Symmetry plane correlations in Pb–Pb collisions at √sNN = 2.76 TeV
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Symmetry plane correlations in Pb–Pb collisions at √sNN = 2.76 TeV © CERN for the benefit of the ALICE collaboration 2023 Published version ALICE Collaboration ALICE Collaboration. (2023). Symmetry plane correlations in Pb–Pb collisions at √sNN = 2.76 TeV. European Physical Journal C, 83, Article 576. https://doi.org/10.1140/epjc/s10052-02311658-w 2023
Eur. Phys. J. C (2023) 83:576 https://doi.org/10.1140/epjc/s10052-023-11658-w Regular Article - Experimental Physics Symmetry plane correlations in Pb–Pb collisions at √sNN =2.76 TeV ALICE Collaboration CERN, 1211 Geneva 23, Switzerland Received: 16 February 2023 / Accepted: 23 May 2023 © CERN for the benefit of the ALICE collaboration 2023 Abstract A newly developed observable for correlations between symmetry planes, which characterize the direction of the anisotropic emission of produced particles, is measured in Pb–Pb collisions at √sNN = 2.76 TeV with ALICE. This so-called Gaussian Estimator allows for the first time the study of these quantities without the influence of correlations between different flow amplitudes. The centrality dependence of various correlations between two, three and four symmetry planes is presented. The ordering of magnitude between these symmetry plane correlations is discussed and the results of the Gaussian Estimator are compared with measurements of previously used estimators. The results utilizing the new estimator lead to significantly smaller correlations than reported by studies using the Scalar Product method. Furthermore, the obtained symmetry plane correlations are compared to state-of-the-art hydrodynamic model calculations for the evolution of heavy-ion collisions. While the model predictions provide a qualitative description of the data, quantitative agreement is not always observed, particularly for correlators with significant non-linear response of the medium to initial state anisotropies of the collision system. As these results provide unique and independent information, their usage in future Bayesian analysis can further constrain our knowledge on the properties of the QCD matter produced in ultrarelativistic heavy-ion collisions. 1 Introduction One of the most important discoveries in the physics of heavy-ion collisions at ultrarelativistic energies is the observation of a deconfined state of nuclear matter dubbed quark– gluon plasma (QGP). This extreme state is produced during the heavy-ion collision evolution, and its properties resemble the properties of a perfect liquid. Unprecedentedly large data sets collected at the LHC enable the most quantitative description of the QGP to date. Given the complexity of the system produced in heavy-ion collisions, an important e-mail: [email protected] program in the field is the development of observables that provide new and independent information inaccessible with previous measurements [1–8]. The intersecting volume of two heavy ions is anisotropic in coordinate space, either due to collision geometry (particularly in non-central collisions with large values of impact parameter) or due to fluctuations of positions of participating nucleons (most significant in central head-on collisions). Anisotropic pressure gradients, which develop in this volume containing the strongly interacting nuclear matter, transfer the initial-state spatial anisotropies into finalstate anisotropies in momentum space. This phenomenon is known as anisotropic flow and it is a sensitive probe of all stages in the heavy-ion collision evolution [9]. Anisotropic flowmeasurements areusedtoconstrainthetransportproperties of the QGP, for instance ratios of shear and bulk viscositiestoentropydensity [4,7,10–13].Theanisotropicemission of particles in the plane transverse to the beam direction is quantified with amplitudes vnand symmetry planes nby usingtheFourier seriesdecomposition ofthe azimuthalangle (ϕ) distribution of produced particles [14] f(ϕ) =1 2π1+2∞ n=1 vncos[n(ϕ −n)].(1) A detailed discussion of properties of vnand ncan be found in Ref. [15]. The symmetry plane nhas a simple geometrical interpretation when the anisotropic distribution can be parameterized only with one harmonic n, since then it can be shown that f(n+ϕ) =f(n−ϕ), i.e. symmetry plane n is the plane for which it is equally probable for a particle to be emitted above or below it. Historically, the emphasis was on studying the amplitudes vn, but the symmetry planes also carry a very important information about different stages in heavy-ion collision evolution. Unlike the flow amplitudes vn, a single symmetry plane ncannot be estimated directly in an experiment using correlation techniques – the simplest available observables are symmetry plane correlations (SPC), for instance 0123456789().: V,-vol 123
576 Page 2 of 21 Eur. Phys. J. C (2023) 83:576 cos 4(4−2)[15,16]. Such correlations are the subject of this study. Inthe early anisotropicflowanalyses,the goal wastomeasure vnwith respect to the reaction plane (a plane spanned by the beam axis and impact parameter vector), and it was assumedthatallsymmetryplanesareapproximatelythesame and equal to the orientation of the reaction plane. Therefore, the first flow measurements were exclusively of flow amplitudes vn. The first experimental results for SPC can be traced back to the E877 experiment [17]. These initial measurements were performed by the standard event plane method with the subevent technique [18]. The first measurementsofSPCinvolvingtwosymmetryplanesintheRHICera were obtained by PHENIX in Refs. [19,20]. An alternative approach was pursued by NA49 and STAR using 3-particle mixed-harmonic correlations, which by definition have contributions from SPC [21,22]. In the first flow studies at LHC energies, the ALICE Collaborationdemonstratedin Ref. [23] that the symmetry planes 2and 3fluctuate independently inall consideredcentralities.Finally, the mostdetailed experimental results to date were published by the ATLAS Collaboration in Ref. [24], where also for the first time the strength of correlations among three symmetry planes was presented. ATLAS systematically studied the centrality dependence of SPC both in the initial and final state using the analysis techniquefromRefs.