Measurement of dijet azimuthal decorrelations in pp collisions at ﬃﬃ s p = 8 TeV with the ATLAS detector and determination of the strong coupling

A measurement of the rapidity and transverse momentum dependence of dijet azimuthal decorrelations is presented, using the quantity R Δ ϕ . The quantity R Δ ϕ specifies the fraction of the inclusive dijet events in which the azimuthal opening angle of the two jets with the highest transverse momenta is less than a given value of the parameter Δ ϕ max . The quantity R Δ ϕ is measured in proton-proton collisions at ﬃﬃﬃ s p ¼ 8 TeV as a function of the dijet rapidity interval, the event total scalar transverse momentum, and Δ ϕ max . The measurement uses an event sample corresponding to an integrated luminosity of 20 . 2 fb − 1 collected with the ATLAS detector at the CERN Large Hadron Collider. Predictions of a perturbative QCD calculation at next-to-leading order in the strong coupling with corrections for nonperturbative effects are compared to the data. The theoretical predictions describe the data in the whole kinematic region. The data are used to determine the strong coupling α S and to study its running for momentum transfers from 260 GeV to above 1.6 TeV. Analysis that combines data at all momentum transfers results in α S ð m Z Þ ¼ 0 . 1127 þ 0 . 0063 − 0 . 0027 . DOI: 10.1103/PhysRevD.98.092004


I. INTRODUCTION
In high-energy particle collisions, measurements of the production rates of hadronic jets with large transverse momentum p T relative to the beam direction can be employed to test the predictions of perturbative quantum chromodynamics (pQCD). The results can also be used to determine the strong coupling α S , and to test the pQCD predictions for the dependence of α S on the momentum transfer Q (the "running" of α S ) by the renormalization group equation (RGE) [1,2]. Previous tests of the RGE through α S determinations in hadronic final states have been performed using data taken in ep collisions (5 < Q < 60 GeV) [3][4][5], in e þ e − annihilation (10 < Q < 210 GeV) [6,7], in pp collisions (50 < Q < 400 GeV) [8][9][10], and in pp collisions (130 < Q < 1400 GeV) [11][12][13][14][15]. The world average value is currently α S ðm Z Þ ¼ 0.1181 AE 0.0011 [16].
Recent α S results from hadron collisions are limited by theoretical uncertainties related to the scale dependence of the fixed-order pQCD calculations. The most precise α S ðm Z Þ result from hadron collision data is α S ðm Z Þ ¼ 0.1161 þ0.0041 −0.0048 [9], obtained from inclusive jet cross-section data, using pQCD predictions beyond the next-to-leading order (NLO). However, when the cross-section data are used in α S determinations, the extracted α S results are directly affected by our knowledge of the parton distribution functions (PDFs) of the proton, and their Q dependence. The PDF parametrizations depend on assumptions about α S and the RGE in the global data analyses in which they are determined. Therefore, in determinations of α S and its Q dependence from cross-section data the RGE is already assumed in the inputs. Such a conceptual limitation when using crosssection data can largely be avoided by using ratios of multijet cross sections in which PDFs cancel to some extent. So far, the multijet cross-section ratios R ΔR [10] and R 3=2 [11] have been used for α S determinations at hadron colliders. In this article, α S is determined from dijet azimuthal decorrelations, based on the multijet cross-section ratio R Δϕ [17]. The RGE predictions are tested up to Q ¼ 1.675 TeV.
The decorrelation of dijets in the azimuthal plane has been the subject of a number of measurements at the Fermilab Tevatron Collider [18] and the CERN Large Hadron Collider (LHC) [19,20]. The variable Δϕ dijet investigated in these analyses is defined from the angles in the azimuthal plane (the plane perpendicular to the beam direction) ϕ 1;2 of the two highest-p T jets in the event as Δϕ dijet ¼ jϕ 1 − ϕ 2 j. In exclusive high-p T dijet final states, the two jets are correlated in the azimuthal plane with Δϕ dijet ¼ π. Deviations from this (Δϕ dijet < π) are due to additional activity in the final state, as described in pQCD by processes of higher order in α S . Due to kinematic constraints, the phase space in 2 → 3 processes is restricted to Δϕ dijet > 2π=3 [21] and lower Δϕ dijet values are only accessible in 2 → 4 processes. Measurements of dijet production with 2π=3 < Δϕ dijet < π (Δϕ dijet < 2π=3) therefore test the pQCD matrix elements for three-jet (fourjet) production.
The quantity R Δϕ is defined as the fraction of all inclusive dijet events in which Δϕ dijet is less than a specified value Δϕ max . This quantity can be exploited to extend the scope of the previous analyses towards studies of the rapidity dependence of dijet azimuthal decorrelations. Since R Δϕ is defined as a ratio of multijet cross sections for which the PDFs cancel to a large extent, it is well suited for determinations of α S and for studies of its running.
The quantity R Δϕ has so far been measured in pp collisions at a center-of-mass energy of ffiffi ffi s p ¼ 1.96 TeV at the Fermilab Tevatron Collider [22]. This article presents the first measurement of R Δϕ in pp collisions, based on data at ffiffi ffi s p ¼ 8 TeV taken with the ATLAS detector during 2012 at the LHC, corresponding to an integrated luminosity of 20.2 AE 0.4 fb −1 [23]. The data are corrected to "particle level" [24], and are used to extract α S and to study its running over a range of momentum transfers of 262 < Q < 1675 GeV.

