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A measurement of the decorrelation of azimuthal angles between the two jets with the largest transverse momenta is presented for seven regions of leading jet transverse momentum up to 2.2 TeV. The analysis is based on the proton-proton collision data collected with the CMS experiment at a centre-of-mass energy of 8 TeV corresponding to an integrated luminosity of 19.7 fb-1. The dijet azimuthal decorrelation is caused by the radiation of additional jets and probes the dynamics of multijet production. The results are compared to fixed-order predictions of perturbative quantum chromodynamics (QCD), and to simulations using Monte Carlo event generators that include parton showers, hadronization, and multiparton interactions. Event generators with only two outgoing high transverse momentum partons fail to describe the measurement, even when supplemented with next-to-leading-order QCD corrections and parton showers. Much better agreement is achieved when at least three outgoing partons are complemented through either next-to-leading-order predictions or parton showers. This observation emphasizes the need to improve predictions for multijet production.
Hadronic jets with large transverse momenta pT are produced in high-energy proton-proton collisions when two partons interact with high momentum transfer via the strong force. At leading order (LO) in perturbative quantum chromodynamics (pQCD), two final-state partons are produced back-to-back in the transverse plane. For this case, the azimuthal angular separation between the two leading pT jets in the transverse plane, Δϕdijet = |ϕjet1 - ϕjet2|, equals π. The nonperturbative effects of multiparton interactions or hadronization disturb this correlation only mildly, and Δϕdijet ≈ π still holds. However, the production of a third high-pT jet leads to a decorrelation in azimuthal angle. The smallest achievable value of Δϕdijet = 2π/3 occurs in a symmetric star-shaped 3-jet configuration. Fixed-order calculations in pQCD for 3-jet production with up to four outgoing partons provide next-to-leading-order (NLO) predictions for the region of 2π/3 ≤ Δϕdijet < π. If more than three jets are produced, the azimuthal angle between the two leading jets can approach zero, although very small angular separations are suppressed because of the finite jet sizes for a particular jet algorithm. The measurement of the dijet azimuthal angular decorrelation is an interesting tool to gain insight into multijet production processes without measuring jets beyond the leading two.
This paper reports the measurement of the normalized dijet differential cross section as a function of the dijet azimuthal angular separation,
for seven regions of the leading jet pT, , within a rapidity region of |y| < 2.5. Experimental and theoretical uncertainties are reduced by normalizing the Δϕdijet distribution to the total dijet cross section σdijet within each region of . For the first time, azimuthal angular separations Δϕdijet over the full phase space from 0 to π are covered. Comparisons are made to fixed-order predictions up to NLO for 3-jet production, and to NLO and LO dijet as well as to tree-level multijet production, each matched with parton showers and complemented with multiparton interactions and hadronization.
The measurement is performed using data collected during 2012 with the CMS experiment at the CERN LHC, corresponding to an integrated luminosity of 19.7 fb-1 of proton-proton collisions at . Previous measurements of dijet azimuthal decorrelation were reported by the D0 Collaboration in collisions at at the Tevatron [1, 2], and by the CMS and ATLAS Collaborations in pp collisions at at the LHC [3, 4].
A detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. . The central feature of the CMS detector is a superconducting solenoid, 13 m in length and 6 m in inner diameter, providing an axial magnetic field of 3.8 T. Within the field 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. Charged particle trajectories are measured by the tracker with full azimuthal coverage within pseudorapidities |η| < 2.5. The ECAL, which is equipped with a preshower detector in the endcaps, and the HCAL cover the region |η| < 3. In addition to the barrel and endcap detectors, CMS has extensive forward calorimetry, which extends the coverage up to |η| < 5. Finally, muons are measured up to |η| < 2.4 by gas-ionization detectors embedded in the steel flux-return yoke outside the solenoid.
This measurement uses data samples that were collected with single-jet high-level triggers (HLT) . Four such single-jet HLTs were considered that require at least one jet in the event to have pT > 140, 200, 260, and 320 GeV, respectively. All triggers were prescaled during the 2012 run except the highest-threshold trigger. The integrated luminosity ℒ for the four trigger samples is shown in Table 1. The trigger efficiency is estimated using triggers with lower pT thresholds. Using these four jet-energy thresholds gives 100 % trigger efficiencies in the corresponding four momentum regions , , , and .