[16,25].ItwasconcludedthatSPCoriginate both from correlated fluctuations in the initial geometry and from the non-linear mixing between different flow harmonics in the final state. Subsequent experimental publications which used SPC to constrain the details of the non-linear hydrodynamic response can be found in Refs. [26–30]. In theoretical studies, SPC can be obtained directly both in coordinate and in momentum space [16,24,25,31–38]. State-of-the-art modeling of heavy-ion collisions covers all stages of its evolution starting from the initial conditions to the final free streaming of produced particles. The SPC in the initial state can be obtained event-by-event directly from the underlying model of the collision geometry using for instance energy density distribution or nucleon positions, while in the final state SPC are the event-by-event output of the model used to describe all subsequent stages in the evolution. Therefore, in theoretical studies it is not, in general, necessary to build an estimator for SPC from the azimuthal angles of final-state particles, like it is done in an experiment. In order to ease the comparison between theoretical and experimental results, azimuthal correlators were used to indirectly estimate SPC also in Refs. [10,39–44]. Other types of theoretical studies involving symmetry planes can be found in Refs. [45–50]. Several experimental difficulties associated with the SPC render their measurements particularly challenging. Even in thesimplestrealisation,itisnecessarytoconstructnon-trivial estimators for SPC to resolve these issues. Unlike the flow amplitudes vn, each symmetry plane ntaken individually is not invariant under rotations of the coordinate system in the laboratory frame in which azimuthal angles are measured (see Eq. 1). Therefore, the simplest rotationally-invariant physical observable involving symmetry planes is the difference of two symmetry planes. In an actual experiment such rotations are unavoidable as a direct consequence of random event-by-event fluctuations of the direction of the impact parameter vector. Only symmetry planes that are different, apart for trivial periodicity, carry independent information, and therefore any dependence on periodicity must be removed from all SPC observables by definition. The widely used technique to suppress systematic biases from shortrange nonflow correlations by introducing pseudorapidity gaps in the measured azimuthal correlators which are used to estimate SPC is not applicable due to decorrelations of symmetry planes as a function of pseudorapidity [50–56]. Moreover, it has been shown recently that the effect of flow magnitude correlations, which have been either completely [24]or partially [27] neglected in the existing measurements, may overshadow the correlations of symmetry planes in the analysis with the Scalar Product (SP) method [15]. The new and improvedestimatorforSPCfromRef.[15],whichovercomes these limitations, is introduced next. The starting point is the following relation between vnand n, and multiparticle azimuthal correlations [15,39,57]: va1 n1···vak nkei(a1n1n1+···+aknknk)=ei(n1ϕ1+···+nlϕl).(2) Inthis equation, angularbracketsindicatean average overthe azimuthal angles of all distinct sets of lparticles measured in the same event. The coefficients aiare positive integers which ensure that all harmonics niand symmetry planes niare unique on the left-hand side in the above expression. These coefficients can be understood in the following way: aicounts how many timesaharmonicniappearsintheazimuthal correlatoronthe right-hand side of Eq. (2) (harmonics with positive and negative signs are counted separately). The total number of particles, i.e. the order of the multiparticle azimuthal correlator, is given by iai. The index kon the left-hand side labels only unique harmonics in the original set n1,n2,...,nl, therefore k≤l. As an example, for the correlator ei(2ϕ1−ϕ2−ϕ3) it follows that n1=2,a1=1,n2=n3=−1,a2=2. The advantage of this generalized notation is that now niand ai decouple naturally either into a subscript or into an exponent when associated with flow amplitudes vniin Eq. (2), which enables their distinct physical interpretation. Finally, solely from the definition of the Fourier series in Eq. (1) one can prove that v−n=vnand −n=n, which is used in the rest of the paper. Due to this property, the final acoefficient for harmonic nin Eq. (2)isasuman+a−n. 123
Eur. Phys. J. C (2023) 83:576 Page 3 of 21 576 Taking into account all these technical considerations, the simplest definition of SPC observables is provided by the following expression [24,25,39]: ei(a1n1n1+···+aknknk), k i aini=0,(3) where all aiare positive and all niare unique integers. Angular brackets indicate here an average over all events. Defined this way, SPC observables are rotationally invariant and therefore invariant with respect to random event-byevent fluctuations of the impact parameter vector, while the periodicity of each individual symmetry plane is accounted for by definition. Experimentally, Eq. (2)isusedasastarting point for an estimator for SPC. However, to isolate the true SPC part, the prefactor va1 n1···vak nkhas to be divided out. The importance of this technical detail was neglected in all previously used SPC estimators. The new and improved SPC estimator, named the