Azimuthal anisotropy of charged particles with transverse momentum up to 100 GeV / c in PbPb collisions at

The Fourier coeﬃcients v 2 and v 3 characterizing the anisotropy of the azimuthal distribution of charged particles produced in PbPb collisions at √ s NN = 5 . 02 TeV are measured with data collected by the CMS experiment. The measurements cover a broad transverse momentum range, 1 < p T < 100 GeV / c . The analysis focuses on the p T > 10 GeV / c range, where anisotropic azimuthal distributions should reﬂect the path-length dependence of parton energy loss in the created medium. Results are presented in several bins of PbPb collision centrality, spanning the 60% most central events. The v 2 coeﬃcient is measured with the scalar product and the multiparticle cumulant methods, which have different sensitivities to initial-state ﬂuctuations. The values from both methods remain positive up to p T ∼ 60–80 GeV / c , in all examined centrality classes. The v 3 coeﬃcient, only measured with the scalar product method, tends to zero for p T (cid:2) 20 GeV / c . Comparisons between theoretical calculations and data provide new constraints on the path-length dependence of parton energy loss in heavy ion collisions and highlight the importance of the initial-state ﬂuctuations. ©


Introduction
Several observations made at RHIC in AuAu collisions at centerof-mass energy per nucleon pair √ s NN = 200 GeV [1][2][3][4] and at the LHC in PbPb collisions at √ s NN = 2.76 and 5.02 TeV [5][6][7][8][9][10] establish that high-energy partons lose a significant fraction of their energy while traversing the hot and dense medium created in these collisions. Measurements of the nuclear modification factor (R AA ), a ratio that quantifies the modification of particle spectra between pp and heavy ion collisions, show a large suppression of high transverse-momentum (p T ) charged hadrons at RHIC [11][12][13][14][15][16] and at LHC [7][8][9][10]. Also, a strong asymmetry is observed in the energies of the two jets in dijet events in PbPb collisions [5,6]. These observations have triggered much progress in the understanding of jet quenching phenomena, but do not provide sufficient information for a detailed understanding of how the parton energy loss depends on the distance traversed by the partons in the medium. The study of anisotropies in the azimuthal angle distributions of high-p T hadrons can provide revealing information that is complementary to previous measurements. These anisotropies are characterized by the v n coefficients of a Fourier expansion in E-mail address: cms-publication-committee-chair@cern.ch. the distributions of azimuthal angle measured with respect to the event plane, defined by the direction of maximum particle density in the transverse plane [17]. Such studies have been performed at RHIC [18] and at the LHC [19][20][21] up to p T ≈ 10 and 60 GeV/c, respectively. Most jet quenching models are unable to simultaneously reproduce the R AA and v 2 measurements [22][23][24]. Nevertheless, recent attempts to solve this puzzle have shown promise by considering initial-state collision geometry asymmetries and fluctuations [25,26], which are predicted to strongly affect the high-p T v n coefficients, but not the R AA values. In particular, the fluctuations generate odd harmonics [27] and the measurement of the v 3 coefficient up to very high p T is expected to clarify the importance of considering initial-state fluctuations in the modeling of parton energy loss [25,26]. In this Letter, the azimuthal anisotropy of charged particles produced in PbPb collisions at √ s NN = 5.02 TeV is measured up to p T ≈ 100 GeV/c. The scalar product (SP) method [28,29] is used to determine the v 2 and v 3 coefficients as a function of p T and collision centrality in the pseudorapidity range |η| < 1. The unprecedented statistical reach of the √ s NN = 5.02 TeV PbPb sample

The CMS detector
The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter providing a 3.8 T field. Within the solenoid volume there are a silicon pixel and strip tracker detector, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter (HCAL), each composed of a barrel and two endcap sections. Muons are measured in gas-ionization detectors embedded in the steel flux-return yoke outside the solenoid. The silicon tracker measures charged particles within |η| < 2.5 and provides a p T resolution of about 1.5% for 100 GeV charged particles. Furthermore, the track impact parameter resolution is about 25-90  μm in the transverse (longitudinal) dimension, depending on η and p T [31]. Iron and quartz-fiber Cherenkov hadron forward (HF) calorimeters cover the range 2.9 < |η| < 5.2 on either side of the interaction region. The granularity of the HF towers is η × φ = 0.175 × 0.175 radians, allowing an accurate reconstruction of the heavy ion event plane. A more 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. [32]. The detailed Monte Carlo simulation of the CMS detector response is based on Geant4 [33].

