Measurement and QCD analysis of double-differential inclusive jet cross sections in proton-proton collisions at s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{s} $$\end{document} = 13 TeV

A measurement of the inclusive jet production in proton-proton collisions at the LHC at s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{s} $$\end{document} = 13 TeV is presented. The double-differential cross sections are measured as a function of the jet transverse momentum pT and the absolute jet rapidity |y|. The anti-kT clustering algorithm is used with distance parameter of 0.4 (0.7) in a phase space region with jet pT from 97 GeV up to 3.1 TeV and |y| < 2.0. Data collected with the CMS detector are used, corresponding to an integrated luminosity of 36.3 fb−1 (33.5 fb−1). The measurement is used in a comprehensive QCD analysis at next-to-next-to-leading order, which results in significant improvement in the accuracy of the parton distributions in the proton. Simultaneously, the value of the strong coupling constant at the Z boson mass is extracted as αS(mZ) = 0.1170±0.0019. For the first time, these data are used in a standard model effective field theory analysis at next-to-leading order, where parton distributions and the QCD parameters are extracted simultaneously with imposed constraints on the Wilson coefficient c1 of 4-quark contact interactions.


Introduction
Quantum chromodynamics (QCD) is the theory describing strong interactions among partons (quarks and gluons), the fundamental constituents of hadrons. In high-energy proton-proton (pp) collisions, partons from both colliding protons interact, producing energetic collimated sprays of hadrons (jets) in the final state. Inclusive jet production in p + p → jet + X, consisting of events with at least one jet, is a key process to test QCD - predictions at the highest achievable energy scales. At the CERN LHC, inclusive jet production in pp collisions has been extensively measured by both the ATLAS and CMS Collaborations at several centre-of-mass energies √ s. The present status of the measurements with corresponding integrated luminosities is summarised in table 1; the earlier 7 TeV measurements [1-3] with lower integrated luminosities are omitted. In ref.
[4], the first data collected by the CMS Collaboration at 13 TeV were analysed.
These measurements were compared with fixed-order predictions in perturbative QCD (pQCD) at next-to-leading order (NLO). Predictions at next-to-leading-logarithmic order (NLL) [13] and at next-to-next-to-leading order (NNLO) are available [14,15] and describe the LHC data [12] well using the transverse momentum p T of an individual jet as the renormalisation and factorisation scales.
Inclusive jet production at high momenta probes the proton structure in the kinematic range of high fraction x of the proton momentum carried by the parton. In particular, it is directly sensitive to the gluon distribution in the proton at high x. Measurements of the inclusive jet cross sections provide additional constraints on the parton distribution functions (PDFs) and the value of the strong coupling constant α S , as demonstrated e.g. in previous CMS publications [11,16].
In this paper, the data collected by the CMS experiment in 2016, corresponding to an integrated luminosity of up to 36.3 fb −1 , are analysed. The measurement of the doubledifferential inclusive jet cross sections is presented as a function of the jet p T and jet rapidity |y|. The jets are clustered with the anti-k T jet algorithm [17], as implemented in the FastJet package [18]. Two jet distance parameters, R = 0.4 and 0.7, are used: • R = 0.4 is a default in most recent analyses both in ATLAS and CMS at 13 TeV.
• R = 0.7 is chosen for most of QCD analyses using jet measurements because the effects of out-of-cone radiation are smaller. In particular, this value is used in the analogous analysis [11] with CMS data at 8 TeV.
The size of the nonperturbative (NP) corrections as a function of R is discussed in ref. [19]. Experimentally, the impact of the jet radius on the inclusive jet cross sections is studied by the CMS Collaboration in ref. [20]. The impact of the present measurements on the proton PDFs is illustrated in a QCD analysis, where the measured double-differential cross sections of inclusive jet production are used together with data from deep inelastic scattering (DIS) at HERA [21]. In addition, the CMS measurements [22] of normalised triple-differential top quark-antiquark -2 -

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(tt) production cross sections are used, which provide additional sensitivities to the gluon distribution, α S , and the top quark mass m t .
Furthermore, the effect of beyond the standard model (BSM) particle exchanges between the quarks is studied, using the model of contact interactions (CI) [23,24], added to the standard model (SM) process via effective couplings. The earlier searches for CI by the CMS Collaboration were performed using inclusive jet [25] and dijet [26][27][28] production. In those analyses, fixed values for the Wilson coefficients were assumed and the limits on the scale of the new interactions were set based on a comparison of data to the SM+CI prediction. However, the PDFs used in the SM prediction are derived assuming the validity of the SM at high jet p T , where the effects of new physics are expected to be most pronounced. Consequently, the BSM effects might be absorbed into PDFs and the interpretation of high-p T jet data may be biased. In the present analysis, the CI Wilson coefficient c 1 is a free parameter in the effective field theory (EFT)-improved SM (SMEFT) fit and is extracted simultaneously with the PDFs. The scenarios investigated correspond to purely left-handed, vector-and axial vector-like CI.
The paper is organised as follows. In section 2, a brief description of the detector is given. In section 3, the measurement of the double-differential cross sections is detailed. Theoretical predictions are explained in section 4. Experimental and theoretical cross sections are compared in section 5. Finally, the QCD interpretation is given in section 6. The paper is summarised in section 7.

The CMS detector
The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter (HCAL), each composed of a barrel and two endcap sections. Forward calorimeters extend the pseudorapidity η coverage provided by the barrel and endcap detectors. Muons are detected in gas-ionisation chambers embedded in the steel flux-return yoke outside the solenoid.
The ECAL consists of 75 848 lead tungstate crystals, which provide coverage in |η| < 1.48 in the barrel region and 1.48 < |η| < 3.00 in two endcap regions. Preshower detectors consisting of two planes of silicon sensors interleaved with a total of 3X 0 of lead are located in front of each ECAL endcap detector.
In the region |η| < 1. 74, the HCAL cells have widths of 0.087 in η and 0.087 in azimuth (φ). In the η-φ plane, and for |η| < 1. 48, the HCAL cells map on to 5×5 arrays of ECAL crystals to form calorimeter towers projecting radially outwards from close to the nominal interaction point. For |η| > 1. 74, the coverage of the towers increases progressively to a maximum of 0.174 in ∆η and ∆φ. Within each tower, the energy deposits in ECAL and HCAL cells are summed to define the calorimeter tower energies.
The reconstructed vertex with the largest value of summed physics-object p 2 T is taken to be the primary pp interaction vertex.

