Anisotropic flow phenomena are a key probe of the existence of Quark-Gluon Plasma. Several new observables associated with correlations between anisotropic flow harmonics are developed, which are expected to be sensitive to the initial fluctuations and transport properties of the created matter in heavy-ion collisions. I review recent developments of correlations of anisotropic flow harmonics. The experimental measurements, together with the comparisons to theoretical model calculations, open up new opportunities of exploring novel QCD dynamics in heavy-ion collisions.
Danish Council for Independent ResearchDanmarks Grundforskningsfond1. Introduction
One of the fundamental questions in the phenomenology of Quantum Chromo Dynamics (QCD) is, what are the properties of matter at extreme densities and temperatures where quarks and gluons are in a new state of matter, the so-called Quark-Gluon Plasma (QGP)? [1, 2]. Collisions of high-energy heavy ions, at the Brookhaven Relativistic Heavy Ion Collider (RHIC) and the CERN Large Hadron Collider (LHC), allow us to create and study the properties of the QGP matter in the laboratory. This matter expands under large pressure gradients, which transfer the inhomogeneous initial conditions into azimuthal anisotropy of produced particles in momentum space. This anisotropy of produced particles is one of the probes of the properties of the QGP [3, 4]. It can be characterized by an expansion of the single-particle azimuthal distribution P(φ):(1)Pφ=12π∑n=-∞+∞Vn→e-inφ,where φ is the azimuthal angle of emitted particles, Vn→ is the nth order flow vector defined as Vn→=vneinΨn, its magnitude vn is the nth order anisotropic flow harmonic, and its orientation is symmetry plane (participant plane) angle Ψn. Alternatively, this anisotropy can be generally given by the joint probability density function (PDF) in terms of vn and Ψn as(2)Pvm,vn,…,Ψm,Ψn,…=1NeventdNeventvmvn⋯dvmdvn⋯dΨmdΨn⋯. In the last decade, the experimental measurements of anisotropic flow vn [5–55], combined with theoretical advances from calculations made in a variety of frameworks [56–62], have led to broad and deep knowledge of initial conditions and properties of the created hot/dense QCD matter. In particular, the precision anisotropic flow measurements based on the huge data collected at the LHC experiments and the successful description from hydrodynamic calculations demonstrate that the QGP created in heavy-ion collisions behaves like a strongly coupled liquid with a very small specific shear viscosity η/s [63–68], which is close to a quantum limit 1/4π [69].
It has been investigated into great details of event-by-event fluctuations of single flow harmonic. Based on the measurements of higher-order cumulants of anisotropic flow [43, 48, 51, 74, 75] and the event-by-event vn distributions [40], it was realized that the newly proposed Elliptic-Power function [76–78] gives the best description of underlying PDF of single harmonic vn distributions [72, 79, 80]. On the other hand, it has been known for a while that both the flow harmonic (magnitude) vn and its symmetry plane (orientation) Ψn of the flow vector Vn→ fluctuate event-by-event [81–83], but only recently pT and η dependent flow angle (Ψn) and magnitude (vn) were predicted by hydrodynamic calculations [84, 85]. Many indications were quickly obtained in experiments by looking at the deviations from unity of vn[2]/vn{2} [86] and factorization ratio rn [52, 55, 86]. These measurements were nicely predicted or reproduced by hydrodynamic calculations and are found to be sensitive to the initial state density fluctuations and/or the shear viscosity of the expanding fireball medium [84, 85, 87]. Most of these above-mentioned studies are focused on the fluctuations of single flow harmonics and their corresponding symmetry planes, as a function of collisions centrality, transverse momentum pT, and pseudorapidity η. Results of correlations between symmetry planes [28, 41] reveal a new type of correlations between different order flow vectors, which was investigated in the observable of v2n/Ψn before [88–90]. In particular, some of the symmetry planes correlations show quite different centrality dependence from the initial state and final state, and this characteristic sign change during system evolution is correctly reproduced by theoretical calculations [62, 82, 91], thus confirming the validity of hydrodynamic framework in heavy-ion collisions and further yielding valuable additional insights into the fluctuating initial conditions and hydrodynamic response [62, 82, 92].