II. DEFINITION OF R Δϕ AND THE ANALYSIS PHASE SPACE
The definitions of the quantity R Δϕ and the choices of the variables that define the analysis phase space are taken from the proposal in Ref. [17]. Jets are defined by the antik t jet algorithm as implemented in FASTJET [25,26]. The anti-k t jet algorithm is a successive recombination algorithm in which particles are clustered into jets in the E-scheme (i.e., the jet four-momentum is computed as the sum of the particle four-momenta). The radius parameter is chosen to be R ¼ 0.6. This is large enough for a jet to include a sufficient amount of soft and hard radiation around the jet axis, thereby improving the properties of pQCD calculations at fixed order in α S , and it is small enough to avoid excessive contributions from the underlying event [27]. An inclusive dijet event sample is extracted by selecting all events with two or more jets, where the two leading-p T jets have p T > p T min . The dijet phase space is further specified in terms of the variables y boost and y Ã , computed from the rapidities, y 1 and y 2 , of the two leading-p T jets as y boost ¼ ðy 1 þ y 2 Þ=2 and y Ã ¼ jy 1 − y 2 j=2, respectively. 1 In 2 → 2 processes, the variable y boost specifies the longitudinal boost between the dijet and the proton-proton center-of-mass frames, and y Ã (which is longitudinally boost invariant) represents the absolute value of the jet rapidities in the dijet center-of-mass frame. The dijet phase space is restricted to jy boost j < y max boost and y Ã < y Ã max . The variable H T is defined as the scalar sum of the jet p T for all jets i with p Ti > p T min and jy i − y boost j < y Ã max . Furthermore, the leading-p T jet is required to have p T1 > H T =3. The values of the parameters p T min , y max boost , and y Ã max ensure that jets are well measured in the detector within jyj < 2.5 and that contributions from nonperturbative corrections and pileup (additional protonproton interactions within the same or nearby bunch crossings) are small. The requirement p T1 > H T =3 ensures (for a given H T ) a well-defined minimum p T1 which allows single-jet triggers to be used in the measurement. It also reduces the contributions from events with four or more jets, and therefore pQCD corrections from higher orders in α S . The values of all parameters are specified in Table I. The quantity R Δϕ is defined in this inclusive dijet event sample as the ratio where the denominator is the inclusive dijet cross section in the phase space defined above, in bins of the variables H T and y Ã . The numerator is given by the subset of the denominator for which Δϕ dijet of the two leading-p T jets obeys Δϕ dijet < Δϕ max . The measurement of the y Ã dependence of R Δϕ allows a test of the rapidity dependence of the pQCD matrix elements. The value of Δϕ max is directly related to the hardness of the jet(s) produced in addition to the two leading-p T jets in the event. The transverse momentum sum H T is one possible choice that can be related to the scale at which α S is probed. The measurement is made as a function of H T in three different y Ã regions and for four different values of Δϕ max (see Table II).

III. THEORETICAL PREDICTIONS
The theoretical predictions in this analysis are obtained from perturbative calculations at fixed order in α S with additional corrections for nonperturbative effects.
The pQCD calculations are carried out using NLOJET++ [28,29] interfaced to FASTNLO [30,31] based on the matrix elements for massless quarks in the MS scheme [32]. The renormalization and factorization scales are set to In inclusive dijet production at leading order (LO) in pQCD this choice is equivalent to other common choices: μ 0 ¼p T ¼ ðp T1 þ p T2 Þ=2 and μ 0 ¼ p T1 . The evolution of α S is computed using the numerical solution of the next-to-leading-logarithmic (2-loop) approximation of the RGE.
The pQCD predictions for the ratio R Δϕ are obtained from the ratio of the cross sections in the numerator and denominator in Eq. (1), computed to the same relative order (both either to NLO or to LO). The pQCD predictions for the cross section in the denominator by NLOJET++ are available up to NLO. For Δϕ max ¼ 7π=8; 5π=6; 3π=4 (2π=3) the numerator is a three-jet (four-jet) quantity for which the pQCD predictions in NLOJET++ are available up to NLO (LO) [21].
The PDFs are taken from the global analyses MMHT2014 (NLO) [33,34], CT14 (NLO) [35], and NNPDFv2.3 (NLO) [36]. 2 For additional studies, the PDF sets ABMP16 (NNLO) [38] 3 and HERAPDF 2.0 (NLO) [39] are used, which were obtained using data from selected processes only. All of these PDF sets were obtained for a series of discrete α S ðm Z Þ values, in increments of Δα S ðm Z Þ ¼ 0.001 (or Δα S ðm Z Þ ¼ 0.002 for NNPDFv2.3). In all calculations in this paper, the PDF sets are consistently chosen to correspond to the value of α S ðm Z Þ used in the matrix elements. The extraction of α S from the experimental R Δϕ data requires a continuous dependence of the pQCD calculations on α S ðm Z Þ. This is obtained by cubic interpolation (linear extrapolation) for α S ðm Z Þ values inside (outside) the ranges provided by the PDF sets. The central predictions that are compared to the data use α S ðm Z Þ ¼ 0.118, which is close to the current world average, and the MMHT2014 PDFs. The MMHT2014 PDFs also provide the largest range of α S ðm Z Þ values (0.108 ≤ α S ðm Z Þ ≤ 0.128). For these reasons, the MMHT2014 PDFs are used to obtain the central results in the α S determinations.
The uncertainties of the perturbative calculation are estimated from the scale dependence (as an estimate of missing higher-order pQCD corrections) and the PDF uncertainties. The former is evaluated from independent variations of μ R and μ F between μ 0 =2 and 2μ 0 (with the restriction 0.5 ≤ μ R =μ F ≤ 2.0). The PDF-induced uncertainty is computed by propagating the MMHT2014 PDF uncertainties. In addition, a "PDF set" uncertainty is included as the envelope of the differences of the results obtained with CT14, NNPDFv2.3, ABMP16, and HERAPDF 2.0, relative to those obtained with MMHT2014.
The pQCD predictions based on matrix elements for massless quarks also depend on the number of quark flavors, in gluon splitting (g → qq), n f , which affects the tree-level matrix elements and their real and virtual corrections, as well as the RGE predictions and the PDFs obtained from global data analyses. The central results in this analysis are obtained for a consistent choice n f ¼ 5 in all of these contributions. Studies of the effects of using n f ¼ 6 in the matrix elements and the RGE, as documented in Appendix A, show that the corresponding effects for R Δϕ are between −1% and þ2% over the whole kinematic range of this measurement. Appendix A also includes a study of the contributions from the tt production process, concluding that the effects on R Δϕ are less than 0.5% over the whole analysis phase space.
The corrections due to nonperturbative effects, related to hadronization and the underlying event, were obtained in Ref.
For this analysis, the central results are taken to be the average values obtained from PYTHIA with tunes AMBT1 and DW. The corresponding uncertainty is taken to be half of the difference (the numerical values are provided in Ref. [47]). The results obtained with PYTHIA tunes A and S-Global as well as HERWIG are used to study systematic uncertainties.