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The particle-flow (PF) algorithm [29] reconstructs and identifies each individual particle in an event, with an optimised combination of information from the various elements of the CMS detector. The energy of charged hadrons is determined from a combination of their momentum measured in the tracker and the matching ECAL and HCAL energy deposits, corrected for zero-suppression effects and for the response function of the calorimeters to hadronic showers. Finally, the energy of neutral hadrons is obtained from the corresponding corrected ECAL and HCAL energies. Jets are reconstructed offline from PF objects using the anti-k T algorithm [17,18] with R of 0.4 and 0.7.
Jet momentum is the vector sum of all PF candidate momenta in the jet, and is determined from simulation to be, on average, within 5-10% of the true momentum over the entire p T spectrum and detector acceptance. Additional pp interactions within the same or nearby bunch crossings (pileup) can contribute additional tracks and calorimetric energy depositions that increase the detector-level jet momentum. To mitigate this effect, tracks identified as originating from pileup vertices are discarded and an offset correction is applied to correct for remaining contributions. Jet energy corrections are derived from simulation studies so the average measured response of jets becomes identical to that of particle-level jets. In situ measurements of the momentum balance in dijet, photon+jet, Z +jet, and multijet events are used to determine any residual differences between the jet energy scale (JES) in data and in simulation, and appropriate corrections are derived [30]. Additional selection criteria are applied to each jet to remove jets potentially affected by instrumental effects or reconstruction failures [31].

Data analysis
The inclusive jet double-differential cross section, as a function of the individual jet p T and y is defined as follows: where L corresponds to the integrated luminosity, ∆p T (∆y) to the bin width of the jet p T (y), and N eff jets to the effective number of jets per bin estimated at the particle level, i.e. after corrections for detector effects. The binning scheme coincides with the one chosen in former publications [4,8,9,11].
The details of the data and simulation are described in the following: the selection of the events and jets (section 3.1), the triggers (section 3.2), the calibrations (section 3.3), corrections for efficiencies, misidentification, and migrations due to the limited resolution of the detector (section 3.4), and the experimental uncertainties (section 3.5).

Event selection
The data samples recorded in 2016 correspond to an integrated luminosity of 36.3 fb −1 (33.5 fb −1 ) for events with jets clustered with R = 0.4 (0.7) [36]. Triggers for the larger jet distance parameter were activated after 2.8 fb −1 of data had been taken, which explains the difference in the integrated luminosities.
The detector response in simulations is modelled with Geant4 [37]. The simulations in table 2 include a simulation of the pileup produced with pythia 8+CUETP8M1, which correspond to the pileup conditions in the data.
The primary vertex (PV) must satisfy |z PV | < 24 cm and ρ PV < 2 cm, where z PV (ρ PV ) corresponds to the longitudinal (radial) distance from the nominal interaction point. Event filters mentioned in section 2 are applied to reduce the noise from the detector.
In addition, the jets must satisfy quality criteria based on the jet constituents to remove the effect of detector noise, and must be reconstructed within |y| < 2.5, corresponding to the tracker acceptance. Jets reconstructed in regions of the detector corresponding to defective zones in the calorimeters are excluded from the measurement and recovered later in the unfolding procedure; the effect is of the order of a percent and is uniform as a function of p T .

Triggers
The prescaled single-jet triggers are used, requiring at least one jet in the event with jet p HLT T larger than a certain threshold. All triggers are prescaled, except the one with  Table 3. The HLT ranges and effective integrated luminosities used in the jet cross section measurement for R = 0.4. The first (second) row shows the p T threshold for the HLT (offline PF) reconstruction; the third row corresponds to the effective luminosity of each trigger L.
the highest threshold, and correspond to different effective integrated luminosities. Two different series of triggers are used for each value of the R parameter, shown in tables 3-4. The jets are weighted event by event in contrast to previous measurements [4,8,9,11] of inclusive jet production by the CMS Collaboration, where the whole contribution of each trigger is normalised with respect to its effective luminosity. This makes a difference, especially in terms of smoothness of the spectrum even after unfolding, since the trigger rate is nonlinear as a function of the instantaneous luminosity and the JES corrections generally vary with time and with pileup conditions. An event is considered for the measurement only if the leading jet reconstructed with the offline PF algorithm is matched to an HLT jet. The data contain events selected with a combination of triggers in different, exclusive intervals of the leading jet p T . The edges of each interval are determined in such a way that the corresponding trigger has an efficiency above 99.5% in all p T and |y| bins. The efficiency of each trigger is determined from the data set recorded by the next single-jet trigger (with lower but closest threshold), except for the most inclusive trigger. To determine the region of efficiency of the most inclusive trigger and to cross-check the efficiencies obtained for the other triggers, a tagand-probe method is applied to dijet topologies, which counts the leading and subleading offline PF jets that can be matched to HLT jets. The measured spectrum is corrected for the residual trigger inefficiency as a function of p T and |y|. Finally, to control possible steps in the distribution caused by passing the trigger thresholds, the spectrum is fitted in |y| regions using a truncated Taylor expansion with Chebyshev polynomials of the first kind as basis [45]; the χ 2 /N dof (where dof is degree of freedom) is compatible or close to one within statistical uncertainties.
The additional trigger inefficiency due to the prefiring effect, mentioned in section 2, is corrected event by event in the data before the unfolding procedure, using maps of prefiring probability in 2.0 < |η| < 3.0 as a function of p T and η. The total event weight is obtained as the product of the nonprefiring probability of all jets. The resulting effect is typically 2% for the spectrum at |y| < 2.0.