Particles are reconstructed and identified using a particle-flow (PF) algorithm, which combines the information from the individual subdetectors [7, 8]. The four-vectors of particle candidates, reconstructed by the above technique, are used as input to the jet-clustering algorithm. Jets are reconstructed using the infrared- and collinear-safe anti-kT clustering algorithm with a distance parameter R = 0.7 . The clustering is performed with the FastJet package  using four-momentum summation.
The reconstructed jets require small additional energy corrections to account for various reconstruction inefficiencies in tracks and clusters in the PF algorithm. These jet energy corrections  are derived using (1) simulated events, generated with pythia 6.4.22  with tune Z2* [13, 14] and processed through the CMS detector simulation based on Geant4 , and (2) measurements containing dijet, photon+jet, and Z+jet events. The jet energy corrections, which depend on the η and pT of the jet, are applied to the jet four-momentum vectors as multiplicative factors . The overall factor is typically 1.2 or smaller, approximately uniform in η, and is 1.05 or smaller for jets having pT > 100 GeV. An offset correction is applied to take into account the extra energy clustered into jets from additional proton-proton interactions within the same or neighbouring bunch crossings (in-time and out-of-time pileup) . Pileup effects are important only for jets with low pT and become negligible for jets with pT > 200 GeV. The current measurement is, therefore, insensitive to pileup effects on jet energy calibration.
Each event is required to have at least one vertex reconstructed offline  with a position along the beam line that is within 24 cm of the nominal interaction point. To suppress nonphysical jets, i.e. jets resulting from noise in the ECAL and/or HCAL calorimeters, stringent criteria  are applied for identifying jets: each jet should contain at least two particles, one of which is a charged hadron, and the jet energy fraction carried by neutral hadrons and photons should be less than 90 %. The efficiency for identifying physical jets using these criteria is greater than 99 %.
The two leading jets, which define Δϕdijet, are selected by considering all jets in the event with pT > 100 GeV and an absolute rapidity |y| < 5. Events are selected in which the leading jet pT exceeds 200 GeV and the rapidities y1 and y2 of the two leading jets lie within the tracker coverage of |y| < 2.5.
To reduce the background from and heavy vector boson production, the variable ET//∑ ET is used. The sum of the transverse energies is ∑ ET = ∑iEisinθi, and the missing transverse energy , where θ is the polar angle and the sum runs over all PF candidates in the event. A noticeable fraction of high-pT jet events with large ET/ emerges from production with semileptonically decaying b quarks. In addition, Z/W+jet(s) events with Z decays to neutrinos and W decays into charged leptons with neutrinos have high ET/ values. The distributions of the variable ET//∑ ET are shown in Fig. 1 for the two regions Δϕdijet < π/2 (top) and π/2 < Δϕdijet < π (bottom). The data (points) are compared to simulated events (stacked), using MadGraph 18.104.22.168  matched to pythia6  for event generation. Although some deviations of the simulation with respect to the data are visible in Fig. 1 (cf. Ref. ), the distributions allow a selection criterion to be optimized with respect to the ratio of signal over background. Events with ET//∑ ET > 0.1 are rejected in both regions of Δϕdijet considered in Fig. 1, which corresponds to about 0.7 % of the data sample. Negligible background fractions of ≈ 1 % and ≈ 0.1 % remain for the two regions Δϕdijet < π/2 and π/2 < Δϕdijet < π, respectively.
The normalized dijet cross section differential in Δϕdijet (Eq. 1) is corrected for detector smearing effects and unfolded to the level of stable (decay length cτ > 1 cm) final-state particles. In this way, a direct comparison of the measurement with corresponding results from other experiments and with QCD predictions can be made.