IV. ATLAS DETECTOR
ATLAS is a general-purpose detector consisting of an inner tracking detector, a calorimeter system, a muon spectrometer, and magnet systems. A detailed description of the ATLAS detector is given in Ref. [47]. The main components used in the R Δϕ measurement are the inner detector, the calorimeters, and the trigger system. The position of the pp interaction is determined from charged-particle tracks reconstructed in the inner detector, located inside a superconducting solenoid that provides a 2 T axial magnetic field. The inner detector, covering the region jηj < 2.5, consists of layers of silicon pixels, silicon microstrips, and transition radiation tracking detectors.
During 2012, for pp collisions, the ATLAS trigger system was divided into three levels, labeled L1, L2, and the event filter (EF) [48,49]. The L1 trigger is hardware-based, while L2 and EF are software based and impose increasingly refined selections designed to identify events of interest. The jet trigger identifies electromagnetically and hadronically interacting particles by reconstructing the energy deposited in the calorimeters. The L1 jet trigger uses a sliding window of Δη × Δϕ ¼ 0.8 × 0.8 to find jets and requires these to have transverse energies E T above a given threshold, measured at the electromagnetic scale. Jets triggered by L1 are passed to the L2 jet trigger, which reconstructs jets in the same region using a simple cone jet algorithm with a cone size of 0.4 in (η, ϕ) space. Events are accepted if a L2 jet is above a given E T threshold. In events which pass L2, a full event reconstruction is performed by the EF. The jet EF constructs topological clusters [50] from which jets are then formed, using the anti-k t jet algorithm with a radius parameter of R ¼ 0.4. These jets are then calibrated to the hadronic scale. Events for this analysis are collected either with single-jet triggers with different minimum E T requirements or with multijet triggers based on a single high-E T jet plus some amount of H T (the scalar E T sum) of the multijet system. The trigger efficiencies are determined relative to fully efficient reference triggers, and each trigger is used above an H T threshold where it is more than 98% efficient. The triggers used for the different H T regions in the offline analysis are listed in Table III.
Single-jet triggers select events if any jet with jηj < 3.2 is above the E T thresholds at L1, L2, and the EF. Due to their high rates, the single-jet triggers studied are highly prescaled during data-taking. Multijet triggers select events if an appropriate high-E T jet is identified and the H T value, summed over all jets at the EF with jηj < 3.2 and E T > 45 GeV, is above a given threshold. The additional H T requirement significantly reduces the selected event rate, and lower prescales can be applied. The integrated luminosity of the data sample collected with the highest threshold triggers is 20.2 AE 0.4 fb −1 .
The detector response for the measured quantities is determined using a detailed simulation of the ATLAS detector in GEANT 4 [51,52]. The particle-level events, subjected to the detector simulation, were produced by the PYTHIA event generator [53] (version 8.160) with CT10 PDFs. The PYTHIA parameters were set according to the AU2 tune [54]. The "particle-level" jets are defined based on the four-momenta of the generated stable particles (as recommended in Ref. [24], with a proper lifetime τ satisfying cτ > 10 mm, including muons and neutrinos from hadron decays). The "detector-level" jets are defined based on the four-momenta of the simulated detector objects.