Gaussian Estimator (GE), was developed recently in Ref. [15]. Its key improvement amounts to using the following expression to estimate SPC: cos a1n1n1+···+aknknkGE =π 4va1 n1··· vak nkcos a1n1n1+···+aknknk v2a1 n1··· v2ak nk , (4) which was derived by approximating multi-harmonic flow fluctuations with a two-dimensional Gaussian distribution. Both the numerator and denominator on the right-hand side in the above expression can be estimated by using Eq. (2) with suitably chosen harmonics ni. Further explanations of the technical details of the GE based on the example cos [4(4−2)]are provided in Appendix A. The main improvement of this new estimator can be found in the denominator where the GE has the joined multivariate moment of different flow amplitudes, v2a1 n1··· v2ak nk.This is in contrast to the previously used SP estimator, defined as [43] cos a1n1n1+···+aknknkSP =va1 n1··· vak nkcos a1n1n1+···+aknknk v2a1 n1···v2ak nk ,(5) which uses instead v2a1 n1 ··· v2ak nkin the denominator and therefore assumes that event-by-event fluctuations of flow amplitudes are mutually independent. This assumption is in contradiction with recent experimental results which reported strong and non-trivial correlated fluctuations of different flow amplitudes, both at RHIC and LHC energies, and across different collisions systems [26,58–61]. These shortcomings of the previous SPC results are the main motivation forthecurrentwork.AsitwaspointedoutinRef.[15],theGE does not account for cross-correlations between flow amplitudes and symmetry planes. However, the study in Ref. [15] showed that the contribution by these cross-correlations is minor when compared to the correlations between flow amplitudes. The rest of the article is organized as follows. In Sect. 2 the ALICE detector is introduced, together with the analyzed datasetand analysis details, such as theevent and trackselection criteria. In Sect. 3the SPC results using the GE are presented, comparisons with previous experimental results are discussed, and confrontation with state-of-the-art theoretical models is displayed. The article concludes in Sect. 4with the summary. A more detailed discussion about the technical details of the GE can be found Appendix A. 2 Data analysis The data set consists of Pb–Pb collisions at a center-ofmass energy per nucleon pair √sNN =2.76 TeV recorded by ALICE in 2010. A detailed description of the apparatus and its performance is given in Refs. [62,63]. The silicon pixel detector (SPD), which comprises the two innermost layers of the inner tracking system (ITS) [64,65], and both V0 detectors [66] were used for triggering. The latter consists of two arrays of scintillator counters, the V0A and V0C, covering a pseudorapidity range of 2.8<η<5.1 and −3.7<η<−1.7, respectively. The SPD covers pseudorapidities of |η|<2.0 for its inner and |η|<1.4 for its outer layer. Minimum bias collisions were selected by requiring a signal in at least two out of the three following: two chips in the outer layer of the SPD, the V0A, and the V0C. For this analysis, only events with a primary vertex within ±10 cm of the nominal interaction point along the beam axis were used. The centrality of the collisions [67] was estimated with the SPD. Backgrounds events due to beam–gas interactions and parasitic beam–beam interactions were removed by using V0 and Zero Degree Calorimeter [68] timing information. Overall, after the event selection the used data set consists of 7.36 ×106reconstructed collisions for the centrality range 0–50%. The reconstruction of charged particle trajectories was performed using only information from the time projection chamber (TPC) [69] due to its uniform acceptance in azimuth. This analysis used tracks with transverse momenta 0.2<pT<5.0GeV/cand in a pseudorapidity range of |η|<0.8, while covering the full azimuth. The lower boundaryof thetransverse momentumselectionensured alargeand stable tracking efficiency in the TPC, while the upper cutoff decreases the contribution from jets which in general have 123
576 Page 4 of 21 Eur. Phys. J. C (2023) 83:576 larger momenta. The charged tracks were accepted for the analysis if they had a minimum of 70 out of a maximum of 159 space points in the TPC. The χ2per space point from the track fit was set to be within 0.1<χ 2/NDF <4.0. The distance of closest approach (DCA) of the extrapolated tracks to the primary vertex was required to be at maximum 2.4cm in the transverse plane and 3.2cm in the beam direction. Daughter tracks with a reconstructed secondary weak-decay kink topology (i.e. tracks with an abrupt change of direction) were discarded. The contamination from secondaries as well as the reconstruction efficiency with this track selection can be found in Ref. [70]. The pT-dependentreconstructionefficiencywascorrected using particle weights according to Ref. [57]. These weights were obtained with the HIJING (Heavy-Ion Jet INteraction Generator) Monte Carlo generator [71] by comparison of generated and reconstructed tracks. For the latter, a GEANT3 [72] detector simulation and event reconstruction was used in addition to HIJING. At the same time, weights to correct for non-uniform acceptance in azimuthal angle did not have to be applied due to the uniform acceptance of the TPC over the whole azimuth in the analyzed data set. Nonflow contributions, i.e. correlations between a few particles unrelated to collective anisotropic flow, were investigated with HIJING for the numerator and denominator of the GE in Eq. (4) separately. For all SPC combinations, both the numerator and denominator were found to be consistent with zero in all considered centrality ranges, demonstrating that the analyzed SPC observables are not influenced by most important sources of nonflow correlations such as jets or resonance decays. The statistical uncertainties of