Measurement of the inclusive jet cross-sections in proton--proton collisions at $\sqrt{s}= $8 TeV with the ATLAS detector

TL;DR

This paper reports measurements of inclusive jet cross-sections in proton-proton collisions at 8 TeV with the ATLAS detector, comparing results to QCD predictions and identifying some discrepancies.

Contribution

First measurement of inclusive jet cross-sections at 8 TeV with detailed comparison to theoretical models and parton distribution functions.

Findings

Measured cross-sections agree with QCD predictions within uncertainties.

Observed tensions suggest potential areas for refinement in theoretical models.

Data provides constraints on proton parton distribution functions.

Abstract

Inclusive jet production cross-sections are measured in proton--proton collisions at a centre-of-mass energy of $\sqrt{s}=$8 TeV recorded by the ATLAS experiment at the Large Hadron Collider at CERN. The total integrated luminosity of the analysed data set amounts to $20.2$ fb$^{-1}$. Double-differential cross-sections are measured for jets defined by the anti-$k_{t}$ jet clustering algorithm with radius parameters of $R=0.4$ and $R=0.6$ and are presented as a function of the jet transverse momentum, in the range between 70 GeV and 2.5 TeV and in six bins of the absolute jet rapidity, between 0 and 3.0. The measured cross-sections are compared to predictions of quantum chromodynamics, calculated at next-to-leading order in perturbation theory, and corrected for non-perturbative and electroweak effects. The level of agreement with predictions, using a selection of different parton distribution functions for the proton, is quantified. Tensions between the data and the theory predictions are observed.

Tables

Table 1. Table 1: Summary of Monte Carlo generator configurations used for the evaluation of the non-perturbative corrections. The name of the generator and the soft physics model tune as well as the PDF set used when deriving the tune is specified.

Generator	Tune	PDF
Pythia 8	4C [78]	CTEQ6L [74]
	Monash [76]	NNPDF2.3L [79, 80]
	AU2 [34]	CT10 [35]
	AU2 [34]	CTEQ6L [74]
	A14 [77]	NNPDF2.3L [79, 80]
	A14 [77]	MRSTW2008lo∗∗ [81]
	A14 [77]	CTEQ6L [74]
Herwig++	UE-EE-5 [82, 83]	CTEQ6L [74]
	UE-EE-5 [82, 83]	MRSTW2008lo∗∗[81]
	UE-EE-4 [82, 83]	CTEQ6L [74]

Table 2. Table 2: Observed P obs subscript 𝑃 obs P_{\mathrm{obs}} values evaluated for the NLO QCD predictions corrected for non-perturbative and electroweak effects and the measured inclusive jet cross-section of anti- k t subscript 𝑘 𝑡 {k_{t}} jets with R = 0.4 𝑅 0.4 R=0.4 and R = 0.6 𝑅 0.6 R=0.6 . Only measurements with p T > 100 subscript 𝑝 T 100 p_{\text{T}}{}>100   GeV are included. The predictions are evaluated for various PDF sets. The default scale choice p T j ​ e ​ t , m ​ a ​ x superscript subscript 𝑝 T j 𝑒 𝑡 𝑚 𝑎 𝑥 {p_{\text{T}}^{\mathrm{j}et,max}} is used.

	$P_{\mathrm{obs}}$
Rapidity ranges	CT14	MMHT2014	NNPDF3.0	HERAPDF2.0
Anti-kt jets $R=0.4$
	$44\%$	$28\%$	$25\%$	$16\%$
	$43\%$	$29\%$	$18\%$	$18\%$
	$44\%$	$47\%$	$46\%$	$69\%$
	$3.7\%$	$4.6\%$	$7.7\%$	$7.0\%$
	$92\%$	$89\%$	$89\%$	$35\%$
	$4.5\%$	$6.2\%$	$16\%$	$9.6\%$
Anti-kt jets $R=0.6$
	$6.7\%$	$4.9\%$	$4.6\%$	$1.1\%$
	$1.3\%$	$0.7\%$	$0.4\%$	$0.2\%$
	$30\%$	$33\%$	$47\%$	$67\%$
	$12\%$	$16\%$	$15\%$	$3.1\%$
	$94\%$	$94\%$	$91\%$	$38\%$
	$13\%$	$15\%$	$20\%$	$8.6\%$

Table 3. Table 3: Summary of χ 2 superscript 𝜒 2 \chi^{2} /ndf obtained from the comparison of the inclusive jet cross-section and the NLO QCD prediction for various PDF sets and scale choices for anti- k t subscript 𝑘 𝑡 {k_{t}} jets with R = 0.4 𝑅 0.4 R=0.4 and R = 0.6 𝑅 0.6 R=0.6 , for several p T subscript 𝑝 T p_{\text{T}} cuts, using all | y | 𝑦 |y| bins. All the corresponding p 𝑝 p -values are ≪ 10 − 3 much-less-than absent superscript 10 3 \ll 10^{-3} .

${\chi }^2$/ndf	$p_{\text{T}}^{jet,max}$		$p_{\text{T}}^{jet}$
	$R=0.4$	$R=0.6$	$R=0.4$	$R=0.6$
$p_{\text{T}}>70$ GeV
CT14	349/171	398/171	340/171	392/171
HERAPDF2.0	415/171	424/171	405/171	418/171
NNPDF3.0	351/171	393/171	350/171	393/171
MMHT2014	356/171	400/171	354/171	399/171
\IfSubStrptcut1ptcut$p_{\text{T}}>70$ GeV \IfSubStrptcut1ptcut1$p_{\text{T}}>100$ GeV \IfSubStrptcut1ptcut2 GeV \IfSubStrptcut1ptcut3 GeV \IfSubStrptcut1ptcut4 GeV \IfSubStrptcut1ptcut5 GeV \IfSubStrptcut1ptcut6 GeV \IfSubStrptcut1ptcut7 GeV \IfSubStrptcut1ptcut8 GeV \IfSubStrptcut1ptcut9$p_{\text{T}}>952$ GeV
CT14	321/159	360/159	313/159	356/159
HERAPDF2.0	385/159	374/159	377/159	370/159
NNPDF3.0	333/159	356/159	331/159	356/159
MMHT2014	335/159	364/159	333/159	362/159
\IfSubStrptcut2ptcut$p_{\text{T}}>70$ GeV \IfSubStrptcut2ptcut1$p_{\text{T}}>100$ GeV \IfSubStrptcut2ptcut2 GeV \IfSubStrptcut2ptcut3 GeV \IfSubStrptcut2ptcut4 GeV \IfSubStrptcut2ptcut5 GeV \IfSubStrptcut2ptcut6 GeV \IfSubStrptcut2ptcut7 GeV \IfSubStrptcut2ptcut8 GeV \IfSubStrptcut2ptcut9$p_{\text{T}}>952$ GeV
CT14	272/134	306/134	262/134	301/134
HERAPDF2.0	350/134	331/134	340/134	326/134
NNPDF3.0	289/134	300/134	285/134	299/134
MMHT2014	292/134	311/134	284/134	308/134
\IfSubStrptcut3ptcut$p_{\text{T}}>70$ GeV \IfSubStrptcut3ptcut1$p_{\text{T}}>100$ GeV \IfSubStrptcut3ptcut2 GeV \IfSubStrptcut3ptcut3 GeV \IfSubStrptcut3ptcut4 GeV \IfSubStrptcut3ptcut5 GeV \IfSubStrptcut3ptcut6 GeV \IfSubStrptcut3ptcut7 GeV \IfSubStrptcut3ptcut8 GeV \IfSubStrptcut3ptcut9$p_{\text{T}}>952$ GeV
CT14	128/72	149/72	118/72	145/72
HERAPDF2.0	148/72	175/72	141/72	170/72
NNPDF3.0	119/72	141/72	115/72	139/72
MMHT2014	132/72	143/72	122/72	140/72