In addition to all these observables, the (anti)correlations between anisotropic flow harmonics vm and vn are found to be extremely interesting [45, 62, 70, 71, 93]. A completely new set of information on the joint probability density function (PDF) can be obtained from the rich correlation pattern observed in experiments. On the other hand, no existent theoretical calculations [62, 70, 71, 93] could provide quantitative descriptions of data [36]. Thus, it is crucial to investigate in depth the relationship between different flow harmonics: whether they are correlated, anticorrelated, or not correlated from both experimental and theoretical points of view.
It is found recently that the relationship between different order flow harmonics can be used to probe the initial state conditions and the hydrodynamic response of the QGP [36, 71, 93–95]. In order to better understand the event-by-event P(φ) distribution, it is critical to investigate the relationship between vm and vn. Considering the naive ellipsoidal shape of the overlap region in noncentral heavy-ion collisions generating nonvanishing even flow harmonics v2n, the correlations between the even flow harmonics are expected. However, it is not straightforward to use geometrical argument to explain the relationship between even flow harmonics for central collisions, where all the harmonics are driven by fluctuations instead of geometry, and to explain the relationship between even and odd flow harmonics for central and noncentral collisions [80]. A linear correlation function c(vm,vn) was proposed to study the relationship between vm and vn [83]. It is defined as(3)cvm,vn=vm-vmevvn-vnevσvmσvnev,where σvm is the standard deviation of the quantity vm; c(vm,vn) is 1 (or −1) if vm and vn are linearly (antilinearly) correlated and is 0 if they are not correlated. It was shown in Figure 1 that there is an anticorrelation between v2 and v3, while a correlation was observed between v2 and v4. In addition, it was demonstrated that c(v2,v4) depends on both the initial conditions and η/s, while c(v2,v3) is only sensitive to η/s [83]. Nevertheless, it cannot be accessible easily in experimental measurements, which rely on two-particle and multiparticle correlations techniques. Thus, it is critical to find an observable which studies the relationship between flow harmonics without contributions from symmetry plane correlations and can be accessed with observable techniques from experiments. Two different approaches, named EventShapeEngineering and SymmetricCumulant, are discussed in the following section.
pT dependence of c(v2,v3) (a) and c(v2,v4) (b) in centrality 20–30% in Pb–Pb collisions at sNN=2.76 TeV. Figures are taken from [83].
2.1. Event Shape Engineering (ESE)
The first experimental attempt was made by ATLAS Collaboration [45], using the Event Shape Engineering (ESE) [96]. This is a technique to select events according to the magnitude of reduced flow vector Vn→. Figure 2 shows the performance of event shape selection on V2 (a) and V3 (b) in ATLAS detector. For each centrality the data sample is divided into several event classes according to V2 or V3 distributions. Then v2 and v3 relationship was investigated by measurements of v2 and v3 in each event class from ESE selection. Without using ESE selection, a boomerang-like pattern was observed for the centrality dependence of v2-v3 correlation. This is mainly due to the fact that v3 has weaker centrality dependence than v2. By using ESE, it was observed in Figure 3(b) that, for event class with the same centrality (shown as the same color), v3 decreases as v2 increases. It suggests that v2 is anticorrelated with v3. Considering the linear hydrodynamic response of v2 and v3 from eccentricity ε2 and triangularity ε3, the anticorrelation between v2 and v3 might reveal the anticorrelation between ε2 and ε3 of the initial geometry. This indication of initial anticorrelations between ε2 and ε3 is observed in model calculations [96, 97].
Distributions of V2 (a) and V3 (b) calculated with ATLAS forward calorimeter for centrality interval 0-1%. Figures are taken from [45].
The correlation of v2 (x-axis) with v3 (y-axis) measured in 0.5 <pT<2 GeV/c. (a) shows v2 and v3 values for fourteen 5% centrality intervals over the centrality range of 0–70% without event shape selection. (b) shows v2 and v3 values in 15 q2 intervals in seven centrality ranges (markers) with larger v2 value corresponding to larger q2 value. Figures are taken from [45].