Event and track selections
The analysis of PbPb collisions is based on a data set corresponding to an integrated luminosity of 404 μb −1 , collected in 2015. Events were collected with several trigger algorithms, composed of a hardware-based level 1 (L1) trigger, followed by a software-based high-level trigger (HLT). The p T region up to 14 GeV/c is covered by a minimum-bias trigger, which requires energy deposits in both HF calorimeters above a predefined threshold of approximately 1 GeV. This minimum-bias trigger was prescaled during data taking. To extend the measurement to higher order coefficients and higher p T (e.g., up to 100 GeV/c), a dedicated trigger that selects events containing a high-p T particle was used. The L1 trigger requirement was based on the transverse energy (E T ) of the highest E T calorimeter region ( η × φ = 0.348 × 0.348) in the barrel region (|η| < 1.044). In the HLT farm, a fast version of the offline tracking algorithms was employed and the highest p T track was required to pass the strict selection criteria described hereafter, resulting in a trigger efficiency of nearly 100%. Different E T and p T thresholds [10] were used at L1 and HLT, respectively, to enrich the data sample with events that contain high-p T tracks.
In the offline analysis, an additional selection of hadronic collisions is applied by requiring at least three towers with an energy deposit of more than 3 GeV per tower in each of the HF detectors. The events are required to have a reconstructed primary vertex, formed by two or more tracks and required to have a distance from the nominal interaction point of less than 15 cm along the beam axis and less than 0.15 cm in the transverse plane. The collision centrality in PbPb events, i.e. the degree of overlap of the two colliding nuclei, is determined from the E T deposited in both HF calorimeters. Collision centrality bins are given in percentage ranges of the total hadronic cross section, 0-5% corresponding to the 5% of collisions with the largest overlap of the two nuclei [34].
A standard CMS high-purity track selection [31,35] is used to select primary tracks (tracks associated with the primary vertex). Additional requirements are applied to enhance the purity of these primary tracks. The track must be consistent with originating from the primary vertex by less than 3 standard deviations when estimating both the longitudinal and transverse distances of closest approach. The relative uncertainty of the p T measurement, σ (p T )/p T , must be less than 10%. To ensure high tracking efficiency and reduce the rate of misreconstructed tracks, primary tracks are restricted to the |η| < 1 and p T > 1 GeV/c region. Furthermore, tracks with p T > 20 GeV/c are required to match a compatible energy deposit in the calorimeters (ECAL + HCAL).
The tracking efficiency and detector acceptance in PbPb collisions are evaluated using simulated hydjet 1.9 [36] minimum bias and hydjet-embedded pythia [37] dijet events. The combined geometrical acceptance and efficiency for primary track reconstruction, for p T > 1 GeV/c and |η| < 1, is 60-75%, depending on centrality. Finally, the rate of misreconstructed tracks reaches its maximum in the most central events, where it approaches 10%.

