Measurement of inclusive jet and dijet production in pp collisions at \sqrt{s}=7 TeV using the ATLAS detector

Measurement of inclusive jet and dijet production in collisions
at  TeV using the ATLAS detector

The ATLAS Collaboration
August 28, 2019
Abstract

Inclusive jet and dijet cross sections have been measured in proton-proton collisions at a centre-of-mass energy of 7 TeV using the ATLAS detector at the Large Hadron Collider. The cross sections were measured using jets clustered with the anti- algorithm with parameters and . These measurements are based on the 2010 data sample, consisting of a total integrated luminosity of 37 pb. Inclusive jet double-differential cross sections are presented as a function of jet transverse momentum, in bins of jet rapidity. Dijet double-differential cross sections are studied as a function of the dijet invariant mass, in bins of half the rapidity separation of the two leading jets. The measurements are performed in the jet rapidity range , covering jet transverse momenta from 20 GeV to 1.5 TeV and dijet invariant masses from 70 GeV to 5 TeV. The data are compared to expectations based on next-to-leading order QCD calculations corrected for non-perturbative effects, as well as to next-to-leading order Monte Carlo predictions. In addition to a test of the theory in a new kinematic regime, the data also provide sensitivity to parton distribution functions in a region where they are currently not well-constrained.

pacs:
10, 12.38.Qk, 13.87.Ce
preprint: CERN-PH-EP-2011-192preprint: Submitted to Physical Review D

I Introduction

At the Large Hadron Collider (LHC), jet production is the dominant high transverse-momentum (\pt) process. Jet cross sections serve as one of the main observables in high-energy particle physics, providing precise information on the structure of the proton. They are an important tool for understanding the strong interaction and searching for physics beyond the Standard Model (see, for example, Refs. UA1 Collaboration (1984); L3 Collaboration (1990); UA2 Collaboration (1991); ZEUS Collaboration (2002a, b); ALEPH Collaboration (2003); DELPHI Collaboration (2005); ZEUS Collaboration (2005); OPAL Collaboration (2006); CDF Collaboration (2007); D0 Collaboration (2008); CDF Collaboration (2008); D0 Collaboration (2009a); D0 Collaboration (2009b); H1 Collaboration (2010a, b); ZEUS Collaboration (2010); D0 Collaboration (2010); CMS Collaboration (2011a); CMS Collaboration (2011b)).

The ATLAS Collaboration has published a first measurement of inclusive jet and dijet production at  TeV, using an integrated luminosity of 17 nb ATLAS Collaboration (2011a). This measurement considered only jets with transverse momentum larger than 60 GeV and in a rapidity interval 111ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the -axis along the beam pipe. The -axis points from the IP to the centre 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, referred to the -axis. The pseudorapidity is defined in terms of the polar angle with respect to the beamline as . When dealing with massive jets and particles, the rapidity is used, where is the jet energy and is the -component of the jet momentum..

The analysis presented here extends the previous measurement using the 2010 data sample of (, an integrated luminosity more than 2000 times larger than that of the previous study. This more than doubles the kinematic reach at high jet transverse momentum and large dijet invariant mass. There are strong physics reasons to extend the measurement to jets of lower transverse momentum and larger rapidity as well. Jets at lower are more sensitive to non-perturbative effects from hadronisation and the underlying event, and forward jets may be sensitive to different dynamics in QCD than central jets. Moreover, LHC experiments have much wider rapidity coverage than those at the Tevatron, so forward jet measurements at the LHC cover a phase space region that has not been explored before.

The kinematic reach of this analysis is compared to that of the previous ATLAS study in Fig. 1. This data sample extends the existing inclusive jet measurement from 700 GeV to 1.5 TeV and the existing dijet mass measurement from 1.8 TeV to 5 TeV. Thus this analysis probes next-to-leading order (NLO) perturbative QCD (pQCD) and parton distribution functions (PDFs) in a new kinematic regime. The results span approximately in , the fraction of the proton momentum carried by each of the partons involved in the hard interaction.

Figure 1: Kinematic reach of the inclusive jet cross section measured in this analysis compared to that of the previous study ATLAS Collaboration (2011a) for jets identified using the \AKTalgorithm with . The kinematic limit for the center-of-mass energy of 7 TeV is also shown.

Ii The ATLAS Detector

The ATLAS detector is described in detail in Ref. ATLAS Collaboration (2008). In this analysis, the tracking detectors are used to define candidate collision events by constructing vertices from tracks, and the calorimeters are used to reconstruct jets.

The inner detector used for tracking and particle identification has complete azimuthal coverage and spans the region . It consists of layers of silicon pixel detectors, silicon microstrip detectors, and transition radiation tracking detectors, surrounded by a solenoid magnet that provides a uniform field of 2 T.

The electromagnetic calorimetry is provided by the liquid argon (LAr) calorimeters that are split into three regions: the barrel (), the endcap () and the forward (FCal: ) regions. The hadronic calorimeter is divided into four distinct regions: the barrel (), the extended barrel (), both of which are scintillator/steel sampling calorimeters, the hadronic endcap (HEC; ), which has LAr/Cu calorimeter modules, and the hadronic FCal (same -range as for the EM-FCal) which uses LAr/W modules. The total calorimeter coverage is .

Iii Cross Section Definition

Jet cross sections are defined using the \AKT jet algorithm Cacciari et al. (2008) implemented in the \fastjet Cacciari and Salam (2006) package. Two different values are used for the clustering parameter (0.4 and 0.6), which can be seen intuitively as the radius of a circular jet in the plane of azimuthal angle and rapidity. The jet cross section measurements are corrected for all experimental effects, and thus are defined for the “particle-level” final state of a proton-proton collision Buttar et al. (2008). Particle-level jets in the Monte Carlo simulation are identified using the algorithm and are built from stable particles, which are defined as those with a proper lifetime longer than 10 ps. This definition includes muons and neutrinos from decaying hadrons.

Inclusive jet double-differential cross sections are measured as a function of jet in bins of , in the region . The term “inclusive jets” is used in this paper to indicate that all jets in each event are considered in the cross section measurement. Dijet double-differential cross sections are measured as a function of the invariant mass of the two leading (highest ) jets, which is given as , where and are the energies and momenta of the two leading jets. The cross sections are binned in the variable , defined as half the rapidity difference of the two leading jets, . The quantity is the rapidity in the two-parton centre-of-mass frame (in the massless particle limit), where it is determined by the polar scattering angle with respect to the beamline, :

(1)

For the dijet measurement, the two leading jets are selected to lie in the region, where the leading jet is required to have and the sub-leading jet . Restricting the leading jet to higher improves the stability of the NLO calculation Frixione and Ridolfi (1997).

Theory calculations are used in the same kinematic range as the measurement.

Iv Monte Carlo Samples

The \pythia 6.423 generator Sjostrand et al. (2006) with the MRST LO* PDF set Sherstnev and Thorne (2008) was used to simulate jet events in proton-proton collisions at a centre-of-mass energy of  TeV and to correct for detector effects. This generator utilizes leading-order perturbative QCD matrix elements (ME) for processes, along with a leading-logarithmic, -ordered parton shower (PS), an underlying event simulation with multiple parton interactions, and the Lund string model for hadronisation. Samples were generated using the ATLAS Minimum Bias Tune 1 (AMBT1) set of parameters ATLAS Collaboration (2010a), in which the model of non-diffractive scattering has been tuned to ATLAS measurements of charged particle production at  GeV and  TeV.

The particle four-vectors from these generators were passed through a full simulation ATLAS Collaboration (2010b) of the ATLAS detector and trigger that is based on GEANT4 Agostinelli et al. (2003). Finally, the simulated events were reconstructed and jets were calibrated using the same reconstruction chain as the data.

V Theoretical Predictions

v.1 Fixed-Order calculations

v.1.1 NLO Predictions

The measured jet cross sections are compared to fixed-order NLO pQCD predictions, with corrections for non-perturbative effects applied. For the hard scattering, both the \nlojet 4.1.2 Nagy (2003) package and the \powheggenerator Alioli et al. (2011a); Nason (2007) were used, the latter in a specific configuration where the parton shower was switched off and calculations were performed using NLO matrix elements. The two programs have been used with the CT10 Lai et al. (2010a) NLO parton distribution functions, and the same value of normalisation and factorisation scale, corresponding to the transverse momentum of the leading jet, :

(2)

For \powheg, is evaluated at leading order and is denoted . Using this scale choice, the cross section results of the two NLO codes are compatible at the few percent level for inclusive jets over the whole rapidity region. They are also consistent for dijet events where both jets are in the central region, while they differ substantially when the two leading jets are widely separated in rapidity (). In these regions, \nlojetgives an unstable and much smaller cross section than POWHEG that is even negative for some rapidity separations. \powhegremains positive over the whole region of phase space. It should be noted that the forward dijet cross section predicted by \nlojetin this region has a very strong scale dependence, which however is much reduced for larger values of scale than that of Eq. 2.