Figure 4 shows the investigation of relationship between v2 and v4. A boomerang-like pattern, although weaker than that for v2-v3 relationship shown in Figure 3(a), is observed in Figure 4(a), prior to the ESE selection. After the ESE selection, it is found in Figure 4(b) that v4 increases with increasing v2. This suggests a correlation between the two harmonics and it can be understood by the interplay between linear and nonlinear collective dynamics in the system evolution [45]. This nonlinear contribution of v4 from v2 is further investigated by fitting the correlation pattern using v4=c02+(c1v22)2, where c0 and c1 denote the linear and nonlinear components. It is found that the linear component has weak centrality dependence, while the nonlinear component, increasing dramatically with collision centrality, becomes the dominant contribution in the most peripheral collisions [45].
The correlation of v2 (x-axis) with v4 (y-axis) measured in 0.5 <pT<2 GeV/c. (a) shows v2 and v3 values for fourteen 5% centrality intervals over the centrality range of 0–70% without event shape selection. (b) shows v2 and v4 values in 15 q2 intervals in seven centrality ranges (markers) with larger v2 value corresponding to larger q2 value. Figures are taken from [45].
These (anti)correlation patterns between vm and vn observed in experiments open a new window to the understanding of the collectivity phenomena in heavy-ion collisions. However, it was also noticed that these measurements were based on 2-particle correlations, which might be suffered by nonflow effects, and they require subdividing such calculations and modeling resolutions associated with ESE due to finite event-wise multiplicities. Considering the computational constraints, this approach cannot be performed easily in hydrodynamic calculations which usually are based on limited statistics compared to experimental data.
2.2. Symmetric Cumulants (SC)
A new type of observables for the analyses of flow harmonic correlations, symmetric cumulants (originally named Standard Candles (SC) in [93]), was proposed as SC(m,n)=⟪cos(mφ1+nφ2-mφ3-nφ4)⟫c. If m≠n, the isotropic part of the corresponding four-particle cumulant is given by (4)⟪cosmφ1+nφ2-mφ3-nφ4⟫c=⟪cosmφ1+nφ2-mφ3-nφ4⟫-⟪cosmφ1-φ2⟫⟪cosnφ1-φ2⟫=vm2vn2-vm2vn2.For a detector with uniform acceptance in azimuthal direction, the asymmetric terms, forexample, ⟪cos(mφ1-nφ2)⟫, are averaged to zero. The single event 4-particle correlation ⟪cos(mφ1+nφ2-mφ3-nφ4)⟫ could be calculated as(5)cosmφ1+nφ2-mφ3-nφ4=1MM-1M-2M-3Vm2Vn2-2ReVm+nVm∗Vn∗-2ReVmVm-n∗Vn∗+Vm+n2+Vm-n2-M-4Vm2+Vn2+MM-6.And the single event 2-particle correlation ⟪cos[m(φ1-φ2)]⟫ could be obtained as(6)cosmφ1-φ2=1MM-1Vm2-M.Then, the weights of M(M-1) and M(M-1)(M-2)(M-3) are used to get the event-averaged 2-particle and 4-particle correlations, as introduced in [93]. Due to the definition, this new type of 4-particle cumulants SC(m,n) is independent of the symmetry planes Ψm and Ψn and is expected to be less sensitive to nonflow correlations, which should be strongly suppressed in 4-particle cumulants. This was confirmed by SC(m,n) calculation using HIJING model [98, 99] which does not include anisotropic collectivity but, for example, azimuthal correlations due to jet production. It is observed that both ⟪cos(mφ1+nφ2-mφ3-nφ4)⟫ and ⟪cos[m(φ1-φ2)]⟫⟪cos[n(φ1-φ2)]⟫ are nonzero, while SC(m,n) are compatible with zero in HIJING simulations [36]. This confirms that SC(m,n) measurements are nearly insensitive to nonflow correlations. Therefore, it is believed that SC(m,n) is nonzero if there is (anti)correlations of vn and vm. The investigation of SC(m,n) will allow us to know whether finding vm larger than vm in an event will enhance or reduce the probability of finding vn larger than vn in that event, which provides unique information for the event-by-event simulations of anisotropic flow harmonics.
Figure 5 shows the first calculation of SC(4,2) (solid markers) and SC(3,2) (open markers) as a function of centrality from AMPT model [93]. Nonzero values for both SC(4,2) and SC(3,2) are observed. Positive SC(4,2) suggests a correlation between the event-by-event fluctuations of v2 and v4, which indicates that finding v2 larger than 〈v2〉 in an event enhances the probability of finding v4 larger than 〈v4〉 in that event. On the other hand, the negative results of SC(3,2) imply that finding v2 larger than 〈v2〉 enhances the probability of finding v3 smaller than 〈v3〉 [93].