Analysis technique
The anisotropies of the particle azimuthal angle distributions are characterized by the v n Fourier coefficients, determined by the expansion dN/dφ ∼ 1 + 2 n v n cos[n(φ − n )], where N is the number of particles and n is the nth harmonic symmetry plane angle. Event-by-event variations in the initial energy density of the collision lead to the measured event plane fluctuations about the (experimentally inaccessible) symmetry plane [38]. The SP method is used to measure azimuthal correlations and extract Fourier coefficients. In this method, the v n coefficients can be expressed in where M represents the number of tracks or HF towers with energy above a certain threshold in each event, φ k is the azimuthal angle of the kth track or HF tower, and ω k is a weighting factor equal to unity for Q n , p T for the tracks ( Q nC ), and E T for the HF towers ( Q n A and Q nB ). The angular brackets denote averages over all events. The Q n vector is based on the particles of interest, i.e., tracks with |η| < 1. The Q n A and Q nB vectors are determined from the two HF calorimeters, covering the range 3 < |η| < 5, while the Q nC vector is obtained using tracks with |η| < 0.75.
If the particle of interest comes from the positive-η side of the tracker, then Q n A is calculated using the negative-η side of HF, and vice versa. The large η gap imposed between Q n A and Q n suppresses few-particle correlations, such as those induced by high-p T jets and particle decays, which do not depend on the event plane direction EP n . The real part is taken for all averages of Q -vector products over the events. Azimuthal asymmetries that arise from the acceptance and other detector-related effects are taken into account using a two-step process, where the Q -vector is first recentered and subsequently flattened [39]. These corrections and their effects on the results are negligible for the CMS detector. Since the measurements include correlations between low-and high-p T particles, the recently established event-plane decorrelation effect [40] cannot be neglected. It is expected to reduce the v n values in comparison to those determined if the event planes would be established exclusively using high-p T particles. The model calculations that include fluctuations in the initial state take into account this effect [26].
The multiparticle cumulant method [30,41] is also used to measure v 2 from genuine 4-, 6-, and 8-particle correlations, with the advantage of being less sensitive to few-particle correlations, e.g., jet fragmentation. The cumulants are expressed in terms of the corresponding Q n vectors. We first define the 2-, 4-, 6-, and 8-particle correlators as where the double average symbol indicates that the average is taken over all particle combinations and for all events. The unbiased estimators of the reference multiparticle cumulants, c n {}, are defined as [41][42][43] c n {4} = 4 − 2 2 2 , c n {6} = 6 − 9 4 2 + 12 2 3 , In order to perform a measurement differential in p T in the multiparticle cumulant framework, one of the particles in Eq. (3) is restricted to belong to a certain p T bin. Denoting by 2 , etc., the modified particle correlators, the differential multiparticle cumulants are defined in Ref. [43] and can be derived as described in Ref. [41], Finally, with respect to the reference multiparticle cumulants, the differential 4-, 6-, and 8-particle v n (p T , η) coefficients are de- The statistical uncertainties are evaluated with a data-driven method, as previously employed in Ref. [42]. The data set is divided into 10 subsets with roughly equal numbers of events and the standard deviation of the resulting distribution of the cumulant is used to estimate the uncertainties.

Systematic uncertainties
At low p T , the relative systematic uncertainties for v 2 {SP} and v 3 {SP} are found to be similar. At high p T , the v 3 {SP} statistical uncertainties are too large to properly disentangle statistical fluctuations from systematic effects. Therefore, the v 2 systematic uncertainties, expressed in terms of relative values in %, are applied to v 3 , with the exception of the uncertainties due to the few-particle correlations, discussed below. The systematic uncertainties due to the vertex position selection and to the p T dependence of the tracking efficiency corrections are common to the SP and cumulant analyses. They are found to be less than 1% and independent of p T and centrality. The systematic uncertainties due to misreconstructed tracks are derived by changing the track selection criteria. The results are found to depend on p T but not centrality, and are also different for the cumulant and SP methods. The track selection uncertainties have been found to gradually increase from ∼ 2% at low p T to ∼ 50% for p T > 60 GeV/c for the SP method, and from ∼ 2% to ∼ 2% for the cumulant analysis. The SP results have an additional uncertainty arising from few-particle correlations. This uncertainty is determined by varying the η gap and contributes differently to the v 2 and v 3 measurements. It is found to depend on both p T and centrality, and ranges in absolute value from 0 to 0.022 for v 2 and from 0 to 0.030 for v 3 .

