Measurements of azimuthal anisotropies of jet production in Pb+Pb collisions at $\sqrt{s_{NN}} =$ 5.02 TeV with the ATLAS detector

The azimuthal variation of jet yields in heavy-ion collisions provides information about the path-length dependence of the energy loss experienced by partons passing through the hot, dense nuclear matter known as the quark-gluon plasma. This paper presents the azimuthal anisotropy coefficients $v_2$, $v_3$, and $v_4$ measured for jets in Pb+Pb collisions at $\sqrt{s_{NN}} =$ 5.02 TeV using the ATLAS detector at the LHC. The measurement uses data collected in 2015 and 2018, corresponding to an integrated luminosity of 2.2 nb$^{-1}$. The $v_n$ values are measured as a function of the transverse momentum of the jets between 71 GeV and 398 GeV and the event centrality. A nonzero value of $v_2$ is observed in all but the most central collisions. The value of $v_2$ is largest for jets with lower transverse momentum, with values up to 0.05 in mid-central collisions. A smaller, nonzero value of $v_3$ of approximately 0.01 is measured with no significant dependence on jet $p_T$ or centrality, suggesting that fluctuations in the initial state play a small but distinct role in jet energy loss. No significant deviation of $v_4$ from zero is observed in the measured kinematic region.


Introduction
The primary physics aim of the heavy-ion program at the Large Hadron Collider (LHC) is to produce and study the quark-gluon plasma (QGP), the high-temperature state of quantum-chromodynamic matter in which quarks and gluons are no longer confined within protons and neutrons (for a recent review, see Ref. [1]). Measurements of jets produced in the early stages of heavy-ion collisions provide information about the short-distance-scale interactions of high-energy partons with the QGP. The overall rate of jets in central Pb+Pb collisions at a given transverse momentum T is found to be about a factor of two lower than expectations based on collisions, up to a T of approximately 1 TeV [2,3]. This suppression can be explained by the downward slope of the jet T spectrum and the reduction in parton T due to energy loss while traversing the QGP. The energy loss from partons is expected to depend on the length of the QGP region that the parton traverses. The geometry of the overlapping nuclei in mid-central collisions leads to shorter average path lengths if the jet is oriented along the direction of the collision impact parameter vector 1 than if the jet is oriented in the perpendicular direction. This should lead to a dependence of the jet yield on the azimuthal angle [4][5][6].
One key observable in understanding the path-length dependence of energy loss is the azimuthal anisotropy of jets. The azimuthal distribution of jets is described via a Fourier expansion: where the and Ψ are the magnitude and orientation of the th -order anisotropy, and is the azimuthal angle of jets. The Ψ , or event-plane angles, are oriented such that a jet produced in-plane, or along the direction of the event-plane angle, will traverse on average less QGP than a jet produced out-of-plane, or perpendicular to the event-plane angle. Similar Fourier expansions are often used to describe the azimuthal variation of the yield of soft particles, which is typically associated with hydrodynamic flow (see Ref. [7]). It is important to note that at high T , hydrodynamic flow is not expected to be the source of azimuthal variation. Measurements of the for high-T particles have been performed at the Relativistic Heavy Ion Collider (RHIC) [8,9]. The first measurement of the 2 for fully reconstructed jets was reported in Ref. [10] for Pb+Pb collisions at √ NN = 2.76 TeV. The measured 2 values were found to be positive for jets with transverse momentum 45-160 GeV. The 2 values were found to be smaller in the most central and most peripheral collisions. This is expected because the second-order eccentricity of the initial state is small in the most central collisions, while in the most peripheral collisions there is little energy loss in any direction. A measurement by ALICE using jets reconstructed from charged particles obtained similar results [11]. Related measurements by CMS and ATLAS have been performed with charged particles at high T in 5.02 TeV Pb+Pb collisions [12,13]. Reference [12] reported positive 2 values for charged particles with T up to 60-80 GeV. Until now, there have been no measurements of jet 2 in √ NN = 5.02 TeV Pb+Pb collisions and no measurements of the higher-order anisotropies, such as 3 and 4 , of jets in any collision system. Such measurements could provide new information about how the energy loss depends on path length and the initial collision geometry.