The centrality dependence of symmetric cumulants SC(4,2) and SC(3,2) at sNN. Figures are taken from [72, 93].
Several configurations of the AMPT model have been investigated to better understand the results based on AMPT simulations [93]. Partonic interactions can be tweaked by changing the partonic cross section: the default value is 10 mb, while using 3 mb generates weaker partonic interactions in ZPC [100, 101]. One can also change the hadronic interactions by controlling the termination time in ART. Setting NTMAX = 3, where NTMAX is a parameter which controls the number of time steps in ART (rescattering time), will effectively turn off the hadronic interactions [100, 101]. For SC(4,2) and SC(3,2) calculations for three different scenarios, (a) 3 mb, (b) 10 mb, and (c) 10 mb, no rescattering is presented in Figure 5. It is found that when the partonic cross section is decreasing from 10 mb (lower shear viscosity) to 3 mb (higher shear viscosity), the strength of SC(4,2) decreases. Additionally, the “10 mb, no rescattering” setup seems to give slightly smaller magnitudes of SC(4,2) and SC(3,2).
Further studies have been performed in AMPT initial conditions, based on the observable of SC(m,n)ε which is defined as εm2εn2-εm2εn2 [72]. The centrality dependence of SC(4,2)ε and SC(3,2)ε is presented as red circles and blue diamonds in Figure 5(b). Positive and increasing trend from central to peripheral collisions has been observed for SC(4,2)ε. In contrast, negative and decreasing trend was observed for SC(3,2)ε in the AMPT initial conditions. This shows that finding ε2 larger than 〈ε2〉 in an event enhances the probability of finding ε4 larger than 〈ε4〉, while in parallel enhancing the probability of finding ε3 smaller than 〈ε3〉 in that event. Same conclusions were obtained using MC-Glauber initial conditions [75].
Based on AMPT calculations, it seems that the signs of SC(m,n)v (for m,n=2,3,4) in the final state are determined by the correlations of SC(m,n)ε in the initial state, while its magnitude also depends on the properties of the created system. This clearly suggests that SC(m,n)v is a new promising observable to constrain the initial conditions and the transport properties of the system.
The first experimental measurements of centrality dependence of SC(4,2) (red squares) and SC(3,2) (blue circles) are presented in Figure 6(a). Positive values of SC(4,2) are observed for all cases of centrality. This confirms a correlation between the event-by-event fluctuations of v2 and v4. On the other hand, the measured negative results of SC(3,2) show the anticorrelation between v2 and v3 magnitudes. The same measurements are performed using the like-sign technique, which is another powerful approach to estimate nonflow effects [27]. It was found that the difference between correlations for like-sign and all charged combinations, which might be mainly due to nonflow effects, is much smaller compared to the magnitudes of SC(m,n) itself. This further proves that nonzero values of SC(m,n) measured in experiments cannot be explained by nonflow effects solely.
The centrality dependence of symmetric cumulants SC(4,2) (red markers) and SC(3,2) (blue markers) at sNN=2.76 TeV Pb–Pb collisions. Figures are taken from [36].
In addition, the comparison between experimental data and the event-by-event perturbative-QCD+saturation+hydro (“EKRT”) calculations [62], which incorporate both initial conditions and hydrodynamic evolution, is shown in Figure 6. It was shown that this model can capture quantitatively the centrality dependence of individual v2, v3, and v4 harmonics in central and mid-central collisions [62]. However, it can only qualitatively but not quantitatively predict SC(m,n) measurements by ALICE. For given η/s(T) parameterization tuned by individual flow harmonic, the calculation cannot describe SC(4,2) and SC(3,2) simultaneously for any single centrality. Experimental measurements are also compared to the VISH2+1 model calculations (see Figure 7), using various combinations of initial conditions (IC) from (a) MC-Glb, (b) MC-KLN, and (c) MC-AMPT with η/s= 0.08 and 0.20. It is noticed that the one with MC-Glb IC and η/s= 0.08 is compatible with SC(4,2) measurement and the calculation with MC-AMPT IC and η/s= 0.08 can describe SC(3,2) measurement [70]. However, just like EKRT calculations, none of these combinations is able to describe SC(4,2) and SC(3,2) simultaneously. Thus, it is concluded that the new SC(m,n) observables provide better handle on the initial conditions and η/s(T) than each of the individual harmonic measurements alone.
The centrality dependence of symmetric cumulants SC(4,2) (red markers) and SC(3,2) (blue markers) at sNN=2.76 TeV Pb–Pb collisions by VISH2+1 simulations. Figures are taken from [70].
After being presented for the first time at Quark Matter 2015 conference, preliminary results of SC(4,2) and SC(3,2) gained a lot of attention [102]. One of the key suggestions was to normalize SC(m,n) by dividing with the products vm2vn2 in order to get rid of influences from individual flow harmonics. The results are shown in Figure 6(b), with normalized SC(3,2) and SC(4,2) observables by dividing with the products v32v22 and v42v22, respectively [36]. The 2-particle correlations vm2 and vn2 are obtained with a pseudorapidity gap of Δη>1.0 to suppress contributions from nonflow effects. It was shown in Figure 8(a) that the normalized SC(4,2) observable exhibits clear sensitivity to different η/s parameterizations and the initial conditions, which provides a unique opportunity to discriminate between various possibilities of the detailed setting of η/s(T) of the produced QGP and the initial conditions used in hydrodynamic calculations. On the other hand, normalized SC(3,2) is independent of the setting of η/s(T). In addition, it was demonstrated in Figure 9 that the normalized SC(3,2), also named NSCv(3,2) in the following text, is compatible with its corresponding observable SCε(3,2) in the initial state. Thus, NSCv(3,2) could be taken as golden observable to directly constrain initial conditions without demands for precise knowledge of transport properties of the system [70]. Furthermore, none of existing theoretical calculations can reproduce the data; there is still a long way to go for the development of hydrodynamic calculations.
The centrality dependence of normalized symmetric cumulants NSC(m,n) at sNN=2.76 TeV Pb–Pb collisions by VISH2+1 simulations. Figures are taken from [70].
The centrality dependence of NSC(3,2) (a, b, and c) and C(v32,v22) (d, e, and f) and the corresponding observables in the initial conditions at sNN=2.76 TeV Pb–Pb collisions from VISH2+1. Figures are taken from [70].
Predictions of relationship between other harmonics are provided in [70] and shown in Figure 8. Besides different sensitivities to IC and η/s as seen above, the centrality dependence of the relationship between flow harmonics seems quite different. For instance, despite the differences in the initial conditions, a maximum value of SC(5,3) is observed in central collision using η/s=0.20, while the maximum value is seen in more peripheral collision if η/s=0.08 is used.
Compared to the previous measurements of relationship between flow harmonics investigated using the ESE technique, SC(m,n) observable provides a quantitative measure of these correlation strengths. Further investigations on relationship between flow harmonics using list of observables in Table 1 could be performed as a function of centrality, transverse momentum, and pseudorapidity et al., which is clearly nontrivial. Although one did not use the information of symmetry planes in both ESE and SC studies, recent study just reveals that flow harmonic correlations might not be completely independent on symmetry plane correlations [73]. The proportionality relations between symmetric cumulants involving higher harmonics v4 or v5 and symmetry plane correlations are derived, which seem to build the bridge between flow harmonic correlations and flow angle correlations (symmetry plane correlations). This might point out to a new direction of investigations of correlations between flow vectors and will shed a new light on the nature of fluctuating initial conditions and η/s of the created QGP in heavy-ion collisions.
List of observables for correlations of flow harmonics, including all combinations of symmetric 2-harmonic 4-particle cumulants (up to v6).
Observables
Equations
Number of particles
Exp.