V. MEASUREMENT PROCEDURE
The inclusive dijet events used for the measurement of R Δϕ were collected between April and December 2012 by the ATLAS detector in proton-proton collisions at ffiffi ffi s p ¼ 8 TeV. All events used in this measurement are required to satisfy data-quality criteria which include stable beam conditions and stable operation of the tracking systems, calorimeters, solenoid, and trigger system. Events that pass the trigger selections described above are included in the sample if they contain at least one primary collision vertex with at least two associated tracks with p T > 400 MeV, in order to reject contributions due to cosmic-ray events and beam background. The primary vertex with highest P p 2 T of associated tracks is taken as the event vertex.
Jets are reconstructed offline using the anti-k t jet algorithm with a radius parameter R ¼ 0.6. Input to the jet algorithm consists of locally calibrated threedimensional topological clusters [50] formed from sums of calorimeter cell energies, corrected for local calorimeter response, dead material, and out-of-cluster losses for pions. The jets are further corrected for pileup contributions and then calibrated to the hadronic scale, as detailed in the following. The pileup correction is applied to account for the effects on the jet response from additional interactions within the same proton bunch crossing ("in-time pileup") and from interactions in bunch crossings preceding or following the one of interest ("out-of-time pileup"). Energy is subtracted from each jet, based upon the energy density in the event and the measured area of the jet [55]. The jet energy is then adjusted by a small residual correction depending on the average pileup conditions for the event. This calibration restores the calorimeter energy scale, on average, to a reference point where pileup is not present [56]. Jets are then calibrated using an energy-and η-dependent correction to the hadronic scale with constants derived from data and Monte Carlo samples of jets produced in multi-jet processes. A residual calibration, based on a combination of several in situ techniques, is applied to take into account differences between data and Monte Carlo simulation. In the central region of the detector, the uncertainty in the jet energy calibration is derived from the transverse momentum balance in Z þ jet, γ þ jet or multijet events measured in situ, by propagating the known uncertainties of the energies of the reference objects to the jet energies. The energy uncertainties for the central region are then propagated to the forward region by studying the transverse momentum balance in dijet events with one central and one forward jet [57]. The energy calibration uncertainty in the high-p T range is estimated using the in situ measurement of the response to single isolated hadrons [58]. The jet energy calibration's total uncertainty is decomposed into 57 uncorrelated contributions, of which each is fully correlated in p T . The corresponding uncertainty in jet p T is between 1% and 4% in the central region (jηj < 1.8), and increases to 5% in the forward region (1.8 < jηj < 4.5). The jet energy resolution has been measured in the data using the bisector method in dijet events [59-61] and the Monte Carlo simulation is seen to be in good agreement with the data. The uncertainty in the jet energy resolution is affected by selection parameters for jets, such as the amount of nearby jet activity, and depends on the η and p T values of the jets. Further details about the determinations of the jet energy scale and resolution are given in Refs. [58,59,62].
The angular resolution of jets is obtained in the Monte Carlo simulation by matching particle-level jets with detector-level jets, when their distance in ΔR ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðΔy 2 þ Δϕ 2 Þ p is smaller than the jet radius parameter. The jet η and ϕ resolutions are obtained from a Gaussian fit to the distributions of the difference between the detector-level and particle-level values of the corresponding quantity. The difference between the angular resolutions determined from different Monte Carlo simulations is taken as a systematic uncertainty for the measurement result, which is about 10%-15% for p T < 150 GeV and decreases to about 1% for p T > 400 GeV. The bias in jet η and ϕ is found to be negligible.
All jets within the whole detector acceptance, jηj < 4.9, are considered in the analysis. Data-quality requirements are applied to each reconstructed jet according to its properties, to reject spurious jets not originating from hard-scattering events. In each H T bin, events from a single trigger are used and the same trigger is used for the numerator and the denominator of R Δϕ . In order to test the stability of the measurement results, the event sample is divided into subsamples with different pileup conditions. The R Δϕ results for different pileup conditions are compatible within the statistical uncertainties without any systematic trends. The measurement is also tested for variations resulting from loosening the requirements on the event-and jetdata-quality conditions, and the observed variations are also consistent within the statistical uncertainties.
The distributions of R Δϕ ðH T ; y Ã ; Δϕ max Þ are corrected for experimental effects, including detector resolutions and inefficiencies, using the simulation. To ensure that the simulation describes all relevant distributions, including the p T and y distributions of the jets, the generated events are reweighted, based on the properties of the generated jets, to match these distributions in data, and to match the H T dependence of the observed inclusive dijet cross section as well as the R Δϕ distributions and their H T dependence. To minimize migrations between H T bins due to resolution effects, the bin widths are chosen to be larger than the detector resolution. The bin purities, defined as the fraction of all reconstructed events that are generated in the same bin, are 65%-85% for Δϕ max ¼ 7π=8 and 5π=6, and 50%-75% for Δϕ max ¼ 3π=4 and 2π=3. The bin efficiencies, defined as the fraction of all generated events that are reconstructed in the same bin, have values in the same ranges as the bin purities. The corrections are obtained bin by bin from the generated PYTHIA events as the ratio of the R Δϕ results for the particle-level jets and the detector-level jets. These corrections are typically between 0% and 3%, and never outside the range from −10% to þ10%. Uncertainties in these corrections due to the modeling of the migrations by the simulation are estimated from the changes of the correction factors when varying the reweighting function. In most parts of the phase space, these uncertainties are below 1%. The results from the bin-by-bin correction procedure were compared to the results when using a Bayesian iterative unfolding procedure [63], and the two results agree within their statistical uncertainties.
The uncertainties of the R Δϕ measurements include two sources of statistical uncertainty and 62 sources of systematic uncertainty. The statistical uncertainties arise from the data and from the correction factors. The systematic uncertainties are from the correction factors (two independent sources, related to variations of the reweighting of the generated events), the jet energy calibration (57 independent sources), the jet energy resolution, and the jet η and ϕ resolutions. To avoid double counting of statistical fluctuations, the H T dependence of the uncertainty distributions is smoothed by fitting either linear or quadratic functions in logðH T =GeVÞ. From all 62 sources of experimental correlated uncertainties, the dominant systematic uncertainties are due to the jet energy calibration. For Δϕ max ¼ 7π=8 and 5π=6 the jet energy calibration uncertainties are typically between 1.0% and 1.5% and always less than 3.1%. For smaller values of Δϕ max they can be as large as 4% (for Δϕ max ¼ 3π=4) or 9% (for Δϕ max ¼ 2π=3).