Equations

$${\mathcal{N}}_i^{\mathrm{part}}=\sum _j{\mathcal{N}}_j^{\mathrm{reco}}\cdot {\mathcal{P}}_j\cdot {\mathcal{A}}_{ij}/{\mathcal{E}}_i,$$

$$\begin{array}{rl}& f_1(p_{\text{T}},y)=C(p_{\text{T}},y)\cdot c_1/\log \left(M(y)/p_{\text{T}}\right),\\ & f_2(p_{\text{T}},y)=C(p_{\text{T}},y)\cdot c_2\cdot y^2/\log \left(M(y)/p_{\text{T}}\right),\\ & f_3(p_{\text{T}},y)=C(p_{\text{T}},y)\cdot c_3,\\ & f_4(p_{\text{T}},y)=C(p_{\text{T}},y)\cdot c_4\cdot y^2,\\ & f_5(p_{\text{T}},y)=C(p_{\text{T}},y)\cdot c_5\cdot \log \left(15p_{\text{T}}/M(y)\right),\\ & f_6(p_{\text{T}},y)=C(p_{\text{T}},y)\cdot c_6\cdot y^2\cdot \log \left(15p_{\text{T}}/M(y)\right),\end{array}$$

Full text

\AtlasTitle

Measurement of the inclusive jet cross-sections in proton–proton collisions at $\sqrt{s}=8\leavevmode\nobreak\ \text{Te\kern-1.00006ptV}{}$ with the ATLAS detector

\PreprintIdNumberCERN-EP-2017-043 \AtlasDate \AtlasJournalJHEP \AtlasJournalRefJHEP 09 (2017) 020 \AtlasDOI10.1007/JHEP09(2017)020 \AtlasAbstractInclusive jet production cross-sections are measured in proton–proton collisions at a centre-of-mass energy of $\sqrt{s}=8\leavevmode\nobreak\ \text{Te\kern-1.00006ptV}{}$ recorded by the ATLAS experiment at the Large Hadron Collider at CERN. The total integrated luminosity of the analysed data set amounts to $20.2$ fb*-1*. Double-differential cross-sections are measured for jets defined by the anti-${k_{t}}$ jet clustering algorithm with radius parameters of $R=0.4$ and $R=0.6$ and are presented as a function of the jet transverse momentum, in the range between $70$GeV and $2.5$TeV and in six bins of the absolute jet rapidity, between [math] and $3.0$. The measured cross-sections are compared to predictions of quantum chromodynamics, calculated at next-to-leading order in perturbation theory, and corrected for non-perturbative and electroweak effects. The level of agreement with predictions, using a selection of different parton distribution functions for the proton, is quantified. Tensions between the data and the theory predictions are observed.

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Contents

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toc

1 Introduction

The Large Hadron Collider (LHC) [1] at CERN, colliding protons on protons, provides a unique opportunity to explore the production of hadronic jets in the TeV energy range. In Quantum Chromodynamics (QCD), jet production can be interpreted as the fragmentation of quarks and gluons produced in a short-distance scattering process. The inclusive jet production cross-section ($p+p\rightarrow{\mathrm{j}et}+X$) gives valuable information about the strong coupling constant ($\alpha_{\mathrm{s}}$) and the structure of the proton. It is also among the processes directly testing the experimentally accessible space-time distances.

Next-to-leading-order (NLO) perturbative QCD calculations [2, 3] give quantitative predictions of the jet production cross-sections. Progress in next-to-next-to-leading-order (NNLO) QCD calculations has been made over the past several years [4, 5, 6, 7, 8, 9]. After the completion of the first calculations of some sub-processes [10, 11], the complete NNLO QCD inclusive jet cross-section calculation was published recently [12].

As fixed-order QCD calculations only make predictions for the quarks and gluons associated with the short-distance scattering process, corrections for the fragmentation of these partons to particles need to be applied. The measurements can also be compared to Monte Carlo event generator predictions that directly simulate the particles entering the detector. These event generators can be based on calculations with leading-order (LO) or NLO accuracy for the description of the short-distance scattering process as well as additional QCD radiation, hadronisation and multiple parton interactions [13].

Inclusive jet production cross-sections have been measured in proton–antiproton collisions at the Tevatron collider at various centre-of-mass energies. The latest and most precise measurements at $\sqrt{s}=1.96\text{Te\kern-1.00006ptV}$ can be found in Refs. [14, 15]. At the LHC, the ALICE, ATLAS and CMS collaborations have measured inclusive jet cross-sections in proton–proton collisions at centre-of-mass energies of $\sqrt{s}=2.76\text{Te\kern-1.00006ptV}$ [16, 17, 18] and $\sqrt{s}=7\leavevmode\nobreak\ \text{Te\kern-1.00006ptV}$ [19, 20, 21, 22, 23], and recently the CMS Collaboration has also measured them at $\sqrt{s}=8\leavevmode\nobreak\ \text{Te\kern-1.00006ptV}$ [24] and $\sqrt{s}=13\leavevmode\nobreak\ \text{Te\kern-1.00006ptV}$ [25].

This paper presents the measurement of the inclusive jet cross-sections in proton–proton collisions at a centre-of-mass energy of $\sqrt{s}=8$ TeV using data collected by the ATLAS experiment in $2012$ corresponding to an integrated luminosity of $20.2$ fb*-1*. The cross-sections are measured double-differentially and presented as a function of the jet transverse momentum, $p_{\text{T}}$, in six equal-width bins of the absolute jet rapidity, $|y|$. Jets are reconstructed using the anti-${k_{t}}$ jet clustering algorithm [26] with radius parameters of $R=0.4$ and $R=0.6$. The measurement is performed for two jet radius parameters, since the uncertainties in the theoretical predictions are different. The kinematic region of \mbox{70}\text{Ge\kern-1.00006ptV}{}\leq p_{\text{T}}{}\leq\mbox{2.5}\text{Te\kern-1.00006ptV}{} and $|y|<3$ is covered.

The measurements explore a higher centre-of-mass energy than the previous ATLAS measurements and are also more precise due to the higher integrated luminosity and the better knowledge of the jet energy measurement uncertainties. Fixed-order NLO QCD predictions calculated for a suite of proton parton distribution function (PDF) sets, corrected for non-perturbative (hadronisation and underlying event) and electroweak effects, are quantitatively compared to the measurement results, unfolded for detector effects. The results are also compared to the predictions of a Monte Carlo event generator based on the NLO QCD calculation for the short-distance scattering process matched with parton showers, followed by hadronisation. The measurement is performed with two different jet radius parameters to test the sensitivity to perturbative (higher-order corrections and parton shower) and non-perturbative effects.

The outline of the paper is as follows. A brief description of the ATLAS detector is given in Section 2. The inclusive jet production cross-section is defined in Section 4. Section 3 gives an overview of the data set and Monte Carlo simulations used. The details of the experimental measurement are presented in the next sections. Section 5 describes the event and jet selection for the measurement. The jet energy calibration and the uncertainties associated with the jet energy measurements are outlined in Section 6. The procedure to unfold the detector effects is detailed in Section 7 and the propagation of the systematic uncertainties in the measurements is explained in Section 8. The theoretical predictions are described in Section 9. The results together with a quantitative comparison of the measurements to the theory predictions are presented in Section 10.

2 ATLAS detector

The ATLAS experiment [27] at the LHC is a multipurpose particle detector with a forward-backward symmetric cylindrical geometry and a near $4\pi$ coverage in solid angle.111ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the $z$-axis along the beam pipe. The $x$-axis points from the IP to the centre of the LHC ring, and the $y$-axis points upwards. Cylindrical coordinates $(r,\phi)$ are used in the transverse plane, $\phi$ being the azimuthal angle around the $z$-axis. The pseudorapidity is defined in terms of the polar angle $\theta$ as $\eta=-\ln\tan(\theta/2)$. Angular distance is measured in units of $\Delta R\equiv\sqrt{(\Delta y)^{2}+(\Delta\phi)^{2}}$, where $y$ is the jet rapidity. It consists of an inner tracking detector surrounded by a thin superconducting solenoid providing a $2\text{\,}\mathrm{T}$ axial magnetic field, electromagnetic and hadron calorimeters, and a muon spectrometer. The inner tracking detector covers the pseudorapidity range $|\eta|<2.5$ and is made of silicon pixel, silicon microstrip, and transition-radiation tracking detectors. Lead/liquid-argon (LAr) sampling calorimeters provide electromagnetic (EM) energy measurements with high granularity. A hadron (steel/scintillator-tile) calorimeter covers the central pseudorapidity range ($|\eta|<1.7$). The endcap and forward regions are instrumented with LAr calorimeters for EM and hadronic energy measurements up to $|\eta|=4.9$. The muon spectrometer surrounds the calorimeters and is based on three large air-core toroid superconducting magnets with eight coils each. Its bending power ranges between $2.0$ and $6.0\text{\,}\mathrm{T}\text{\,}\mathrm{m}$ for most of the detector.