Th.
cos2φ1+3φ2-2φ3-3φ4c
〈v22v32〉-v22v32
4
[36]
[70–72]
cos2φ1+4φ2-2φ3-4φ4c
〈v22v42〉-v22v42
4
[36]
[70–73]
cos2φ1+5φ2-2φ3-5φ4c
〈v22v52〉-v22v52
4
[70, 71, 73]
cos2φ1+6φ2-2φ3-6φ4c
〈v22v62〉-v22v62
4
cos3φ1+4φ2-3φ3-4φ4c
〈v32v42〉-v32v42
4
[70]
cos3φ1+5φ2-3φ3-5φ4c
〈v32v52〉-v32v52
4
[70, 71, 73]
cos3φ1+6φ2-3φ3-6φ4c
〈v32v62〉-v32v62
4
cos4φ1+5φ2-4φ3-5φ4c
〈v42v52〉-v42v52
4
cos4φ1+6φ2-4φ3-6φ4c
〈v42v62〉-v42v62
4
cos5φ1+6φ2-5φ3-6φ4c
〈v52v62〉-v52v62
4
⋯
⋯
6
3. Summary
In the past two decades, the underlying PDF of each single harmonic P(vn) was investigated in great detail. However, at the moment, how the joint underlying PDF, including different order symmetry planes and harmonics, is described is an open question, especially if these correlations between different flow harmonics modify the single harmonics P(vn). New observables discussed here begin to answer these open questions. Nevertheless, many more investigations between different flow harmonics, including higher-order cumulants and higher harmonics, are necessary to reasonably constrain the joint PDF and ultimately lead to new insights into the nature of fluctuation of the created matter in heavy-ion collisions. How to turn the multitude of measured and possibly measurable in future relationships between anisotropic flow harmonics into a focused search for correct initial conditions and detailed setting of η/s is an exciting challenge for the theory community.
Competing Interests
The author declares that there are no competing interests regarding the publication of this paper.
Acknowledgments
The author thanks J. J. Gaardhøje, K. Gajdošová, L. Yan, J. Y. Ollitrault, and H. Song for the comments on the manuscript and fruitful discussions. The author is supported by the Danish Council for Independent Research, Natural Sciences, and the Danish National Research Foundation (Danmarks Grundforskningsfond).
LeeT. D.Feynman rules of quantum chromodynamics inside a hadronShuryakE. V.Quantum chromodynamics and the theory of superdense matterOllitraultJ.-Y.Anisotropy as a signature of transverse collective flowVoloshinS.ZhangY.Flow study in relativistic nuclear collisions by Fourier expansion of azimuthal particle distributionsAltC.AnticicT.BaatarB.Directed and elliptic flow of charged pions and protons in Pb + Pb collisions at 40A and 158 A GeVAckermannK. H.AdamsN.AdlerC.Elliptic flow in Au+Au collisions at sNN=130 GeVAdlerC.AhammedZ.AllgowerC.Identified particle elliptic flow in Au + Au collisions at sNN=130 GeVAdlerC.AhammedZ.AllgowerZ.Elliptic flow from two- and four-particle correlations in Au+Au collisions at sNN=130 GeVAdamsJ.AggarwalM. M.AhammedZ.Azimuthal anisotropy in Au+Au collisions at sNN=200 GeVAdamsJ.AdamczykL.AdkinsJ. K.Multistrange Baryon Elliptic Flow in Au + Au Collisions at sNN=200 GeVAbelevB. I.AggarwalM. M.AhammedZ.Centrality dependence of charged hadron and strange hadron elliptic flow from sNN=200 GeV Au+Au collisionsAdamczykL.AdkinsJ. K.AgakishievG.Elliptic flow of identified hadrons in Au+Au collisions at sNN=7.7–62.4 GeVAdcoxK.AdlerS. S.AjitanandN. N.Flow measurements via two-particle azimuthal correlations in Au+Au collisions at sNN=130 GeVAdlerS. S.AfanasievS.AidalaC.Elliptic flow of identified hadrons in Au+Au collisions at sNN=200 GeVAdareA.AfanasievS.AidalaC.Scaling properties of azimuthal anisotropy in Au + Au and Cu + Cu collisions at sNN=200 GeVAfanasievS.AidalaC.AjitanandN. N.Systematic studies of elliptic flow measurements in Au + Au collisions at sNN=200 GeVAdareA.AfanasievS.AidalaC.Elliptic and hexadecapole flow of charged hadrons