” [Rehermann:2010vq] is used. It is defined as , where is the lepton transverse momentum and the sum is over all good-quality tracks (excluding the lepton track) that have \GeV and a distance from the lepton . The parameter is set to 10 \GeV and the isolation requirement is applied for both the electrons and muons. An isolation cone that decreases in size with increasing improves the selection efficiency of the decay of high- 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- jets. In order to improve the reconstruction efficiency in the highly boosted topology, the same overlap removal procedure as used in Ref. [Aad:2015fna] has been adopted. First, jets close to electrons, with 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 -subtracted jet is retained if it satisfies the jet selection criteria listed above, otherwise it is rejected. After this procedure, electrons that lie within from a small- 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 around a small- jet axis.

The missing transverse momentum is the magnitude of the vector sum of the transverse energy of all calorimeter cells [Aad:2012re]. 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  \GeV. The events are required to contain exactly one reconstructed lepton candidate with . The transverse mass of the lepton and is defined as , where is the azimuthal angle between the lepton and . Events are retained if \GeV and 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- jet close to the lepton () and the existence of a reconstructed large- trimmed jet with mass  \GeV. To improve the rejection of background jets, originating from light quarks or gluons, a cut on the splitting scale [Ellis:1993tq, Catani:1993hr] of the large- jets is made. The splitting scale is calculated by reclustering the large- jet with the -clustering algorithm, and taking the distance between the two subjets of the final clustering step to be , where and are the transverse momenta of the two subjets and 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- jet has  \GeV. The large- jet must be well separated from the lepton () and from the small- jet associated with the lepton (). The leading- trimmed large- jet satisfying these requirements is referred to as the top-jet candidate. Finally, at least one of the two top quark candidates must be -tagged. This implies that either the highest- small- jet close to the lepton () or at least one small- jet close to the large- jet () is -tagged333The reconstruction of a large- jet does not prevent the reconstruction of small- jets overlapping with it..

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 are expected to be semileptonic events.

## 5 Background estimation

After the event selection the background is composed primarily, in order of importance, of +jets,  dilepton, single top, and QCD multijet events. The +jets background is obtained from MC simulation with normalization and heavy-flavor content adjusted in data control regions. The dilepton background is determined as a fraction of the full sample predicted by MC simulation. QCD multijet events are estimated with a fully data-driven method. Single top production as well as minor backgrounds (+jets and diboson) are determined from MC simulation normalized to the best available theoretical calculation of their cross-sections.

The +jets background estimate uses as a starting point the Alpgen+Pythia samples normalized to the inclusive boson NNLO cross-section. The normalization and heavy-flavor fraction of the +jets background have large theoretical uncertainties, and are then determined from data. The overall +jets normalization is obtained by exploiting the expected charge asymmetry in the production of and bosons at a collider [Aad:2012hg, ATLAS:2012an]. This asymmetry is predicted precisely by theory, and other processes in the sample are symmetric in charge except for a small contamination from single top and events, which is corrected by MC simulation. The total number of +jets events in the sample can thus be estimated with the following equation:

 NW++NW−=(rMC+1rMC−1)(D+−D−), (1)

where is the ratio of the number of events with positive leptons to the number with negative leptons in the MC simulation, and and are the number of events with positive and negative leptons in the data, respectively. The signal sample has too few events to apply Eq. (1) directly. Instead a sample enhanced in +jets events is obtained by removing the -tagging, , jet mass, and requirements. The heavy-flavor fraction scale factors correct for potential mismodeling in the generator of the fractions of production associated with different flavor components (, , ). They are estimated in a sample with the same lepton and selections as the signal selection, but with only two small- jets and no -tagging requirements. The -jet multiplicity, in conjunction with knowledge of the -tagging and mistag efficiency, is used to extract the heavy-flavor fraction in this sample. A common scale factor is used for the and 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 +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 [ATLAS-CONF-2014-058]. 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: {linenomath}

 Nmultijet=(ϵ−1)fϵ−fNT+ϵfϵ−fNL,

where and are the efficiencies for leptons that passed the loose selections to also pass the tight (signal) selections, for real and fake leptons respectively, is the number of events with a tight lepton, and is the number of events with a loose lepton that failed the tight cuts. The efficiency 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 ) 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 events in each bin is estimated using the same MC sample used to model the signal.