A three-level trigger system is used to select events. The first-level trigger is implemented in hardware and uses a subset of the detector information to reduce the accepted event rate to at most $75\text{\,}\mathrm{kHz}$. This is followed by two software-based trigger levels that together reduce the accepted event rate to $400\text{\,}\mathrm{Hz}$ on average depending on the data-taking conditions during 2012.

The relevant systems used to select events with jets are the minimum-bias trigger scintillators (MBTS), located in front of the endcap cryostats covering $2.1<|\eta|<3.8$, as well as calorimeter-based jet triggers covering $|\eta|<3.2$ for central jets [28].

3 Data set and Monte Carlo simulations

The measurement uses proton-proton collision data at a centre-of-mass energy of $\sqrt{s}=8$ TeV collected by the ATLAS detector during the data-taking period of the LHC in $2012$. The LHC beams were operated with proton bunches organised in "bunch trains", with bunch-crossing intervals (or bunch spacing) of $50$ $\mathrm{n}s$.

The absolute luminosity measurement is derived from beam-separation scans performed in November $2012$ and corresponds to $20.2$ fb*-1* with an uncertainty of $1.9\%$. The uncertainty in the luminosity is determined following the technique described in Refs. [29]. The average number of interactions per bunch crossing, $\left<\mu\right>$, was $10\leq\left<\mu\right>\leq 36$. All data events considered in this analysis have good detector status and data quality.

For the simulation of the detector response to scattered particles in proton–proton collisions, events are generated with the Pythia 8 [30] (v8.160) Monte Carlo event generator. It uses LO QCD matrix elements for $2\rightarrow 2$ processes, along with a leading-logarithmic (LL) $p_{\text{T}}$-ordered parton shower [31] including photon radiation, underlying-event simulation with multiple parton interactions [32], and hadronisation with the Lund string model [33]. The MC event generator’s parameter values are set according to the AU2 underlying event tune [34] and the CT10 PDF set [35] is used.

The stable particles from the generated events are passed through the ATLAS detector simulation [36] based on the Geant4 software toolkit [37] and are reconstructed using the same version of the ATLAS software as used to process the data. Effects from multiple proton–proton interactions in the same and neighbouring bunch crossings (pile-up) are included by overlaying inclusive proton–proton collision events (minimum bias), which consist of single-, double- and non-diffractive collisions generated by the Pythia 8 event generator using the A2 tune [34] based on the MSTW2008 LO PDF set [38]. The Monte Carlo events are weighted such that the distribution of the generated mean number of proton–proton collisions per bunch crossing matches that of the corresponding data-taking period. The particles from additional interactions are added before the signal digitisation and reconstruction steps of the detector simulation, but are not considered a signal and are therefore not used in the definition of the cross-section measurement defined in Section 4.

For the evaluation of non-perturbative effects, the Pythia 8  [30] (v8.186) and Herwig++ [39] (v2.7.1) [40] event generators are also employed as described in Section 9.3. The latter also uses LO matrix elements for the $2\rightarrow 2$ short-distance process together with a LL angle-ordered parton shower [41]. It implements an underlying-event simulation based on an eikonal model [42] and the hadronisation process based on the cluster model [43].

The Powheg [44, 45, 46] method provides MC event generation based on an NLO QCD calculation matched to LL parton showers using the Powheg Box 1.0 package [47]. In this simulation the CT10 PDF set [35] is used. The simulation of parton showers, the hadronisation and the underlying event is based on Pythia 8 [30] using the AU2 tune [34]. These predictions are refered to as the Powheg predictions in the following.

The renormalisation and factorisation scales for the fixed-order NLO prediction are set to the transverse momentum of each of the outgoing partons of the $2\to 2$ process, $p_{\text{T}}^{\mathrm{Born}}$. In addition to the hard scatter, Powheg also generates the hardest partonic emission in the event using the LO $2\to 3$ matrix element or parton showers. The radiative emissions in the parton showers are limited by the matching scale $\mu_{\text{M}}$ provided by Powheg.

4 Inclusive jet cross-section definition

Jets are identified with the anti-${k_{t}}$ [26] clustering algorithm using the four-momentum recombination scheme, implemented in the FastJet [48] library, using two values of the jet radius parameter, $R=0.4$ and $R=0.6$. Throughout this paper, the jet cross-section measurements refer to jets built from stable particles defined by having a proper mean decay length of $c\tau>10$ mm. Muons and neutrinos from decaying hadrons are included in this definition. More information about the particle definition can be found in Ref. [49]. These jets are called "particle-level" jets in the following.

The inclusive jet double-differential cross-section, $\mathrm{d}^{2}\sigma/$d$p_{\text{T}}$d$y$, is measured as a function of the jet transverse momentum $p_{\text{T}}$ in bins of rapidity $y$. In this context, "inclusive" cross-section means that all reconstructed jets in accepted events contribute to the measurement in the bins corresponding to their $p_{\text{T}}$ and $y$ values.

The kinematic range of the measurement is \mbox{70}\leavevmode\nobreak\ \text{Ge\kern-1.00006ptV}\leq p_{\text{T}}\leq\mbox{2.5}\leavevmode\nobreak\ \text{Te\kern-1.00006ptV} and $|y|<3$.

5 Event and jet selection

A set of single-jet triggers with various $p_{\text{T}}$ thresholds are used to preselect events to be recorded. The highest threshold trigger accepts all events passing the threshold. To keep the trigger rate to an acceptable level, the triggers with lower $p_{\text{T}}$ thresholds are only read out for a fraction of all events.

A $p_{\text{T}}$-dependent trigger strategy is adopted in order to optimise the statistical power of the measurement. Trigger efficiencies are studied using the trigger decisions in samples selected by lower-threshold jet triggers. The efficiency of the lowest $p_{\text{T}}$ jet trigger is determined with an independent trigger based on the MBTS scintillators. For each measurement bin, the trigger is chosen such that the highest effective luminosity (i.e. the lowest prescale factor) is obtained and the trigger is fully efficient. This procedure is performed separately for each of the jet radius parameters and for each jet rapidity bin.

At least one reconstructed vertex with at least two associated well-reconstructed tracks is required. Jet quality criteria are applied to reject jets from beam–gas events, beam–halo events, cosmic-ray muons and calorimeter noise bursts following the procedure described in Ref. [50].

In the $2012$ data set the central hadron calorimeter had a few modules turned off for certain long time periods or suffered from power-supply trips that made them non-operational for a few minutes. The energy deposited in these modules is estimated using the energy depositions in the neighbouring modules [50]. This correction overestimates the true deposited energy. Therefore, events where a jet with $p_{\text{T}}{}\geq 40\leavevmode\nobreak\ \text{Ge\kern-1.00006ptV}{}$ points to such a calorimeter region are rejected both in data and simulation.

6 Jet energy calibration and resolution

6.1 Jet reconstruction

Jets are defined with the anti-${k_{t}}$ clustering algorithm with the jet radius parameters $R=0.4$ and $R=0.6$. The input objects for the jet algorithm are three-dimensional topological clusters (topoclusters) [51, 52] built from the energy deposits in calorimeter cells. A local cluster weighting calibration (LCW) based on the topology of the calorimeter energy deposits is then applied to each topocluster to improve the energy resolution for hadrons impinging on the calorimeter [51, 52]. The four-momentum of the LCW-scale jet is defined as the sum of four-momenta of the locally calibrated clusters in the calorimeter treating each cluster as a four-momentum with zero mass.

6.2 Jet energy calibration

Jets are calibrated using the procedure described in Refs. [50, 51]. The jet energy is corrected for the effect of multiple proton-proton interactions (pile-up) both in collision data and in simulated events. Further corrections depending on the jet energy and the jet pseudorapidity ($\eta$) are applied to achieve a calibration that matches the energy of jets composed of stable particles in simulated events. Fluctuations in the particle content of jets and in hadronic calorimeter showers are reduced with the help of observables characterising internal jet properties. These corrections are applied sequentially (Global Sequential Calibration). Differences between data and Monte Carlo simulation are evaluated using ${\mathrm{i}nsitu}$ techniques exploiting the $p_{\text{T}}$ balance of a jet and a well-measured object such as a photon ($\gamma$+jet balance), a $Z$ boson ($Z$+jet balance) or a system of jets (multijet balance). These processes are used to calibrate the jet energy in the central detector region, while the $p_{\text{T}}$ balance in dijet events is used to achieve an intercalibration of jets in the forward region with respect to central jets (dijet balance).