Analysis procedure
This analysis uses the event-plane method to determine coefficients as described in Ref. [29] and used in previous measurements [30,31]. The geometry of the initial collision can be characterized by a series of observed event-plane angles, Ψ obs , 3 determined by the azimuthal variation of transverse energy in the forward calorimeters. Only the range | | > 4.0 of the forward calorimeters is used in this analysis to reduce any bias of the event-plane determination from jets in the FCal. The resolution of the event-plane angles, Res{Ψ }, is determined by comparing in each event the values calculated in the forward and backward sides of the detector as detailed in Ref. [13]. The resolution is determined for each bin in centrality, and ranges from approximately 0.6-0.9 for Ψ 2 , 0.3-0.6 for Ψ 3 , and 0.2-0.3 for Ψ 4 .
The jet reconstruction procedures follow those used by ATLAS for previous jet measurements in Pb+Pb collisions [2, 10]. Jets are reconstructed using the anti-algorithm [16] implemented in the FastJet software package [32]. Jets with = 0.2 are formed by clustering calorimetric towers of spatial size Δ × Δ = 0.1 × /32. The energies in the towers are obtained by summing the energies of calorimeter cells at the electromagnetic energy scale [33] within the tower boundaries. A background subtraction procedure is applied to estimate within each event the underlying event (UE) average transverse energy density, ( , ), where the dependence is due to global azimuthal correlations in the particle production from hydrodynamic flow [13]. The modulation accounts for the contribution to the UE of the second-, third-, and fourth-order azimuthal anisotropy harmonics characterized by values of flow coefficients UE [13]. Any potential residual effect of the azimuthal variation of the underlying event on the jet reconstruction is accounted for by the systematic uncertainties described in Section 5. The UE is also corrected for -and -dependent nonuniformities of the detector response by correction factors derived in MB Pb+Pb data.
An iterative procedure is used to remove the impact of jets on the estimated and UE values. The first estimate of the average transverse energy density of the UE, ( ), is evaluated in 0.1 intervals of , excluding those which overlap with "seed" jets. In the first subtraction step, the seeds are defined to be a union of = 0.2 jets and = 0.4 track-jets. Track-jets are reconstructed by applying the antialgorithm with = 0.4 to charged particles with T > 4 GeV. The = 0.2 jets must pass a cut on the value of the tower energy, while the track-jets are required to have T > 7 GeV. The background is then subtracted from each tower constituent and jet kinematics are recalculated. After the first iteration, the and values are updated by excluding from the UE determination the regions within Δ = 0.4 of both the track-jets and the newly reconstructed = 0.2 jets with T > 25 GeV. The updated and UE values are used to update the jet kinematic properties in the second iteration.
Jet -and T -dependent correction factors derived in simulations are applied to the measured jet energy to correct for the calorimeter energy response [34]. An additional correction based on in situ studies of jets recoiling against photons, bosons, and jets in other regions of the calorimeter is applied [35]. This calibration is followed by a "cross-calibration" which relates the jet energy scale (JES) of jets reconstructed by the procedure outlined in this section to the JES in 13 TeV collisions [36].
So-called "truth jets" are defined in the MC sample before detector simulation by applying the antialgorithm with = 0.2 to stable particles with a proper lifetime greater than 30 ps, but excluding muons and neutrinos, which do not leave significant energy deposits in the calorimeter. 3 The observed event-plane angles are defined as Ψ = 1 tan −1  The JES and jet energy resolution (JER) for = 0.2 jets are shown in Figure 1 as a function of truth T . They are derived by matching each truth jet to the closest reconstructed and calibrated jet from the MC overlay sample within an angular distance of Δ = 0.15. The JES and JER are taken to be the means and standard deviations of the reco T / truth T distributions, respectively. The JES differs from unity by approximately 1% at 70 GeV and 2.5% at 400 GeV; this deviation is due to isolation cuts used in the determination of the jet calibration and is corrected for by the unfolding procedure described below. The JES has no significant centrality dependence. The JER improves with increasing T and from central to peripheral collisions. Figure 2 shows the JES and JER for = 0.2 jets as a function of the angle between the jet and the observed second-order event-plane angle. The dependence of the JES on this angle is smaller than its dependence on T , with variations up to approximately 0.5% between in-plane and out-of-plane jets. The rapidity range used in this measurement, | | < 1.2, is selected to minimize the JES dependence on the angle with respect to the event plane. The JER also shows a small dependence on the angle between the jet and the second-order event-plane angle, with the resolution of in-plane jets up to 0.5% larger than that for out-of-plane jets.