The forward dijet cross section for \nlojetis much more stable if instead of a scale fixed entirely by , a scale that depends on the rapidity separation between the two jets is used. The values chosen for each -bin follow the formula:

(3)

and are indicated by the histogram in Fig. 2. These values are motivated by the formula (shown by the dot-dashed curve):

(4)

that is suggested in Ref. Ellis et al. (1992), and are in a region where the cross section predictions are more stable as a function of scale (they reach a “plateau”). At small , the scale in Eq. 3 reduces to the leading jet (dotted line), which is used for the inclusive jet predictions. With this scale choice, \nlojetis again in reasonable agreement with \powheg, which uses the scale from Eq. 2. The \nlojetpredictions are used as a baseline for both inclusive jet and dijet calculations, with the scale choice from Eq. 2 for the former and that from Eq. 3 for the latter. The \powhegscale used for both inclusive jets and dijets, , is given by Eq. 2 but evaluated at leading order. Despite using different scale choices, the dijet theory predictions from \nlojetand \powhegare stable with respect to relatively small scale variations and give consistent results.

Figure 2: The histogram indicates the values of the renormalisation and factorisation scales (denoted by ) used for the dijet predictions obtained using \nlojet, as a function of , half the rapidity separation between the two leading jets. This is motivated by the scale choice suggested in Ref. Ellis et al. (1992) (dot-dashed line), and is also compared to the scale choice used for the inclusive jet predictions (dotted line).

The results are also compared with predictions obtained using the MSTW 2008 Martin et al. (2009), NNPDF 2.1 (100) Ball et al. (2010); Forte et al. (2010) and HERAPDF 1.5 H1 and ZEUS Collaborations PDF sets.

The main uncertainties on the NLO prediction come from the uncertainties on the PDFs, the choice of factorisation and renormalisation scales, and the uncertainty on the value of the strong coupling constant . To allow for fast and flexible evaluation of PDF and scale uncertainties, the \applgrid Carli et al. (2010) software was interfaced with \nlojetin order to calculate the perturbative coefficients once and store them in a look-up table. The PDF uncertainties are defined at 68% CL and evaluated following the prescriptions given for each PDF set. They account for the data uncertainties, tension between input data sets, parametrisation uncertainties, and various theoretical uncertainties related to PDF determination.

To estimate the uncertainty on the NLO prediction due to neglected higher-order terms, each observable was recalculated while varying the renormalisation scale by a factor of two with respect to the default choice. Similarly, to estimate the sensitivity to the choice of scale where the PDF evolution is separated from the matrix element, the factorisation scale was separately varied by a factor of two. Cases where the two scales are simultaneously varied by a factor 2 in opposite directions were not considered due to the presence of logarithmic factors in the theory calculation that become large in these configurations. The envelope of the variation of the observables was taken as a systematic uncertainty. The effect of the uncertainty on the value of the strong coupling constant, , is evaluated following the recommendation of the CTEQ group Lai et al. (2010b), in particular by using different PDF sets that were derived using the positive and negative variations of the coupling from its best estimate.

Electro-weak corrections were not included in the theory predictions and may be non-negligible Moretti et al. (2006).

Figure 3: Non-perturbative correction factors for inclusive jets identified using the \AKTalgorithm with distance parameters and in the rapidity region , derived using various Monte Carlo generators. The correction derived using \pythia6.425 with the AUET2B CTEQ6L1 tune is used for the fixed-order NLO calculations presented in this analysis.

v.1.2 Non-Perturbative Corrections

The fixed-order NLO calculations predict parton-level cross sections, which must be corrected for non-perturbative effects to be compared with data. This is done by using leading-logarithmic parton shower generators. The corrections are derived by using \pythia6.425 with the AUET2B CTEQ6L1 tune ATLAS Collaboration (2011b) to evaluate the bin-wise ratio of cross sections with and without hadronisation and the underlying event. Each bin of the parton-level cross section is then multiplied by the corresponding correction. The uncertainty is estimated as the maximum spread of the correction factors obtained from \pythia6.425 using the AUET2B LO, AUET2 LO, AMBT2B CTEQ6L1, AMBT1, Perugia 2010, and Perugia 2011 tunes (PYTUNE_350), and the \pythia 8.150 tune  ATLAS Collaboration (2011c, b); Skands (2009, 2010), as well as those obtained from the \herwigpp 2.5.1 Bahr et al. (2008) tune UE7000-2 ATLAS Collaboration (2011b). The AMBT2B CTEQ6L1 and AMBT1 tunes, which are based on observables sensitive to the modeling of minimum bias interactions, are included to provide a systematically different estimate of the underlying event activity.

The corrections depend strongly on the jet size; therefore separate sets of corrections and uncertainties were derived for jets with and . The correction factors and their uncertainties depend on the interplay of the hadronisation and the underlying event for the different jet sizes, and they have a significant influence at low and low dijet mass. For , the correction factors are dominated by the effect of hadronisation and are approximately 0.95 at jet  GeV, increasing closer to unity at higher . For , the correction factors are dominated by the underlying event and are approximately 1.6 at jet  GeV, decreasing to between 1.0-1.1 for jets above  GeV. Fig. 3 shows the non-perturbative corrections for inclusive jets with rapidity in the interval , for jet clustering parameters and . The correction factors for the other rapidity bins become closer to unity as the jet rapidity increases, as can be seen in Fig. 20 in Appendix A.

Non-perturbative corrections have been evaluated for the dijet measurement as well, as a function of the dijet mass and the rapidity interval , for each of the two jet sizes. These follow a similar behaviour to those for inclusive jets, with the corrections becoming smaller for large invariant masses and rapidity differences.

v.2 NLO Matrix Element + Parton Shower

The measured jet cross sections are also compared to \powheg Alioli et al. (2011b), an NLO parton shower Monte Carlo generator that has only recently become available for inclusive jet and dijet production. \powheguses the \powhegboxpackage Nason (2004); Frixione et al. (2007); Alioli et al. (2010) and allows one to use either \pythiaor \herwig Corcella et al. (2001) + \jimmyButterworth et al. (1996) to shower the partons, hadronise them, and model the underlying event. The ATLAS underlying event tunes, AUET2B for \pythiaand AUET2 ATLAS Collaboration (2010c) for \herwig, are derived from the standalone versions of these event generators, with no optimisation for the \powhegpredictions. The showering portion of \powheguses the PDFs from \pythiaor \herwigas part of the specific tune chosen.

In the \powhegalgorithm, each event is built by first producing a QCD partonic scattering. The renormalisation and factorisation scales are set to be equal to the transverse momentum of the outgoing partons, , before proceeding to generate the hardest partonic emission in the event.222Technical details of the \powheggeneration parameters, which are discussed below, are given in Refs. Alioli et al. (2011a); Nason (2007). The folding parameters used are 5-10-2. A number of different weighting parameters are used to allow coverage of the complete phase space investigated: 25 GeV, 250 GeV and 400 GeV. The minimum Born is 5 GeV. For all the samples, the leading jet transverse momentum is required to be no more than seven times greater than the leading parton’s momentum. The of any additional partonic interactions arising from the underlying event is required to be lower than that of the hard scatter generated by \powheg. The parameters used in the input file for the event generation are bornktmin = 5 \GeV, bornsuppfact = 2, 250, 400 \GeV, foldcsi = 5, foldy = 10, and foldphi = 2. The CT10 NLO PDF set is used in this step of the simulation. Then the event is evolved to the hadron level using a parton shower event generator, where the radiative emissions in the parton showers are required to be softer than the hardest partonic emission generated by \powheg.

The coherent simulation of the parton showering, hadronisation, and the underlying event with the NLO matrix element is expected to produce a more accurate theoretical prediction. In particular, the non-perturbative effects are modeled in the NLO parton shower simulation itself, rather than being derived separately using a LO parton shower Monte Carlo generator as described in Sec. V.1.2.