Calibration
The JES corrections are applied according to the standard CMS procedure [30]. An additional smoothing procedure is applied to the JES corrections (originally parameterised with linear splines in bins of p T ) to ensure and preserve the smoothness of the cross sections using the same fit method described in section 3.2.   Table 4. The HLT ranges and effective integrated luminosities used in the jet cross section measurement for R = 0.7. The first (second) row shows the p T threshold for the HLT (offline PF) reconstruction; the third row corresponds to the effective luminosity of each trigger L.
The detector-level p T (p rec T ) is rescaled such that the jet energy resolution (JER) in the simulated samples matches the JER in the data; this procedure is also known as JER smearing. A matching between the particle-and detector-level jets is performed for each event. The particle-level jets are ordered by decreasing p T . Each particle-level jet is matched to the highest-p T detector-level jet present in a cone with ∆R = 0.2 (0.35) for jets clustered with R = 0.4 (0.7); a particle-level jet may be matched to only one detector-level jet. The response is fitted from matched jets using a double Crystal-Ball function [46,47] to account for the presence of non-Gaussian tails. The width of the resolution is extracted and fitted as a function of p rec T , in bins of η and the variable ρ defined in ref. [48]. A modified NSC function is used, where the second term in eq. (8.10) of ref.
[30] is extended by p d T with d being an additional fit parameter. The p rec T is then rescaled as a function of the response with a scale factor; if no matching can be performed in the Gaussian core of the double Crystal-Ball function, the response is estimated with a Gaussian of width obtained from the modified NSC function.
The simulation of the pileup also includes the signal, but with a lower event count. To avoid double-counting and ensure a good statistical precision over the whole phase space, events with a hard scatter from the pileup simulation and events with anomalously large weights are discarded from the simulated samples. To ensure correct normalisation, the remaining events are reweighted to restore the originally generated spectrum. Finally, the pileup profile used in the simulation is corrected to match that in data.

Correction for detector effects
The measured detector-level distribution is unfolded to the particle level using corrections derived from the simulated events.
Sequentially, corrections for the background, migrations and inefficiencies are applied. To estimate the different effects, the same matching algorithm as described in the context of the JER smearing is used. Successfully matched jets, both in the Gaussian core and in the tails of the response, are used to estimate the response matrix, whereas unmatched jets at the particle (detector) level are used to estimate the inefficiencies (background). The background contribution is at 1-2%-level at low p T and is negligible at medium at high p T , while the maximum inefficiency reaches 2-5% at low and high p T .
Various types of backgrounds and inefficiencies are considered: the migrations in/out of the phase space, and the unmatched jets that correspond to either pileup or objects wrongly identified as jets by the reconstruction algorithm.

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The probability matrix (PM), shown in figure 1 for jets clustered with R = 0.7, is obtained by normalising the response matrix row by row. It contains the probability for a given particle-level jet to be reconstructed as a given detector-level jet. Assuming a PM A and the sum of the backgrounds b, both obtained from the simulation, and given a measured detector-level distribution y, the particle-level distribution x is determined by minimising the following objective function: where V is the covariance matrix of the detector-level data describing the statistical uncertainties associated with the data (including correlations), as well as with the backgrounds. The detector-level distribution has twice the number of bins as in the particle-level distribution. The matrix condition number of the PM, i.e. the ratio of the highest and lowest eigenvalues, is close to 4, assuring that the PM is not ill-conditioned; therefore no additional regularisation is applied. The whole procedure is performed with the TUnfold package [49], version 17.9. Finally, the residual inefficiencies are obtained from the rate of particle-level jets that cannot be matched to any detector-level jets passing the event selection, and corrected bin-by-bin on the particle-level distribution x.

Uncertainties
The measurement is affected by systematic and statistical uncertainties. The contributions of the various uncertainties are shown in figure 2, where the coloured band indicates the bin-to-bin fully correlated uncertainties and the vertical error bars indicate the uncorrelated uncertainties.
The bin-to-bin fully correlated uncertainties are determined as follows: • Variations of the JES corrections and of the prefiring correction are applied to the data at the detector level and are mapped to the particle level by repeating the unfolding procedure. The JES uncertainties are the dominant uncertainties in this measurement.
• Systematic effects related to the JER and to the pileup profile correction are varied in the simulated sample and propagated to the particle level by repeating the unfolding procedure.
• The normalisation of the estimates of the inefficiencies and backgrounds, obtained from the Monte Carlo (MC) simulation, are varied separately within a conservative estimate of 5%, covering a potential model dependence in migrations in the phase space and an impact from the matching algorithm in the unfolding procedure. Simulation CMS 13 TeV Figure 1. The probability matrix, estimated with a simulated sample based on pythia 8, for jets clustered using the anti-k T algorithm with R = 0.7. The horizontal (vertical) axis corresponds to jets at the particle (detector) level. The global 5 × 5 structure corresponds to the bins of rapidity y of the jets, indicated by the labels in the uppermost row and rightmost column; the horizontal and vertical axes of each cell correspond to the transverse momentum p T of the jets. The colour range covers a range from 10 −6 to 1 and the rows are normalised to unity, indicating the probability for a particle-level jet generated with values of p gen T and |y| gen to be reconstructed at the detector level with values of p rec T and |y| rec . Migrations outside of the phase space are not included; migrations across rapidity bins only occur among adjacent rapidity bins. The dashed lines indicate the diagonal bins in each rapidity cell. spectrum is corrected to match the data. The modified version is obtained by applying a smooth correction as a function of the p T , y, and jet multiplicity. The effect is strongest for |y| > 1.5, where pythia 8 does not describe the data well. Alternatively to pythia 8, MadGraph5_amc@nlo simulation is used and agrees well with the pythia 8 results.  Relative uncertainties in the double-differential cross section, as functions of jet transverse momentum (x axis) and rapidity (cells), for jets clustered using the anti-k T algorithm with R = 0.7. The systematic uncertainties are shown in different, noncumulative colour bands: the red bands correspond to JES uncertainties, the yellow bands to the JER uncertainties, and the blue bands to all other sources, including the integrated luminosity uncertainty, the model uncertainty, uncertainties in the migrations in and out of the phase space, and uncertainties in various inefficiencies and backgrounds. The vertical error bars include the statistical uncertainties from the data and from the pythia 8 simulated sample used for the unfolding, as well as the binwise systematic uncertainties, all summed in quadrature. The total uncertainty, shown in green, includes all systematic and statistical uncertainties summed in quadrature.
• A fully correlated 1.2% uncertainty in the integrated luminosity calibration is applied to the nominal variation of the unfolded spectrum [36].
Bin-to-bin fluctuations in the systematic variations are removed by applying a smoothing procedure based on Chebyshev polynomials following the method described in section 3.2. Uncorrelated and partly correlated uncertainties among p T and y bins arise from various origins: • The inclusive jet measurement is based on multiple jets recorded in each event. This is the dominant contribution to the uncorrelated uncertainties shown in figure 2.
• An additional uncorrelated systematic uncertainty of 0.2% is added before the unfolding to account for differences in alternative methods of determining the trigger efficiency.
• Statistical fluctuations in the simulated distributions.
After unfolding, the different sources are no longer distinguished from one another; in addition, the unfolding procedure introduces anti-correlations among directly neighbouring bins. The correlation matrix after the unfolding is shown in figure 3.