The calibration procedure that establishes the jet energy scale (JES) and the associated systematic uncertainty is given in more detail in the following:

Pile-up correction

Jets are corrected for the contributions from additional proton-proton interactions within the same (in-time) or nearby (out-of-time) bunch crossings [53]. First, for each event a correction based on the jet area and the median $p_{\text{T}}$ density $\rho$ [54, 55] is calculated. The jet area is a measure of the susceptibility of the jet to pile-up and is determined for each jet. The density, $\rho$, is a measure of the pile-up activity in the event. Subsequently, an average offset subtraction is performed based on the number of additional interactions and reconstructed vertices ($N_{\text{P}V}$) in the event. It is derived by comparing reconstructed calorimeter jets, with the jet-area correction applied, to particle jets in simulated inclusive jet events.

The correction for contributions from additional proton–proton interactions can also remove part of the soft physics contributions, e.g. the contribution from the underlying event. This contribution is restored on average by the MC-based jet energy scale correction discussed below. The impact of pile-up subtraction on the jet energy resolution is corrected for in the unfolding step (see Section 7).

Jet energy scale

The energy and the direction of jets are corrected for instrumental effects (non-compensating response of the calorimeter, energy losses in dead material, and out-of-cone effects) and the jet energy scale is restored on average to that of the particles entering the calorimeter using an inclusive jet Monte Carlo simulation [56]. These corrections are derived in bins of energy and the pseudorapidity of the jet.

Global sequential correction

The topology of the calorimeter energy deposits and of the tracks associated with jets can be exploited to correct for fluctuations in the jet’s particle content [51, 57]. The measured mean jet energy depends on quantities such as the number of tracks, the radial extent of the jets as measured from the tracks in the jets, the longitudinal and lateral extent of the hadronic shower in the calorimeter and the hits in the muon detector associated with the jet. A correction of the jet energy based on these quantities can therefore improve the jet resolution and reduce the dependence on jet fragementation effects. The correction is constructed from a MC sample based on one generator such that the jet energy scale correction is unchanged for the inclusive jet sample, but the jet energy resolution is improved and the sensitivity to jet fragmentation effects such as differences between quark- or gluon-induced jets is moderated. The dependence of this correction on the MC generator is treated as uncertainty.

Correction for difference between data and Monte Carlo simulation

A residual calibration is applied to correct for remaining differences between the jet energy response in data and simulation. This correction is derived ${\mathrm{i}nsitu}$ by comparing the results of $\gamma$+jet, $Z$+jet, dijet and multijet $p_{\text{T}}$-balance techniques [56, 58, 59]. The level of agreement between the jet energy response in the Monte Carlo simulation and the one in the data is evaluated by exploiting the $p_{\text{T}}$ balance between a photon or a $Z$ boson and a jet. In the $p_{\text{T}}$ range above about $800$ GeV, which cannot be reached by $\gamma$+jet events, the recoil system of low-$p_{\text{T}}$ jets in events with more than two jets is used (multijet balance).

This correction is applied to the central detector region. The relative response in all detector regions is equalised using an intercalibration method that uses the $p_{\text{T}}$ balance in dijet events where one jet is central and one jet is in the forward region of the detector ($\eta$-intercalibration).

In the region above $p_{\text{T}}$ = $1.7$ TeV, where the ${\mathrm{i}nsitu}$ techniques do not have sufficient statistical precision, the uncertainty in the jet energy measurement is derived from single-hadron response measurements [60, 61].

6.3 Jet energy scale uncertainties

The jet calibration corrections are combined following the procedure described in Ref. [56]. The systematic and statistical uncertainties of each of the above mentioned corrections contribute to the total JES uncertainty as independent systematic components.

The ${\mathrm{i}nsitu}$ techniques are based on various processes leading to jets with different fragmentation patterns. Differences in the calorimeter response to jets initiated by quarks or gluons in the short-distance processes lead to an additional uncertainty. Limited knowledge of the exact flavour composition of the analysed data sample is also considered as an uncertainty. An estimation of flavour composition based on the Pythia and the Powheg + Pythia Monte Carlo simulations is used in order to reduce this uncertainty.

A systematic uncertainty needs to be assigned to the correction, based on the muon hits behind the jet, that corrects jets with large energy deposition behind the calorimeter (punch-through).

In total, $66$ independent systematic components uncorrelated among each other and fully correlated across $p_{\text{T}}$ and $\eta$, constitute the full JES uncertainty in the configuration with the most detailed description of correlations [56]. A simplification is performed in this standard configuration: the $\eta$-intercalibration statistical uncertainty being treated as one uncertainty component fully correlated between the jet rapidity and $p_{\text{T}}$ bins for which the $\eta$-intercalibration was performed. However, at the level of precision achieved in this analysis a detailed description of the statistical uncertainties of the $\eta$-intercalibration calibration procedure is important. For this reason, in this measurement, the total statistical uncertainty of the $\eta$-intercalibration in the standard configuration is replaced by $240$ ($250$) uncertainty components for jets with $R=0.4$ ($R=0.6$), propagated from the various bins of the ${\mathrm{i}nsitu}$ $\eta$-intercalibration analysis [58].

The total uncertainty in the JES is below $1\%$ for $100\leavevmode\nobreak\ \text{Ge\kern-1.00006ptV}{}<p_{\text{T}}{}<1500\leavevmode\nobreak\ \text{Ge\kern-1.00006ptV}{}$ in the central detector region ($|\eta|\leq 0.8$) rising both towards lower and higher $p_{\text{T}}$ and larger $|\eta|$ [56].

6.4 Jet energy resolution and uncertainties

The fractional uncertainty in the jet $p_{\text{T}}$ resolution (JER) is derived using the same ${\mathrm{i}nsitu}$ techniques as used to determine the JES uncertainty from the width of the ratio of the $p_{\text{T}}$ of a jet to the $p_{\text{T}}$ of a well-measured particle such as a photon or a $Z$ boson [59]. In addition, the balance between the jet transverse momenta in events with two jets at high $p_{\text{T}}$ can be used ($\eta$-intercalibration) [58]. This method allows measurement of the JER at high jet rapidities and in a wide range of transverse momenta. The results from individual methods are combined similarly to those for the JES [56]. This JER evaluation includes a correction for physics effects such as radiation of extra jets which can also alter the $p_{\text{T}}$ ratio width. This correction is obtained from a Monte Carlo simulation.

The JER uncertainty has in total $11$ systematic uncertainty components. Nine systematic components are obtained by combining the systematic uncertainties associated with the ${\mathrm{i}nsitu}$ methods. The last two are the uncertainty due to the electronic and pile-up noise measured in inclusive proton-proton collisions and the absolute JER difference between data and MC simulation as determined with the ${\mathrm{i}nsitu}$ methods. The latter is non-zero only for low-$p_{\text{T}}$ jets in forward rapidity regions. In the rest of the phase-space region the JER in MC simulation is better than in data and this uncertainty is eliminated by smearing the jet $p_{\text{T}}$ in simulation such that the resulting resolution matches closely the one in data. Each JER systematic component describes an uncertainty that is fully correlated in jet $p_{\text{T}}$ and pseudorapidity. The $11$ JER components are treated independently from each other.

6.5 Jet angular resolution and uncertainties

The jet angular resolution (JAR) is estimated from comparisons of the polar angles of a reconstructed jet and the matched particle-level jet using the Monte Carlo simulation. This estimate is cross-checked by comparing the standard jets using calorimeter energy deposits as inputs to the ones using tracks in the inner detector [50, 51]. A relative uncertainty of $10$% is assigned to the JAR to account for possible differences between data and MC simulation.

7 Unfolding of detector effects

The reconstructed jet spectra in data are unfolded to correct for detector inefficiencies and resolution effects to obtain the inclusive jet cross-section that refers to the stable particles entering the detector. The detector unfolding is based on Monte Carlo simulation and is performed in three consecutive steps, namely, a correction for the matching impurity at reconstruction level, the unfolding for resolution effects and a correction for the matching inefficiency at particle level, as explained below. In order to account for migrations from lower $p_{\text{T}}$ into the region of interest, this study is performed in a wider $p_{\text{T}}$ range than the one for the final result.