## 6 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.

### 6.1 Detector modeling

The uncertainty on the large- jet energy scale (JES), jet mass scale (JMS), and splitting scale is obtained using two different data-driven methods. For \GeV for JES, and for all for the JMS and splitting scale, the ratio of the large- jets kinematic variables reconstructed from the calorimeter clusters to those from inner-detector tracks is compared between data and MC simulation. For \GeV for JES, the of large- jets are compared to the well-calibrated of photons in a large sample of photon+jets events. An additional MC-based uncertainty, referred to as large- 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 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- jet energy resolution (JER) is determined by smearing the jet energy such that the resolution is degraded by 20% [ATLAS:2012am, Aad:2012ag] and evaluating the effect on the final result. The same smearing procedure is applied to determine the uncertainty due to the large- jet mass resolution (JMR). The uncertainties on the large- jets JES are the dominant contribution to the total uncertainty of this measurement, in particular the topology and photon+jet calibration uncertainties.

The small- jet energy scale uncertainty is derived using a combination of simulations, test beam data, and in situ measurements [Aad:2014bia, Aad:2011he, Aad:2012vm]. 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 balance in dijet events [JER].

The efficiency to tag -jets and mistag light jets is corrected in Monte Carlo events by applying -tagging scale factors, extracted in 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 [ATLAS-CONF-2014-004, CON-2012-043, CONF-2012-040]. The -jet calibration is performed for jets with 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 -jet 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 enriched control regions. The lepton trigger and reconstruction efficiency scale factors, energy scale, and energy resolution are varied within their uncertainties [Aad:2011mk, Aad:2014rra].

The uncertainty associated with is calculated by propagating the energy scale and resolution systematic uncertainties on all physics objects to the calculation. Additional uncertainties arising from energy deposits not associated with any reconstructed objects are also included [Aad:2012re].

The uncertainty on the integrated luminosity is 2.8% and is derived following a methodology similar to that defined in Ref. [Aad:2013ucp].

### 6.2 Signal and background modeling

The parton shower and hadronization uncertainty is computed by comparing the results obtained with Powheg+Pythia (without electroweak corrections applied) and Powheg+Herwig. The 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 [Botje:2011sn] using the MC@NLO+Herwig sample with three different PDF sets (CT10 [Lai:2010vv], MSTW [Martin:2009iq] and NNPDF [Ball:2012cx]). 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 +jets shape uncertainty is extracted by varying the renormalization and matching scales in Alpgen. The +jets MC statistical uncertainty is also computed and its contribution to the cross-section uncertainty increases with the top-jet candidate from approximately 1% to 6%. A new set of +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 +jets background, 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 [Kidonakis:2011wy, Kidonakis:2010ux, Kidonakis:2010tc]. A generator uncertainty is included for the 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 and processes is also included. A conservative uncertainty of 50% is applied to the normalization of the subdominant +jets and diboson backgrounds.

## 7 Data and MC comparison at detector level

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. LABEL:fig:reco_all. 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 . The distribution of the top-jet candidate constitutes the input to the unfolding procedure and is studied in more detail in the following sections.

## 8 Differential cross-section determination

Differential cross-sections are measured as a function of the \pt of the top-jet candidate at particle level and the 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 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 cross-sections with MC predictions.

### 8.1 Combination of decay channels

The +jets and +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.

### 8.2 Particle- and parton-levels fiducial region definitions

Particle-level corrections to the data are derived from leptons and jets in simulated \ttbar events that are constructed using stable particles, with a mean lifetime greater than seconds, which result directly from the hard-scattering interaction or from subsequent decays of particles with a shorter lifetime.

All leptons (, , ,, ) not from hadron decays are considered as prompt isolated leptons. The leptons from decays are accepted only if the parent is not a hadron decay product itself. The four-momenta of photons within a cone of = 0.1 around the electron or muon direction are added to those of the leptons (dressed leptons). Both the small- and large- jets are reconstructed using all stable particles except for the selected dressed leptons. The trimming procedure applied to detector-level jets is also applied to particle-level jets. A small- jet with \pt  25 \GeV and is considered to be “-tagged” if there exists at least one -hadron with \GeV clustered in the jet.444The -hadrons are not stable and do not contribute to the total four-vector of the jet, only their decay products do. However, they are clustered with their energy set to a negligible value to check that they match the jet geometrically [Cacciari:2008gn].

The missing transverse momentum 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 25 \GeV, 2.5.

• \Etmiss

20 \GeV and \Etmiss+\mtw 60 \GeV.

• At least one small- jet with 25 \GeV, 2.5, and a distance 1.5 from the lepton. If there is more than one such jet, the one with the largest \pt is considered to be the leptonic -jet candidate (the -jet associated to the leptonic top quark decay).