Theoretical predictions
In the following, the fixed-order pQCD predictions, the electroweak (EW) corrections, and the NP corrections are described.
-10 -JHEP02(2022)142 The correlation matrix at the particle level, for jets clustered using the anti-k T algorithm with R = 0.7. It contains contributions from the data and from the pythia 8 sample used to perform the unfolding. The global 4 × 4 structure corresponds to the bins of rapidity y of the jets, indicated by the labels in the uppermost row and rightmost column; the horizontal and vertical axes of each cell correspond to the transverse momentum p T of the jets. The colour range covers a range from −1 to 1 and indicates correlations in blue shades and anti-correlations in red shades, except for values between −0.1 and 0.1. Correlations across rapidity bins reach significant values mostly at the edges of the p T range.

Fixed-order predictions
Fixed-order pQCD predictions for the inclusive jet production are available at NLO and NNLO accuracy, obtained using the NLOJet++ [50,51] and NNLOJET (rev5918) [14,15,52] programs, with NLO calculations implemented in fastNLO [53]. The calculations are performed for five active massless quark flavours. The renormalisation (µ r ) and factorisation (µ f ) scales are set to the individual jet p T . Alternative prediction using µ r and µ f set to the scalar sum of the parton p T (H T parton) is used for comparison. In ref. [15] JHEP02(2022)142 the individual jet p T was found to be a better choice for the scale than the transverse momentum of the leading-jet p max T . Furthermore in refs. [54][55][56] it was discussed that NNLO calculations with jet distance parameter of the anti-k T clustering algorithm R = 0.7 are more stable than those with R = 0.4.
To estimate possible uncertainty due to missing higher-order contributions, the scales are varied independently by a factor of 2 up and down, avoiding cases with µ f /µ r = 4 ±1 . The largest deviation of the cross section from the result obtained with the central scale choice is used as an estimate of the scale uncertainty. In general, the scale uncertainties for R = 0.7 are larger than for R = 0.4.
The NLO QCD prediction is improved to NLO+NLL accuracy with a simultaneous jet radius and threshold resummation k-factor for each bin i: where the singular NLO terms σ NLO sing. and the resummed contributions σ NLL are obtained using the NLL-Jet calculation, provided by the authors of ref.
[13]. Following their approach, the σ NLO for the resummation factor is computed using the modified Ellis Kunszt Soper (MEKS) code version 1.0 [61], and choosing the renormalisation and factorisation scales µ r = µ f = p max T .

Electroweak corrections
The EW effects, which arise from the virtual exchange of the massive W and Z gauge bosons are calculated to NLO accuracy [62] and are applied to the fixed-order QCD predictions. In the high-p T region, these EW effects grow to 11%, as illustrated in figure 4. No uncertainty associated with these corrections is available yet. The contribution of real production of EW boson production in association with jets is estimated at NLO using mcfm [63][64][65] program to be at most at percent level which is negligible for the present analysis.

Nonperturbative corrections
The NP corrections are defined for each bin i as where PS stands for parton shower, HAD for hadronisation, and MPI for multiparton interaction. The NP factors correct for the hadronisation and the MPI effects that are not included in the fixed-order pQCD predictions. At low p T , the NP corrections are dominated by MPI, which increases the radiation in the jet cone by a constant offset. This is especially T Anti-k |y| < 0.5 0.5 < |y| < 1.0 1.0 < |y| < 1.5 1.5 < |y| < 2.0 Figure 4. The EW corrections for inclusive jet cross sections, as reported in ref. [62]. The values for jets clustered using the anti-k T algorithm with R = 0.4 (0.7) are shown on the left (right); each curve corresponds to a rapidity bin. important for R = 0.7. On the other hand, hadronisation plays a role at smaller R. The effects of perturbative radiation are partially considered in the higher-order predictions; for this reason the PS simulation is included in both the numerator and the denominator. It is stronger for smaller R, where out-of-cone radiation plays a larger role, which NLL corrections can account for.
To define final NP corrections, pythia 8 CP1 tune [66] and herwig++ EE5C tune [40] are fitted with a smooth function a 0 +a 1 /p a 2 T . The correction is obtained from the resulting envelope with the central value taken in the middle of the envelope and the uncertainties from its edges.