The jet yield is determined as a function of T , centrality, and Δ . For each centrality and Δ selection, the jet T spectra are unfolded to correct for jet energy scale and resolution effects using a one-dimensional Bayesian unfolding [37] as implemented in the RooUnfold package [38]. The response matrices are filled using spatially matched truth jet and reconstructed jet pairs from the MC overlay sample. The response matrices are reweighted in truth T by the ratio of the T spectra in data to that in the reconstructed MC sample, such that the T spectra in the response matrices better represent those in the data. The reweighting is done separately in each Δ bin, such that the response matrices include the same modulation as seen in the raw data. The unfolding is performed using three iterations, which was found to minimize the combination of the statistical uncertainty and relative bin migration for subsequent iterations. The data are not unfolded to correct for the angular resolution of the jets, which is found to be small compared to the size of the Δ binning.
For each selection in T , centrality and harmonic value , a function is fitted to the unfolded Δ distributions to extract the obs values. The fit function is: (1 + 2 obs cos( Δ )) 0 0  where the overall normalization and the value of obs are the free parameters in the fitting procedure. The fitted obs values are then corrected for the finite event-plane resolution as described in Ref.
[29], where = obs /Res{Ψ }. In addition to the measurements differential in jet T , the values are also obtained in an inclusive T bin for jets with 71 < T < 398 GeV, following the same procedure as used in the differential measurement.

Systematic uncertainties
The systematic uncertainties in this measurement arise from the JES and JER, the unfolding procedure, and the biasing of the event plane by a forward-produced jet correlated with the jet of interest. The systematic uncertainties presented in this section are given in terms of the absolute change to the measured values. For each uncertainty component the entire analysis procedure is repeated with the variation under consideration and the uncertainty contributions are added in quadrature to obtain the total systematic uncertainty in the measurement.
The systematic uncertainty in the JES has six parts. First, a centrality-independent baseline component is determined from in situ studies of the calorimeter response to jets reconstructed with the procedure used in 13 TeV collisions [33,39]. A second, centrality-independent component accounts for the relative energy scale difference between the jet reconstruction procedures used in this analysis and those in 13 TeV collisions [36]. Potential inaccuracies in the MC sample in the description of the relative abundances of jets initiated by quarks and gluons and of the calorimetric response to quark and gluon jets are accounted for by the third component. The fourth, centrality-dependent, component accounts for modifications of the parton shower due to quenching and thus possibly a different detector response to jets in Pb+Pb collisions that is not modeled by the MC simulation. It is evaluated by the method used for 2015 and 2011 data [36], which compares the jet T measured in the calorimeter and the sum of the transverse momenta of charged particles within the jet, in both the data and MC samples. The charged particles are selected with T > 4 GeV to remove effects of the UE. The size of the centrality-dependent uncertainty in the JES reaches 1.2% in the most central collisions. An additional, centrality-independent component of 0.5% is included to account for potential year-to-year differences observed between the peripheral Pb+Pb data taken in 2018 and the collision data taken in 2017 which is used for the calibration. The systematic uncertainties from the JES discussed above are derived for = 0.4 jets. The fifth component does not depend on collision centrality and it accounts for the potential difference in uncertainties between = 0.4 and = 0.2 jets. This uncertainty is assessed by comparing the ratio of T for matched = 0.2 and = 0.4 jets measured in data and the MC sample. The size of this JES uncertainty is approximately 1%. Each component is varied separately by ±1 standard deviation in MC samples, applied as a function of T and , and the response matrices are recomputed. The data are then unfolded with the modified matrices. Because the measurement is sensitive only to the relative variation in yields as a function of Δ , the measured values are insensitive to these JES uncertainties that do not depend on Δ and therefore these are subdominant uncertainties.
The sixth uncertainty in the JES comes from a potential variation of the scale as a function of the angle between the jet and the event plane. The maximum size of the variations is determined by comparing the jet T measured in the calorimeter and the sum of the transverse momenta of charged particles within the jet, as a function of Δ , in both the data and MC samples. The due to potential variations in the JES, JES , is determined by modifying the jets in the MC sample for different values of Δ using the comparison of the calorimeter and track measurements and measuring the resulting . The data in each Δ bin are then scaled by 1 + 2 JES cos( Δ ) and fit to extract the systematic variation. Because this measurement is only sensitive to the relative jet yields as a function of Δ and not the overall scale of the yields, systematic variations that vary as a function of Δ will result in a larger uncertainty in the than variations which only depend on the T of a jet, such as those described above. Therefore, the uncertainty in the variation of the scale as a function of the angle between the jet and the event plane is the dominant uncertainty in the JES for this measurement. In 20-40% central collisions for jets with 71 < T < 79 GeV where the measured 2 is largest, this uncertainty accounts for approximately 95% of the total uncertainty on the 2 due to the JES. For jets in the same centrality collisions with 316 < T < 398 GeV, this uncertainty accounts for approximately 80% of the total uncertainty on the 2 due to the JES. In 0-5%, 5-10%, and 10-20% central collisions, this uncertainty accounts for >80% of the total uncertainty on the 3 and 4 due to the JES for the full kinematic range of the measurement.