• At least one trimmed large- jet with 300 \GeV, mass 100 \GeV, 40 \GeV, and 2, well separated from both the lepton ( 2.3) and the leptonic -jet candidate ( 1.5). The jet mass is reconstructed from the four-vector sum of the particles constituting the jet. If more than one large- jet satisfies these criteria, the one with largest \pt is chosen. The jet passing this selection is referred to as the particle-level top-jet candidate.

• At least one -tagged small- jet such that and/or the leptonic -jet candidate is -tagged.

The particle-level event selection is summarized in Table 1. Fiducial particle-level corrections are determined by using only simulated \ttbar events in which exactly one of the bosons, resulting from the decay of the \ttbar pair, decays to an electron or a muon either directly or through a lepton decay. All other \ttbar events are not used. The cross-section is then determined as a function of the particle-level top-jet candidate transverse momentum, \ptptcl.

For the parton level, the top quark that decays to a hadronically decaying boson is considered just before the decay and after QCD radiation, selecting events in which the momentum of such a top quark, \ptptn, is larger than 300 \GeV. Parton-level corrections are determined by using only simulated \ttbar events in which exactly one of the boson decays to an electron or a muon or a lepton (including hadronic decays). The correction to the full parton-level phase space defined above is obtained by accounting for the branching ratio of \ttbar pairs to the \ljetschannel.

### 8.3 Unfolding to particle and parton levels

The procedure to unfold the distribution of \ptreco, the \pt of the detector-level leading- trimmed large- jet, to obtain the differential cross-section as a function of \ptptcl is composed of several steps, outlined in:

 \dsigma\ttbard\ptptcl(\ptptcli) = NiptclΔ\ptptcli\lum (2) = 1Δ\ptptcli\lumfiptcl!reco⋅∑jM−1ijfjreco!ptclf\ttbar,ℓ+jets(Njreco−Njreco,bgnd),

where is the number of observed events in bin of \ptreco with the detector-level selection applied, is the total number of events in bin of \ptptcl that meet the fiducial region selection, is the size of bin of \ptptcl, and is the integrated luminosity of the data sample. The corrections that are applied to \ptreco are all extracted from the nominal Powheg+Pythia sample.

First, the non- background contamination, , is subtracted from the observed number of events in each \ptreco bin. The contribution from non-\lplus \ttbar events is taken into account by the multiplicative correction , which represents the fraction of +jets events extracted from the nominal Powheg+Pythia sample.

In a second step the correction factor , also referred to as acceptance correction, corrects the \ptreco spectrum for the events that pass the detector-level selection but fail the particle-level selection. For each \ptreco bin , 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 is shown in Fig. LABEL:fig:fakecorr(a) for various MC generators.

The third step corrects for detector resolution effects. A migration matrix is constructed to correlate the \ptreco-binned distribution to the \ptptcldistribution. The matrix represents the probability for an event with \ptptcl in bin to have a \ptreco in bin . This matrix is shown in Fig. LABEL:fig:migration(a). It shows that approximately 50% to 85% of events have values of \ptptcl and of \ptreco that fall in the same bin.

The inversion of the migration matrix to correct \ptreco 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 [svd]. 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. [svd]. In parallel the bin size for the \ptptcl (and \ptreco) 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 \ptreco in \ttbar simulated events ranges from 7% to 3% for \ptreco values between 250 \GeV and 1.2 \TeV. Finally, the optimization requires the unfolding to be unbiased, i.e., that a given input \ptptcl 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 = 1535 \GeV falls outside this region in the \muplus sample, and none in the \eplus sample.

The fourth step is to apply a bin-by-bin correction factor , also referred to as efficiency correction, which restores the contribution of \ttbar 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 is shown in Fig. LABEL:fig:eff(a).

The ability of the full correction procedure to recover a distribution that is significantly different from the nominal sample is tested. Simulated \ttbar events are reweighted such that the \ptreco distribution matches the data. The corresponding \ptptcl spectrum of the distorted \ptreco input spectrum is recovered with subpercent accuracy after unfolding.

The differential cross-section as a function of \ptptn is then derived according to:

 \dsigma\ttbard\ptptn(\ptptnk) = NkpartonBΔ\ptptnk\lum (3) = 1BΔ\ptptnk\lumfkparton!ptcl⋅∑j^M−1jkfjptcl!partonNjptcl.