The unfolding method is based on the matrix inversion algorithm implemented in the software package RooUnfold . Unfolding uses a response matrix that maps the distribution at particle-level onto the measured one. The response matrix is derived from a simulation that uses the true dijet cross section distribution from pythia6 with tune Z2*  as input, and introduces the smearing effects by taking into account the Δϕdijet resolution. As a cross-check, the response matrix was filled from event samples that have been passed through a detector simulation. No significant difference was observed. The unfolded distributions differ from the raw distributions by 3–4 % for Δϕdijet < π/2 and by less than 3 % for π/2 < Δϕdijet < π. A two-dimensional unfolding based on the iterative D’Agostini algorithm , which corrects for the smearing effects by taking into account both Δϕdijet and pT resolutions, gives almost identical results.
The main systematic uncertainties arise from the estimation of the jet energy scale (JES) calibration, the jet pT resolution, and the unfolding correction. The JES uncertainty is estimated to be 1.0–2.5 % for PF jets, depending on the jet pT and η [11, 16, 23]. The resulting uncertainties in the normalized Δϕdijet distributions range from 7 % at Δϕdijet ≈ 0 via 3 % at π/2 to 1 % at π.
The jet pT resolution is determined from a full detector simulation using events generated by pythia6 with tune Z2*, and is scaled by factors derived from data . The effect of the jet pT resolution uncertainty is estimated by varying it by one standard deviation up and down, and comparing the Δϕdijet distributions before and after the changes. This results in a variation in the normalized Δϕdijet distributions ranging from 5 % at Δϕdijet ≈ 0 via 3 % at π/2 to 0.5 % at π.
The uncertainty in the unfolding correction factors is estimated by checking the dependence of the response matrix on the choice of the Monte Carlo (MC) generator. An alternative response matrix is built using the herwig++ 2.5.0  event generator with the default tune of version 2.3. The observed effect is less than 1 %. An additional systematic uncertainty obtained by varying the Δϕdijet resolution by ±10 % to determine the unfolding correction factors is estimated to be of the order of 1 %. This variation of the Δϕdijet resolution by ±10 % is motivated by the observed difference between data and simulation in the Δϕdijet resolution. A total systematic unfolding uncertainty of 1 % accounts for the choice of the MC generator in building the response matrix and the Δϕdijet resolution.
The unfolded dijet cross section differential in Δϕdijet and normalized by the dijet cross section integrated over the entire phase space is shown in Fig. 2 for seven regions. Each region is scaled by a multiplicative factor for presentation purposes. The Δϕdijet distributions are strongly peaked at π and become steeper with increasing . Overlaid on the data for Δϕdijet > π/2 are predictions from pQCD, presented in more detail in the next section, using parton distribution functions (PDF) of the CT10 PDF set.
The theoretical predictions for the normalized dijet cross section differential in Δϕdijet are based on a 3-jet calculation at NLO. The correction of nonperturbative (NP) effects, which account for multiparton interactions (MPI) and hadronization, is studied using event samples simulated with the pythia6 (tune Z2*) and herwig++ (tune 2.3) event generators. Small NP effects are expected, since this measurement deals with a normalized distribution. These corrections are found to be of the order of 1 %, roughly at the limit of the accuracy of the MC simulations. Therefore NP corrections are considered to be negligible and are not applied.
The fixed-order calculations are performed using the NLOJet++ program version 4.1.3 [25, 26] within the framework of the fastNLO package version 2.3.1 . The differential cross section is calculated for 3-jet production at NLO, i.e. up to terms of order , with three or four partons in the final state. This calculation has LO precision in the region π/2 ≤ Δϕdijet < 2π/3 and NLO precision for 2π/3 ≤ Δϕdijet < π. The bin including Δϕdijet = π is computed from the NLO dijet cross section within this bin. For each region in , the differential cross section is normalized to the dijet cross section calculated at LO for π/2 ≤ Δϕdijet < 2π/3 and at NLO, i.e. up to terms proportional to , for 2π/3 ≤ Δϕdijet ≤ π. The use of the LO dijet cross section for the normalization in the region π/2 ≤ Δϕdijet < 2π/3 leads to an improved description of the data and avoids artificially increased scale uncertainties as described in Refs. [28, 29]. Of course, this difference in normalization leads to a discontinuity proportional to at Δϕdijet = 2π/3.