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The NP corrections defined in eq. (4.2) are shown in figure 5. The corrections are larger for R = 0.7 than for R = 0.4, since a larger cone size includes more effects from the underlying event.

Results
In figure 6, the inclusive jet cross sections are presented as functions of the jet p T and |y| for R = 0.4 and 0.7. The cross sections are shown for four absolute rapidity intervals: |y| < 0.5, 0.5 < |y| < 1.0, 1.0 < |y| < 1.5, and 1.5 < |y| < 2.0 with jet p T > 97 GeV. The data are compared with fixed-order NNLO QCD predictions using CT14 PDF, corrected for NP and EW effects. The data cover a wide range of the jet p T from 97 GeV up to 3.1 TeV.
In figure 7 (figure 8), the ratios of the measured cross sections to the NNLO QCD predictions using different scale choices and to the NLO+NLL predictions with various PDFs sets are shown for jets with R = 0.4 (0.7). In general at high p T , smaller (higher) cross sections are predicted than experimentally measured for the central (forward) rapidities. The theoretical uncertainties are larger at high p T and are dominated by the PDF uncertainties. The scale uncertainties are significantly smaller at NNLO as compared to NLO+NLL. The prediction using jet p T as renormalisation and factorisation scale results in a harder p T spectrum than in case of scale set to H T parton. The NLO+NLL calculations predict harder p T spectrum than the NNLO calculations. All predictions describe the data well within the experimental and theory uncertainties.
The CT14, NNPDF 3.1, and MMHT 2014 PDF sets include CMS and ATLAS measurements of inclusive jet cross sections at 7 TeV, whereas ABMP 16 does not include LHC jet measurements and HERAPDF 2.0 is based only on the HERA DIS data. Predictions obtained with these PDFs are similar at low p T , although significant differences at high p T are observed. These differences arise from differences in the gluon distribution at high x in the available PDFs and point to high sensitivity of the present measurement to the proton PDFs.
Additional comparisons to theoretical predictions are available in Appendix A.

The QCD analysis
The sensitivity of the presented measurement of the inclusive jet production to the proton PDFs and to the value of the strong coupling constant at the mass of the Z boson α S (m Z ) is investigated in a comprehensive QCD analysis. The jet cross section for R = 0.7 is used because of reduced out-of-cone radiation and a better description of the measurement by the pQCD predictions. The QCD analysis is performed either assuming only the standard model or applying effective corrections to the QCD calculation of the inclusive jet production to include 4-quark contact interactions.

Data sets used
In this QCD analysis, the double-differential inclusive jet production cross section with R = 0.7 is used together with the charged-and neutral-current DIS cross sections of   . The double-differential cross section of inclusive jet production, as a function of p T and |y|, for jets clustered using the anti-k T algorithm with R = 0.4, presented as ratios to the QCD predictions. The data points are shown by filled circles, with statistic uncertainties shown by vertical error bars, while the total experimental uncertainty is centred at one and is presented by the orange band. In the upper panel, the data are divided by the NNLO prediction, corrected for NP and EW effects, using CT14nnlo PDF and with renormalisation and factorisation scales jet p T and, alternatively H T (blue solid line). In the lower panel, the data are shown as ratio to NLO+NLL prediction, calculated with CT14nlo PDF, and corrected for NP and EW effects. The scale (PDF) uncertainties are shown by red solid (dashed) lines. NLO+NLL predictions obtained with alternative PDF sets are displayed in different colours as a ratio to the central prediction using CT14nlo.
HERA [21]. In addition, the normalised triple-differential tt cross section [22] from CMS is used. As demonstrated in ref. [22], the top quark pole mass m pole t , the gluon distribution, and the value of α S (m Z ) are closely correlated in the triple-differential tt production cross section, thus providing additional sensitivity to the gluon distribution and to α S (m Z ). In the CMS tt and jet measurements, the common systematic sources associated with the JES uncertainties are taken as 100% correlated.

Theoretical calculations used in QCD analysis
The SM theoretical predictions for the inclusive jet production cross section at NLO and NNLO are obtained as described in section 4.1 and are corrected for NP and EW effects. The NNLO calculation is approximated by k-factors, obtained as a ratio of fixed-order NNLO to NLO calculations using CT14nnlo PDF in each bin in p T and |y|. These are applied to the NLOJet++ prediction interfaced to xFitter using fast-grid techniques of fastNLO [53]. In a similar way, the NLO prediction is improved to NLO+NLL as explained in section 4. The QCD prediction for the normalised triple-differential cross section of the tt production is available only at NLO and is described in detail in ref. [22]. The renormalisation and factorisation scales are set to the four-momentum transfer Q for the DIS data and to the individual jet p T for inclusive jet cross section measurements.

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Following ref. [22], in the case of tt production, the scales are set to µ r = µ f = 1 2 i m Ti . The sum over i covers the final-state partons t, t, and at most three light partons in a tt + 2 jets scenario. The transverse mass m Ti ≡ m 2 i + p T 2 i is computed using the mass m i and transverse momentum p Ti of the partons [22].
The QCD analysis at NLO is extended into a SMEFT study by adding dimension-6 operators for colour-charged fermions to the SM Lagrangian L SM [23,24], so that Here, the c n are Wilson coefficients and Λ is the scale of new physics. For the 4-quark CI, the nonrenormalisable operators O n are where the sums in c and d run over generations, whereas i, j, k, l are colour indices. The L and R subscripts denote the handedness of the quarks. The operators in eqs. (6.2)-(6.4) correspond to having integrated out a colour-singlet BSM exchange between two quark lines. The colour-singlet exchanges are dominant in quark compositeness [77] or Z models [78]. The operators commonly denoted O 2 , O 4 and O 6 in literature involve a product of Gell-Mann matrices in place of δ ij δ kl and cause colour-octet; they are not considered here. The CI studied in the SMEFT fits is either purely left-handed, vector-like, or axial vector-like. The only free Wilson coefficient is c 1 , which multiplies the operator O 1 in eq. (6.1). The coefficients c 3 and c 5 are determined from c 1 in accordance with how the quark-line handedness may change in the interaction. In the left-handed singlet model there are CI only between two left-handed lines, and hence c 3 = c 5 = 0. Vector-like and axial vector-like exchanges allow interactions also between right-handed quarks, giving c 5 = c 1 in both cases. For interactions between quark lines of different handedness, the vector-like exchange implies c 3 = 2c 1 , whereas the axial vector-like model has c 3 = −2c 1 . Further details of the theoretical model are given in ref. [23].
In the QCD analysis, the SMEFT prediction for the double-differential cross section of the inclusive jet production reads where k NLO+NLL is given in eq. (4.1) and EW and NP are explained in sections 4.2-4.3; the CI term is computed at NLO using the cijet software [24] interfaced to xFitter.