Similarly to Eq. (2), is the total number of events in bin of \ptptcl that enter the particle-level fiducial region described in Sec. 8.2, is the number of events in bin of \ptptn in the full phase space, is the size of bin of the parton-level \ptptn (and of \ptptcl), is the total integrated luminosity of the data sample, and =0.438 [BRRef] is the branching ratio for \ttbar events with exactly one of the bosons, from the decay of the \ttbar pair, decaying to an electron or a muon or a lepton.

The corrections that are applied to the \ptptcl variable are derived following steps similar to the ones described to derive . They are also extracted from the nominal Powheg+Pythia sample. First, the factor corrects the \ptptcl spectrum for the events that pass the particle-level selection but fail the parton-level selection, shown in Fig. LABEL:fig:fakecorr(b). Effects relating \ptptn to \ptptclare corrected with the same matrix unfolding procedure used for detector effects. This migration matrix is shown in Fig. LABEL:fig:migration(b). A final correction factor is applied in bins of \ptptn to correct the result from the particle level to the partonic phase space, shown in Fig. LABEL:fig:eff(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.

### 8.4 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 \ptreco 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 [bootstrap], 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 \ptreco 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 \ttbar generator, 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  LABEL:fig:syst shows the effect of the statistical and systematic uncertainties on and . The total uncertainty generally increases with the measured 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- jet energy scale, in particular its components due to the topology uncertainty at low and the uncertainty from balance in photon+jet events at high . 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 modeling uncertainties. The parton shower or generator uncertainties are dominant for nearly all 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 \ptreco 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 \ptreco 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 \ttbar 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 values employing the covariance matrix and by inferring corresponding -values. The are evaluated using: {linenomath}

 χ2=VT⋅Cov−1⋅V,

where is the vector of differences between measured differential cross-section values and predictions, and is the inverse of the covariance matrix.

## 9 Results and interpretation

The unfolding procedure is applied to the observed top-jet candidate distribution. The cross-sections are provided in Table 4 and Fig. 1 for the particle-level cross-section, and in Table 5 and Fig. 2 for the parton-level cross-section. The higher efficiency of reconstruction techniques for highly boosted top quarks allows measurement of the top quark 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 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 spectrum that is harder than the one observed, with a difference that generally increases with . 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 and -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 -values of 0.11 (for Powheg+Pythia without electroweak corrections), 0.41, and 0.14 are obtained, respectively. Only the prediction of Alpgen+Herwig is significantly disfavored by the data at the particle level with a -value of .

The measured differential cross-sections are compared in Fig. LABEL:fig:ewkcorr to the predictions of Powheg+Pythia with and without the electroweak corrections applied. The electroweak corrections lead to a slightly softer spectrum, increasing the particle-level -value from 0.11 to 0.28 without and with the corrections, respectively. The measured differential cross-sections are also compared in Fig. LABEL:fig:hdamp_herapdf to Powheg+Pythia predictions using either the HERAPDF [Aaron:2009aa] or CT10 PDF sets, and two different values of the Powheg parameter, the nominal value and one with , which increases the amount of hard radiation and yields a lower -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 , corresponding to a -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 [Campbell:2010ff], which is interfaced with Applgrid [Carli:2010rw] 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 and factorization scale are dynamic: , where is the average 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, , and . The predictions are compared to the measured parton-level cross-section in Fig. 3. All predictions are in good agreement with the measured cross-section within the quoted uncertainties, which are dominated by systematic uncertainties correlated between bins.

## 10 Conclusions

The differential production cross-section in \rts = 8 \TeV  collisions has been measured as a function of the hadronically decaying top quark in a high- regime, using a dataset corresponding to an integrated luminosity of 20.3

## Acknowledgments

We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently.

We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC, Australia; BMWFW and FWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP, Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and NSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic; DNRF, DNSRC and Lundbeck Foundation, Denmark; IN2P3-CNRS, CEA-DSM/IRFU, France; GNSF, Georgia; BMBF, HGF, and MPG, Germany; GSRT, Greece; RGC, Hong Kong SAR, China; ISF, I-CORE and Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; FOM and NWO, Netherlands; RCN, Norway; MNiSW and NCN, Poland; FCT, Portugal; MNE/IFA, Romania; MES of Russia and NRC KI, Russian Federation; JINR; MESTD, Serbia; MSSR, Slovakia; ARRS and MIZŠ, Slovenia; DST/NRF, South Africa; MINECO, Spain; SRC and Wallenberg Foundation, Sweden; SERI, SNSF and Cantons of Bern and Geneva, Switzerland; MOST, Taiwan; TAEK, Turkey; STFC, United Kingdom; DOE and NSF, United States of America. In addition, individual groups and members have received support from BCKDF, the Canada Council, CANARIE, CRC, Compute Canada, FQRNT, and the Ontario Innovation Trust, Canada; EPLANET, ERC, FP7, Horizon 2020 and Marie Sk0̆142odowska-Curie Actions, European Union; Investissements d’Avenir Labex and Idex, ANR, Region Auvergne and Fondation Partager le Savoir, France; DFG and AvH Foundation, Germany; Herakleitos, Thales and Aristeia programmes co-financed by EU-ESF and the Greek NSRF; BSF, GIF and Minerva, Israel; BRF, Norway; the Royal Society and Leverhulme Trust, United Kingdom.