VI. MEASUREMENT RESULTS
The measurement results for R Δϕ ðH T ; y Ã ; Δϕ max Þ are corrected to the particle level and presented as a function of H T , in different regions of y Ã and for different Δϕ max requirements. The results are listed in Appendix B in Tables VI-IX, and displayed in Fig. 1, at the arithmetic center of the H T bins. At fixed (y Ã , Δϕ max ), R Δϕ ðH T ; y Ã ; Δϕ max Þ decreases with increasing H T and increases with increasing y Ã at fixed (H T , Δϕ max ). At fixed (H T , y Ã ), R Δϕ decreases with decreasing Δϕ max .
Theoretical predictions based on NLO pQCD (for Δϕ max ¼ 7π=8, 5π=6, and 3π=4) or LO (for Δϕ max ¼ 2π=3) with corrections for nonperturbative effects, as described in Sec. III, are compared to the data. The ratios of data to the theoretical predictions are displayed in Fig. 2.
To provide further information about the convergence of the pQCD calculation, the inverse of the NLO K-factors are also shown (defined as the ratio of predictions for R Δϕ at NLO and LO, . In all kinematical regions, the data are described by the theoretical predictions, even for Δϕ max ¼ 2π=3, where the predictions are only based on LO pQCD and have uncertainties of about 20% (dominated by the dependence on μ R and μ F ). The data for Δϕ max ¼ 7π=8 and 5π=6 allow the most stringent tests of the theoretical predictions, since for these Δϕ max values the theoretical uncertainties are typically less than AE5%.

VII. SELECTION OF DATA POINTS FOR THE α S EXTRACTION
The extraction of α S ðQÞ at different scales Q ¼ H T =2 is based on a combination of data points in different kinematic regions of y Ã and Δϕ max , with the same H T . The data points are chosen according to the following criteria.
(1) Data points are used only from kinematic regions in which the pQCD predictions appear to be most reliable, as judged by the renormalization and factorization scale dependence, and by the NLO K-factors.
(2) For simplicity, data points are only combined in the α S extraction if they are statistically independent, i.e., if their accessible phase space does not overlap. (3) The preferred data points are those for which the cancellation of the PDFs between the numerator and the denominator in R Δϕ is largest. (4) The experimental uncertainty at large H T is limited by the sample size. If the above criteria give equal preference to two or more data sets with overlapping phase space, the data points with smaller statistical uncertainties are used to test the RGE at the largest possible momentum transfers with the highest precision. Based on criterion (1), the data points obtained for Δϕ max ¼ 2π=3 are excluded, as the pQCD predictions in NLOJET++ are only available at LO. Furthermore, it is 0.0 < y* < 0.5 0.5 < y* < 1.0 1.0 < y* < 2.0

MMHT2014 PDFs
FIG. 1. The measurement of R Δϕ ðH T ; y Ã ; Δϕ max Þ as a function of H T in three regions of y Ã and for four choices of Δϕ max . The inner error bars indicate the statistical uncertainties, and the sum in quadrature of statistical and systematic uncertainties is displayed by the total error bars. The theoretical predictions, based on pQCD at NLO (for Δϕ max ¼ 7π=8, 5π=6, and 3π=4) and LO (for Δϕ max ¼ 2π=3) are shown as solid and dashed lines, respectively. The shaded bands display the PDF uncertainties and the scale dependence, added in quadrature.
observed that the points for Δϕ max ¼ 3π=4 have a large scale dependence, which is typically between þ15% and −10%. For the remaining data points with Δϕ max ¼ 7π=8 and 5π=6 at larger y Ã (1 < y Ã < 2), the NLO corrections are negative and (with a size of 5%-23%) larger than those at smaller y Ã , indicating potentially larger corrections from not yet calculated higher orders. The conclusion from criterion (1) is therefore that the pQCD predictions are most reliable in the four kinematic regions 0 < y Ã < 0.5 and 0.5 < y Ã < 1, for Δϕ max ¼ 7π=8 and Δϕ max ¼ 5π=6, where the NLO K-factors are typically within AE5% of unity.
The requirement of statistically independent data points according to criterion (2) means that the data points from different y Ã regions can be combined, but not those with different Δϕ max . The choice whether to use the data with Δϕ max ¼ 7π=8 or 5π=6 (in either case combining the data for 0 < y Ã < 0.5 and 0.5 < y Ã < 1) is therefore based on criteria (3) and (4).
The cancellation of the PDFs, as addressed in criterion (3), is largest for those data points for which the phase space of the numerator in Eq. (1) is closest to that of the denominator. Since the numerator of R Δϕ is a subset of the denominator, this applies more to the data at larger values of Δϕ max . For those points, the fractional contributions from different partonic subprocesses (gg → jets, gq → jets, qq → jets), and the ranges in the accessible proton momentum fraction x are more similar for the numerator and denominator, resulting in a larger cancellation of PDFs in R Δϕ . This argument, based on the third criterion, leads to the same conclusion as the suggestion of criterion (4), to use the data set with smallest statistical uncertainty.
Based on the four criteria, α S is therefore extracted combining the data points in the rapidity regions 0 < y Ã < 0.5 and 0.5 < y Ã < 1 for Δϕ max ¼ 7π=8. Extractions of α S from the data points in other kinematical regions in y Ã and Δϕ max are used to investigate the dependence of the final results on those choices.