the measured SPC were obtained via propagation of uncertainties of the numerator and denominator in Eq. (4). Systematic uncertainties were evaluated by varying the default event and track selections. All variations were performed one at a time and only those with a difference larger than 2σ, where σis the uncertainty of thedifference,withrespecttothedefaultselectionweretaken into account for the final systematic uncertainty. All individual systematic variations were considered independent and combined in quadrature to obtain the total systematic uncertainty. Regarding the event selection criteria, the position of the primary vertex along the beam line was varied to ±6cm and ±8 cm, where a relative effect on the measured observables of up to 5% was found. A systematic uncertainty of up to6% from the centralityestimationwas determinedby using the V0 instead of the SPD. To evaluate the uncertainty due to the track selection, the number of TPC clusters used in the track reconstruction was varied to a required minimum of 80, 90 and 100 compared to the default 70. This resulted in a systematic uncertainty of up to 6%. The sensitivity of the results to the track quality was checked by varying the χ2/NDF to 0.3<χ 2/NDF <4.0 and 0.1<χ 2/NDF <3.5, which led to an additional uncertainty of up to 6%. Two variations were performed regarding the DCA by changing the upper limit in the transverse direction to 1cm and in the longitudinal direction to 2cm. The variation of the DCA changes the contributionfromsecondariesintheanalysisastheseparticlesusually have a larger DCA than primary particles. The DCA variation in the transverse plane led to a systematic uncertainty of about 3–10%, while the check along the beam axis had a relative variation of about 4%. Additionally, an independent analysis was performed by using a different track reconstruction procedure, which employs combined information from both the TPC and the ITS. This led to an uncertainty in the range of 5–10%. 3 Results The centrality dependence of the correlations between different combinations of two and three symmetry planes, as well as the first measurement of a correlation between four planes, are presented in Fig. 1. In the case of two symmetry planes shown in Fig. 1a, the strongest correlation is observed for cos [4(4−2)]GE, while the correlation strength gets weaker for cos [6(6−2)]GE and cos [6(6−3)]GE.Theresultsforcos [6(2−3)]GE are compatible with zero within uncertainties. A hierarchy, cos [4(4−2)]GE >cos [6(6−3)]GE > cos [6(6−2)]GE, holds for the centrality range 5–50%, with an exception of cos [6(6−3)]GE and cos [6(6−2)]GE being comparable at centralities above 20%. The details of the centrality dependence vary for the different combinations of symmetry planes. While cos [4(4−2)]GE and cos [6(6−2)]GE are increasing non-linearly from central to semicentral collisions, cos [6(6−3)]GE shows a weak centrality dependence. The observed zero signal for cos [6(2−3)]GE indicates that no correlation is present within the current uncertainties for the final-state planes 2and 3,whilev2 and v3are anti-correlated [26,44,58,59,73–75]. This result justifies the necessity of measuring separately correlations of symmetry planes and flow magnitudes, because these measurements can be used to independently constrain properties of the matter produced in heavy-ion collisions. The different magnitudes of correlations are also observed forthreesymmetryplanesasshowninFig.1b.Themagnitude and details of the centrality dependence vary for different combinationsof flow harmonics. Thecos [6(2−3)]GE Fiveobservable exhibits the strongest correlations and cos [22−63+44]GE shows the weaker signal. The SPC cos [22−63+44]GE is the only correlator with a negative sign, which will be discussed later on in more detail. The cos [82−33−55]GE observable is consistent with zero within uncertainties, similar to 123
Eur. Phys. J. C (2023) 83:576 Page 5 of 21 576 (a) cos[4(Ψ4−Ψ2)]GE cos[6(Ψ2−Ψ3)]GE cos[6(Ψ6−Ψ2)]GE cos[6(Ψ6−Ψ3)]GE (b) cos[2Ψ2−6Ψ3+4Ψ 4]GE cos[2Ψ2+4Ψ 4−6Ψ6]GE cos[2Ψ2−3Ψ3−4Ψ4+5Ψ 5]GE cos[2Ψ2+3Ψ 3−5Ψ5]GE cos[8Ψ2−3Ψ3−5Ψ5]GE ALICE Pb–Pb √sNN = 2.76 TeV 0.2<p T<5.0 GeV/c |η|<0.8 Fig. 1 Comparison of the extracted correlations between different combinations of two symmetry planes (a) and between three and four planes (b) using the GE in Eq. (4). Statistical (systematic) uncertainties are shown as lines (boxes) cos [6(2−3)]GE. The correlation between four planes, cos [22−33−44+55]GE showsthestrongestcentrality dependence among all harmonic combinations and increases towards peripheral collisions. The magnitudes of SPC are ordered approximately based on the corresponding order of the particle correlations. The two largest SPCs, cos [4(4−2)]GE and cos [22+33−55]GE, are both measured with threeparticle correlators. In contrast, the smallest ones are cos [6(2−3)]GE and cos [82−33−55]GE, which are fiveand six-particles correlations, respectively. One possible explanation is the following: the flow vector fluctuations encoded in the observed correlations are mainly attributed to the fluctuation of the initial state. Also, the initial state fluctuation is attributed to the fluctuation of a finite amount of “sources” produced at the degrees of freedom collision points, namely protons and neutrons, in the collision region. The central limit theorem (CLT) states that for independent random variables (here, the position of sources), the sample average tends toward a Gaussian distribution when the number of sampling increases. A clear example of such behavior was studied for initial ellipticity in Ref. [76], where it was shown how the ellipticity fluctuation distribution changes from elliptic-power distribution with large skewness to a Gaussian distribution at a large number of sources. The order of particle correlations corresponds to