Vi Data Selection and Calibration

vi.1 Dataset

The inclusive jet and dijet cross section measurements use the full ATLAS 2010 data sample from proton-proton collisions at  TeV.

For low- jets, only the first 17 nb of data taken are considered since the instantaneous luminosity of the accelerator was low enough that a large data sample triggered with a minimum bias trigger (see Sec. VI.2) could be recorded. This provides an unbiased sample for reconstructing jets with between 20-60 GeV, below the lowest jet trigger threshold. In addition, during this period there were negligible contributions from “pile-up” events, in which there are multiple proton-proton interactions during the same or neighbouring bunch crossings. Thus this period provides a well-measured sample of low- jets. The first data taking period was not used for forward jets with and  GeV because the forward jet trigger was not yet commissioned.

For all events considered in this analysis, good operation status was required for the first-level trigger, the solenoid magnet, the inner detector, the calorimeters and the luminosity detectors, as well as for tracking and jet reconstruction. In addition, stable operation was required for the high-level trigger during the periods when this system was used for event rejection.

vi.2 Trigger

Three different triggers have been used in this measurement: the minimum bias trigger scintillators (MBTS); the central jet trigger, covering ; and the forward jet trigger, spanning . The MBTS trigger requires at least one hit in the minimum bias scintillators located in front of the endcap cryostats, covering , and is the primary trigger used to select minimum-bias events in ATLAS. It has been demonstrated to have negligible inefficiency for the events of interest for this analysis ATLAS Collaboration (2010d) and is used to select events with jets having transverse momenta in the range 20-60 GeV. The central and forward jet triggers are composed of three consecutive levels: Level 1 (L1), Level 2 (L2) and Event Filter (EF). In 2010, only L1 information was used to select events in the first 3 pb of data taken, while both the L1 and L2 stages were used for the rest of the data sample. The jet trigger did not reject events at the EF stage in 2010.

The central and forward jet triggers independently select data using several thresholds for the jet transverse energy (), each of which requires the presence of a jet with sufficient at the electromagnetic (EM) scale.333The electromagnetic scale is the basic calorimeter signal scale for the ATLAS calorimeters. It has been established using test-beam measurements for electrons and muons to give the correct response for the energy deposited in electromagnetic showers, while it does not correct for the lower response of the calorimeter to hadrons. For each L1 threshold, there is a corresponding L2 threshold that is generally 15 GeV above the L1 value. Each such L1+L2 combination is referred to as an L2 trigger chain. Fig. 4 shows the efficiency for L2 jet trigger chains with various thresholds as a function of the reconstructed jet for jets with for both the central and forward jet triggers. Similar efficiencies are found for jets with , such that the same correspondence between transverse momentum regions and trigger chains can be used for the two jet sizes. The highest trigger chain does not apply a threshold at L2, so its L1 threshold is listed.

Figure 4: Combined L1+L2 jet trigger efficiency as a function of reconstructed jet for \AKTjets with in the central region (a), the barrel-endcap transition region (b) and the FCal region (c) for the different L2 trigger thresholds used in the analysis. The trigger thresholds are at the electromagnetic scale, while the jet is at the calibrated scale (see Sec. VI.3). The highest trigger chain used for does not apply a threshold at L2, so its L1 threshold is listed. The efficiency in the rapidity range is not expected to reach 100% due to the presence of a dead FCal trigger tower that spans 0.9% of the -acceptance. This inefficiency is assigned as a systematic uncertainty on the trigger efficiency in the measurement.

As the instantaneous luminosity increased throughout 2010, it was necessary to prescale triggers with lower thresholds, while the central jet trigger with the highest threshold remained unprescaled. As a result, the vast majority of the events where the leading jet has transverse momentum smaller than about 100 GeV have been taken in the first period of data-taking, under conditions with a low amount of pile-up, while the majority of the high- events have been taken during the second data-taking period, with an average of 2-3 interactions per bunch crossing. For each -bin considered in this analysis, a dedicated trigger chain is chosen that is fully efficient () while having as small a prescale factor as possible. For inclusive jets fully contained in the central or in the forward trigger region, only events taken by this fully efficient trigger are considered. For inclusive jets in the HEC-FCal transition region , neither the central nor the forward trigger is fully efficient. Instead, the logical OR of the triggers is used, which is fully efficient at sufficiently high jet (see Fig. 5).

Figure 5: Efficiencies for the central and forward jet triggers with a L1 threshold of 10 GeV, and for their logical OR, as a function of the rapidity of the reconstructed jet in the transition region between the two trigger systems. The logical OR is used for the inclusive jet measurement to collect data in the rapidity slice.

A specific strategy is used to account for the various prescale combinations for inclusive jets in the HEC-FCal transition region, which can be accepted either by the central jet trigger only, by the forward jet trigger only, or by both. A similar strategy is used for dijet events in a given -bin, which can be accepted by several jet triggers depending on the transverse momenta and pseudorapidities of the two leading jets. Events that can be accepted by more than one trigger chain have been divided into several categories according to the trigger combination that could have accepted the events. For inclusive jets in the transition region, these correspond to central and forward triggers with a similar threshold; for dijets the trigger combination depends on the position and transverse momenta of the two leading jets, each of which is “matched” to a trigger object using angular criteria. Corrections are applied for any trigger inefficiencies, which are generally below 1%. The equivalent luminosity of each of the categories of events is computed based on the prescale values of these triggers throughout the data-taking periods, and all results from the various trigger combinations are combined together according to the prescription given in Ref. Lendermann et al. (2009).

vi.3 Jet Reconstruction and Calibration

Jets are reconstructed at the electromagnetic scale using the \AKT algorithm. The input objects to the jet algorithm are three-dimensional topological clusters Lampl et al. (2008) built from calorimeter cells. The four-momentum of the uncalibrated, EM-scale jet is defined as the sum of the four-momenta of its constituent calorimeter energy clusters. Additional energy due to multiple proton-proton interactions within the same bunch crossing (“pile-up”) is subtracted by applying a correction derived as a function of the number of reconstructed vertices in the event using minimum bias data. The energy and the position of the jet are next corrected for instrumental effects such as dead material and non-compensation. This jet energy scale (JES) correction is calculated using isolated jets444An isolated jet is defined as a jet that has no other jet within , where is the clustering parameter of the jet algorithm. in the Monte Carlo simulation as a function of energy and pseudorapidity of the reconstructed jet. The JES correction factor ranges from about 2.1 for low-energy jets with  GeV in the central region to less than 1.2 for high-energy jets in the most forward region . The corrections are cross-checked using in-situ techniques in collision data (see below) ATLAS Collaboration (2011d).

vi.4 Uncertainties in Jet Calibration

The uncertainty on the jet energy scale is the dominant uncertainty for the inclusive jet and dijet cross section measurements. Compared to the previous analysis ATLAS Collaboration (2011a), this uncertainty has been reduced by up to a factor of two, primarily due to the improved calibration of the calorimeter electromagnetic energy scale obtained from events ATLAS Collaboration (2011e), as well as an improved determination of the single particle energy measurement uncertainties from in-situ and test-beam measurements ATLAS Collaboration (2009). This improvement is confirmed by independent measurements, including studies of the momenta of tracks associated to jets, as well as the momentum balance observed in +jet, dijet, and multijet events ATLAS Collaboration (2011d).

In the central barrel region (), the dominant source of the JES uncertainty is the knowledge of the calorimeter response to hadrons. This uncertainty is obtained by measuring the response to single hadrons using proton-proton and test-beam data, and propagating the uncertainties to the response for jets. Additional uncertainties are evaluated by studying the impact on the calorimeter response from varying settings for the parton shower, hadronization, and underlying event in the Monte Carlo simulation. The estimate of the uncertainty is extended from the central calorimeter region to the endcap and forward regions, the latter of which lies outside the tracking acceptance, by exploiting the transverse momentum balance between a central and a forward jet in events where only two jets are produced.