VIII. DETERMINATION OF α S
The R Δϕ measurements in the selected kinematic regions are used to determine α S and to test the QCD predictions for its running as a function of the scale Q ¼ H T =2. The α S results are extracted by using MINUIT [64], to minimize the χ 2 function specified in Appendix C. In this approach, the experimental and theoretical uncertainties that are correlated between all data points are treated in the Hessian method [65] by including a nuisance parameter for each uncertainty source, as described in Appendix C. The only exceptions are the uncertainties due to the PDF set and the μ R;F dependence of the pQCD calculation. These uncertainties are determined from the variations of the α S results, when repeating the α S extractions for different PDF sets and for variations of the scales μ R;F as described in Sec. III. Results of α S ðQÞ (with Q ¼ H T =2, taken at the arithmetic centers of the H T bins) are determined from the R Δϕ data for Δϕ max ¼ 7π=8, combining the data points in the two y Ã regions of 0 < y Ã < 0.5 and 0.5 < y Ã < 1.0. Nine α S ðQÞ values are determined in the range 262 < Q ≤ 1675 GeV. A single χ 2 minimization provides the uncertainties due to the statistical uncertainties, the experimental correlated uncertainties, the uncertainties due to the nonperturbative corrections, and the MMHT2014 PDF uncertainty. Separate χ 2 minimizations are made for variations of μ R and μ F (in the ranges described in Sec. III), and also for the CT14, NNPDFv2.3, ABMP16, and HERAPDF 2.0 PDF sets. The largest individual variations are used to quantify the uncertainty due to the scale dependence and the PDF set, respectively. The so-defined PDF set uncertainty may partially double count some of the uncertainties already taken into account by the MMHT2014 PDF uncertainties, but it may also include some additional systematic uncertainties due to different approaches in the PDF determinations. The α S ðQÞ results are displayed in Fig. 3 and listed in Table IV. In addition, assuming the validity of the RGE, all 18 data points in 0 < y Ã < 0.5 and 0.5 < y Ã < 1.0 for Δϕ max ¼ 7π=8 are used to extract a combined α S ðm Z Þ result. The combined fit (for MMHT2014 PDFs at the default scale) gives χ 2 ¼ 21.7 for 17 degrees of freedom and a result of α S ðm Z Þ ¼ 0.1127 (the uncertainties are detailed in Table V). The fit is then repeated for the CT14, NNPDFv2.3, ABMP16, and HERAPDF 2.0 PDF sets, for which the α S ðm Z Þ results differ by þ0.0001, þ0.0022, þ0.0026, and þ0.0029, respectively. Fits for various choices of μ R and μ F result in variations of the α S ðm Z Þ results between −0.0019 and þ0.0052.
Further dependence of the α S results on some of the analysis choices is investigated in a series of systematic studies.
(i) Changing the Δϕ max requirement.-Based on the criteria outlined in Sec. VII it was decided to use the data for Δϕ max ¼ 7π=8 in the α S analysis. If, instead, the data with Δϕ max ¼ 5π=6 are used, the α S ðm Z Þ result changes by þ0.0052 to α S ðm Z Þ ¼ 0.1179, with an uncertainty of þ0.0065 and −0.0045 due to the scale dependence. (ii) Extending the y Ã region.-For the central α S results, the data points with 1 < y Ã < 2 are excluded. If α S ðm Z Þ is determined only from the data points for 1 < y Ã < 2 (with Δϕ max ¼ 7π=8) the α S ðm Z Þ result changes by −0.0018, with an increased scale dependence, to α S ðm Z Þ ¼ 0.1109 þ0.0071 −0.0031 with χ 2 ¼ 13.8 for 7 degrees of freedom. If the data points for 1 < y Ã < 2 are combined with those  The results for α S ðQÞ determined from the R Δϕ data for Δϕ max ¼ 7π=8 with 0 < y Ã < 0.5 and 0.5 < y Ã < 1.0. All uncertainties have been multiplied by a factor of 10 3 . for 0 < y Ã < 0.5 and 0.5 < y Ã < 1, the result is It is seen that the largest upward variation (of þ0.0052) is obtained for the correlated variation is obtained for the anti-correlated variation x μ R ¼ 0.5 and x μ F ¼ 2.0, which is, however, outside the range 0.5 ≤ x μ R =x μ F ≤ 2. The lowest variation within this range (−0.0014) is obtained for x μ R ¼ 0.5 and x μ F ¼ 1.0. (viii) Effects of the Hessian method.-In the Hessian approach, a fit can explore the multidimensional uncertainty space to find the χ 2 minimum at values of the nuisance parameters associated to the sources of systematic uncertainties, that do not represent the best knowledge of the corresponding sources. While in this analysis the shifts of the nuisance parameters are all small, it is still interesting to study their effects on the α S fit results. Therefore, the α S ðm Z Þ extraction is repeated, initially including the uncorrelated (i.e., statistical) uncertainties only. Then, step by step, the experimental correlated uncertainties, the uncertainties of the nonperturbative corrections, and the PDF uncertainties are included. These fits produce α S ðm Z Þ results that differ by less than AE0.0004 from the central result. These systematic studies show that the α S results are rather independent of the analysis choices and demonstrate the stability of the α S extraction procedure. These variations are not treated as additional uncertainties because their resulting effects are smaller than the other theoretical uncertainties. The largest variation of the α S ðm Z Þ result, TABLE V. Fit result for α S ðm Z Þ, determined from the R Δϕ data for Δϕ max ¼ 7π=8 with 0.0 < y Ã < 0.5 and 0.5 < y Ã < 1.0. All uncertainties have been multiplied by a factor of 10 3 . by þ0.0052, is obtained when using the data with Δϕ max ¼ 5π=6 instead of Δϕ max ¼ 7π=8. This difference may be due to different higher-order corrections to the NLO pQCD results for different Δϕ max values. This assumption is consistent with the observed scale dependence of the α S ðm Z Þ results, within which the results for both choices of Δϕ max agree (0.1127 þ 0.0052 versus 0.1179 − 0.0045 for Δϕ max ¼ 5π=6 and 7π=8, respectively). It is therefore concluded from the systematic studies that no further uncertainties need to be assigned. The final result from the combined fit is α S ðm Z Þ ¼ 0.1127 þ0.0063 −0.0027 with the individual uncertainty contributions given in Table V. This result and the corresponding RGE prediction are also shown in Fig. 3. For all α S results in Tables IV and V, the uncertainties are dominated by the μ R dependence of the NLO pQCD calculation.
Within the uncertainties, the α S ðm Z Þ result is consistent with the current world average value of α S ðm Z Þ ¼ 0. −0.0071 ), with comparable uncertainties. The compatibility of the results of this analysis, based on the measurements of R Δϕ , with the world average value of α S ðm Z Þ is demonstrated in Appendix D.
The individual α S ðQÞ results are compared in Fig. 4 with previously published α S results obtained from jet measurements [4][5][6][7][9][10][11][12][13][14][15] and with the RGE prediction for the combined α S ðm Z Þ result obtained in this analysis. The new results agree with previous α S ðQÞ results in the region of overlap, and extend the pQCD tests to momentum transfers up to 1.6 TeV, where RGE predictions are consistent with the α S ðQÞ results, as discussed in Appendix E.