the order of the cumulants of the underlying flow vector fluctuation. To see the clear connection, correlations should be written in a Cartesian notation rather than polar notation (see Refs. [77–79] for the relation between skewness and Kurtosis of flow vector distribution to the particle correlations). Only the second-order cumulant, namely the width of the distribution, is nonvanishing for a Gaussian distribution. As a result, higher-order cumulants (skewness, kurtosis, etc.) are small for distributions close to Gaussian. These studies are done for flow amplitudes with only one harmonic, but the logic is true for more than one harmonic as well. The observed ordering of magnitudes in Fig. 1indicates that the contribution of higher-order cumulants is smaller compared to lower ones in general, meaning the lowest-order cumulants have the dominant role in deviation from Gaussianity. A crossing between cos [6(6−3)]GE (a three-particle correlation) and cos [6(6−2)]GE (a four-particle correlation) is observed with centralities above 25% where the number of final state particles is lower. The same is true for cos [22+44−66]GE (a three-particle correlation) and cos [22−33−44+55]GE (a four-particle correlation). The effect of non-Gaussianity is expected to be more dominant in this centrality region since the system size is smaller and less number of sources are expected. At a finite number of sources, the actual ordering of the correlation magnitudes depends on the details of the underlying source fluctuation that needs a separate study. In Figs. 2and 3the experimental data for SPC estimated with the GE are compared with the results obtained from ATLAS [24] and ALICE [27] using the SP method. While the analysisoftheSPmethodbyALICEusedthesamekinematic range as the work presented in this article, the analysis by ATLAS was performed in a wider range of 0.5GeV/c<pT and |η|<2.5. Despite this difference in kinematic regions, theSPCextractedbytheSPmethodfromALICEandATLAS agree within uncertainties. In general, the obtained data from the GE are significantly smaller than the estimates performed with the SP method for centralities larger than 10%. This difference is mainly attributed to the fact that correlations between flow amplitudes were not removed in the SP method as it was demonstrated in Ref. [15]. For the SPC cos [4(4−2)]andcos [6(6−2)]showninFig.2, 123
576 Page 6 of 21 Eur. Phys. J. C (2023) 83:576 (a) cos[4(Ψ4−Ψ2)] ALICE GE, 0.2<p T<5.0 GeV/c,|η|<0.8 (b) cos[6(Ψ6−Ψ3)] ALICE (PLB 773 68 (2017)) SP, 0.2<p T<5.0GeV/c,|η|<0.8 ATLAS (PRC 90 024905 (2014)) SP, 0.5GeV/c < pT,|η|<2.5 (c) cos[6(Ψ2−Ψ3)](d) cos[6(Ψ6−Ψ2)] Pb–Pb √sNN = 2.76 TeV Fig. 2 Experimental data of correlations between two symmetry planes obtained with the GE compared with measurements from ATLAS [24] and ALICE [27] using the SP method. Statistical and systematic uncertainties are represented by lines and boxes, respectively (a) cos[2Ψ2+3Ψ 3−5Ψ5](b) cos[2Ψ2+4Ψ 4−6Ψ6] (c) cos[8Ψ2−3Ψ3−5Ψ5] ALICE GE, 0.2<p T<5.0 GeV/c,|η|<0.8 ALICE (PLB 773 68 (2017)) SP, 0.2<p T<5.0 GeV/c,|η|<0.8 ATLAS (PRC 90 024905 (2014)) SP, 0.5 GeV/c < pT,|η|<2.5 (d) cos[2Ψ2−6Ψ3+4Ψ 4] Pb–Pb √sNN = 2.76 TeV Fig. 3 Correlations between three symmetry planes obtained with the GE compared with measurements from ATLAS [24]andALICE[27]using the SP method. Statistical and systematic uncertainties are shown as lines and boxes, respectively 123
Eur. Phys. J. C (2023) 83:576 Page 7 of 21 576 the GE and SP method are compatible only in 0–5% centrality, while for cos [6(6−3)]the GE differs in all centrality intervals when compared to the SP method by ATLAS. For centralities larger than 5%, a clear splitting between all of the previously mentioned SPC is visible with significantly smaller values obtained by the GE. For cos [6(2−3)] the experimental data of the GE are compatible with zero within the uncertainties in all considered centrality intervals. In contrast to that, the results of the SP method show a small, but non-zero value. However, the results obtained with the GE show larger uncertainties when compared to the SP for this particular SPC. Future studies with larger data sets will show whether the SPC cos [6(2−3)]remains compatible with zero within uncertainties when using the GE or if a small non-zero correlation exists which cannot be resolved within the present uncertainties. In the latter case, the results ofthe GEwill nonethelessleadto significantlysmaller values than reported by the SP method. Similarly, the experimental results of the GE and the SP method are compared to each other for SPC between three planes. The results are presented in Fig. 3. For the combinations cos [22+33−55],cos [22+44−66] and cos [22−63+44]the GE again leads to significantly smaller values than the SP method for centralities larger than 10%. For cos [22+33−55]it has to be noted that the observables previously employed by ALICE [27] and ATLAS [24] differ in the denominator. ATLAS uses a fully factorized denominator v2 2v2 3v2 5as in the definition of the SP method (5), while the denominator in the ALICE measurement is only partially factorized v2 2v2 3v2 5and thus is not defined exactly as in Eq. (5). To ease the notation in Fig. 3we still label cos [22+33−55]measured by ALICE [27]asSP method.TheSPC cos [82−33−55]is the onlycombination where the estimates by the GE and the SP method are compatible with each other within uncertainties in all considered centralities, as the results from the SP method are already close to zero. The difference in physical interpretation between the two SPC involving 2,3and 5is discussed later. The new measurements of SPC with the GE are compared withMonteCarlosimulationswiththeTRENTo+VISH(2+1) +UrQMD event generator [80–84]. In this article, the maximuma posteriori (MAP)estimationobtained intheBayesian analysis in