The general QCD analysis strategy and PDF uncertainties
The procedure for determining the PDFs follows the approach of HERAPDF [21,79]. The contributions of charm and beauty quarks are treated in the Thorne-Roberts [80][81][82] variable-flavour number scheme at NLO. The values of heavy quark masses are set to m c = 1.47 GeV and m b = 4.5 GeV. The m pole t and α S (m Z ) are free parameters in the PDF fits. The DIS data are restricted to high Q 2 by setting Q 2 min = 7.5 GeV 2 . The parameterised PDFs are the gluon distribution xg(x), the valence quark distributions xu v (x), and xd v (x), as well as xU(x) for the up-and xD(x) for the down-type antiquark distributions. At the starting scale of QCD evolution Q 2 0 = 1.9 GeV 2 , the general form of the parameterisation for a PDF f is  [21].
The D f and E f parameters probe the sensitivity of the results to the specific selected functional form. In general, the parameterisation is obtained by first setting all D and E parameters to zero and then including them in the fit, one at a time. The improvement in the χ 2 of a fit is monitored and the procedure is stopped when no further improvement is observed. Differences in the data sets or theoretical predictions lead to differences in the resulting parameterisation.
In the full QCD fit, the uncertainties in the individual PDFs and in the extracted non-PDF parameters are estimated similarly to the approach of HERAPDF [21,79], which accounts for the fit, model, and parameterisation uncertainties as follows.
Fit uncertainties: originate from the uncertainties in the used measurements and are obtained by using the Hessian method [83] implying the tolerance criterion ∆χ 2 = 1, which corresponds to the 68% confidence level (CL). Alternatively, the fit uncertainties are estimated by using the MC method [84,85], where MC replicas are created by randomly fluctuating the cross section values in the data within their statistical and systematic uncertainties. For each fluctuation, the fit is performed and the central values for the fitted parameters and their uncertainties are estimated using the mean and the root mean square values over the replicas.
Parameterisation uncertainty: is estimated by extending the functional forms of all PDFs with additional parameters D and E, which are added, independently, one at a time.

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The resulting uncertainty is constructed as an envelope, built from the maximal differences between the PDFs (or non-PDF parameters) resulting from all the parameterisation variations and the results of the central fit. The total PDF uncertainty is obtained by adding in quadrature the fit and the model uncertainties, while the parameterisation uncertainties are added linearly.
The QCD analysis is performed in few steps. First, the impact of the CMS data on global PDF set CT14 [57] is investigated by using the profiling technique [86][87][88], as implemented in xFitter. The available implementation does not allow for a simultaneous profiling of the PDF and non-PDF parameters such as α S (m Z ), m pole t or c 1 . Therefore, those parameters are profiled individually. Next, the HERA DIS and CMS jet data are used in a full QCD fit in SM at NNLO, where the PDFs and the value of α S (m Z ) are determined simultaneously. Further, the full QCD fit in SMEFT at NLO is performed using the HERA data and the CMS measurements of inclusive jet and tt production at √ s = 13 TeV, where the PDFs, α S (m Z ), m pole t , and c 1 are obtained at the same time. The individual steps of the QCD interpretation are described in detail in the following.