The unfolding of the detector resolution in jet $p_{\text{T}}$ is based on a modified Bayesian technique, the Iterative Dynamically Stabilised (IDS) method [62]. This unfolding method uses a transfer matrix describing the migrations of jets across the $p_{\text{T}}$ bins, between the particle level and the reconstruction level. A minimal number of iterations in the IDS unfolding method is chosen such that the residual bias, evaluated through a data-driven closure test (see below), is within a tolerance of $1\%$ in the bins with less than $10\%$ statistical uncertainty. In this measurement this is achieved after one iteration.

The transfer matrix used in the unfolding is derived by matching a particle level jet with a reconstructed jet in Monte Carlo simulations, when both are closer to each other than to any other jet and lie within a radius of $\Delta R=0.3$.

The matching purity, $\mathcal{P}$, is defined as the ratio of the number of matched reconstructed jets to the total number of reconstructed jets. The matching efficiency, $\mathcal{E}$, is defined as the ratio of the number of matched particle jets to the total number of particle jets. If jets migrate to other rapidity bins, they are considered together with the jets that are completely unmatched. In this way the migrations across rapidity bins are effectively taken into account by bin-to-bin corrections.

The final result is given by

[TABLE]

where $i$ and $j$ are the bin indices of the jets at particle- and reconstructed-levels and $\mathcal{N}^{\mathrm{part}}$ and $\mathcal{N}^{\mathrm{reco}}$ are the number of particle-level and reconstructed jets in a given bin. The symbol $\mathcal{A}$ denotes the unfolding matrix obtained by the IDS method from the transfer matrix. The element $\mathcal{A}_{ij}$ describes the probability for a reconstructed jet in $p_{\text{T}}$ bin $j$ to originate from particle-level $p_{\text{T}}$ bin $i$.

The precision of the unfolding technique is assessed using a data-driven closure test [62, 20]. The particle-level $p_{\text{T}}$ spectrum in the MC simulation is reweighted such that the reweighted reconstructed spectrum and the data agree. The reconstructed spectrum in this reweighted MC simulation is then unfolded using the same procedure as for the data. The ratio of the unfolded spectrum to the reweighted particle-level spectrum provides an estimate of the unfolding bias. The residual bias is taken into account as a systematic uncertainty. After one IDS iteration, this uncertainty is of the order of a few per mille in the whole phase-space region, except for the very high $p_{\text{T}}$ bins in each of the rapidity bins, where it grows to a few percent (up to $15\%$ in certain cases).

The statistical and systematic uncertainties are evaluated by repeating the unfolding as explained in Section 8.

8 Propagation of the statistical and systematic uncertainties

The statistical uncertainties are propagated through the unfolding procedure using an ensemble of pseudo-experiments. For each pseudo-experiment in the ensemble, a weight fluctuated according to a Poisson distribution with a mean value equal to one is applied to each event in data and simulation. This procedure takes into account the correlation between jets produced in the same event. The unfolding is performed for each pseudo-experiment. An ensemble of $10000$ pseudo-experiments is used to calculate a covariance matrix for the cross-section in each jet rapidity bin. The total statistical uncertainty is obtained from the covariance matrix, where bin-to-bin correlations are also encoded. The separate contributions from the data and from the MC statistics are obtained from the same procedure by fluctuating only either the data or the simulated events. Furthermore, an overall covariance matrix is constructed to describe the full statistical covariance among all analysis bins.

To propagate the JES uncertainties to the measurement, the jet $p_{\text{T}}$ is scaled up and down by one standard deviation of each of the components (see Section 6) in the MC simulation. The resulting $p_{\text{T}}$ spectra are unfolded for detector effects using the nominal unfolding matrix. The difference between the nominal unfolded cross-section and the one with the jet $p_{\text{T}}$ scaled up and down is taken as a systematic uncertainty.

The uncertainty in the JER is the second largest individual source of systematic uncertainty. The effect of each of the $11$ JER systematic uncertainty components is evaluated by smearing the energy of the reconstructed jets in the MC simulation such that the resolution is degraded by the size of each uncertainty component. A new transfer matrix is constructed using the smeared jets and is used to unfold the data spectra. The difference of the cross-sections unfolded with the jet-energy-smeared transfer matrix and the nominal transfer matrix is taken as a systematic uncertainty. The JER uncertainty is applied symmetrically as an upward and downward variation.

The JAR is propagated to the cross-section in the same way as for the JER.

The uncertainty associated with the residual model dependence in the unfolding procedure is described in Section 7. The systematic uncertainties propagated through the unfolding are evaluated using a set of pseudo-experiments for each component, as in the evaluation of the statistical uncertainties.

The use of pseudo-experiments for the evaluation of the systematic uncertainties allows an evaluation of the statistical fluctuations. The statistical fluctuations of the systematic uncertainties are reduced using a smoothing procedure. For each component, the $p_{\text{T}}$ bins are combined until the propagated uncertainty value in the bin has a Poisson statistical significance larger than two standard deviations. A Gaussian kernel smoothing [50] is used to restore the original fine bins.

An uncertainty for the jet cleaning procedure described in Section 5 is estimated from the relative difference between the efficiencies obtained from the distributions with and without the jet quality cut in data and simulation.

The uncertainty in the luminosity measurement of $1.9$% [29] is propagated as being correlated across all measurement bins.

An uncertainty in the beam energy of $0.1\%$ [63] is considered when comparing data with the theory prediction at a fixed beam energy. The induced uncertainty at the cross-section level is evaluated by comparing the theory predictions at the nominal and shifted beam energies. It amounts for $0.2\%$ at low $p_{\text{T}}$ and $1\%$ at high $p_{\text{T}}$ in the central region and rises up to $3\%$ at highest $p_{\text{T}}$ and high rapidity. This uncertainty is similar for jets with $R=0.4$ and $R=0.6$.

The individual systematic uncertainty sources are treated as uncorrelated with each other for the quantitative comparison of the data and the theory prediction. When shown in figures the individual uncertainties are added in quadrature to obtain the total systematic uncertainty. The shape of the systematic uncertainties follows a log-normal distribution, as in the analysis of inclusive jet production at 7 TeV[19]. The systematic uncertainties in the inclusive jet cross-section measurement are shown in Figure 1 for representative rapidity regions for anti-${k_{t}}$ jets with $R=0.4$ and $R=0.6$. In the central (forward) region the total uncertainty is about $5$% ($10$%) at medium $p_{\text{T}}$ of $300$–$600$ GeV. The uncertainty increases towards both lower and higher $p_{\text{T}}$ reaching to $15\%$ at low $p_{\text{T}}$ and $50$% at high $p_{\text{T}}$. The JES and JER uncertainties for jets with different sizes are rather similar at the jet level. However, at the cross-section level differences occur due to the different slopes of the distributions.

The dominant systematic uncertainty source for the measurement of the inclusive jet cross-sections is related to the jet energy measurement. The jet energy scale uncertainty is larger than the jet energy resolution uncertainty.

9 Theoretical predictions

9.1 Next-to-leading-order QCD calculation

The NLOJet++ [64] (v4.1.3) software program is used to calculate the NLO QCD predictions for the $2\rightarrow 2$ processes for the inclusive jet cross-sections. The renormalisation and factorisation scales are set to the $p_{\text{T}}$ of the leading jet in the event, i.e. $\mu_{\text{R}}=\mu_{\text{F}}={p_{\text{T}}^{\mathrm{j}et,max}}$. For fast and flexible calculations with various PDFs as well as different renormalisation and factorisation scales, the APPLGRID software [65] is interfaced with NLOJet++.

The inclusive jet cross-sections are presented for the CT14 [66], MMHT2014 [67], NNPDF3.0 [68], HERAPDF2.0 [69] PDF sets provided by the LHAPDF6 [70] library. The value for the strong coupling constant $\alpha_{\mathrm{s}}$ is taken from the corresponding PDF set.

Three sources of uncertainty in the NLO QCD calculation are considered: the PDFs, the choice of renormalisation and factorisation scales, and the value of $\alpha_{\mathrm{s}}$. The PDF uncertainty is defined at $68\%$ confidence level (CL) and is evaluated following the prescriptions given for each PDF set, as recommended by the PDF4LHC group for PDF-sensitive analyses [71]. The scale uncertainty is evaluated by varying the renormalisation and factorisation scales by a factor of two with respect to the original choice in the calculation. The envelope of the cross-sections with all possible combinations of the scale variations, except the ones when the two scales are varied in opposite directions, is considered as a systematic uncertainty. An alternative scale choice, \mbox{\mu_{\mathrm{R}}}=\mbox{\mu_{\mathrm{F}}}={p_{\text{T}}^{\mathrm{j}et}}, the $p_{\text{T}}$ of each individual jet that enters the cross-section calculation, is also considered. This scale choice is proposed in Ref. [72]. The difference with respect to the prediction obtained for the ${p_{\text{T}}^{\mathrm{j}et,max}}$ scale choice is treated as an additional uncertainty. The uncertainty from $\alpha_{\mathrm{s}}$ is evaluated by calculating the cross-sections using two PDF sets determined with two different values of $\alpha_{\mathrm{s}}$ and then scaling the cross-section difference corresponding to an $\alpha_{\mathrm{s}}$ uncertainty $\Delta\alpha_{\mathrm{s}}=0.0015$ as recommended in Ref. [71].

The uncertainties in the NLO QCD cross-section predictions obtained with the CT14 PDF set are shown in Figure 2 for representative phase-space regions. The renormalisation and factorisation scale uncertainty is the dominant uncertainty in most phase-space regions, rising from around $5-10$% at low $p_{\text{T}}$ in the central rapidity bin to about $50$% in the highest $p_{\text{T}}$ bins in the most forward rapidity region. This uncertainty is asymmetric and it is larger for anti-${k_{t}}$ jets with $R=0.6$ than for jets with $R=0.4$. The alternative scale choice, ${p_{\text{T}}^{\mathrm{j}et}}$, leads to a similar inclusive jet cross-section at the highest jet $p_{\text{T}}$, but gives an increasingly higher cross-section when the jet $p_{\text{T}}$ decreases. For $p_{\text{T}}=70$ GeV this difference is about $10$%. The PDF uncertainties vary from $5$% to $50$% depending on the jet $p_{\text{T}}$ and rapidity. The $\alpha_{\mathrm{s}}$ uncertainty is about $3$% and is rather constant in the considered phase-space regions.

9.2 Electroweak corrections

The NLO QCD predictions are corrected for electroweak effects derived using an NLO calculation in the electroweak coupling ($\alpha$) and based on a LO QCD calculation  [73]. The CTEQ6L1 PDF set is used [74]. This calculation includes tree-level effects on the cross-section of $O(\alpha\alpha_{S},\alpha^{2})$ as well as effects of loops of weak interactions at $O(\alpha\alpha_{\mathrm{s}}^{2})$. Effects of photon or $W$/$Z$ radiation are not included in the corrections. Real $W$/$Z$ radiation may affect the cross-section by a few percent at $p_{\text{T}}\sim 1$ TeV [75].

The correction factors were derived in the phase space considered for the measurement presented here and are provided by the authors of Ref. [73] through a private communication. No uncertainty associated with these corrections is presently estimated.

Figure 3 shows the electroweak corrections for jets with $R=0.4$ and $R=0.6$. The correction reaches more than $10\%$ for the highest $p_{\text{T}}$ in the lowest rapidity bin, but decreases rapidly as the rapidity increases. It is less than $3\%$ for jets with $|y|>1$.

9.3 Non-perturbative corrections

In order to compare the fixed-order NLO QCD calculations to the measured inclusive jet cross-sections, corrections for non-perturbative (NP) effects need to be applied. Each bin of the NLO QCD cross-section is multiplied by the corresponding correction for non-perturbative effects.

The corrections are derived using LO Monte Carlo event generators complemented by the leading-logarithmic parton shower by evaluating the bin-wise ratio of the cross-section with and without the hadronisation and the underlying event processes.

The MC event generators are run twice, once with the hadronisation and underlying event switched on and again with these two processes switched off. The inclusive jet cross-sections are built either from the stable particles or from the last partons in the event record, i.e. the partons after the parton showers finished and before the hadronisation process starts. These partons are the ones that are used in the Lund string model and the cluster fragmentation model to form the final-state hadrons. The bin-by-bin ratios of the inclusive jet cross-sections are taken as an estimate for the non-perturbative corrections.

The nominal correction is obtained from the Pythia 8 event generator [30] with the AU2 tune using the CT10 PDF [35], i.e. the same configuration as used to correct the data for detector effects (see Section 3). The uncertainty is estimated as the envelope of the corrections obtained from a series of alternative Monte Carlo event generator configurations as shown in Table 1.

The correction factors are shown in Figure 4 in representative rapidity bins for anti-${k_{t}}$ jets with $R=0.4$ and $R=0.6$ as a function of the jet $p_{\text{T}}$.

The nominal correction increases the cross-section by $4\%$ ($15\%$) for $p_{\text{T}}=70$ GeV for anti-${k_{t}}$ jets with $R=0.4$ ($R=0.6$). The large differences between the two jet sizes result from the different interplay of hadronisation and underlying-event effects. While for anti-${k_{t}}$ jets with $R=0.4$ the contribution from the hadronisation tends to cancel with the one from the underlying event, for anti-${k_{t}}$ jets with $R=0.6$ the effect from the underlying event becomes dominant. At large $p_{\text{T}}$ the non-perturbative correction factor is close to $1$. There is only a small dependence of the non-perturbative corrections on the jet rapidity.

The nominal correction is larger than the correction from other MC configurations. The corrections based on Pythia 8 with the Monash [76] or the A14 [77] tunes give correction factors that are closer to $1$. The corrections based on Herwig++ give corrections that are much lower than the one based on Pythia 8. The correction based on Herwig++ is $-10\%$ ($1\%$) for $p_{\text{T}}=70$ GeV for anti-${k_{t}}$ jets with $R=0.4$ ($R=0.6$).

9.4 NLO QCD matched with parton showers and hadronisation

The measured inclusive jet cross-section can be directly compared to predictions based on the Powheg Monte Carlo generator where an NLO QCD calculation for the hard scattering $2\to 2$ process is matched to parton showers, hadronisation and underlying event.

A procedure to estimate the effect of the matching of the hard scattering and the parton shower is not yet well established. Therefore, no uncertainties are shown for the Powheg predictions. The Powheg prediction’s uncertainty due to PDF is expected to be similar to that in fixed-order NLO calculations, whereas the uncertainty due to $\alpha_{\mathrm{s}}$ is expected to be larger, and the uncertainty due to the renormalisation and factorisation scales smaller.

The simulation using a matched parton shower has a more coherent treatment of the effect of parton showers and hadronisation than the approach using a fixed-order NLO QCD calculation corrected for non-perturbative effects. However, ambiguities in the matching procedure and the tuning of the parton shower parameters based on processes simulated only at leading order by Pythia 8 may introduce additional theoretical uncertainties. Therefore, quantitative comparisons using theoretical uncertainties based on Powheg are not performed in this paper.

10 Results

10.1 Qualitative comparisons of data to NLO QCD calculations

The measured double-differential inclusive jet cross-sections are shown in Figure 5 and Figure 6 as a function of the jet $p_{\text{T}}$ for anti-${k_{t}}$ jets with $R=0.4$ and $R=0.6$ for each jet rapidity bin. The cross-section covers $11$ orders of magnitude in the central rapidity region and $9$ orders of magnitude in the forward region. Jet transverse momenta above $p_{\text{T}}=2$ TeV are observed. In the most forward region the jet $p_{\text{T}}$ reaches about $500$ GeV. Tabulated values of all observed results, with full details of uncertainties and their correlations, are also provided in the Durham HEP database [84].

The measurement is compared to an NLO QCD prediction using the MMHT2014 PDF set [67] based on NLOJet++ corrected for non-perturbative and electroweak effects. The shaded band shows the total theory uncertainty as explained in Section 9.1. This theory prediction describes the gross features in the data.

The ratio of NLO QCD calculations to data corrected for non-perturbative and electroweak effects for various PDF sets is shown in Figure 7 and Figure 8 for anti-${k_{t}}$ jets $R=0.4$ and $R=0.6$, respectively. At low $p_{\text{T}}$ the level of agreement is very sensitive to non-perturbative effects. When using Pythia 8 as the nominal non-perturbative correction, the NLO QCD prediction is typically about $10$–$20\%$ above the data at low $p_{\text{T}}$, whereas the NLO QCD prediction corrected with Herwig++ follows the data well for anti-${k_{t}}$ jets with $R=0.4$, while it is $5$–$10\%$ below the data for anti-${k_{t}}$ jets with $R=0.6$.

The comparison is also influenced by the nominal choice of renormalisation and factorisation scales in the NLO QCD calculation. Setting the scale to ${p_{\text{T}}^{\mathrm{j}et}}$ instead of ${p_{\text{T}}^{\mathrm{j}et,max}}$ (see Section 9.1) leads to an NLO QCD prediction that is at low jet $p_{\text{T}}$ higher than the prediction using the ${p_{\text{T}}^{\mathrm{j}et,max}}$ scale (about $8\%$ at $p_{\text{T}}=100$ GeV for all pseudorapidity regions). With this scale setting the deviation from the data at low $p_{\text{T}}$ is larger.

The recent calculation of NNLO QCD inclusive jet cross-sections at $\sqrt{s}=7$ TeV is higher than in NLO QCD at low jet $p_{\text{T}}$ for all jet rapidity regions [12]. For instance, for $p_{\text{T}}=100$ GeV the increase from NLO to NNLO is about $10\%$. For both the NNLO and the NLO QCD calculations the ${p_{\text{T}}^{\mathrm{j}et,max}}$ scale is used. Therefore, it is expected that the NNLO QCD prediction at $\sqrt{s}=8$ TeV would deviate from the data more strongly than the NLO QCD calculation. This deviation might need to be accommodated by an adjustment of the PDFs.

Towards higher $p_{\text{T}}$ the NLO QCD predictions get closer to the data while for $p_{\text{T}}>1$ TeV they rise with respect to the data. For the highest $p_{\text{T}}$ at central rapidities they are typically up to $10$–$20\%$ higher than data. The behaviour of the CT14, NNPDF3.0 and MMHT2014 PDF sets is similar. The NLO QCD predictions based on the HERAPDF2.0, however, are significantly lower than data in the region $300<p_{\text{T}}<1000$ GeV.

In the most forward region, $|y|>2$, all PDF sets give predictions close to the data at low $p_{\text{T}}$ for anti-${k_{t}}$ jets with $R=0.4$ and $R=0.6$. However, towards higher $p_{\text{T}}$ and in particular for $p_{\text{T}}{}>400$ GeV the CT14, NNPDF3.0 and MMHT2014 PDF sets give predictions much higher than the data. The prediction for the HERAPDF2.0 is lower than for the other PDF sets and also falls below the data. In this region, both the experimental and the theoretical uncertainties become large.

Overall, the NLO QCD prediction based on the CT14 PDF set gives the best qualitative agreement, while HERAPDF2.0 gives the worst agreement over a wide jet $p_{\text{T}}$ range. However, the central values from the HERAPDF2.0 PDF set are more consistent with the data in the forward region at high $p_{\text{T}}$. This indicates that this measurement has sensitivity to constrain PDFs.

10.2 Quantitative comparison of data to NLO QCD calculations

A quantitative comparison of the NLO QCD predictions, corrected for non-perturbative and electroweak effects, to the measurement is performed using the method described in Ref. [85]. The $\chi^{2}$ value and the corresponding observed $p$-value, $P_{\mathrm{obs}}$, are computed taking into account the asymmetries and the correlations of the experimental and theoretical uncertainties. The individual experimental and theoretical uncertainty components are assumed to be uncorrelated among one another and fully correlated across the $p_{\text{T}}$ and $y$ bins. The correlation of the statistical uncertainties across $p_{\text{T}}$ and rapidity bins are taken into account using covariance matrices derived from $10000$ pseudo-experiments obtained by fluctuating the data and the MC simulation (see Section 7).

For the theoretical prediction, the uncertainties related to the scale variations, the alternative scale choice, the PDF eigenvectors, the non-perturbative corrections and the strong coupling constant are treated as separate uncertainty components. In the case of the NNPDF3.0 PDF set, the replicas are used to evaluate a covariance matrix, from which the eigenvectors are then determined.

Table 2 shows the evaluated $P_{\mathrm{obs}}$ for the NLO QCD predictions corrected for non-perturbative and electroweak effects for each rapidity bin considered individually. In this case, only cross-section measurements with \IfSubStrptcut1ptcut$p_{\text{T}}>70$ GeV \IfSubStrptcut1ptcut1$p_{\text{T}}>100$ GeV \IfSubStrptcut1ptcut2$100<p_{\text{T}}<900$ GeV \IfSubStrptcut1ptcut3$100<p_{\text{T}}<400$ GeV \IfSubStrptcut1ptcut4$70<p_{\text{T}}<100$ GeV \IfSubStrptcut1ptcut5$100<p_{\text{T}}<240$ GeV \IfSubStrptcut1ptcut6$240<p_{\text{T}}<408$ GeV \IfSubStrptcut1ptcut7$408<p_{\text{T}}<642$ GeV \IfSubStrptcut1ptcut8$642<p_{\text{T}}<952$ GeV \IfSubStrptcut1ptcut9$p_{\text{T}}>952$ GeV are included in the quantitative comparison of data and theory to reduce the influence of non-perturbative corrections.

For anti-${k_{t}}$ jets with $R=0.4$, $P_{\mathrm{obs}}$ values larger than about $4\%$ are found for all cross-sections and PDF sets. This indicates a satisfactory description of the data by the theory. The lowest $P_{\mathrm{obs}}$ values are found in the jet rapidity region $1.5\leq|y|<2.0$ and $2.5\leq|y|<3.0$. For anti-${k_{t}}$ jets with $R=0.6$ good agreement is found in the regions with $|y|>1$. Here, the $P_{\mathrm{obs}}$ values are larger than about $10\%$. However, in the central region $|y|<1$ the agreement is worse than for jets with $R=0.4$ resulting in $P_{\mathrm{obs}}$ values of the order of a percent or lower.

Similar studies were performed, for each rapidity bin, in various $p_{\text{T}}$ ranges: $p_{\text{T}}>70$ GeV, \IfSubStrptcut2ptcut$p_{\text{T}}>70$ GeV \IfSubStrptcut2ptcut1$p_{\text{T}}>100$ GeV \IfSubStrptcut2ptcut2$100<p_{\text{T}}<900$ GeV \IfSubStrptcut2ptcut3$100<p_{\text{T}}<400$ GeV \IfSubStrptcut2ptcut4$70<p_{\text{T}}<100$ GeV \IfSubStrptcut2ptcut5$100<p_{\text{T}}<240$ GeV \IfSubStrptcut2ptcut6$240<p_{\text{T}}<408$ GeV \IfSubStrptcut2ptcut7$408<p_{\text{T}}<642$ GeV \IfSubStrptcut2ptcut8$642<p_{\text{T}}<952$ GeV \IfSubStrptcut2ptcut9$p_{\text{T}}>952$ GeV , \IfSubStrptcut3ptcut$p_{\text{T}}>70$ GeV \IfSubStrptcut3ptcut1$p_{\text{T}}>100$ GeV \IfSubStrptcut3ptcut2$100<p_{\text{T}}<900$ GeV \IfSubStrptcut3ptcut3$100<p_{\text{T}}<400$ GeV \IfSubStrptcut3ptcut4$70<p_{\text{T}}<100$ GeV \IfSubStrptcut3ptcut5$100<p_{\text{T}}<240$ GeV \IfSubStrptcut3ptcut6$240<p_{\text{T}}<408$ GeV \IfSubStrptcut3ptcut7$408<p_{\text{T}}<642$ GeV \IfSubStrptcut3ptcut8$642<p_{\text{T}}<952$ GeV \IfSubStrptcut3ptcut9$p_{\text{T}}>952$ GeV . In all these cases, a similar level of agreement is observed between the measurement and the theory prediction, with a general trend of $P_{\mathrm{obs}}$ values decreasing with the increasing number of bins (i.e. when considering wider phase-space regions).

In addition to the quantitative comparisons of the theory and data cross-sections in individual jet rapidity bins, all data points can be considered together. Table 10.2 shows the $\chi^{2}$ values for each PDF set, $R$ value and scale choice, when using all the $|y|$ bins together. Various $p_{\text{T}}$ ranges are tested. All the corresponding $P_{\mathrm{obs}}$ are much smaller than $10^{-3}$. If the statistical uncertainty of the $\eta$-intercalibration were treated as a single component (see Section 6), the $\chi^{2}$ values computed in Table 10.2 would be strongly enhanced (by even more than $200$ units for some configurations).

Further quantitative comparisons using all the $|y|$ bins together were performed in more restricted $p_{\text{T}}$ ranges ( \IfSubStrptcut4ptcut$p_{\text{T}}>70$ GeV \IfSubStrptcut4ptcut1$p_{\text{T}}>100$ GeV \IfSubStrptcut4ptcut2$100<p_{\text{T}}<900$ GeV \IfSubStrptcut4ptcut3$100<p_{\text{T}}<400$ GeV \IfSubStrptcut4ptcut4$70<p_{\text{T}}<100$ GeV \IfSubStrptcut4ptcut5$100<p_{\text{T}}<240$ GeV \IfSubStrptcut4ptcut6$240<p_{\text{T}}<408$ GeV \IfSubStrptcut4ptcut7$408<p_{\text{T}}<642$ GeV \IfSubStrptcut4ptcut8$642<p_{\text{T}}<952$ GeV \IfSubStrptcut4ptcut9$p_{\text{T}}>952$ GeV , \IfSubStrptcut5ptcut$p_{\text{T}}>70$ GeV \IfSubStrptcut5ptcut1$p_{\text{T}}>100$ GeV \IfSubStrptcut5ptcut2$100<p_{\text{T}}<900$ GeV \IfSubStrptcut5ptcut3$100<p_{\text{T}}<400$ GeV \IfSubStrptcut5ptcut4$70<p_{\text{T}}<100$ GeV \IfSubStrptcut5ptcut5$100<p_{\text{T}}<240$ GeV \IfSubStrptcut5ptcut6$240<p_{\text{T}}<408$ GeV \IfSubStrptcut5ptcut7$408<p_{\text{T}}<642$ GeV \IfSubStrptcut5ptcut8$642<p_{\text{T}}<952$ GeV \IfSubStrptcut5ptcut9$p_{\text{T}}>952$ GeV , \IfSubStrptcut6ptcut$p_{\text{T}}>70$ GeV \IfSubStrptcut6ptcut1$p_{\text{T}}>100$ GeV \IfSubStrptcut6ptcut2$100<p_{\text{T}}<900$ GeV \IfSubStrptcut6ptcut3$100<p_{\text{T}}<400$ GeV \IfSubStrptcut6ptcut4$70<p_{\text{T}}<100$ GeV \IfSubStrptcut6ptcut5$100<p_{\text{T}}<240$ GeV \IfSubStrptcut6ptcut6$240<p_{\text{T}}<408$ GeV \IfSubStrptcut6ptcut7$408<p_{\text{T}}<642$ GeV \IfSubStrptcut6ptcut8$642<p_{\text{T}}<952$ GeV \IfSubStrptcut6ptcut9$p_{\text{T}}>952$ GeV , \IfSubStrptcut7ptcut$p_{\text{T}}>70$ GeV \IfSubStrptcut7ptcut1$p_{\text{T}}>100$ GeV \IfSubStrptcut7ptcut2$100<p_{\text{T}}<900$ GeV \IfSubStrptcut7ptcut3$100<p_{\text{T}}<400$ GeV \IfSubStrptcut7ptcut4$70<p_{\text{T}}<100$ GeV \IfSubStrptcut7ptcut5$100<p_{\text{T}}<240$ GeV \IfSubStrptcut7ptcut6$240<p_{\text{T}}<408$ GeV \IfSubStrptcut7ptcut7$408<p_{\text{T}}<642$ GeV \IfSubStrptcut7ptcut8$642<p_{\text{T}}<952$ GeV \IfSubStrptcut7ptcut9$p_{\text{T}}>952$ GeV , \IfSubStrptcut8ptcut$p_{\text{T}}>70$ GeV \IfSubStrptcut8ptcut1$p_{\text{T}}>100$ GeV \IfSubStrptcut8ptcut2$100<p_{\text{T}}<900$ GeV \IfSubStrptcut8ptcut3$100<p_{\text{T}}<400$ GeV \IfSubStrptcut8ptcut4$70<p_{\text{T}}<100$ GeV \IfSubStrptcut8ptcut5$100<p_{\text{T}}<240$ GeV \IfSubStrptcut8ptcut6$240<p_{\text{T}}<408$ GeV \IfSubStrptcut8ptcut7$408<p_{\text{T}}<642$ GeV \IfSubStrptcut8ptcut8$642<p_{\text{T}}<952$ GeV \IfSubStrptcut8ptcut9$p_{\text{T}}>952$ GeV and \IfSubStrptcut9ptcut$p_{\text{T}}>70$ GeV \IfSubStrptcut9ptcut1$p_{\text{T}}>100$ GeV \IfSubStrptcut9ptcut2$100<p_{\text{T}}<900$ GeV \IfSubStrptcut9ptcut3$100<p_{\text{T}}<400$ GeV \IfSubStrptcut9ptcut4$70<p_{\text{T}}<100$ GeV \IfSubStrptcut9ptcut5$100<p_{\text{T}}<240$ GeV \IfSubStrptcut9ptcut6$240<p_{\text{T}}<408$ GeV \IfSubStrptcut9ptcut7$408<p_{\text{T}}<642$ GeV \IfSubStrptcut9ptcut8$642<p_{\text{T}}<952$ GeV \IfSubStrptcut9ptcut9$p_{\text{T}}>952$ GeV ), for the CT14 PDF set. While good agreement is observed in the range \IfSubStrptcut4ptcut$p_{\text{T}}>70$ GeV \IfSubStrptcut4ptcut1$p_{\text{T}}>100$ GeV \IfSubStrptcut4ptcut2$100<p_{\text{T}}<900$ GeV \IfSubStrptcut4ptcut3$100<p_{\text{T}}<400$ GeV \IfSubStrptcut4ptcut4$70<p_{\text{T}}<100$ GeV \IfSubStrptcut4ptcut5$100<p_{\text{T}}<240$ GeV \IfSubStrptcut4ptcut6$240<p_{\text{T}}<408$ GeV \IfSubStrptcut4ptcut7$408<p_{\text{T}}<642$ GeV \IfSubStrptcut4ptcut8$642<p_{\text{T}}<952$ GeV \IfSubStrptcut4ptcut9$p_{\text{T}}>952$ GeV , for both jet radii $R$ values, the $P_{\mathrm{obs}}$ values for the other ranges are small (often below $0.1\%$). For the same five restricted $p_{\text{T}}$ ranges above $100$ GeV, considering this time pairs of consecutive $|y|$ bins, good agreement between data and theory is observed in most cases. Good agreement is also observed when considering pairs of one central and one forward (i.e. first–last) $|y|$ bins. These tests show that the source of the low $P_{\mathrm{obs}}$ values discussed above is not localised in a single rapidity bin, nor due to some possible tension between the central and the forward regions.

Since the difference between the non-perturbative corrections with two Monte Carlo generators is taken as a systematic uncertainty, the result of the quantitative comparison has little sensitivity to which correction is chosen as the nominal one. Even using the correction that brings the fixed-order NLO QCD to the Powheg prediction, i.e. including an additional correction for parton shower effects, does not alter the $P_{\mathrm{obs}}$ values. It is therefore expected that an explicit correction of parton shower effects as suggested in Ref. [86] has a similar effect. The quantitative comparison is also not very sensitive to the choice of nominal renormalisation and factorisation scales in the NLO calculations.

A set of $\chi^{2}$ values were also evaluated for the ABM11 PDF set [87], for $R=0.4$ and $R=0.6$, for the ${p_{\text{T}}^{\mathrm{j}et,max}}$ and ${p_{\text{T}}^{\mathrm{j}et}}$ scale choices, in the full $p_{\text{T}}$ range, for individual $|y|$ bins, as well as all the $|y|$ bins together. In this case, tension between data and the theory prediction is observed even in individual $|y|$ bins, with $P_{\mathrm{obs}}$ values below $10^{-3}$ for both $|y|<0.5$ and $0.5\leq|y|<1.0$. When using all the $|y|$ bins together, the $\chi^{2}$ is significantly larger than for other PDF sets, by up to 152 – 232 units compared to the results obtained for CT14.