Results
Fig . 1 shows the v 2 and v 3 results obtained from the SP method as a function of p T , up to about 100 GeV/c, in seven collision centrality ranges. From low-to high-p T , the v 2 and v 3 values first increase with increasing p T , up to a maximum near p T ≈ 3 GeV/c, before decreasing again. In most centrality ranges, v 2 remains positive up to p T ∼ 60-80 GeV/c, becoming consistent with zero at higher p T . Positive v 3 values are found up to p T ≈ 20 GeV/c over the 0-40% centrality range. At higher p T , the measured v 3 value is consistent with zero within the experimental uncertainties. Given the systematic uncertainties, the measured values are compatible with zero. Some negative v 3 values are seen at high p T in the 40-50% centrality range, but such peripheral events are the most contaminated by back-to-back jet correlations. This is confirmed by studying the η gap dependence of the results in both measured and simulated events, where the latter include dijets embedded into hydjet events with zero input anisotropy. In the centrality range 50-60%, v 3 is only measured up to 20 GeV/c because of lack of events containing higher p T particles.
The v 2 and v 3 results are compared to the CUJET3.0 [44] and SHEE [25] models for several centrality bins. A key difference between these two models is that the SHEE framework includes initial-state geometry fluctuations, while CUJET3.0 uses a smooth hydrodynamic background. The CUJET3.0 model uses perturbative quantum chromodynamics (pQCD) calculations to describe the hard parton interactions in the quark-gluon plasma (QGP), complemented by a perfect-fluid hydrodynamic expansion of the medium. The SHEE calculations use viscous hydrodynamics including eventby-event fluctuations in the soft sector [26,45,46], in addition to an energy loss model [26,47,48]. They are performed with a low shear viscosity to entropy density ratio (η/s), less than or equal to 0.12 (although higher values do not affect the high-p T predictions), a chemical freezout temperature of 160 MeV, and a linear path-length dependence of the energy loss inspired by pQCD, similar to that in CUJET3.0. In addition, both model calculations are only valid for p T > 10 GeV/c.
Over the full centrality range, the CUJET3.0 calculations describe qualitatively the trend observed in the v 2 data for p T > 10 GeV/c, but fail to quantitatively reproduce the results. For instance, in the centrality range 0-30% and for 10 < p T < 40 GeV/c, v 2 is overestimated by 10-50%, while the model largely underestimates it in the peripheral bins. The SHEE calculations of both v 2 and v 3 are in good agreement with the data for p T > 10 GeV/c over the full centrality range. The success of the SHEE framework suggests that modeling the initial-state fluctuations may be a crucial ingredient to describe the experimental data related to parton energy loss. Although not shown in the figure, a scenario in the SHEE framework with a quadratic path-length dependence of the energy loss, inspired by gauge-gravity duality [49,50], was also tested and seen to disagree with the data. As just one example, this alternative pathlength dependence is found to overestimate the data by 30-40% for p T > 20 GeV/c in the 20-30% centrality range.
The v 2 values are also obtained from 4-, 6-, and 8-particle cumulant analyses, as shown in Fig. 2, where the SP v 2 results are also included for comparison. For p T < 3 GeV/c, the results follow the expectation from Bessel-Gaussian or elliptic power v 2 distribu- The v 2 and v 3 results from the SP method as a function of p T , in seven collision centrality ranges from 0-5% to 50-60%. The vertical bars (shaded boxes) represent the statistical (systematic) uncertainties. The curves represent calculations made with the CUJET3.0 [44] and the SHEE models [26] (see text).

Fig. 2.
Comparison between the v 2 results from the SP and the 4-, 6-, and 8-particle cumulant methods, as a function of p T , in six centrality ranges from 0-5% to 50-60%. The vertical bars (shaded boxes) represent the statistical (systematic) uncertainties. [51][52][53]. The observation that the multiparticle cumulant values remain simi- gests that the azimuthal anisotropy is strongly affected by the initial-state geometry and its event-by-event fluctuations [25,26]. At higher p T , the difference between SP and multiparticle cumulant results shows a tendency to decrease. Nevertheless, the uncertainties are too large to draw a firm conclusion. This tendency might be due to p T dependence of flow vector fluctuations, which depends on the shear viscosity over entropy density ratio of the medium [26,54]. Therefore, the results presented in Fig. 2 provide important information to constrain the QGP shear viscosity in PbPb collisions.  to be compatible within statistical uncertainties and to decrease when selecting higher p T particles. This suggests that the initialstate geometry and its fluctuations are likely to be the common causes of the observed particle azimuthal anisotropies at both low and high p T .

Summary
The azimuthal anisotropy of charged particles produced in PbPb collisions at √ s NN = 5.02 TeV has been studied using data collected by the CMS experiment. The v 2 and v 3 coefficients are determined, as a function of collision centrality, over the widest transverse momentum range studied to date (from 1 up to 100 GeV/c). For the first time, the multiparticle cumulant method is used for p T > 20 GeV/c. Over the measured centrality range, positive v 2 values are found up to p T ∼ 60-80 GeV/c, while the v 3 values are consistent with zero for p T > 20 GeV/c. For , consistent with a collective behavior arising from the hydrodynamic expansion of a quark-gluon plasma. The similarity of v 2 {SP}, v 2 {4}, v 2 {6}, and v 2 {8} at high p T suggests that v 2 originates from the path-length dependence of parton energy loss associated with an asymmetric initial collision geometry. In addition, a common trend in the centrality dependence of v 2 is observed over the full p T range, further supporting a common connection to the initial-state geometry and its fluctuations. A model calculation (SHEE) incorporating initialstate fluctuations with a linear path-length dependence of parton energy loss is found to be in good agreement with the data, over the wide p T and centrality ranges probed in this analysis.

Acknowledgements
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 centers and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses.