Correlated event-by-event fluctuations of flow harmonics in Pb-Pb collisions at $\sqrt{s_{_{\rm NN}}}=2.76$ TeV

We report the measurements of correlations between event-by-event fluctuations of amplitudes of anisotropic flow harmonics in nucleus-nucleus collisions, obtained for the first time using a new analysis method based on multiparticle cumulants in mixed harmonics. This novel method is robust against systematic biases originating from non-flow effects and by construction any dependence on symmetry planes is eliminated. We demonstrate that correlations of flow harmonics exhibit a better sensitivity to medium properties than the individual flow harmonics. The new measurements are performed in Pb-Pb collisions at the centre-of-mass energy per nucleon pair of $\sqrt{s_{_{\rm NN}}}=2.76$ TeV by the ALICE experiment at the Large Hadron Collider (LHC). The centrality dependence of correlation between event-by-event fluctuations of the elliptic, $v_2$, and quadrangular, $v_4$, flow harmonics, as well as of anti-correlation between $v_2$ and triangular, $v_3$, flow harmonics are presented. The results cover two different regimes of the initial state configurations: geometry-dominated (in mid-central collisions) and fluctuation-dominated (in the most central collisions). Comparisons are made to predictions from MC-Glauber, viscous hydrodynamics, AMPT and HIJING models. Together with the existing measurements of individual flow harmonics the presented results provide further constraints on initial conditions and the transport properties of the system produced in heavy-ion collisions.

The properties of an extreme state of matter, the Quark-Gluon Plasma (QGP), are studied by colliding heavy ions at BNL's Relativistic Heavy Ion Collider (RHIC) and at CERN's Large Hadron Collider (LHC).One of the most widely utilized physical phenomena in the exploration of QGP properties is collective anisotropic flow [1,2].The large elliptic flow discovered at RHIC energies [3], which at the LHC energy of 2.76 TeV is 30% larger [4] and recently reported in [5] to increase even further at 5.02 TeV, demonstrated that the QGP behaves like a strongly coupled liquid with a very small ratio of the shear viscosity to entropy density (η/s), which is close to a universal lower bound of 1/4π [6].
Anisotropic flow is traditionally quantified with harmonics v n and corresponding symmetry plane angles ψ n in the Fourier series decomposition of particle azimuthal distribution (parameterized with azimuthal angle ϕ) in the plane transverse to the beam direction [7]: The shape of the intersecting zone of two identical heavy ions in non-central collisions is approximately ellipsoidal.This initial anisotropy is transferred via interactions among constituents and the pressure gradients developed in the QGP medium to an observable final-state anisotropic emission of particles with respect to the symmetry plane(s) of the intersecting zone.The resulting anisotropic flow for such an idealized ellipsoidal geometry is determined solely by even Fourier harmonics v 2n , and only one symmetry plane (the reaction plane) exists.Recently the importance of flow fluctuations and related additional observables have been identified.This has led to new concepts such as: non-vanishing odd harmonics v 2n−1 at midrapidity [8], non-identical symmetry plane angles ψ n and their inter-correlations [9][10][11][12][13][14], stochastic nature of harmonic v n and its probability density function P(v n ) [15][16][17][18][19][20], and, finally, the importance of higher order flow moments v k n (where the angular brackets denote an average over all events, and k ≥ 2) [21].Two distinct regimes for anisotropic flow development are nowadays scrutinized separately: geometry-dominated (in mid-central collisions) and fluctuation-dominated (in the most central collisions) [11].
Anisotropic flow is generated by the initial anisotropic geometry and its fluctuations coupled with an expansion of the produced medium.The initial coordinate space anisotropy can be quantified in terms of eccentricity coefficients ε n and corresponding symmetry plane angles Φ n [8,15,22].A great deal of effort is being invested to understand the relations between momentum space Fourier harmonics v n and symmetry planes ψ n on one side, and their spatial counterparts, ε n and Φ n , on the other side.These relations describe the response of the produced system to the initial coordinate space anisotropies, and therefore provide a rich repository of constraints for the system properties.In the early studies it was regularly assumed that, for small eccentricities, the harmonics v n respond linearly to the eccentricities ε n of the same order, v n ∝ ε n , and that ψ n Φ n [8,10,23,24].However, for sizable eccentricities recent studies argue that the anisotropies in momentum and coordinate space are related instead with the matrix equation connecting a set of anisotropic flow harmonics {v n } and a set of eccentricity coefficients {ε n }; it was demonstrated that the hydrodynamic response is both non-diagonal and non-linear, and that in general ψ n = Φ n [9,11,25,26].The first realization led to the conclusion that a relationship between event-by-event fluctuations of amplitudes of two different flow harmonics v m and v n can exist.This is hardly surprising for even flow harmonics in non-central collisions because the ellipsoidal shape generates non-vanishing values for all even harmonics v 2n [27], not only for elliptic flow.However, this simple geometrical argument cannot explain the possible relation between even and odd flow harmonics in non-central collisions, and the argument is not applicable in the central collisions, where all initial shapes are equally probable since they originate solely from fluctuations.Recently a linear correlation coefficient c(a, b) was defined in this context, which becomes 1 (-1) if observables a and b are fully linearly (antilinearly) correlated and zero in the absence of correlation [25].Model calculations of this new observable showed that neither v 2 and v 3 nor v 3 and v 4 are linearly correlated in non-central collisions.Most importantly, it was demonstrated that c(v 2 , v 4 ) depends strongly both on the η/s of the QGP and on the value of c(ε 2 , ε 4 ), which quantifies the relationship between corresponding eccentricities in the initial state [25].Therefore it was concluded that new observables c(v n , v m ), depending on the choice of flow harmonics v n and v m , are sensitive both to the fluctuations of the initial conditions and to the transport properties of the QGP, with the potential to discriminate between the two respective contributions when combined with a measurement of individual flow harmonics [25].
In this Letter we study the relationship between event-by-event fluctuations of magnitudes of two different flow harmonics of order n and m by using a recently proposed 4-particle observable [28]: with the condition m = n for two positive integers m and n.We refer to these new observables as Symmetric 2-harmonic 4-particle Cumulant, and use notation SC(m, n), or just SC.The double angular brackets indicate that the averaging procedure has been performed in two steps -first over all distinct particle quadruplets in an event, and then in the second step the single-event averages were weighted with 'number of combinations'.The latter for single-event average 4-particle correlations is mathematically equivalent to a unit weight for each individual quadruplet when the multiplicity differs event-by-event [29].In both 2-particle correlators above all distinct particle pairs are considered in each case.The four-particle cumulant in Eq. ( 2) is less sensitive to non-flow correlations than any 2-or 4-particle correlator on the right-hand side taken individually [30,31].The last equality is true only in the absence of non-flow effects [32].The observable in Eq. ( 2) is zero in the absence of flow fluctuations, or if the magnitudes of harmonics v m and v n are uncorrelated [28].It is also unaffected by relationship between symmetry plane angles ψ m and ψ n .The four-particle cumulant in Eq. ( 2) is proportional to the linear correlation coefficient c(a, b) introduced in [25] and discussed above, with a = v 2 m and b = v 2 n .Experimentally it is more reliable to measure the higher order moments of flow harmonics v k n (k ≥ 2) with 2-and multiparticle correlation techniques [31,33,34], than to measure the first moments v n with the event plane method, due to systematic uncertainties involved in the event-by-event estimation of symmetry planes [35,36].Therefore, we have used the new multiparticle observable in Eq. ( 2) as meant to be the least biased measure of the correlation between event-by-event fluctuations of magnitudes of two different harmonics v m and v n [28].
The dataset used in this analysis was obtained with the ALICE detector [38,39].It consists of minimumbias Pb-Pb collisions recorded during 2010 LHC Pb-Pb run at √ s NN = 2.76 TeV.With the default event and track selection criteria described below, we have obtained in total about 1.8 × 10 5 events per 1% centrality bin width.All individual systematic variations were combined in quadrature to obtain the final uncertainty.
The centrality was determined with the V0 detector [40][41][42].As a part of systematic checks centrality was determined independently with the Time Projection Chamber (TPC) [43] and the Silicon Pixel Detector (SPD) [44,45], which have slightly worse resolution [42].A systematic difference of up to 3% was observed in SC(m, n) results when using different centrality estimations.Charged particles were reconstructed with the TPC and the Inner Tracking System (ITS) [44,45] immersed in a 0.5 T solenoidal field.The TPC is capable of detecting charged particles in the transverse momentum range 0.1 < p T < 100 GeV/c, with a p T resolution of less than 6% for tracks below 20 GeV/c.Due to TPC dead zones between neighboring sectors, the track finding efficiency is about 75% for p T = 200 MeV/c and then it saturates at about 85% for p T > 1 GeV/c in Pb-Pb collisions.The TPC covers full azimuth and has a pseudorapidity coverage of |η| < 0.9.Tracks reconstructed using the TPC and ITS are referred to as global, while tracks reconstructed only with the TPC are referred to as TPC-only.
For online triggering, the V0 and SPD detectors were used [39].The reconstructed primary vertex is required to lie within ±10 cm of the nominal interaction point in the longitudinal direction along the beam axis.The cut on the position of the primary vertex along the beam axis was varied from ±12 cm to ±6 cm, the resulting SC measurements are consistent with those obtained with the default cut.
The main analysis was performed with global tracks selected in the transverse momentum interval 0.2 < p T < 5.0 GeV/c and pseudorapidity region |η| < 0.8.With this choice of low p T cut-off we are reducing event-by-event biases from smaller reconstruction efficiency at lower p T , while the high p T cut-off was introduced to reduce the contribution to the anisotropies from jets.Reconstructed tracks were required to have at least 70 TPC space points (out of a maximum of 159).Only tracks with a transverse distance of closest approach (DCA) to the primary vertex less than 3 mm are accepted to reduce the contamination from secondary tracks.Tracks with kinks (the tracks that appear to change direction due to multiple scattering, K ± decays) were rejected.
An independent analysis was performed with TPC-only and hybrid tracks (see below).For TPC-only tracks, the DCA cut was relaxed to 3 cm, providing different sensitivity to contamination from secondary tracks.Both the azimuthal acceptance and the reconstruction efficiency as a function of transverse momentum differ between TPC-only and global tracks.The resulting difference between independent analyses with global and TPC-only tracks was found to be 1-5% in all the centrality ranges studied, both for SC (3,2) and SC(4, 2).In another independent analysis with hybrid tracks, three different types of tracks were combined, in order to overcome the non-uniform azimuthal acceptance due to dead zones in SPD, and to achieve the best transverse momentum resolution [39].In this analysis the DCA cut was set to 3.2 cm in the longitudinal and to 2.4 cm in the transverse direction.The results between global and hybrid tracks differ by 3 to 5%, depending on the observable considered.
One of the largest contributions to the systematic uncertainty originates from the non-uniform reconstruction efficiency as a function of transverse momentum.For the observables SC(3, 2) and SC(4, 2) the uncertainty is 7% and 8%, respectively.In order to correct the measurements of these azimuthal correlators for various detector inefficiencies, we have constructed particle weights as a function of azimuthal angle ϕ and transverse momentum p T , and used the prescription outlined in [28].In particular, p T -weights were constructed as a ratio of transverse momentum distribution obtained from Monte Carlo generated tracks and from tracks reconstructed after they have passed through the detector simulated with GEANT3 [46].
We have used four Monte Carlo models in this paper.The HIJING model [47,48] was utilized to obtain the p T -weights [28].Secondly, the HIJING model was used to estimate the strength of nonflow correlations (typically few-particle correlations insensitive to the collision geometry).We have evaluated the observables of interest in coordinate space by modeling the initial conditions with a MC-Glauber model [49].We have compared the centrality dependence of our observables with theoretical model from [50], where the initial energy density profiles are calculated using a next-to-leading order perturbative-QCD+saturation model [51,52].The subsequent spacetime evolution is described by relativistic dissipative fluid dynamics with different parametrizations for the temperature dependence of the shear viscosity to entropy density ratio η/s(T ).Each of the η/s(T ) parametrizations is adjusted to reproduce the measured v n from central to mid-peripheral collisions.Finally, we provide an independent estimate of the centrality dependence of our observables by utilizing the AMPT model [53].
The centrality dependence of SC(4, 2) (red squares) and SC(3, 2) (blue circles) are presented in Fig. 1.Positive values of SC(4, 2) are observed for all centralities.This suggests a correlation between the eventby-event fluctuations of v 2 and v 4 , which indicates that finding v 2 larger than v 2 in an event enhances the probability of finding v 4 larger than v 4 in that event.On the other hand, the negative results of SC (3,2) show the anti-correlation between v 2 and v 3 magnitudes, which further imply that finding v 2 larger than v 2 enhances the probability of finding v 3 smaller than v 3 .We have calculated the SC observables using HIJING which does not include anisotropic collectivity but e.g.azimuthal correlations due to jet production [47,48].It is found that in HIJING both cos(mϕ ] are non-zero.However, the calculation of SC observables from HIJING are compatible with zero for all centralities, which suggests that the SC measurements are nearly insensitive to non-flow correlations.We have also performed a study using the like-sign technique, which is another powerful approach to estimate the non-flow effects [4].It was found that the difference between correlations for like-sign and all charged combinations are within 10%.This demonstrates that non-zero values of SC measurements cannot be explained by non-flow effects. A study based on the AMPT model showed that the observed (anti-)correlations are also sensitive to the transport properties, e.g. the partonic and hadronic interactions [20,28].Fig. 2 shows the comparison of SC (3,2) and SC(4, 2) to the AMPT calculations which generally predict the correct sign but underestimate their magnitude.The comparison between experimental data and the theoretical calculations [50],   which incorporate both the initial conditions and system evolution, is shown in Fig. 2 (top).The model captures qualitatively the centrality dependence, but not quantitatively.Most notably, there is no single centrality for which a given η/s(T ) parameterization describes simultaneously both SC(4, 2) and SC (3,2).On the other hand, the same theoretical model captures quantitatively the centrality dependence of individual v 2 , v 3 and v 4 harmonics with a precision better than 10% in central and mid-central collisions [50].We therefore conclude that individual flow harmonics v n and new SC(m, n) observables together provide a better handle on the initial conditions and η/s(T ) than each of them alone.This is emphasized in Fig. 2 (middle), where SC(3, 2) and SC(4, 2) observables were divided with the products , respectively, in order to obtain the normalized SC observables (the result for 60-70% is omitted due to large statistical uncertainty).These products were obtained with 2-particle correlations and using a pseudorapidity gap of |∆η| > 1.0 to suppress biases from few-particle non-flow correlations.We have found that the normalized SC(4, 2) observable exhibits much better sensitivity to different η/s(T ) parameterizations than the normalized SC(3, 2) observable, see Fig. 2 (middle), and than the individual flow harmonics [50].These findings indicate that the normalized SC (3,2) observable is sensitive mainly to the initial conditions, while the normalized SC(4, 2) observable is sensitive to both the initial conditions and the system properties, which is consistent with the prediction from [25].
It can be seen in Fig. 1 that SC(4, 2) and SC(3, 2) increase non-linearly up to centrality 60%.Assuming only linear response v n ∝ ε n , we expect that the normalized SC(m, n) evaluated in coordinate space can capture the measurement of centrality dependence of normalized SC(m, n) in the momentum space.The correlations between the n th and m th order harmonics were estimated with calculations of e. a normalized SC observable in the coordinate space, which we denote SC(m, n) ε / ε 2 n ε 2 m .Here the ε n (or ε m ) is the n th (or m th ) order coordinate space anisotropy, following the definition in [8].Different scenarios of the MC-Glauber model, named wounded nucleon (WN) and binary collisions (BC) weights, have been used.An increasing trend from central to peripheral collisions with different sign has been observed in Fig. 2 (bottom) for SC(4, 2) and SC (3,2).A dramatic deviation of SC (4,2) between data and model calculation is observed for non-central collisions.This deviation increases from mid-central to peripheral, which could be understood as the contribution of the non-linear response (ε 2 contributes to v 4 ) increasing as a function of centrality, which is consistent with that reported in [54].Since the normalized SC(3, 2) appears to be sensitive only to initial conditions and not to η/s(T ), see Fig. 2 (middle), MC-Glauber model captures better its centrality dependence than for normalized SC(4, 2) observable, see Fig. 2

(bottom).
The relationship between the flow harmonics v 2 , v 3 , v 4 have also been investigated by the ATLAS Collaboration using the ESE technique [54][55][56].For events with a larger v 2 , the ATLAS Collaboration showed these have a smaller than average v 3 , and a larger than average v 4 .For events with a smaller v 2 , the opposite trend occurred.These observations are consistent with the patterns observed via the SC measurements presented in this Letter.The SC observables, however, provide a compact quantitative measure of these correlations, without fitting correlations between v n and v m .This simplifies the quantitative comparison of the SC observables with hydrodynamical calculations as shown in Fig. 2.
In the most central collisions the anisotropies originate mainly from fluctuations, i.e. the initial ellipsoidal geometry characteristic for mid-central collisions plays little role in this regime.Therefore we have performed a separate analysis for centrality range 0-10% in centrality bins of 1%.The results are presented in Fig. 3.We observe that event-by-event fluctuations of v 2 and v 4 remain correlated, and of v 2 and v 3 anti-correlated, also in this regime.However, the strength of the (anti)-correlations exhibits a different centrality dependence than for the wider centrality range shown in Fig. 1.As seen in Fig. 3 (top) the centrality dependence cannot be linearly extrapolated from the 0-10% region to the full centrality range.Comparison with two different parameterizations of the MC-Glauber initial conditions for normalized SC observables presented in Fig. 3 (bottom) suggests that the BC parameterization (binary collisions weights) is favored by the data in most central collisions.This agreement may suggest the scaling with Correlated event-by-event fluctuations of flow harmonics... Fig. 3: Correlated and anti-correlated event-by-event fluctuations in coordinate (MC-Glauber) and momentum space (data).On the bottom panel we present the normalized SC observables, where pseudorapidity gap |∆η| > 1.0 was applied in both two-particle correlations in denominator used to estimate individual flow harmonics.
In summary, we have measured for the first time the new multiparticle observables, the Symmetric 2harmonic 4-particle Cumulants (SC), which quantify the relationship between event-by-event fluctuations of two different flow harmonics.We have found that fluctuations of v 2 and v 3 are anti-correlated in all centralities, however the details of the centrality dependence differ in the fluctutation-dominated (most central) and the geometry-dominated (mid-central) regimes.Fluctuations of v 2 and v 4 are correlated for all centralities.The SC observables were used to discriminate between the state-of-the-art hydro model calculations with different parameterizations of the temperature dependence of η/s, for all of which the centrality dependence of elliptic, triangular and quadrangular flow has weaker sensitivity at the LHC.In particular, the centrality dependence of SC(4, 2) cannot be captured with the constant η/s.We have also used our results to discriminate between two different parameterizations of initial conditions and have demonstrated that in the fluctuation-dominated regime (in central collisions) MC-Glauber initial conditions with binary collisions weights are favored over wounded nucleon weights.

Fig. 2 :
Fig. 2: AMPT model predictions are shown as hollow symbols in the (top) and (middle) panels.(top) Comparison of observables SC(4,2) (red filled squares) and SC(3,2) (blue filled circles) to theoretical model from [50].Solid lines indicate the predictions with constant η/s, while the dashed lines indicate predictions for different parameterizations of η/s temperature dependence (labeled in the same way as in Fig. 1 in [50]).(middle) Results divided by v 2 m v 2 n .(bottom) Comparison to MC-Glauber using wounded nucleon (WN) and binary collisions (BC) weights.