The crucial computing support from all WLCG partners is acknowledged gratefully, in particular from CERN and the ATLAS Tier-1 facilities at TRIUMF (Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France), KIT/GridKA (Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain), ASGC (Taiwan), RAL (UK) and BNL (USA) and in the Tier-2 facilities worldwide.

## Appendix 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.

\printbibliography

The ATLAS Collaboration

G. Aad, B. Abbott, J. Abdallah, O. Abdinov, R. Aben, M. Abolins, O.S. AbouZeid, H. Abramowicz, H. Abreu, R. Abreu, Y. Abulaiti, B.S. Acharya, L. Adamczyk, D.L. Adams, J. Adelman, S. Adomeit, T. Adye, A.A. Affolder, T. Agatonovic-Jovin, J. Agricola, J.A. Aguilar-Saavedra, S.P. Ahlen, F. Ahmadov, G. Aielli, H. Akerstedt, T.P.A. Åkesson, A.V. Akimov, G.L. Alberghi, J. Albert, S. Albrand, M.J. Alconada Verzini, M. Aleksa, I.N. Aleksandrov, C. Alexa, G. Alexander, T. Alexopoulos, M. Alhroob, G. Alimonti, L. Alio, J. Alison, S.P. Alkire, B.M.M. Allbrooke, P.P. Allport, A. Aloisio, A. Alonso, F. Alonso, C. Alpigiani, A. Altheimer, B. Alvarez Gonzalez, D. Álvarez Piqueras, M.G. Alviggi, B.T. Amadio, K. Amako, Y. Amaral Coutinho, C. Amelung, D. Amidei, S.P. Amor Dos Santos, A. Amorim, S. Amoroso, N. Amram, G. Amundsen, C. Anastopoulos, L.S. Ancu, N. Andari, T. Andeen, C.F. Anders, G. Anders, J.K. Anders, K.J. Anderson, A. Andreazza, V. Andrei, S. Angelidakis, I. Angelozzi, P. Anger, A. Angerami, F. Anghinolfi, A.V. Anisenkov, N. Anjos, A. Annovi, M. Antonelli, A. Antonov, J. Antos, F. Anulli, M. Aoki, L. Aperio Bella, G. Arabidze, Y. Arai, J.P. Araque, A.T.H. Arce, F.A. Arduh, J-F. Arguin, S. Argyropoulos, M. Arik, A.J. Armbruster, O. Arnaez, H. Arnold, M. Arratia, O. Arslan, A. Artamonov, G. Artoni, S. Artz, S. Asai, N. Asbah, A. Ashkenazi, B. Åsman, L. Asquith, K. Assamagan, R. Astalos, M. Atkinson, N.B. Atlay, K. Augsten, M. Aurousseau, G. Avolio, B. Axen, M.K. Ayoub, G. Azuelos, M.A. Baak, A.E. Baas, M.J. Baca, C. Bacci, H. Bachacou, K. Bachas, M. Backes, M. Backhaus, P. Bagiacchi, P. Bagnaia, Y. Bai, T. Bain, J.T. Baines, O.K. Baker, E.M. Baldin, P. Balek, T. Balestri, F. Balli, W.K. Balunas, E. Banas, Sw. Banerjee, A.A.E. Bannoura, L. Barak, E.L. Barberio, D. Barberis, M. Barbero, T. Barillari, M. Barisonzi, T. Barklow, N. Barlow, S.L. Barnes, B.M. Barnett, R.M. Barnett, Z. Barnovska, A. Baroncelli, G. Barone, A.J. Barr, F. Barreiro, J. Barreiro Guimarães da Costa, R. Bartoldus, A.E. Barton, P. Bartos, A. Basalaev, A. Bassalat, A. Basye, R.L. Bates