Results of profiling analysis
The impact of new data on the available PDFs is assessed in a profiling analysis [86][87][88][89]. Here, the PDF profiling is performed at NLO or at NNLO, using the CT14 PDF sets [57] derived at NLO or NNLO, respectively. These PDF sets do not include the CMS tt measurements. The theoretical prediction for the triple-differential tt cross section corresponding to the CMS measurement [22] is available only at NLO.
In the PDF profiling, the strong coupling is fixed to the central value of the CT14 PDF set, α S (m Z ) = 0.118. Once the tt cross sections are used, the m pole t is set to 170.5 GeV, corresponding to the result of ref. [22]. The results of the PDF profiling using the present inclusive jet cross section are shown in figure 9 (figure 10) at NLO (NNLO). According to the sensitivity of the data, the uncertainties in the PDFs are significantly improved by using the CMS jet measurement in the full x range for the gluon and at medium x for the sea quark distributions, whereas the valence distributions remain unchanged.
In addition to profiling the PDFs, the impact of the inclusive jet measurements on the extraction of the strong coupling constant is investigated. For this purpose, the α S -series of the CT14 PDFs at NLO and NNLO is used, where the value of α S (m Z ) was varied from 0.1110 to 0.1220. Note that any possible α S (m Z ) dependence of the k-factors could not be accounted for. The individual profiling is performed for each of the PDF members in the α S series and the resulting χ 2 is shown in figure 11 for both NLO and NNLO. The optimal value of α S (m Z ) and its uncertainty is obtained by a parabolic fit as presented in figure 11. The impact of the scale uncertainty on the result is investigated by varying µ r and µ f in the theoretical predictions for the jet cross section. The for the strong coupling are α S (m Z ) = 0.1170 ± 0.0018 (PDF) ± 0.0035 (scale) at NLO and α S (m Z ) = 0.1130 ± 0.0016 (PDF) ± 0.0014 (scale) at NNLO. The NLO result is in good agreement with the world average [90].
The profiling analysis is repeated by using the triple-differential CMS tt cross section of ref. [22] together with the inclusive jet cross section. Consistent with the available theoretical prediction for the tt measurements, this analysis is performed at NLO. The results are shown in figure 12, where the uncertainty in the profiled gluon distribution is presented in comparison to that of the original CT14 PDF. The reduction of the uncertainty  in the gluon distribution at high x is stronger than in the case when only the CMS inclusive jet cross section is used. This is expected from the additional sensitivity of the tt production to the gluon distribution at high x. Also the α S (m Z ) scan is shown in figure 12, now using both CMS data sets. The resulting NLO value of the strong coupling is α S (m Z ) = 0.1154±0.0009 (PDF)±0.0015 (scale), consistent with the result of ref. [22]. The additional  Figure 13. The profiled Wilson coefficient c 1 for the contact interaction models, assuming the left-handed, vector-like, and axial vector-like scenarios, as obtained in the profiling analysis using NLO+NLL calculation for the jet production and the CT14nlo PDF set. The value of Λ = 10 TeV is assumed. The results are obtained using the CMS measurements of inclusive jet cross section and of normalised triple-differential tt cross section at √ s = 13 TeV. The inner error bar shows the PDF uncertainty at 68% CL, while the outer error bar represents the total uncertainty, obtained from the PDF and scale uncertainties, added in quadrature.
sensitivity of the tt production to the strong coupling becomes visible in the reduced PDF uncertainties. The profiled pole mass of the top quark results in m pole t = 170.3±0.5 (PDF)+ 0.2 (scale) GeV, consistent with the value obtained in ref. [22].
Furthermore, the profiling analysis is repeated assuming the SMEFT prediction for the inclusive jet production cross section. While the results of the profiled PDFs remain unchanged with respect to the SM results described above, the Wilson coefficient c 1 is profiled, assuming the value of the scale of the new interaction Λ = 10 TeV, and the results are summarised in the figure 13.
The resulting values of c 1 are consistent with zero within uncertainties for all investigated CI models, demonstrating a good description of the data by the SM. Since the SMEFT computation is applied only to the inclusive jet production cross section, the c 1 results are independent of the inclusion of tt data. Once the relevant calculations for these data become available, a more global SMEFT interpretation would become possible.

Results of the full QCD fit in SM at NNLO
The present measurement of the inclusive jet production cross section is used together with the inclusive DIS cross section of HERA in a full QCD analysis at NNLO. The PDF parameterisation at the starting scale, resulting from the scan as described in section 6.3 reads:  Table 5. Partial χ 2 per number of data points N dp and the global χ 2 per degree of freedom, N dof , as obtained in the QCD analysis at NNLO of HERA+CMS jet data and HERA-only data. In the DIS data, the proton beam energy is given as E p and the electron energy is 27.5 GeV. which agrees with the previous extractions of the strong coupling constant at NNLO at hadron colliders [91,92], of which it has best precision, to date. The impact of the present CMS jet data in a full QCD fit (HERA+CMS fit) is demonstrated by comparing of the resulting PDFs with an alternative fit, where only the HERA data is used (HERA-only fit). Since the inclusive DIS data have much lower sensitivity to the value of α S (m Z ), it is fixed in the HERA-only fit to the result of the HERA+CMS fit. The comparison of the resulting PDFs is presented in figure 15. The uncertainty is significantly reduced once the CMS jet measurements are included.
The global and partial χ 2 values for each data set, for HERA-only and HERA+CMS fits, are listed in table 5, where the χ 2 values illustrate a general agreement among all the data sets. The somewhat high χ 2 /N dof values for the combined DIS data are very similar to those observed in ref. [21], where they are investigated in detail. CMS measurements of the inclusive jet cross section, the triple-differential normalised CMS tt cross section at 13 TeV, and the HERA DIS cross sections are used in a SMEFT fit. Here, the SM prediction for the inclusive jet cross section is modified to account for CI as described in section 6.2. The parameterisation is reinvestigated, as explained in section 6.3, and results in 16) First, the analysis is performed in the standard model. Then, alternatively, the SMEFT fit is done. Both SM and SMEFT fits are performed at NLO to be consistent with the order of the theoretical prediction for the tt data and for the CI corrections to the SM Lagrangian, although the SM prediction for the inclusive jet cross section is available at NNLO. The partial and global χ 2 values for the SM and SMEFT fits are listed in table 6. The fits with all CI models and various Λ values resulted in very similar χ 2 values.