In the central region (), the uncertainty is lower than 4.6% for all jets with  GeV, which decreases to less than 2.5% uncertainty for jet transverse momenta between 60 and 800 GeV. The JES uncertainty is the largest for low-\pt (20 GeV) jets in the most forward region , where it is about 11-12%. Details of the JES determination and its uncertainty are given in Ref. ATLAS Collaboration (2011d).

vi.5 Offline Selection

vi.5.1 Event Selection

To reject events due to cosmic-ray muons and other non-collision backgrounds, events are required to have at least one primary vertex that is consistent with the beamspot position and that has at least five tracks associated to it. The efficiency for collision events to pass these vertex requirements, as measured in a sample of events passing all selections of this analysis, is well over 99%.

vi.5.2 Jet Selection

For the inclusive jet measurements, jets are required to have  GeV and to be within . They must also pass the specific fully-efficient trigger for each - and -bin, as detailed in Sec. VI.2. For the dijet measurements, events are selected if they have at least one jet with  GeV and another jet with  GeV, both within . Corrections are applied for inefficiencies in jet reconstruction, which are generally less than a few percent.

Jet quality criteria first established with early collision data are applied to reject jets reconstructed from calorimeter signals that do not originate from a proton-proton collision, such as those due to noisy calorimeter cells ATLAS Collaboration (2011d). For this analysis, various improvements to the jet quality selection have been made due to increased experience with a larger data set and evolving beam conditions, including the introduction of new criteria for the forward region.

The main sources of fake jets were found to be: noise bursts in the hadronic endcap calorimeter electronics; coherent noise from the electromagnetic calorimeter; cosmic rays; and beam-related backgrounds.

Quality selection criteria were developed for each of these categories by studying jet samples classified as real or fake energy depositions. This classification was performed by applying criteria on the magnitude and direction of the missing transverse momentum, . Following this, about a dozen events with  GeV were found that pass the standard analysis selection. These events were visually scanned and were generally found to be collision events with mostly low jets and a muon escaping at low scattering angle.

Figure 6: Efficiency for jet quality selection as a function of for \AKTjets with in the rapidity region . The black circles indicate the efficiency measured in-situ using a tag-probe method. The blue squares indicate the fit to the parameterisation used in this analysis, where , , and are fitted constants, and the shaded band indicates the systematic uncertainty on the efficiency obtained by varying the tag jet selection. The turn-on is due to more stringent jet quality selection at low jet .

The efficiency for identifying real jets was measured using a tag-and-probe method. A “probe jet” sample was selected by requiring the presence of a “tag jet” that is within , fulfills the jet quality criteria, and is back-to-back ( 2.6) and well-balanced with a probe jet (, with and where are the transverse momenta of the tag and probe jets). The jet quality criteria were then applied to the probe jet, measuring as a function of its and the fraction of jets that are not rejected.

The efficiency to select a jet is shown in Fig. 6 for an example rapidity region, along with the systematic uncertainty on this efficiency.

The jet quality selection efficiency is greater than 96% for jets with  GeV and quickly increases with jet . The efficiency is above 99% for jet  GeV in all rapidity regions. The inclusive jet and dijet cross sections are corrected for these inefficiencies in regions where the efficiency is less than 99%. The systematic uncertainty on the efficiency is taken as a systematic uncertainty on the cross section.

vi.6 Background, Vertex Position, and Pile-Up

Background contributions from sources other than proton-proton collisions were evaluated using events from cosmic-ray runs, as well as unpaired proton bunches in the accelerator, in which no real collision candidates are expected. Based on the duration of the cosmic-ray runs and the fact that only one event satisfied the selection criteria, the non-collision background rates across the entire data period are considered to be negligible.

The primary vertices span the luminous region around the nominal beamspot. To determine the systematic uncertainty due to possibly incorrect modeling of the event vertex position, the jet spectrum was studied as a function of the position of the primary vertex with the largest of associated tracks. The fraction of events with  mm is 0.06%, and the difference in the spectrum compared to events with  mm is small. Consequently, the uncertainty from mis-modeling of the vertex position was taken to be negligible.

The of each jet is corrected for additional energy from soft pile-up interactions in the event (see Sec. VI.3). An uncertainty associated to this pile-up offset correction is assigned that is dependent on the number of reconstructed primary vertices, as described in Sec. VIII.1. The jet measurements are then compared to the Monte Carlo simulation without pile-up.

vi.7 Luminosity

The integrated luminosity is calculated by measuring interaction rates using several ATLAS devices, where the absolute calibration is derived using van der Meer scans ATLAS Collaboration (2011f). The uncertainty on the luminosity is 3.4% ATLAS Collaboration (2011g). The calculation of the effective luminosity for each bin of the observable for inclusive jets follows the trigger scheme described in Sec. VI.2. The integrated luminosity for each individual trigger is derived using separate prescale factors for each luminosity block (an interval of luminosity with homogeneous data-taking conditions, which is typically two minutes). For dijets, each bin receives contributions from several trigger combinations, for which the luminosity is calculated independently. The luminosity that would be obtained without correction for trigger prescale is (37.3 1.2) pb. Since the central jet trigger with the largest transverse momentum threshold was always unprescaled, this is the effective luminosity taken for jets with transverse momentum above about 220 GeV.

Vii Unfolding

vii.1 Technique used

Aside from the jet energy scale correction, all other corrections for detector inefficiencies and resolutions are performed using an iterative unfolding, based on a transfer matrix that relates the particle-level and reconstruction-level observable, with the same binning as the final distribution. The unfolding is performed separately for each bin in rapidity since the migrations across rapidity bins are negligible compared to those across jet (dijet mass) bins. A similar procedure is applied for inclusive jets and dijets, with the following description applying specifically to the inclusive jet case.

The Monte Carlo simulation described in Sec. IV is used to derive the unfolding matrices. Particle-level and reconstructed jets are matched together based on geometrical criteria and used to derive a transfer matrix. This matrix contains the expected number of jets within each bin of particle-level and reconstructed jet . A folding matrix is constructed from the transfer matrix by normalising row-by-row so that the sum of the elements corresponding to a given particle-level jet is unity. Similarly, an unfolding matrix is constructed by normalising column-by-column so that the sum of the elements corresponding to a specific reconstructed jet is unity. Thus each element of the unfolding matrix reflects the probability for a reconstructed jet in a particular bin to originate from a specific particle-level bin, given the assumed input particle-level jet spectrum. The spectra of unmatched particle-level and reconstructed jets are also derived from the simulated sample. The ratio between the number of matched jets and the total number of jets provides the matching efficiency both for particle-level jets, , and for reconstructed jets, .

The data are unfolded to particle level using a three-step procedure, with the final results being given by the equation:

(5)

where and are the particle-level and reconstructed bin indices, respectively, and is an unfolding matrix refined through iteration, as discussed below.

The first step is to multiply the reconstructed jet spectrum in data by the matching efficiency , such that it can be compared to the matched reconstructed spectrum from the Monte Carlo simulation. In the second step, the iterated unfolding matrix is determined using the Iterative, Dynamically Stabilised (IDS) method Malaescu (2009). This procedure improves the transfer matrix through a series of iterations, where the particle-level distribution is reweighted to the shape of the corrected data spectrum, while leaving the folding matrix unchanged. The main difference with respect to previous iterative unfolding techniques D’Agostini (1995) is that, when performing the corrections, regularisation is provided by the use of the significance of the data-MC differences in each bin. The third step is to divide the spectrum obtained after the iterative unfolding by the matching efficiency at particle level, thus correcting for the jet reconstruction inefficiency.

The statistical uncertainties on the spectrum are propagated through the unfolding by performing pseudo-experiments. An ensemble of pseudo-experiments is created in which each bin of the transfer matrix is varied according to its statistical uncertainty. A separate set of pseudo-experiments is performed where the data spectrum is varied while respecting correlations between jets produced in the same event. The unfolding is then applied to each pseudo-experiment, and the resulting ensembles are used to calculate the covariance matrix of the corrected spectrum.

As a cross-check, the results obtained from the iterative unfolding have been compared to those using a simpler bin-by-bin correction procedure, as well as the “singular value decomposition” (SVD) method implemented in TSVDUnfold Hoecker and Kartvelishvili (1996); Tackmann et al. (2010). These methods use different regularisation procedures and rely to different degrees on the Monte Carlo simulation modelling of the shape of the spectrum. The unfolding techniques have been tested using a data-driven closure test Malaescu (2009). In this test the particle-level spectrum in the Monte Carlo simulation is reweighted and convolved through the folding matrix such that a significantly improved agreement between the data and the reconstructed spectrum from the Monte Carlo simulation is attained. The reweighted, reconstructed spectrum in the Monte Carlo simulation is then unfolded using the same procedure as for the data. The comparison of the result with the reweighted particle-level spectrum from the Monte Carlo simulation provides the estimation of the bias.