The uncertainty due to the JER is evaluated by repeating the unfolding procedure with modified response matrices, where an additional contribution is added to the resolution of the reconstructed T in the MC sample using a Gaussian smearing procedure. The smearing factor is evaluated using an in situ technique in 13 TeV data that involves studies of dĳet energy balance [40,41]. Further, an uncertainty is included to account for differences between the tower-based jet reconstruction and the jet reconstruction used in analyses of 13 TeV data, as well as differences in calibration procedures. Similarly to the JES, an additional uncertainty is assigned to the JER to account for differences between = 0.2 and = 0.4 jets. The resulting uncertainty from the JER is symmetrized.
The final uncertainty in the JER comes from a potential variation of the resolution as a function of the angle between the jet and the event plane due to the increased size of the UE in-plane compared to out-of-plane. The size of the UE is correlated with the size of the fluctuations of the UE which can lead to too small or too large a subtraction and increase the JER. The due to potential variations in the JER, JER , is determined by adding an additional contribution to the JER of the jets in the MC sample for different values of Δ and measuring the resulting . This additional contribution to the JER is determined by correlating the fluctuations in the UE with the size of the UE in data. The unfolded data in each Δ bin are then scaled by 1 + 2 JER cos( Δ ) and fit to extract the systematic variation. The variations in the JER have a minimal effect on the measured values.
The uncertainty in the unfolding procedure was determined by unfolding the data with response matrices that had not been reweighted to match the T spectra in data as described in Section 4 and fitting the unfolded data to obtain new results. The deviation from the nominal unfolding result was symmetrized and taken as the systematic uncertainty contribution.
The uncertainty in the event-plane resolution as determined in Ref.
[31] was found to be negligible in comparison with other uncertainties and is not included. However, it is possible for a jet correlated with the jet of interest to bias the event plane if some of its energy is in the FCal. An estimate of the size of this effect was determined from the MC samples. The MC samples were produced without a correlation between the dĳets in P 8 and the Ψ angles in the overlaid data event. Therefore, the measured of jets coming from the P 8 event should be zero, and any nonzero values are caused by some events having their event-plane determination biased by a jet from the MC sample. The size of the effect that jets biasing the event-plane angles have on the measured in data is estimated using the values found in the MC sample. The azimuthal modulation of the jet yields in the MC sample is subtracted from that in the unfolded data and the resulting values are taken as the systematic variations.
The total systematic uncertainties of the values and the contributions from each source are summarized in Figure 3. The largest uncertainty for 2 is the event-plane bias uncertainty, while for 3 and 4 the uncertainty in the Δ dependence of the JES is largest. The bin-to-bin variations in the unfolding uncertainty are largely statistical in nature. Figure 4 shows the systematic uncertainties of the values measured in the inclusive T bin of 71-398 GeV. The JES and JER uncertainties are smaller than those in the T differential measurements as the variations largely move the jets within the inclusive T bin. Similarly, the unfolding uncertainty becomes smaller as the unfolding is a smaller effect for the inclusive bin. The event-plane bias is the largest uncertainty in 2 , while the JES and event-plane bias are largest uncertainties in 3 Figure 5 shows an example of the angular distribution of jets with respect to the Ψ 2 , Ψ 3 , and Ψ 4 planes, for jets with 71 < T < 79 GeV in the 10-20% centrality bin. For both the Ψ 2 and Ψ 3 dependence there are more jets in-plane than out-of-plane, although for Ψ 3 the angular dependence is smaller. There is no significant dependence of the jet yield on the angle with respect to Ψ 4 .  The 2 values as a function of centrality for different T selections are shown in Figure 6. The 2 values are consistent with zero in the most central collisions, and positive for all other centrality bins over the full T range. For the lower T ranges the 2 values are measured to be as large as 0.05 in mid-central collisions. The 2 shows a decreasing trend with T in mid-central collisions, with a 2 of approximately 0.01-0.02 for jets with T = 200-251 GeV. The value of 2 decreases for jets which have been shown in previous measurements to be less modified by the QGP, namely jets in peripheral collisions and high-T jets. Figure 7 shows the 2 values for 0-5%, 5-10%, and 20-40% centrality collisions as a function of jet T . The value of 2 decreases from the more peripheral 20-40% collisions to the more central collisions, where the path-length difference between in-plane and out-of-plane is the smallest. The dependence of the 2 on T in 5-10% and 20-40% collisions shows qualitatively similar behavior.