Ref. [7] is used for the parameters of the model. In inferring the MAP parameterization, a series of ALICE measurements(two-and four-particle correlations, chargedparticle multiplicities, etc.) were used as inputs into the Bayesian analysis, while the SPC are not included in these studies. Including new observables (e.g. SPC) in the Bayesian analysis can lead to an improvement in the uncertainty of the inferred parameter and resolving the discrepancies [12,13]. If the discrepancy between model and data persists even after including new observables as input, the model itself needs to be revised. In addition to the model predictions of final-state SPC, initial-state participant plane correlations are studied with TRENTo. The participant plane of order ntakes the same role in the initial state as the symmetry plane in the final state. The correlations between participant planes are extracted from the initial state where flow vectors vneinnare replaced first by eccentricities [85], and second by cumulants of the initial energy density [38,86,87]. The eccentricities are defined as neinφn=−{rneinϕ} {rn},n>1,(6) where {···}=rdrdϕε(r,ϕ) stands for the average with respect to the initial energy density ε(r,ϕ) in the transverse direction and (r,ϕ) are the polar coordinates in the transverse plane. Eccentricities are the moments of the initial energy density distribution. The cumulants of the initial energy density distribution, cneinn, are obtained as a combination of eccentricities and the radial moments of the energy density, {rn}. In fact, cumulants are a better measure to study the deformation of a distribution close to a Gaussian. Borrowing a motivating example from Ref. [86], a Gaussian distribution e−x2/2σ2 x−y2/2σ2 yhas infinitely many non-vanishing moments, while only its second order cumulants are non-zero. Following the convention of Ref. [86], the first two cumulants and eccentricities are equivalent, cneinn=neinφnfor n=2,3. Higher order cumulants have non-trivial relations to eccentricities. Here, only the fourth harmonic is shown as an example: c4ei44=4ei4φ4+3{r2}2 {r4}2 2ei4φ2.(7) More details can be found in Refs. [38,86,87]. The comparison with initial and final state SPC demonstrates how much of the observed correlation is inherited from the initial state. This is due to the linear and non-linear response of the medium [88]. For the secondand third-order anisotropies, the linear response is expected to dominate i.e. v2ei22=ω2c2ei22and v3ei33=ω3c3ei33, especially in central and semicentral collisions [73,85,89]. The ωidescribe the linear hydrodynamic coupling constants. For higher orders, non-linear contributions will play a significant role, e.g. in case of the fourth order as v4ei44=ω4c4ei44+ω422c2 2ei42,(8) where ω422 is the non-linear coupling between the secondand fourth-order anisotropies [38,86,87]. As an example of how this impacts the SPC, one can build the quantity v2 2v4ei4(4−2). The real part of its phase corresponds to 123
576 Page 8 of 21 Eur. Phys. J. C (2023) 83:576 (a) cos[4(Ψ4−Ψ2)]GE (b) cos[6(Ψ6−Ψ3)]GE (c) cos[6(Ψ2−Ψ3)]GE ALICE TRENTo (cumulants) TRENTo (eccentricities) TRENTo + VISH2+1 + UrQMD (d) cos[6(Ψ6−Ψ2)]GE Pb–Pb √sNN = 2.76 TeV 0.2<p T<5.0 GeV/c |η|<0.8 Fig. 4 Experimental data for correlations between two symmetry planes compared to theoretical predictions in the initial and final state obtained with TRENToand TRENTo+VISH(2+1)+UrQMD [80– 84], respectively. For cos [6(2−3)]GE (c), the initial state predictions calculated via eccentricities and energy density cumulants, cos[6(φ2−φ3)]GE and cos[6(2−3)]GE, fully overlap. Statistical (systematic) uncertainties of the ALICE data are shown as lines (boxes). The statistical uncertainties of the models are represented by the colored bands the SPC cos [4(4−2)]. Using the linear and non-linear response, one can translate this into the initial state as: v2 2v4ei4(4−2)=ω2ω4c2 2c4ei4(4−2)+ω422ω2 2c2 2.(9) The latter equation shows that the initial and final state SPC cos [4(4−2)]will be equal to each other in the limit of pure linear response, while they will deviate in case of a non-zero non-linear coupling ω422. Figure 4shows the comparison between the model calculations and the experimental data for the correlation between two symmetry planes. For the SPC cos [4(4−2)]GE the initial-state participant plane correlations given by the energy density cumulants overlap with the final-state prediction of the SPC up to 10% in centrality, indicating a vanishing non-linear coupling in the regime dominated by flow fluctuations. For higher centralities, the two curves increasingly deviate, showing the presence of a non-zero non-linear coupling between the secondand fourth-order flow vectors. In particular, it can be observed that the final-state prediction increasingly deviates from the data with increasing centrality. This is expected, since in this regime strong correlations between the second and fourth harmonics can originate fromthe initialellipsoidal geometry. Thenon-linear coupling constant between initial-state ellipticity and v4ei44from TRENTo to iEBE-VISHNU was studied in Ref. [90]. It was demonstrated that this coupling is very small up to 10% and it is positive up to 70% centrality. For cos [6(2−3)]GE the experimental data show a flat centrality behavior and are compatible with zero within the uncertainties. The model predicts small values for cos [6(2−3)]GE with a flat centrality behavior although the predictions for the final state slightly overestimate the data. However, the initial-state correlations decrease to more negative values for centralities above 30%. For cos [6(2−3)]GE, a linear response is expected to be an accurate approximation for harmonics n=2,3 in central and semicentral collisions [73,85,89]. As such, one would expect that the initial state participant plane correlations should be the same as the final state symmetry plane correlations between the secondand third-order harmonics. One possible explanation is that higher-order terms beyond linear response are responsible for decreasing the correlation during the hydrodynamic evolution, and the final value accidentally lands on very small numbers. In this respect, more rigorous study is needed in the future. While the model captures the qualitative behavior of the 123