The bin-by-bin method gives results consistent with those obtained using the IDS technique, but requires the application of an explicit correction for the NLO k-factor to obtain good agreement. A somewhat larger bias is observed for the SVD method.

Figure 7: The jet shape measured using calorimeter energy clusters for \AKTjets with in the rapidity interval , compared to \pythiawith tune AMBT1 (used for unfolding), and for jets with transverse momenta in the range . The statistical error bars are smaller than the size of the markers, while systematic errors are not shown.

vii.2 Cross-check with jet shapes

The use of Monte Carlo simulation to derive the transfer matrix in the unfolding procedure requires that the simulation models the jet properties well. The modelling of the energy flow around the jet core provides a useful test of this. The energy and momentum flow within a jet can be expressed in terms of the differential jet shape, defined for a jet with radius parameter , as the fraction , where is the transverse momentum within a radius of the jet centre, and is the transverse momentum contained within a ring of thickness at a radius from the jet centre.

Jet shape measurements using calorimeter energy clusters and tracks were performed with 3 pb of data ATLAS Collaboration (2011h), and show good agreement with the \pythiaand \herwig+ \jimmyMonte Carlo simulations in the kinematic region  GeV and rapidity . Using the same technique, the uncorrected jet shapes in the forward rapidity region have been studied in the context of the present analysis. As an example, the results for the HEC-FCal transition region , the most difficult detector region to model, are shown in Fig. 7. The maximum disagreement in shape between data and the Monte Carlo simulation is approximately 20%, demonstrating that the distribution of energy within the jets is reasonably well-modeled even in this worst case. Any bias from mis-modeling of the jet shape is included in the unfolding uncertainties described below, so this jet shape study serves only as a cross-check.

Viii Systematic uncertainties and correlations

viii.1 Uncertainty sources from jet reconstruction and calibration

The uncertainty on the jet reconstruction efficiency for (within the tracking acceptance) is evaluated using track jets, which are used to play the role of “truth jets”. In this paper, truth jets are defined to be jets at the particle level, but excluding muons and neutrinos. The efficiency to reconstruct a calorimeter jet given a track jet nearby is studied in both data and the MC simulation. The data versus MC comparison of this efficiency is used to infer the degree to which the calorimeter jet reconstruction efficiency may be mis-modeled in the Monte Carlo simulation. The disagreement was found to be 2% for calorimeter jets with of 20 GeV and less than 1% for those with  GeV. The disagreement for jets with is taken as a systematic uncertainty for all jets in the rapidity range . This is expected to be a conservative estimate in the forward region where the jets have higher energy for a given .

The JES uncertainty was evaluated as described in Sec. VI.4 and in Ref. ATLAS Collaboration (2011d). The jet energy and angular resolutions are estimated from the Monte Carlo simulation using truth jets that have each been matched to a reconstructed calorimeter jet. The jet energy resolution (JER) in the Monte Carlo simulation is compared to that obtained in data using two in-situ techniques, one based on dijet balance and the other using a bisector method ATLAS Collaboration (2010e). In general the two resolutions agree within 14%, and the full difference is taken as a contribution to the uncertainty on the unfolding corrections, which propagates to a systematic uncertainty on the measured cross section as described in Sec. VIII.2. The angular resolution is estimated from the angle between each calorimeter jet and its matched truth-level jet. The associated systematic uncertainty is assessed by varying the requirement that the jet is isolated.

The JES uncertainty due to pile-up is proportional to , where is the number of reconstructed vertices. The total pile-up uncertainty for a given -bin is calculated as the average of the uncertainties for each value of weighted by the relative frequency of that number of reconstructed vertices in the bin.

viii.2 Uncertainty propagation

The uncertainty of the measured cross section due to jet energy scale and jet energy and angular resolutions has been estimated using the Monte Carlo simulation by repeating the analysis after systematically varying these effects. The jet energy scale applied to the reconstructed jets in MC is varied separately for each JES uncertainty source both up and down by one standard deviation. The resulting spectra are unfolded using the nominal unfolding matrix, and the relative shifts with respect to the nominal unfolded spectra are taken as uncertainties on the cross section. The effects of the jet energy and angular resolutions are studied by smearing the reconstructed jets such that these resolutions are increased by one standard deviation of their respective uncertainties (see Sec. VIII.1). For each such variation, a new transfer matrix is constructed, which is used to unfold the reconstructed jet spectrum of the nominal MC sample. The relative shift of this spectrum with respect to the nominal unfolded spectra is taken as the uncertainty on the cross section.

The impact of possible mis-modeling of the cross section shape in the Monte Carlo simulation is assessed by shape variations of the particle-level jet spectra introduced to produce reconstructed-level spectra in agreement with data as discussed in Sec. VII.

The total uncertainty on the unfolding corrections is defined as the sum in quadrature of the uncertainties on the jet energy resolution, jet angular resolution, and the simulated shape. It is approximately 4-5% at low and high (except for the lowest -bin at 20 GeV, where it reaches 20%), and is smaller at intermediate values. This uncertainty is dominated by the component from the jet energy resolution.

viii.3 Summary of the magnitude of the systematic uncertainties

The largest systematic uncertainty for this measurement arises from the jet energy scale. Even with the higher precision achieved recently as described in Sec. VI.4, the very steeply falling jet spectrum, especially for large rapidities, translates even relatively modest uncertainties on the transverse momentum into large changes for the measured cross section.

As described in Sec. VI.7, the luminosity uncertainty is 3.4%. The detector unfolding uncertainties have been discussed in the previous subsection. Various other sources of systematic uncertainties were considered and were found to have a small impact on the results. The jet energy and angular resolutions, as well as the jet reconstruction efficiency, also contribute to the total uncertainty through the unfolding corrections.

The dominant systematic uncertainties for the measurement of the inclusive jet spectrum in representative and regions for jets with are shown in Table 1. Similarly, the largest systematic uncertainties for the dijet mass measurement are given for a few representative and regions in Table 2.

An example of the breakdown of the systematic uncertainties as a function of the jet transverse momentum for the various rapidity bins used in the inclusive jet measurement is shown in Fig. 8.

Figure 8: The magnitude (left) and correlation between \pt-bins (right) of the total systematic uncertainty on the inclusive jet cross section measurement for \AKT jets with in three representative -bins. The magnitudes of the uncertainties from the jet energy scale (JES), the jet energy resolution (JER), and other sources are shown separately. The correlation matrix is calculated after symmetrising the uncertainties. The statistical uncertainty and the 3.4% uncertainty of the integrated luminosity are not shown here.
[GeV]  JES JER Trigger Jet Rec.
20–30 2.1–2.8 17%
20–30 3.6–4.4 13%
80–110 0.3 10% 1%
Table 1: The effect of the dominant systematic uncertainty sources on the inclusive jet cross section measurement, for representative and regions for jets with .
[TeV]  JES JER Trigger Jet Rec.
0.37–0.44 2.0–2.5 7%
2.55–3.04 4.0–4.4 8%
0.21–0.26 0.5 1%
Table 2: The effect of the dominant systematic uncertainty sources on the dijet cross section measurement, for representative and regions for jets with .