40-60%
20-40% 10-20% 5-10% 0-5%  The centrality dependence for the 2 , 3 , and 4 is shown in Figure 8 for the full T range of the measurement, 71-398 GeV. The 2 is nonzero for jets with T < 251 GeV in all but the most central collisions. The 3 is positive and on the order of 0.01 for central and mid-central collisions, and consistent with zero in the most peripheral collisions. The difference of the 3 from 0 is 2.7 for 20-40%, 3.1 for 10-20%, 3.3 for 5-10%, and 1.8 for 0-5% collisions, where is the quadrature sum of the statistical and systematic uncertainties. The value of 4 is compatible with zero. The measurements of 3 and 4 set a limit on the possible impact of initial-state fluctuations on parton energy loss.  The centrality dependence of the measured 3 and 4 values for several T ranges are shown in Figure 9. The 3 shows no significant T or centrality dependence, with larger statistical and systematic uncertainties than in the measurement in the inclusive T bin. The 4 measurement is consistent with zero as a function of both T and centrality, with larger statistical uncertainties due to the poorer event-plane resolution than that for the second-and third-order event-plane angles. 40  [10] and charged-particle jets from Ref.
[11] for the 10-20% and 20-40% centrality bins. The measurement shows good agreement with the previous results, with no significant evidence of a dependence of the values on the collision energy. The 2 and 3 of charged particles from Ref. [12] is also shown. The results show a qualitatively similar T dependence, with the charged-particle distribution shifted to lower T . This is consistent with the expectation that high-T charged particles are likely produced from jets at a higher T . This result improves on both the T reach and precision of previous measurements of high-T jet .  [10] for 20-30% centrality collisions (brown crosses) and 30-40% centrality collisions (cyan X markers) and from Ref.
[11] for 30-50% centrality collisions (green squares) and the 2 and 3 of charged particles in √ NN = 5.02 TeV Pb+Pb collisions from Ref. [12] for 20-30% centrality collisions (blue triangles) and 30-40% centrality collisions (purple diamonds). It is interesting to compare the actual jet yields in-plane versus out-of-plane to study the angular distribution of jets without imposing the cos( Δ ) shape modulation on the data. The ratio of the jet yields in the most in-plane bin, Δ < /8, to the most out-of-plane bin, Δ > 7 /8, is constructed: . These yields must be corrected for the finite event-plane resolution, which is done by assuming that the variation in Δ is dominated by the cos( Δ ) modulation such that 2 corr jet The ratio is further corrected for the effects of the finite bin width by assuming a cos( Δ ) modulation within each bin, and calculating the yields at Δ = 0 and Δ = , and taking the ratio of these values. A similar method was used in Ref.
[10]. This ratio, for = 2 and 3, is shown in Figure 13. A purely cos( Δ ) modulation would cause max to be 1 − 4 /(1 + 2 ) and these calculated values are compared with the max values. No deviation from the cos( Δ ) modulation is observed.

Conclusion
The azimuthal variation of jet quenching is measured in 2.2 nb −1 of Pb+Pb collisions at 5.02 TeV, using the event-plane method to extract the coefficients of jets. The data were collected with the ATLAS detector at the LHC. The 2 is found to be consistent with zero in the most central collisions, with values up to 0.05 in mid-central collisions, and decreasing with increasing T . A first measurement of 3 and 4 of jets is presented. The value of 3 is found to be significantly above zero in mid-central collisions, with a value of approximately 0.01 for central and mid-central collisions. The T -differential measurement of 3 shows no significant T or centrality dependence, while 4 is everywhere consistent with zero. The measured values are consistent with previous measurements of jet and high-T hadron , and improve on both the T reach and precision of these previous results. The positive 2 values in all but the most azimuthally symmetric collisions show the relationship between collision geometry and parton energy loss. Further, the measurements of 3 and 4 will help set limits on the impact of initial-state fluctuations on energy loss. These measurements can be used to constrain models of the path-length dependence of jet quenching.    [48] Star Collaboration, Elliptic flow from two-and four-particle correlations in Au+Au collisions at √ NN = 130 GeV, Phys. Rev. C 66 (3 2002) 034904.