IX. SUMMARY
The multijet cross-section ratio R Δϕ is measured at the LHC. The quantity R Δϕ specifies the fraction of the inclusive dijet events in which the azimuthal opening angle of the two jets with the highest transverse momenta is less than a given value of the parameter Δϕ max . The R Δϕ results, measured in 20.2 fb −1 of pp collisions at ffiffi ffi s p ¼ 8 TeV with the ATLAS detector, are presented as a function of three variables: the total transverse momentum H T , the dijet rapidity interval y Ã , and the parameter Δϕ max . The H T and y Ã dependences of the data are well described by theoretical predictions based on NLO pQCD (for Δϕ max ¼ 7π=8, 5π=6, and 3π=4), or LO pQCD (for Δϕ max ¼ 2π=3), with corrections for nonperturbative effects. Based on the data points for Δϕ max ¼ 7π=8 with 0 < y Ã < 0.5 and 0.5 < y Ã < 1, nine α S results are determined, at a scale of Q ¼ H T =2, over the range of 262 < Q < 1675 GeV. The α S ðQÞ results are consistent with the predictions of the RGE, and a combined analysis results in a value of α S ðm Z Þ ¼ 0.1127 þ0.0063 −0.0027 , where the uncertainty is dominated by the scale dependence of the NLO pQCD predictions.

ACKNOWLEDGMENTS
We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently.  [4][5][6][7][9][10][11][12][13][14][15]. Also shown is the prediction of the RGE for the α S ðm Z Þ result obtained from the R Δϕ data in this analysis. and BNL (USA), the Tier-2 facilities worldwide and large non-WLCG resource providers. Major contributors of computing resources are listed in Ref. [66].

APPENDIX A: EFFECTS OF TOP QUARK CONTRIBUTIONS ON THE pQCD PREDICTIONS
There are two ways in which contributions from top quarks affect the pQCD predictions for R Δϕ . First, the pQCD predictions based on matrix elements for massless quarks also depend on the number of quark flavors in gluon splitting (g → qq), n f , which affects the tree-level matrix elements and their real and virtual corrections, as well as the RGE predictions. The pQCD predictions for the central analysis are obtained for n f ¼ 5. The effects for the measured quantity R Δϕ for the choice n f ¼ 6 are computed in this Appendix. Second, since the decay products of hadronically decaying (anti)top quarks are sometimes reconstructed as multiple jets, the Oðα 2 S Þ tt production process also contributes to three-jet topologies. Since this contribution is of lower order in α S as compared to the pQCD Oðα 3 S Þ three-jet production processes, it is a "superleading" contribution, which is formally more important. This potentially large contribution and the corresponding effects for R Δϕ are also estimated in this Appendix.
In a pQCD calculation in which quark masses are properly taken into account, the contributions from the massive top quark arise naturally at higher momentum transfers, according to the available phase space. In calculations based on matrix elements for massless quarks, n f is a parameter in the calculation. For jet production at the LHC, the alternatives are n f ¼ 5, i.e., ignoring the contributions from g → tt processes (which is the central choice for this analysis), or n f ¼ 6, i.e., treating the top quark as a sixth massless quark. The relative difference between the two alternatives is evaluated from the effects due to the RGE and the matrix elements. For this purpose, the 2-loop solution of the RGE for n f ¼ 5 is replaced by the 2-loop solutions for n f ¼ 5 and n f ¼ 6 with 1-loop matching [67] at the pole mass of the top quark m pole top , assuming that m pole top is equal to the world average of the measured "Monte Carlo mass" of 173.21 GeV [16]. In addition, the matrix elements are recomputed for n f ¼ 6. For a fixed value of α S ðm Z Þ ¼ 0.118, the corresponding effects for the pQCD predictions for R Δϕ are in the range of −1% to þ2%.
The effects on R Δϕ due to the contributions from hadronic decays of tt final states are estimated using POWHEG-BOX [68] (for the pQCD matrix elements) interfaced with PYTHIA (for the parton shower, underlying event, and hadronization) and CTEQ6L1 PDFs [69]. It is seen that the tt process contributes 0.003-0.2% to the denominator of R Δϕ (the inclusive dijet cross section), and 0.006%-0.5% to the numerator (with Δϕ max ¼ 7π=8). The effects for the ratio R Δϕ are 0%-0.5% in the analysis phase space, and there are no systematic trends in the considered distributions within the statistical uncertainties of the generated POWHEG-BOX event sample. Since this effect is about four to eight times smaller than the typical uncertainty due to the renormalization scale dependence, the corresponding effects on α S are not investigated further.