Results of the SMEFT fit at NLO
The PDFs resulting from SM NLO fit are presented in figure 16 demonstrating the contributions of the fit, model, and parameterisation uncertainties. Data sets Partial χ 2 /N dp Partial χ 2 /N dp  Table 6. Partial χ 2 per number of data points N dp and the global χ 2 per degree of freedom, N dof , as obtained in the QCD analysis of HERA DIS data and the CMS measurements of inclusive jet production and the normalised triple-differential tt production at √ s = 13 TeV, obtained in SM and SMEFT analyses.
In the SMEFT fit, the Wilson coefficient c 1 is introduced as a new free parameter, assuming different values for the scale of the new interaction Λ = 5, 10, 13, 20, and 50 TeV. In all SMEFT fits, the χ 2 is reduced by about 10, with just the addition of c 1 as an additional free parameter. Independent of the value of Λ, the strong coupling constant and the top quark mass in these SMEFT fits result to α S (m Z ) = 0.1187 ± 0.0016 (fit) ± 0.0005 (model) ± 0.0023 (scale) ± 0.0018 (param.), and m pole t = 170.4 ± 0.6 (fit) ± 0.1 (model) ± 0.1 (scale) ± 0.2 (param.) GeV. These values agree well with those obtained in the SM fit, and have larger parameterisation uncertainties because of increased flexibility in the SMEFT fit. The PDFs resulting from the SMEFT fits at different values of Λ and for different CI models agree with each other and with the PDFs in the SM fit. In figure 17 the PDFs, which are obtained in the SM fit or in the SMEFT fit using the left-handed CI model, are compared. The PDFs are shown only with their fit uncertainty obtained by using the Hessian method.
To account for possible non-Gaussian tails, the PDF uncertainties are alternatively obtained by using the MC method, based on 800 replicas. The Hessian and the MC , and sea quark (lower right) distributions, shown as functions of x at the scale µ f = m 2 t , resulting from the SM fit using HERA DIS together with the CMS inclusive jet cross section and the normalised tripledifferential cross section of tt production at √ s = 13 TeV. Contributions of fit, model, and parameterisation uncertainties for each PDF are shown. In the lower panels, the relative uncertainty contributions are presented. uncertainties in the SMEFT fit are shown in figure 18. The uncertainties obtained by using the MC method are larger at high x, which might suggest non-Gaussian tails in the PDF uncertainties. However this is not reflected in the uncertainty in c 1 coefficients; the respective uncertainties obtained by Hessian or MC methods agree well.
The Wilson coefficients c 1 are obtained for different assumptions on the value of Λ, as listed in table 7. All SMEFT fits lead to negative c 1 , which would translate into a positive interference with the SM gluon exchange. However, the differences from the SM (c 1 = 0) are not statistically significant. The ratio c 1 /Λ 2 is illustrated for Λ = 50 TeV in figure 19 and is observed to remain constant for various values of Λ.
Conventional searches for CI fix the values of Wilson coefficient to +1 (-1) for a destruc- tive (constructive) interference with the SM gluon exchange, and impose exclusion limits on the scale Λ [90]. The results obtained in the present analysis that indicate negative Wilson coefficients, with |c 1 | close to 1 for Λ = 50 TeV, can be translated into a 95% CL exclusion limit for the left-handed model with constructive interference, corresponding to Λ > 24 TeV. The most stringent comparable result is obtained in the analysis of dijet cross section at √ s = 13 TeV by the ATLAS Collaboration [93], in which the 95% CL exclusion limits for purely left-handed CI of 22 TeV for constructive interference, and of 30 TeV for destructive interference are obtained.
As already observed in the results of profiling, the results of the full fit show agreement between the measurements and the SM prediction. Since the parameterisation and the PDF uncertainties differ in the profiling analysis and in the full fit, a direct quantitative comparison of these results is not possible. The advantage of the full fit with respect to the profiling is in the properly considered, and therefore mitigated, correlations between the QCD parameters and the PDFs. The full SMEFT fit assures that the possible BSM effects are not absorbed in the PDFs and in turn, into the SM prediction, which is the basis for the search for new physics.

Summary
In this paper, the measurement of the double-differential inclusive jet cross sections in proton-proton collisions at √ s = 13 TeV is presented as a function of the jet transverse momentum p T and the jet rapidity |y| for jets reconstructed using the anti-k T clustering -31 -  algorithm with a distance parameter R of 0.4 and 0.7. The phase space covers jet p T from 97 GeV up to 3.1 TeV and jet rapidity up to |y| = 2.0. The measured jet cross sections are compared with predictions of perturbative quantum chromodynamics (pQCD) at next-to-next-to-leading order (NNLO) and next-to-leading order (NLO) with the nextto-leading-logarithmic (NLL) resummation correction, using various sets of parton distribution functions (PDFs). A strong impact of the measurement on determination of the parton distributions is observed, expressed by significant differences among the theoretical -32 -

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predictions using different PDF sets, and by large corresponding uncertainties.
To investigate the impact of the measurements on the PDFs and the strong coupling constant α S , a QCD analysis is performed, where the jet production cross section with R = 0.7 is used together with the HERA measurements of deep inelastic scattering. Significant improvement of the accuracy of the PDFs by using the present measurement in the QCD analysis is demonstrated in a profiling analysis using the CT14 PDF set and in the full PDF fit.
The value of the strong coupling constant at the Z boson mass is extracted in a QCD analysis at NNLO using the inclusive jet cross sections in proton-proton collisions for the first time, and results in α S (m Z ) = 0.1170 ± 0.0014 (fit) ± 0.0007 (model) ± 0.0008 (scale) ± 0.0001 (parametrisation).
The QCD analysis is also performed at NLO, where the CMS measurement of the normalised triple-differential top quark-antiquark production cross section at √ s = 13 TeV is used in addition. In this analysis, the PDFs, the values of the strong coupling constant, and of the top quark pole mass m pole agrees well with the result of the previous CMS analysis using the triple-differential cross section of the top quark-antiquark pair production. Although the inclusive jet production is not directly sensitive to m pole t , the resulting value is improved by the additional constraint on the gluon distribution and on α S (m Z ) provided by the jet measurements.
Furthermore, an alternative QCD analysis is performed with the same data, where the standard model Lagrangian is modified by the introduction of effective terms related to 4-quark contact interactions. In the analysis, the Wilson coefficients for the contact interactions are extracted for different values assumed for the scale Λ of the new interaction. The results are translated into a 95% confidence level exclusion limit for the left-handed model with constructive interference, corresponding to Λ > 24 TeV. These results are compatible with the standard model and the previous limits obtained at the LHC using jet production. The advantage of the present approach is the simultaneous extraction of PDFs, thereby mitigating possible bias in the interpretation of the measurements in terms of physics beyond the standard model.
Tabulated results are provided in the HEPData record for this analysis [94].     Figure 21. Cross sections of inclusive jet production for distance parameter R = 0.4 as a function of transverse momentum of the individual jet in bins of absolute rapidity |y|, compared to the theoretical predictions at NLO using different PDFs (lines of different colors). All results are normalised to the prediction at NLO obtained using CT14nlo PDF (black dashed line). The measurement (solid symbols) is presented with the statistical uncertainties (vertical error bars), while the systematic uncertainty is represented by a yellow filled band, centered at 1.

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. -40 -