in Au+Au collisions at sNN=200 GeVAdareA.AfanasievS.AidalaC.Measurements of higher order flow harmonics in Au + Au collisions at sNN=200 GeVBackB. B.BakerM. D.BartonD. S.Pseudorapidity and centrality dependence of the collective flow of charged particles in Au + Au collisions at sNN=200GeVBackB. B.BakerM. D.BallintijnM.Centrality and pseudorapidity dependence of elliptic flow for charged hadrons in Au+Au collisions at sNN=200 GeVBackB. B.BakerM. D.BallintijnM.Energy dependence of elliptic flow over a large pseudorapidity range in Au+Au collisions at the BNL relativistic heavy ion colliderManlyS.AlverB.BackB. B.System size, energy and pseudorapidity dependence of directed and elliptic flow at RHICBackB. B.BakerM. D.BallintijnM.Energy dependence of directed flow over a wide range of pseudorapidity in Au + Au collisions at the BNL relativistic heavy ion colliderAlverB.BackB. B.BakerM. D.System size, energy, pseudorapidity, and centrality dependence of elliptic flowAlverB.BackB. B.BakerM. D.Event-by-event fluctuations of azimuthal particle anisotropy in Au + Au collisions at sNN=200 GeVAlverB.BackB. B.BakerM. D.Non-flow correlations and elliptic flow fluctuations in Au+Au collisions at sNN=200 GeVAamodtK.AbelevB.Abrahantes QuintanaA.Elliptic flow of charged particles in Pb-Pb collisions at sNN=2.76 TeVAamodtK.AbelevB.AbrahantesA.Higher harmonic anisotropic flow measurements of charged particles in Pb-Pb collisions at sNN=2.76 TeVAbelevB.AdamJ.AdamováD.Anisotropic flow of charged hadrons, pions and (anti-)protons measured at high transverse momentum in Pb-Pb collisions at sNN=2.76 TevAbelevB.AdamJ.AdamováD.Directed flow of charged particles at midrapidity relative to the spectator plane in Pb-Pb collisions at sNN=2.76 TeVAbelevB. B.AdamJ.AdamováD.Multiparticle azimuthal correlations in p-Pb and Pb-Pb collisions at the CERN large hadron colliderAbelevB. B.AdamJ.AdamováD.Elliptic flow of identified hadrons in Pb-Pb collisions at sNN=2.76 TevAdamJ.AdamováD.AggarwalM. M.Event-shape engineering for inclusive spectra and elliptic flow in Pb-Pb collisions at sNN=2.76 TeVAdamJ.AphecetcheL.AudurierB.Charge-dependent flow and the search for the chiral magnetic wave in Pb-Pb collisions at sNN=2.76 TeVAdamJ.AdamováD.AggarwalM. M.Anisotropic flow of charged particles in Pb-Pb collisions at sNN=5.02 TeVAdamJ.MilosevicJ.BiroG.Correlated event-by-event fluctuations of flow harmonics in Pb-Pb collisions at sNN=2.76 TeVhttps://arxiv.org/abs/1604.07663AadG.AbbottB.AbdallahJ.Measurement of the pseudorapidity and transverse momentum dependence of the elliptic flow of charged particles in lead–lead collisions at SNN=2.76 TeV with the ATLAS detectorAadG.AbbottB.AbdallahJ.Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76 Tev lead-lead collisions with the ATLAS detectorAadG.AbajyanT.AbbottB.Measurement with the ATLAS detector of multi-particle azimuthal correlations in p+Pb collisions at sNN=5.02 TeVAadG.TatevikT.AbbottB.Measurement of the distributions of event-by-event flow harmonics in lead-lead collisions at = 2.76 TeV with the ATLAS detector at the LHCAadG.AbatE.AbbottB.Measurement of event-plane correlations in SNN=2.76 TeV lead-lead collisions with the ATLAS detectorAadG.AbbottB.AbdallahJ.Measurement of the centrality and pseudorapidity dependence of the integrated elliptic flow in lead–lead collisions at sNN=2.76 TeV with the ATLAS detectorAadG.AbbottB.AbdallahJ.Measurement of flow harmonics with multi-particle cumulants in Pb+Pb collisions at SNN=2.76 TeV with the ATLAS detectorChatrchyanS.BloomK. A.BockelmanB.Measurement