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576 Page 16 of 21 Eur. Phys. J. C (2023) 83:576 ALICE Collaboration S. Acharya125 , D. Adamová86 , A. Adler69, G. Aglieri Rinella32 , M. Agnello29 , N. Agrawal50 , Z. Ahammed132 , S. Ahmad15 ,S.U.Ahn 70 , I. Ahuja37 , A. Akindinov140 ,M.Al-Turany 97 , D. Aleksandrov140 , B. Alessandro55 , H. M. Alfanda6, R. Alfaro Molina66 ,B.Ali 15 , A. Alici25 , N. Alizadehvandchali114 , A. Alkin32 ,J.Alme20 ,G. Alocco51 ,T.Alt63 ,I. Altsybeev140 ,M. N. Anaam6,C. Andrei45 ,A. Andronic135 , V. Anguelov94 , F. Antinori53 , P. Antonioli50 , N. Apadula74 , L. Aphecetche103 , H. Appelshäuser63 , C. Arata73 , S. Arcelli25 ,M.Aresti 51 , R. Arnaldi55 , J.G.M.C.A.Arneiro 110 , I.C.Arsene 19 , M. Arslandok137 , A. Augustinus32 , R. Averbeck97 ,M.D.Azmi 15 , A. Badalà52 ,J.Bae 104 , Y.W.Baek 40 , X. Bai118 , R. Bailhache63 , Y. Bailung47 , A. Balbino29 , A. Baldisseri128 , B. Balis2, D. Banerjee4, Z. Banoo91 , R. Barbera26 , F. Barile31 , L. Barioglio95 ,M.Barlou 78, G. G. Barnaföldi136 , L. S. Barnby85 , V. Barret125 ,L.Barreto 110 , C. Bartels117 ,K.Barth 32 , E. Bartsch63 , N. Bastid125 ,S.Basu 75 , G. Batigne103 , D. Battistini95 , B. Batyunya141 , D. Bauri46, J. L. Bazo Alba101 , I. G. Bearden83 , C. Beattie137 , P. Becht97 , D. Behera47 , I. Belikov127 , A. D. C. Bell Hechavarria135 , F. Bellini25 ,R.Bellwied 114 , S. Belokurova140 , V. Belyaev140 , G. Bencedi136 , S. Beole24 , A. Bercuci45 , Y. Berdnikov140 , A. Berdnikova94 , L. Bergmann94 , M. G. Besoiu62 ,L.Betev 32 , P. P. Bhaduri132 , A. Bhasin91 , M. A. Bhat4, B. Bhattacharjee41 , L. Bianchi24 , N. Bianchi48 ,J.Bielˇcík35 ,J.Bielˇcíková86 , J. Biernat107 , A.P.Bigot 127 , A. Bilandzic95 ,G.Biro 136 , S. Biswas4,N.Bize 103 ,J.T.Blair 108 ,D.Blau 140 , M. B. Blidaru97 , N. Bluhme38,C.Blume 63 , G. Boca21,54 , F. Bock87 , T. Bodova20 , A. Bogdanov140,S.Boi 22 ,J.Bok 57 , L. Boldizsár136 , M. Bombara37 , P. M. Bond32 , G. Bonomi54,131 , H. Borel128 , A. Borissov140 , A. G. Borquez Carcamo94 , H. Bossi137 , E. Botta24 , Y. E. M. 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Eur. Phys. J. C (2023) 83:576 Page 19 of 21 576 4Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India 5California Polytechnic State University, San Luis Obispo, CA, USA 6Central China Normal University, Wuhan, China 7Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba 8Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico 9Chicago State University, Chicago, IL, USA 10 China Institute of Atomic Energy, Beijing, China 11 Chungbuk National University, Cheongju, Republic of Korea 12 Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava, Bratislava, Slovak Republic 13 COMSATS University Islamabad, Islamabad, Pakistan 14 Creighton University, Omaha, NE, USA 15 Department of Physics, Aligarh Muslim University, Aligarh, India 16 Department of Physics, Pusan National University, Pusan, Republic of Korea 17 Department of Physics, Sejong University, Seoul, Republic of Korea 18 Department of Physics, University of California, Berkeley, CA, USA 19 Department of Physics, University of Oslo, Oslo, Norway 20 Department of Physics and Technology, University of Bergen, Bergen, Norway 21 Dipartimento di Fisica, Università di Pavia, Pavia, Italy 22 Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy 23 Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy 24 Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy 25 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN, Bologna, Italy 26 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN, Catania, Italy 27 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN, Padua, Italy 28 Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo Collegato INFN, Salerno, Italy 29 Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy 30 Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy 31 Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy 32 European Organization for Nuclear Research (CERN), Geneva, Switzerland 33 Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia 34 Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway 35 Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic 36 Faculty of Physics, Sofia University, Sofia, Bulgaria 37 Faculty of Science, P.J. Šafárik University, Kosice, Slovak Republic 38 Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany 39 Fudan University, Shanghai, China 40 Gangneung-Wonju National University, Gangneung, Republic of Korea 41 Department of Physics, Gauhati University, Guwahati, India 42 Helmholtz-Institut für Strahlenund Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany 43 Helsinki Institute of Physics (HIP), Helsinki, Finland 44 High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico 45 Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania 46 Indian Institute of Technology Bombay (IIT), Mumbai, India 47 Indian Institute of Technology Indore, Indore, India 48 INFN, Laboratori Nazionali di Frascati, Frascati, Italy 49 INFN, Sezione di Bari, Bari, Italy 50 INFN, Sezione di Bologna, Bologna, Italy 51 INFN, Sezione di Cagliari, Cagliari, Italy 52 INFN, Sezione di Catania, Catania, Italy 53 INFN, Sezione di Padova, Padua, Italy 54 INFN, Sezione di Pavia, Pavia, Italy 55 INFN, Sezione di Torino, Turin, Italy 56 INFN, Sezione di Trieste, Trieste, Italy 123
576 Page 20 of 21 Eur. Phys. J. C (2023) 83:576 57 Inha University, Incheon, Republic of Korea 58 Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, The Netherlands 59 Institute of Experimental Physics, Slovak Academy of Sciences, Kosice, Slovak Republic 60 Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India 61 Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic 62 Institute of Space Science (ISS), Bucharest, Romania 63 Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany 64 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico 65 Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil 66 Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico 67 iThemba LABS, National Research Foundation, Somerset West, South Africa 68 Jeonbuk National University, Jeonju, Republic of Korea 69 Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany 70 Korea Institute of Science and Technology Information, Daejeon, Republic of Korea 71 KTO Karatay University, Konya, Turkey 72 Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France 73 Laboratoire de Physique Subatomique et de Cosmologie, CNRS-IN2P3, Université Grenoble-Alpes, Grenoble, France 74 Lawrence Berkeley National Laboratory, Berkeley, CA, USA 75 Division of Particle Physics, Department of Physics, Lund University, Lund, Sweden 76 Nagasaki Institute of Applied Science, Nagasaki, Japan 77 Nara Women’s University (NWU), Nara, Japan 78 Department of Physics, School of Science, National and Kapodistrian University of Athens, Athens, Greece 79 National Centre for Nuclear Research, Warsaw, Poland 80 National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India 81 National Nuclear Research Center, Baku, Azerbaijan 82 National Research and Innovation Agency-BRIN, Jakarta, Indonesia 83 Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark 84 Nikhef, National institute for subatomic physics, Amsterdam, The Netherlands 85 Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, UK 86 Nuclear Physics Institute of the Czech Academy of Sciences, Husinecˇ Rež, Czech Republic 87 Oak Ridge National Laboratory, Oak Ridge, TN, USA 88 Ohio State University, Columbus, OH, USA 89 Physics Department, Faculty of science, University of Zagreb, Zagreb, Croatia 90 Physics Department, Panjab University, Chandigarh, India 91 Physics Department, University of Jammu, Jammu, India 92 Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2), Hiroshima University, Hiroshima, Japan 93 Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany 94 Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 95 Physik Department, Technische Universität München, Munich, Germany 96 Politecnico di Bari and Sezione INFN, Bari, Italy 97 Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany 98 Saga University, Saga, Japan 99 Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India 100 School of Physics and Astronomy, University of Birmingham, Birmingham, UK 101 Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru 102 Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria 103 SUBATECH, IMT Atlantique, CNRS-IN2P3, Nantes Université, Nantes, France 104 Sungkyunkwan University, Suwon City, Republic of Korea 105 Suranaree University of Technology, Nakhon Ratchasima, Thailand 106 Technical University of Košice, Kosice, Slovak Republic 123
Eur. Phys. J. C (2023) 83:576 Page 21 of 21 576 107 The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland 108 The University of Texas at Austin, Austin, TX, USA 109 Universidad Autónoma de Sinaloa, Culiacán, Mexico 110 Universidade de São Paulo (USP), São Paulo, Brazil 111 Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil 112 Universidade Federal do ABC, Santo Andre, Brazil 113 University of Cape Town, Cape Town, South Africa 114 University of Houston, Houston, TX, USA 115 University of Jyväskylä, Jyvaskyla, Finland 116 University of Kansas, Lawrence, KS, USA 117 University of Liverpool, Liverpool, UK 118 University of Science and Technology of China, Hefei, China 119 University of South-Eastern Norway, Kongsberg, Norway 120 University of Tennessee, Knoxville, TN, USA 121 University of the Witwatersrand, Johannesburg, South Africa 122 University of Tokyo, Tokyo, Japan 123 University of Tsukuba, Tsukuba, Japan 124 University Politehnica of Bucharest, Bucharest, Romania 125 CNRS/IN2P3, LPC, Université Clermont Auvergne, Clermont-Ferrand, France 126 CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, Université de Lyon, Lyon, France 127 CNRS, IPHC UMR 7178, Université de Strasbourg, 67000 Strasbourg, France 128 Départment de Physique Nucléaire (DPhN), IRFU, Université Paris-Saclay Centre d’Etudes de Saclay (CEA), Saclay, France 129 Università degli Studi di Foggia, Foggia, Italy 130 Università del Piemonte Orientale, Vercelli, Italy 131 Università di Brescia, Brescia, Italy 132 Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India 133 Warsaw University of Technology, Warsaw, Poland 134 Wayne State University, Detroit, MI, USA 135 Institut für Kernphysik, Westfälische Wilhelms-Universität Münster, Münster, Germany 136 Wigner Research Centre for Physics, Budapest, Hungary 137 Yale University, New Haven, CT, USA 138 Yonsei University, Seoul, Republic of Korea 139 Zentrum für Technologie und Transfer (ZTT), Worms, Germany 140 Affiliated with an Institute Covered by a Cooperation Agreement with CERN, Geneva, Switzerland 141 Affiliated with an International Laboratory Covered by a Cooperation Agreement with CERN, Geneva, Switzerland aAlso at: Max-Planck-Institut für Physik, Munich, Germany bAlso at: Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA), Bologna, Italy cAlso at: Department of Applied Physics, Aligarh Muslim University, Aligarh, India dAlso at: Institute of Theoretical Physics, University of Wroclaw, Wrocław, Poland eAlso at: An Institution Covered by a Cooperation Agreement with CERN, Geneva, Switzerland ∗Deceased 123