The number of quark flavours that are assumed to be massless is set to five, and the renormalization and factorization scales, μr and μf, are chosen to be equal to . The PDF sets with NLO evolutions used in the calculations are tabulated in Table 2. The ABM11 PDF set utilizes a fixed flavour number scheme, whereas the rest of the PDF sets use a variable flavour number scheme. The maximum number of flavours is denoted by Nf.
The uncertainties due to the renormalization and factorization scales are evaluated by varying the default choice of between /2 and 2, simultaneously in the differential cross section and in the total cross section, in the following six combinations: , (1 / 2, 1), (1, 1 / 2), (1, 2), (2, 1), and (2, 2). The PDF uncertainties are evaluated according to the prescriptions for the CT10 PDF set in Ref. . The CT10 PDF set employs the eigenvector method with upward and downward variations for each eigenvector. To evaluate the uncertainty due to the value of the strong coupling constant at 68 % confidence level, αS(MZ) is varied by ±0.001 as recommended in Ref. .
The results of fixed-order calculations with the CT10 PDF set are overlaid on the data for Δϕdijet > π/2 in Fig. 2. Figure Figure33 shows the ratio of the normalized dijet cross section differential in Δϕdijet to theory calculated using the CT10 PDF set, together with the combined PDF and αS uncertainty (inner band), and the scale uncertainty (outer band). Also shown are the ratios of theory derived with the alternative PDF sets ABM11 (dashed line), HERAPDF1.5 (dashed–three-dotted line), MSTW2008 (dashed-dotted line), and NNPDF2.1 (dotted line) compared to the prediction with the CT10 PDFs.
The fixed-order calculations agree with the data for azimuthal angular separations larger than 5π/6 except for the highest region, where they exceed the data. For smaller Δϕdijet values between 2π/3 and 5π/6, in particular where the estimate of the theoretical uncertainties becomes small, systematic discrepancies are exhibited that diminish with increasing . In the 4-jet LO region with Δϕdijet < 2π/3, the pattern of increasing deviations towards smaller Δϕdijet and decreasing deviations towards larger is repeated, but with less significance because of the larger scale uncertainty. Similar observations were made in the previous CMS measurement , which exhibited larger discrepancies in the 4-jet region due to the normalization to the NLO dijet cross section instead of a LO one.
The pythia6 , pythia8 , and herwig++  event generators complement LO dijet matrix elements with parton showers to simulate higher-order processes. Both pythia versions, pythia6 with the Z2* tune  and pythia8 with the CUETM1 tune , employ pT-ordered parton showers [38, 39], while herwig++ with the default tune of version 2.3 uses a coherent-branching algorithm with angular ordering of the showers .
The MadGraph program version 22.214.171.124  supplies the results of LO matrix element calculations with two to four outgoing partons that can be matched to the implementations of parton showers, hadronization, and MPI of the event generators. In this analysis, it is interfaced with pythia6 with tune Z2* using the MLM matching procedure  to avoid any double counting between tree-level and parton shower generated parton configurations.
The powheg framework [42–44] provides an NLO dijet calculation  that can also be matched via the parton showers to event generators. Here, powheg is used with the CT10NLO PDF set and is interfaced to pythia8 with the CUET  tune, which employs the LO CTEQ6L1  PDF set. Predictions with parton showers matched to a NLO 3-jet calculation using powheg  or MadGraph5_aMC@NLO  would be even more relevant for a multijet topology. They could not, however, be included within the timescale of this analysis. Approaching azimuthal angular separations close to π, it might also be interesting to compare to predictions employing the technique of pT resummation .
In Fig. 4 the normalized dijet cross section differential in Δϕdijet is compared to the predictions from fixed-order calculations supplemented with parton showers, hadronization, and MPI. The error bars on the data points represent the total experimental uncertainty, which is the quadratic sum of the statistical and systematic uncertainties. Figure 5 shows the ratios of these predictions to the normalized dijet cross section differential in Δϕdijet, for the seven regions. The solid band indicates the total experimental uncertainty and the error bars on the MC points represent the statistical uncertainties in the simulated data.
Among the LO dijet event generators pythia6, pythia8, and herwig++, pythia8 exhibits the smallest deviations from the measurements. pythia6 and herwig++ systematically overshoot the data, particular around Δϕdijet = 5π/6. The best description of the measurement is given by the tree-level multiparton event generator MadGraph interfaced with pythia6 for showering, hadronization, and MPI. The powheg generator (here used only in the NLO dijet mode) matched to pythia8 shows deviations from the data similar to the LO dijet event generators.
A measurement is presented of the normalized dijet cross section differential in the azimuthal angular separation Δϕdijet of the two jets leading in pT for seven regions in the leading-jet transverse momentum . The data set of pp collisions at 8 TeV centre-of-mass energy collected in 2012 by the CMS experiment and corresponding to an integrated luminosity of 19.7 fb-1 is analysed.
The measured distributions in Δϕdijet are compared to calculations in perturbative QCD for 3-jet production with up to four outgoing partons that provide NLO predictions for the range of 2π/3 ≤ Δϕdijet < π and LO predictions for π/2 ≤ Δϕdijet < 2π/3. The NLO predictions describe the data down to values of Δϕdijet ≈ 5π/6, but deviate increasingly when approaching the 4-jet region, starting at Δϕdijet = 2π/3, particularly at low . The pattern of increasing deviations towards smaller Δϕdijet and decreasing deviations towards larger is repeated in the 4-jet LO region with Δϕdijet < 2π/3, but with less significance because of the larger scale uncertainty.
In a comparison of the normalized Δϕdijet distributions to the LO dijet event generators pythia6, pythia8, and herwig++, pythia8 gives the best agreement. pythia6 and herwig++ systematically overshoot the data, particularly for Δϕdijet ≈ 5π/6. A good overall description of the measurement is provided by the tree-level multijet event generator MadGraph in combination with pythia6 for showering, hadronization, and multiparton interactions. The dijet NLO calculations from powheg matched to pythia8 exhibit deviations similar to the LO dijet event generators. Improved multijet predictions can be expected from 3-jet NLO calculations matched to parton showers like from powheg or MadGraph5_aMC@NLO.
Similar observations were reported previously by CMS  and ATLAS , but with less significance because of the smaller data sets. The extension to Δϕdijet values below π/2, the improved LO description in the 4-jet region π/2 ≤ Δϕdijet < 2π/3, and the comparison to dijet NLO calculations matched to parton showers are new results of the present analysis.
We acknowledge discussions and comparisons with P. Sun, C. P. Yuan, and F. Yuan following the approach of . We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centres and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMWFW and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES and CSF (Croatia); RPF (Cyprus); MoER, ERC IUT and ERDF (Estonia); Academy of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); OTKA and NIH (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); MSIP and NRF (Republic of Korea); LAS (Lithuania); MOE and UM (Malaysia); CINVESTAV, CONACYT, SEP, and UASLP-FAI (Mexico); MBIE (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Dubna); MON, RosAtom, RAS and RFBR (Russia); MESTD (Serbia); SEIDI and CPAN (Spain); Swiss Funding Agencies (Switzerland); MST (Taipei); ThEPCenter, IPST, STAR and NSTDA (Thailand); TUBITAK and TAEK (Turkey); NASU and SFFR (Ukraine); STFC (United Kingdom); DOE and NSF (USA). Individuals have received support from the Marie-Curie programme and the European Research Council and EPLANET (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic; the Council of Science and Industrial Research, India; the HOMING PLUS programme of the Foundation for Polish Science, cofinanced from European Union, Regional Development Fund; the OPUS programme of the National Science Center (Poland); the Compagnia di San Paolo (Torino); MIUR project 20108T4XTM (Italy); the Thalis and Aristeia programmes cofinanced by EU-ESF and the Greek NSRF; the National Priorities Research Program by Qatar National Research Fund; the Rachadapisek Sompot Fund for Postdoctoral Fellowship, Chulalongkorn University (Thailand); the Chulalongkorn Academic into Its 2nd Century Project Advancement Project (Thailand); and the Welch Foundation, contract C-1845.