viii.4 Correlations

-bins
Uncertainty Source 0-0.3 0.3-0.8 0.8-1.2 1.2-2.1 2.1-2.8 2.8-3.6 3.6-4.4
JES 1: Noise threshold 1 1 2 3 4 5 6
JES 2: Theory UE 7 7 8 9 10 11 12
JES 3: Theory Showering 13 13 14 15 16 17 18
JES 4: Non-closure 19 19 20 21 22 23 24
JES 5: Dead material 25 25 26 27 28 29 30
JES 6: Forward JES 31 31 31 31 31 31 31
JES 7: response 32 32 33 34 35 36 37
JES 8: selection 38 38 39 40 41 42 43
JES 9: EM+neutrals 44 44 45 46 47 48 49
JES 10: HAD -scale 50 50 51 52 53 54 55
JES 11: High 56 56 57 58 59 60 61
JES 12: bias 62 62 63 64 65 66 67
JES 13: Test-beam bias 68 68 69 70 71 72 73
Unfolding 74 74 74 74 74 74 74
Jet matching 75 75 75 75 75 75 75
Jet energy resolution 76 76 77 78 79 80 81
-resolution 82 82 82 82 82 82 82
Jet reconstruction eff. 83 83 83 83 84 85 86
Luminosity 87 87 87 87 87 87 87
JES 14: Pile-up u u u u u u u
Trigger u u u u u u u
Jet identification u u u u u u u
Table 3: Description of bin-to-bin uncertainty correlation for the inclusive jet measurement. Each number corresponds to a nuisance parameter for which the corresponding uncertainty is fully correlated versus . Bins with the same nuisance parameter are treated as fully correlated, while bins with different nuisance parameters are uncorrelated. The sources indicated by the letter “u” are uncorrelated both between - and -bins. The one-standard-deviation amplitude of the systematic effect associated with each nuisance parameter is detailed in Tables 518 in Appendix B. The JES uncertainties for jets with are determined relative to the JES of jets with . As a consequence, several of the uncertainties that are determined using jets with are also propagated to the more forward rapidities (such as the uncertainties). Descriptions of the JES uncertainty sources can be found in Refs. ATLAS Collaboration (2011d) and ATLAS Collaboration (2011i). All tables are available on \hepdata ATLAS Collaboration (2011j).
-bins
Uncertainty Source 0.0-0.5 0.5-1.0 1.0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0 4.0-4.4
JES 1: Noise threshold 1 1 2 3 4 4 5 6 6
JES 2: Theory UE 7 7 8 9 10 10 11 12 12
JES 3: Theory Showering 13 13 14 15 16 16 17 18 18
JES 4: Non-closure 19 19 20 21 22 22 23 24 24
JES 5: Dead material 25 25 26 27 28 28 29 30 30
JES 6: Forward JES 31 31 31 31 31 31 31 31 31
JES 7: response 32 32 33 34 35 35 36 37 37
JES 8: selection 38 38 39 40 41 41 42 43 43
JES 9: EM+neutrals 44 44 45 46 47 47 48 49 49
JES 10: HAD -scale 50 50 51 52 53 53 54 55 55
JES 11: High 56 56 57 58 59 59 60 61 61
JES 12: bias 62 62 63 64 65 65 66 67 67
JES 13: Test-beam bias 68 68 69 70 71 71 72 73 73
Unfolding 74 74 74 74 74 74 74 74 74
Jet matching 75 75 75 75 75 75 75 75 75
Jet energy resolution 76 76 77 78 79 79 80 81 81
-resolution 82 82 82 82 82 82 82 82 82
Jet reconstruction eff. 83 83 83 83 84 84 85 86 86
Luminosity 87 87 87 87 87 87 87 87 87
JES 14: Pile-up u u u u u u u u u
Trigger u u u u u u u u u
Jet identification u u u u u u u u u
Table 4: Description of bin-to-bin uncertainty correlation for the dijet measurement. Each number corresponds to a nuisance parameter for which the corresponding uncertainty is fully correlated versus dijet mass, . Bins with the same nuisance parameter are treated as fully correlated, while bins with different nuisance parameters are uncorrelated. The sources indicated by the letter “u” are uncorrelated both between - and -bins. The one-standard-deviation amplitude of the systematic effect associated with each nuisance parameter is detailed in Tables 1936 in Appendix C. Descriptions of the JES uncertainty sources can be found in Refs. ATLAS Collaboration (2011d) and ATLAS Collaboration (2011i). All tables are available on \hepdata ATLAS Collaboration (2011j).

The behaviour of various sources of systematic uncertainty in different parts of the detector has been studied in detail in order to understand their correlations across various , and rapidity bins. As shown in Tables 3 and 4, 22 independent sources of systematic uncertainty have been identified, including luminosity, jet energy scale and resolution, and theory effects such as the uncertainty of the modeling of the underlying event and the QCD showering. For example, the sources labeled “JES 7–13” in these tables correspond to the calorimeter response to hadrons, which dominates the JES uncertainty in the central region. After examining the rapidity dependence of all 22 sources, it was found that 87 independent nuisance parameters are necessary to describe the correlations over the whole phase space. The systematic effect on the cross section measurement associated with each nuisance parameter in its range of use is completely correlated in and (dijet mass and ). These parameters represent correlations between the uncertainties of the various bins. Since many of the systematic effects are not symmetric, it is not possible to provide a covariance matrix containing the full information. For symmetric uncertainties corresponding to independent sources, the total covariance matrix is given by:

(6)

where is an index running over the nuisance parameters, and is the one-standard-deviation amplitude of the systematic effect due to source in bin . The full list of relative uncertainties, , where each uncertainty may be asymmetric, is given for all sources and bins of this analysis in Tables 518 and 1936. Fig. 8 shows the magnitude and approximate bin-to-bin correlations of the total systematic uncertainty of the inclusive jet cross section measurement. The correlation matrix is here converted from the covariance matrix, which is obtained using Eq. 6, after symmetrising the uncertainties: . The inclusive jet and dijet data should not be used simultaneously for PDF fits due to significant correlations between the two measurements.

Ix Results and Discussion

ix.1 Inclusive Jet Cross Sections

The inclusive jet double-differential cross section is shown in Figs. 9 and 10 and Tables 518 in Appendix B for jets reconstructed with the \AKTalgorithm with and . The measurement extends from jet transverse momentum of 20 GeV to almost 1.5 TeV, spanning two orders of magnitude in and ten orders of magnitude in the value of the cross section. The measured cross sections have been corrected for all detector effects using the unfolding procedure described in Sec. VII. The results are compared to \nlojetpredictions (using the CT10 PDF set) corrected for non-perturbative effects, where the theoretical uncertainties from scale variations, parton distribution functions, and non-perturbative corrections have been accounted for.

In Figs. 1113, the inclusive jet results are presented in terms of the ratio with respect to the \nlojetpredictions using the CT10 PDF set. Fig. 11 compares the current results to the previous measurements published by ATLAS ATLAS Collaboration (2011a), for jets reconstructed with the algorithm with parameter . This figure is limited to the central region, but similar conclusions can be drawn in all rapidity bins. In particular the two measurements are in good agreement, although the new results cover a much larger kinematic range with much reduced statistical and systematic uncertainties.

Figure 9: Inclusive jet double-differential cross section as a function of jet in different regions of for jets identified using the \AKTalgorithm with . For convenience, the cross sections are multiplied by the factors indicated in the legend. The data are compared to NLO pQCD calculations using \nlojetto which non-perturbative corrections have been applied. The error bars, which are usually smaller than the symbols, indicate the statistical uncertainty on the measurement. The dark-shaded band indicates the quadratic sum of the experimental systematic uncertainties, dominated by the jet energy scale uncertainty. There is an additional overall uncertainty of 3.4% due to the luminosity measurement that is not shown. The theory uncertainty, shown as the light, hatched band, is the quadratic sum of uncertainties from the choice of the renormalisation and factorisation scales, parton distribution functions, , and the modeling of non-perturbative effects, as described in the text.
Figure 10: Inclusive jet double-differential cross section as a function of jet in different regions of for jets identified using the \AKTalgorithm with . For convenience, the cross sections are multiplied by the factors indicated in the legend. The data are compared to NLO pQCD calculations using \nlojetto which non-perturbative corrections have been applied. The theoretical and experimental uncertainties indicated are calculated as described in Fig. 9.
Figure 11: Ratio of inclusive jet cross section to the theoretical prediction obtained using \nlojetwith the CT10 PDF set. The ratio is shown as a function of jet in the rapidity region , for jets identified using the \AKTalgorithm with . The current result is compared to that published in Ref. ATLAS Collaboration (2011a).

Fig. 12 shows the ratio of the measured cross sections to the \nlojettheoretical predictions for various PDF sets. Predictions obtained using CT10, MTSW 2008, NNPDF 2.1, and HERAPDF 1.5, including uncertainty bands, are compared to the measured cross sections, where data and theoretical predictions are normalised to the prediction from the CT10 PDF set. The data show a marginally smaller cross section than the predictions from each of the PDF sets. This trend is more pronounced for the measurements corresponding to the \AKTalgorithm with parameter , compared to .

The description becomes worse for large jet transverse momenta and rapidities, where the MSTW 2008 PDF set follows the measured trend better. However, the differences between the measured cross section and the prediction of each PDF set are of the same order as the total systematic uncertainty on the measurement, including both experimental and theoretical uncertainty sources. A test of the compatibility between data and the PDF curves, accounting for correlations between bins, provides reasonable probabilities for all sets, with non-significant differences between them.555Comparisons to HERAPDF 1.0, CTEQ 6.6 and NNPDF 2.0 were also performed, but they are not shown as they are very similar to those for HERAPDF 1.5, CT10, and NNPDF 2.1, respectively.

The comparison of the data with the \powhegprediction, using the CT10 NLO PDF set, is shown for jets with and in different rapidity regions in Fig. 13. The data are compared with four theory curves, all of which are normalised to the same common denominator of the \nlojetprediction corrected for non-perturbative effects: \powhegshowered with \pythiawith the default AUET2B tune; the same with the Perugia 2011 tune; \powhegshowered with \herwig; and \powhegrun in “pure NLO” mode (fixed-order calculation), without matching to parton shower, after application of soft corrections calculated using \pythiaand the AUET2B tune. Scale uncertainties are not shown for the \powhegcurves, but they have been found to be similar to those obtained with \nlojet.

Figure 12: Ratios of inclusive jet double-differential cross section to the theoretical prediction obtained using \nlojetwith the CT10 PDF set. The ratios are shown as a function of jet in different regions of for jets identified using the \AKTalgorithm with (upper plots) and (lower plots). The theoretical error bands obtained by using \nlojetwith different PDF sets (CT10, MSTW 2008, NNPDF 2.1, HERAPDF 1.5) are shown. Statistically insignificant data points at large are omitted in the ratio.
Figure 13: Ratios of inclusive jet double-differential cross section to the theoretical prediction obtained using \nlojetwith the CT10 PDF set. The ratios are shown as a function of jet in different regions of for jets identified using the \AKTalgorithm with (upper plots) and (lower plots). The ratios of \powhegpredictions showered using either \pythiaor \herwigto the \nlojetpredictions corrected for non-perturbative effects are shown and can be compared to the corresponding ratios for data. Only the statistical uncertainty on the \powhegpredictions is shown. The total systematic uncertainties on the theory and the measurement are indicated. The \nlojetprediction and the \powhegME calculations use the CT10 PDF set. Statistically insignificant data points at large are omitted in the ratio.

Good agreement at the level of a few percent is observed between NLO fixed-order calculations based on \nlojetand \powheg, as described in Sec. V.1.1. However, significant differences reaching are observed if \powhegis interfaced to different showering and soft physics models, particularly at low and forward rapidity, but also at high . These differences exceed the uncertainties on the non-perturbative corrections, which are not larger than 10% for the inclusive jet measurements with , thus indicating a significant impact of the parton shower. The Perugia 2011 tune tends to produce a consistently larger cross section than the standard AUET2B tune over the full rapidity range. The technique of correcting fixed-order calculations for non-perturbative effects remains the convention to define the baseline theory prediction until NLO parton shower generators become sufficiently mature to describe data well. The corrected NLO result predicts a consistently larger cross section than that seen in the data. Good agreement in normalisation is found between the data and the prediction from \powhegshowered with the default tune of \pythia. These results are confirmed by a test of the compatibility of the \powhegresults with the data, where the curve obtained using the \herwigshower results in a much worse after all error correlations have been accounted for.

ix.2 Dijet Cross Sections

The dijet double-differential cross section has been measured as a function of the dijet invariant mass for various bins of the variable , which is the rapidity in the two-parton centre-of-mass frame. The quantity is calculated as half the absolute value of the rapidity difference of the two leading jets, ranging from 0 to 4.4. The results are shown in Figs. 14 and 15 and Tables 1936 in Appendix C for jets with and . The cross section measurements extend from dijet masses of 70 GeV to almost 5 TeV, covering two orders of magnitude in invariant mass and nine orders of magnitude in the cross section. The dijet measurements are fully corrected for detector effects and are compared to \nlojetpredictions calculated using the scale defined in Eq. 3 (see Sec. V.1.1) and the CT10 PDF set, with non-perturbative corrections applied to the theory prediction. The theoretical uncertainties have been assessed as described for the inclusive jet measurements in Sec. IX.1.

The dijet data are also compared with \nlojetpredictions obtained using the MSTW 2008, NNPDF 2.1, and HERAPDF 1.5 PDF sets. Figs. 16 and 17 show the dijet mass spectra for jets with and respectively, where both the data and the predictions from the above-mentioned PDF sets have been normalised to the CT10 prediction. The data for exhibit a slight falling slope with respect to the CT10 prediction and appear to be described better by other PDF sets, a similar behaviour to that observed in the inclusive jet data. However, in all cases, the differences between the data and each PDF set lie well within the systematic and theory uncertainties, indicating a reasonable agreement with the dijet data, particularly in the kinematic region at low .

The data are also compared with \powhegpredictions produced using the CT10 PDF set and showered with different tunes of the \pythiaor \herwiggenerator. These comparisons are shown for and respectively in Figs. 18 and 19, where the data and all theory predictions have been normalised to the \nlojetprediction with CT10. The \nlojetprediction has been corrected for non-perturbative effects calculated using the \pythiaMC with the AUET2B tune. The \powhegpredictions shown are interfaced to the \pythiaparton shower with the AUET2B or Perugia2011 tune, and to the \herwigparton shower using the AUET2 tune. The data are also compared to the \powhegfixed-order NLO prediction (corrected for non-perturbative effects), where the POWHEG prediction has been calculated using a scale choice of .

The data are in best agreement with the \powhegprediction showered with \pythiausing the AUET2B tune. The other \powhegshowered predictions exhibit discrepancies at low dijet mass in all slices, where they predict larger cross sections than are observed in the data.

X Conclusions

Cross section measurements have been presented for inclusive jets and dijets reconstructed with the algorithm using two values of the clustering parameter ( and ). Inclusive jet production has been measured as a function of jet transverse momentum, in bins of jet rapidity. Dijet production has been measured as a function of the invariant mass of the two leading jets, in bins of half their rapidity difference. These results are based on the data sample collected with the ATLAS detector during 2010, which corresponds to () pb of integrated luminosity.

Two different sizes of the jet clustering parameter have been used in order to probe the relative effects of the parton shower, hadronisation, and the underlying event. The measurements have been corrected for all detector effects to the particle level so that they can be compared to any theoretical calculation. In this paper, they have been compared to fixed-order NLO pQCD calculations corrected for non-perturbative effects, as well as to parton shower Monte Carlo simulations with NLO matrix elements. The latter predictions have only recently become available for inclusive jet and dijet production.

The current results reflect a number of significant experimental accomplishments:

  • The cross section measurements extend to 1.5 TeV in jet transverse momentum and 5 TeV in dijet invariant mass, the highest ever measured. These results probe NLO pQCD in a large, new kinematic regime.

  • Using data taken with minimum bias and forward jet triggers, these measurements extend to both the low- region (down to jet transverse momentum of 20 GeV and dijet invariant mass of 70 GeV) and to the forward region (out to rapidities of ). The forward region, in particular, has never been explored before with such precision at a hadron-hadron collider.

  • High-precision measurements of the data collected during LHC beam position scans have determined the uncertainty on the collected luminosity to 3.4%.

  • Detailed understanding of the detector performance has precisely determined systematic uncertainties, in particular those arising from the jet energy scale. In the central region () the JES uncertainty is lower than 4.6% for all jets with  GeV, while for jet transverse momenta between 60 and 800 GeV the JES uncertainty is below 2.5%.

  • The correlations of the cross section measurement across various , , and rapidity bins have been studied for 22 independent sources of systematic uncertainty. These have been provided in the form of 87 nuisance parameters, each of which is fully correlated in and (dijet mass and ), for use in PDF fits.

The experimental uncertainties achieved are similar in size to the theoretical uncertainties in some regions of phase space, thereby providing some sensitivity to different theoretical predictions.

The measurements are compared to fixed-order NLO pQCD calculations, as well as to new calculations in which NLO pQCD matrix elements are matched to leading-logarithmic parton showers. Overall, both sets of calculations agree with the data over many orders of magnitude, although the cross sections predicted by the theory tend to be larger than the measured values at large jet transverse momentum and dijet invariant mass. The matched NLO parton shower calculations predict significant effects of the parton shower in some regions of phase space, in some cases improving and in others degrading the agreement with data with respect to the fixed-order calculations.

These measurements probe and may constrain the largely unexplored area of parton distribution functions at large and high momentum transfer. The results reported here constitute a comprehensive test of QCD across a large kinematic regime.

Xi Acknowledgements

We thank S. Ellis, M. Mangano, D. Soper, and R. Thorne for useful discussions regarding the scales used for the dijet theory predictions; the \powhegauthors for assistance with the \powhegpredictions; and P. Skands for advice regarding the non-perturbative corrections.

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; BMWF, 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; ARTEMIS, European Union; IN2P3-CNRS, CEA-DSM/IRFU, France; GNAS, Georgia; BMBF, DFG, HGF, MPG and AvH Foundation, Germany; GSRT, Greece; ISF, MINERVA, GIF, DIP and Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; FOM and NWO, Netherlands; RCN, Norway; MNiSW, Poland; GRICES and FCT, Portugal; MERYS (MECTS), Romania; MES of Russia and ROSATOM, Russian Federation; JINR; MSTD, Serbia; MSSR, Slovakia; ARRS and MVZT, Slovenia; DST/NRF, South Africa; MICINN, Spain; SRC and Wallenberg Foundation, Sweden; SER, SNSF and Cantons of Bern and Geneva, Switzerland; NSC, Taiwan; TAEK, Turkey; STFC, the Royal Society and Leverhulme Trust, United Kingdom; DOE and NSF, United States of America.

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.

Figure 14: Dijet double-differential cross section as a function of dijet mass, binned in half the rapidity separation between the two leading jets, . The results are shown for jets identified using the \AKTalgorithm with . For convenience, the cross sections are multiplied by the factors indicated in the legend. The data are compared to NLO pQCD calculations using \nlojetto which non-perturbative corrections have been applied. The error bars, which are usually smaller than the symbols, indicate the statistical uncertainty on the measurement. The dark-shaded band indicates the quadratic sum of the experimental systematic uncertainties, dominated by the jet energy scale uncertainty. There is an additional overall uncertainty of 3.4% due to the luminosity measurement that is not shown. The theory uncertainty, shown as the light, hatched band, is the quadratic sum of uncertainties from the choice of the renormalisation and factorisation scales, parton distribution functions, , and the modeling of non-perturbative effects, as described in the text.
Figure 15: Dijet double-differential cross section as a function of dijet mass, binned in half the rapidity separation between the two leading jets, . The results are shown for jets identified using the \AKTalgorithm with . For convenience, the cross sections are multiplied by the factors indicated in the legend. The data are compared to NLO pQCD calculations using \nlojetto which non-perturbative corrections have been applied. The theoretical and experimental uncertainties indicated are calculated as described in Fig. 14.

Figure 16: Ratios of dijet double-differential cross section to the theoretical prediction obtained using \nlojetwith the CT10 PDF set. The ratios are shown as a function of dijet mass, binned in half the rapidity separation between the two leading jets, . The results are shown for jets identified using the \AKTalgorithm with . The theoretical error bands obtained by using \nlojetwith different PDF sets (CT10, MSTW 2008, NNPDF 2.1, HERAPDF 1.5) are shown. The systematic and theoretical uncertainties are calculated as described in Fig. 14.

Figure 17: Ratios of dijet double-differential cross section to the theoretical prediction obtained using \nlojetwith the CT10 PDF set. The ratios are shown as a function of dijet mass, binned in half the rapidity separation between the two leading jets, . The results are shown for jets identified using the \AKTalgorithm with . The theoretical error bands obtained by using \nlojetwith different PDF sets (CT10, MSTW 2008, NNPDF 2.1, HERAPDF 1.5) are shown. The systematic and theoretical uncertainties are calculated as described in Fig. 14.
Figure 18: Ratios of dijet double-differential cross section to the theoretical prediction obtained using \nlojetwith the CT10 PDF set. The ratios are shown as a function of dijet mass, binned in half the rapidity separation between the two leading jets, . The results are shown for jets identified using the \AKTalgorithm with . The ratios of \powhegpredictions showered using either \pythiaor \herwigto the \nlojetpredictions corrected for non-perturbative effects are shown and can be compared to the corresponding ratios for data. Only the statistical uncertainty on the \powhegpredictions is shown. The total systematic uncertainties on the theory and the measurement are indicated. The \nlojetprediction and the \powhegME calculations use the CT10 PDF set.
Figure 19: Ratios of dijet double-differential cross section to the theoretical prediction obtained using \nlojetwith the CT10 PDF set. The ratios are shown as a function of dijet mass, binned in half the rapidity separation between the two leading jets, . The results are shown for jets identified using the \AKTalgorithm with . The ratios of \powhegpredictions showered using either \pythiaor \herwigto the \nlojetpredictions corrected for non-perturbative effects are shown and can be compared to the corresponding ratios for data. Only the statistical uncertainty on the \powhegpredictions is shown. The total systematic uncertainties on the theory and the measurement are indicated. The \nlojetprediction and the \powhegME calculations use the CT10 PDF set.

References

Appendix A Non-perturbative corrections



Figure 20: Non-perturbative correction factors for inclusive jets identified using the \AKTalgorithm with distance parameters and in various rapidity regions, derived using various Monte Carlo generators. The correction derived using \pythia6.425 with the AUET2B CTEQ6L1 tune is used for the fixed-order NLO calculations presented in this analysis.

Appendix B Inclusive Jet Tables

-bin NPC
GeV pb/GeV %
- 0.99(11) 0.86 0.0 0.0 0.0 0.0 0.0 0.70 1.00 0.33
- 0.99(7) 1.33 0.0 0.0 0.0 0.0 0.0 0.0 0.41 1.00 0.22
- 0.99(5) 3.03 0.0 0.0 0.0 0.0 0.27 1.00 0.19
- 0.99(5) 1.10 0.0 0.0 0.0 0.0 0.0 0.27 1.00 0.15
- 0.99(4) 0.68 0.0 0.0 0.0 0.0 0.40 1.00 0.10
- 0.99(3) 0.62 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.35 1.00 0.07
- 1.00(3) 0.69 0.0 0.0 0.0 0.34 1.00 0.06
- 1.00(3) 0.91 0.0 0.0 0.0 0.0 0.37 1.00 0.05
- 1.00(3) 0.86 0.0 0.0 0.0 0.0 0.0 0.28 1.00 0.05
- 1.00(2) 1.03 0.0 0.0 0.0 0.0 0.0 0.26 1.00 0.05
- 1.00(2) 2.02 0.0 0.0 0.0 0.0 0.0 0.0 0.21 1.00 0.05
- 1.00(2) 3.22 0.0 0.0 0.0 0.0 0.0 0.19 1.00 0.05
- 1.00(2) 5.73 0.0 0.0 0.0 0.0 0.0 0.17 1.00 0.21
- 1.00(2) 16.7 0.0 0.0 0.0 0.0 0.0 0.11 1.00 0.21
- 1.00(2) 37.3 0.0 0.0 0.0 0.0 0.0 0.12 1.00 0.21
- 1.00(2) 58.6 0.0 0.0 0.0 0.0 0.08 1.00 0.21
Table 5: Measured jet cross section for , . NPC stands for multiplicative non-perturbative corrections with in brackets, i.e. 1.25(10) means . is the measured cross section. is the statistical uncertainty. and are the correlated and uncorrelated systematic uncertainties, as described in Sec. VIII.4 and Table 3. All uncertainties are given in . An overall luminosity uncertainty of , which is applicable to all ATLAS data samples based on 2010 data, is not shown. All tables are available on \hepdata ATLAS Collaboration (2011j).
-bin NPC
GeV pb/GeV %
- 0.98(11) 0.61 0.0 0.0 0.0 0.0 0.0 0.70 1.00 0.30
- 0.99(7) 1.19 0.0 0.0 0.0 0.0 0.0 0.43 1.00 0.20
- 0.99(6) 2.15 0.0 0.0 0.28 1.00 0.19
- 0.99(5) 0.83 0.0 0.0 0.0 0.0 0.0 0.30 1.00 0.16
- 0.99(4) 0.56 0.0 0.0 0.0 0.0 0.38 1.00 0.11
- 0.99(3) 0.44 0.0 0.0 0.0 0.0 0.0 0.0 0.35 1.00 0.08
- 0.99(3) 0.58 0.0 0.0 0.0 0.0 0.0 0.39 1.00 0.07
- 1.00(3) 0.72 0.0 0.0