APPENDIX B: DATA TABLES
The results of the R Δϕ measurements are listed in Tables VI-IX, together with their relative statistical and systematic uncertainties.  Given is a set of experimental measurement results in bins i of a given quantity with central measurement results d i with statistical and uncorrelated systematic uncertainties σ i;stat and σ i;uncorr , respectively. The experimental measurements are affected by various sources of correlated uncertainties, and δ ij ðϵ j Þ specifies the uncertainty of measurement i due to the source j, where ϵ j is a Gaussian distributed random variable with zero expectation value and unit width. The δ ij ðϵ j Þ specify the dependence of the measured result i on the variation of the correlated uncertainty source j by ϵ j standard deviations, where ϵ j ¼ 0 corresponds to the central value of the measurement (i.e., δ ij ðϵ j ¼ 0Þ ¼ 0), while the relative uncertainties corresponding to plus/minus one standard deviation are given by δ ij ðϵ j ¼ AE1Þ ¼ Δd AE ij . From the central measurement result and the relative uncertainties Δd AE ij , the continuous ϵ j dependence of δ ij ðϵ j Þ can be obtained using quadratic interpolation The theoretical prediction t i ðα S Þ for bin i depends on the value of α S . Furthermore, the theoretical predictions are also affected by sources of correlated uncertainties; δ ik ðλ k Þ specifies the relative uncertainty of t i due to the source k. Like the ϵ j , the λ j are also treated as Gaussian distributed random variables with zero expectation value and unity width. It is assumed that the theoretical predictions can be obtained with statistical uncertainties which are negligible as compared to the statistical uncertainties of the measurements.
The continuous dependence of the relative uncertainty δ ik ðλ k Þ can be obtained through quadratic interpolation between the central result t i and the results t AE ik obtained by variations corresponding to plus/minus one standard deviation due to source k The χ 2 used in the α S extraction is then computed as where i runs over all data points, j runs over all sources of experimental correlated uncertainties, and k over all theoretical correlated uncertainties. The fit result of α S is determined by minimizing χ 2 with respect to α S and the "nuisance parameters" ϵ j and λ k . Table V is lower than the world average value by approximately one standard deviation. In this Appendix, the consistency of the world average of α S ðm Z Þ and the R Δϕ data is investigated using the χ 2 values. The χ 2 values are computed according to Appendix C, using the 18 data points with Δϕ max ¼ 7π=8, and 0.0 < y Ã < 0.5 and 0.5 < y Ã < 1.0. The theoretical predictions are computed for the fixed value of α S ðm Z Þ ¼ 0.1181. The computation of χ 2 uses the Hessian method for the treatment of all uncertainties except for the PDF set uncertainty and the scale dependence, so the χ 2 values do not reflect these theoretical uncertainties. Therefore, a series of χ 2 values is computed for possible combinations of variations of μ R and μ F around the central choice The results are displayed in Table X and compared to the χ 2 values obtained when α S ðm Z Þ is a free fit parameter. When α S ðm Z Þ is fixed to the world average, the χ 2 value for the central scale choice is slightly higher than the one obtained for a free α S ðm Z Þ, and also higher than the expectation of Z Þ is fixed or 17 when it is a free fit parameter. However, the χ 2 definition does not take into account the theoretical uncertainty due to the scale dependence. When the renormalization scale is increased by a factor of two, to μ R ¼ 2μ 0 , lower χ 2 values are obtained, which are similar in size to the ones obtained for a free α S ðm Z Þ, and close to the expectation (the dependence on the factorization scale is rather small). Since these χ 2 values are well within the range of the expectation, it is concluded that, within their uncertainties, the theoretical predictions for the world average value of α S ðm Z Þ are consistent with the R Δϕ data.

APPENDIX E: ON THE COMPATIBILITY OF THE RGE AND THE SLOPE OF THE α S ðQÞ RESULTS
It is natural to ask whether the observed Q dependence (i.e., the running) of the α S ðQÞ results shown in Fig. 3 is described by the RGE or instead exhibits significant deviations at the highest Q values, possibly indicating signals of physics beyond the Standard Model. The consistency of the RGE predictions with the observed slope is investigated in this Appendix. The RGE prediction would be in agreement with the observed Q dependence of the α S ðQÞ results if the latter, when evolved to m Z , give α S ðm Z Þ values that are independent of Q. For this purpose, a linear function in log 10 ðQ=1 GeVÞ, fðQÞ ¼ c þ m · log 10 ðQ=1 GeVÞ, is fitted to the nine α S ðm Z Þ points in Fig. 3 (bottom) and their statistical uncertainties. Here the correlated systematic uncertainties are not taken into account as their correlations are nontrivial since the individual α S ðQÞ results are obtained in separate fits, with different optimizations for the nuisance parameters. The fit results for the slope parameter m and its uncertainty are displayed in Table XI for a fit to the α S ðm Z Þ points at all nine Q values, and also for fits to different subsets of the α S ðm Z Þ points, omitting points either at lower or higher Q.
As documented in Table XI, a fit to all nine α S ðm Z Þ points gives a slope that differs from zero by more than its uncertainty. Fits to groups of data points, however, show that the significance of this slope arises from the two points at lowest Q. Omitting the α S ðm Z Þ point at lowest Q (fitting points Nos. 2-9), or the two points at lowest Q (fitting points # 3-9), both give fit results for which the slope parameter is more consistent with zero, while the α S ðm Z Þ results change by less than AE0.0001. On the other hand, omitting the α S ðQÞ points at highest Q (fitting points Nos. 1-8 or Nos. 1-7) does not affect the significance of the slope. It is therefore concluded that the high-Q behavior of the α S ðQÞ results is consistent with the RGE and that the small differences at lowest Q do not affect the combined α S ðm Z Þ result. TABLE X. The χ 2 values between the 18 data points and the theoretical predictions when α S ðm Z Þ is fixed to the world average value of α S ðm Z Þ ¼ 0.1181 (third column) and when it is a free fitted parameter (fourth column) for variations of the scales μ R and μ F around the central choice   [12] CMS Collaboration, Constraints on parton distribution functions and extraction of the strong coupling constant from the inclusive jet cross section in pp collisions at ffiffi ffi s p ¼ 7 TeV, Eur. Phys. J. C 75, 288 (2015).
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