of higher-order harmonic azimuthal anisotropy in PbPb collisions at SNN=2.76 TeVAadG.AbbottB.AbdallahJ.Measurement of the correlation between flow harmonics of different order in lead-lead collisions at sNN=2.76 TeV with the ATLAS detectorChatrchyanS.KhachatryanV.SirunyanA. M.Centrality dependence of dihadron correlations and azimuthal anisotropy harmonics in PbPb collisions at sNN=2.76 TeVChatrchyanS.AdamováD.AggarwalM. M.Azimuthal anisotropy of charged particles at high transverse momenta in Pb-Pb collisions at sNN=2.76 TeVChatrchyanS.TonoiuD.AzzurriP.Measurement of the elliptic anisotropy of charged particles produced in PbPb collisions at SNN=2.76 TeVChatrchyanS.KhachatryanV.SirunyanA. M.Measurement of the azimuthal anisotropy of neutral pions in Pb-Pb collisions at sNN=2.76 TevChatrchyanS.KhachatryanV.SirunyanA. M.Multiplicity and transverse momentum dependence of two- and four-particle correlations in pPb and PbPb collisionsChatrchyanS.KhachatryanV.SirunyanA. M.Measurement of higher-order harmonic azimuthal anisotropy in PbPb collisions at sNN=2.76 TeVChatrchyanS.KhachatryanV.SirunyanA. M.Studies of azimuthal dihadron correlations in ultra-central Pb-Pb collisions at sNN=2.76 TevKhachatryanV.SirunyanA. M.TumasyanA.Long-range two-particle correlations of strange hadrons with charged particles in pPb and PbPb collisions at LHC energiesKhachatryanV.SirunyanA. M.TumasyanA.Evidence for collective multiparticle correlations in p-Pb collisionsKhachatryanV.SirunyanA. M.TumasyanA.Evidence for transverse-momentum- and pseudorapidity-dependent event-plane fluctuations in PbPb and pPb collisionsHuovinenP.KolbP. F.HeinzU. W.RuuskanenP. V.VoloshinS. A.Radial and elliptic flow at RHIC: further predictionsKolbP. F.HuovinenP.HeinzU.HeiselbergH.Elliptic flow at SPS and RHIC: from kinetic transport to hydrodynamicsLuzumM.RomatschkeP.Conformal relativistic viscous hydrodynamics: applications to RHIC results at sNN=200 GeVLuzumM.RomatschkeP.Viscous hydrodynamic predictions for nuclear collisions at the LHCSongH.HeinzU. W.Causal viscous hydrodynamics in 2 + 1 dimensions for relativistic heavy-ion collisionsSongH.BassS. A.HeinzU.HiranoT.ShenC.200 A GeV Au + Au collisions serve a nearly perfect quark-gluon liquidNiemiH.EskolaK. J.PaatelainenR.Event-by-event fluctuations in a perturbative QCD + saturation + hydrodynamics model: Determining QCD matter shear viscosity in ultrarelativistic heavy-ion collisionsHeinzU.SnellingsR.Collective flow and viscosity in relativistic heavy-ion collisionsLuzumM.PetersenH.Initial state fluctuations and final state correlations in relativistic heavy-ion collisionsHuovinenP.Hydrodynamics at RHIC and LHC: what have we learned?ShuryakE.Heavy ion collisions: achievements and challengeshttps://arxiv.org/abs/1412.8393SongH.Hydrodynamic modelling for relativistic heavy-ion collisions at RHIC and LHCDuslingK.LiW.SchenkeB.Novel collective phenomena in high-energy proton-proton and proton-nucleus collisionsKovtunP.SonD. T.StarinetsA. O.Viscosity in strongly interacting quantum field theories from black hole physicsZhuX.ZhouY.XuH.SongH.Correlations of flow harmonics in 2.76 A TeV Pb-Pb collisionshttp://arxiv.org/abs/1608.05305BhaleraoR. S.OllitraultJ.-Y.PalS.Characterizing flow fluctuations with momentsZhouY.XiaoK.FengZ.LiuF.SnellingsR.Anisotropic distributions in a multiphase transport modelGiacaloneG.YanL.Noronha-HostlerJ.OllitraultJ. Y.Symmetric cumulants and event-plane correlations in Pb + Pb collisionsSnellingsR.Anisotropic flow at the LHC measured with the ALICE detectorZhouY.YanL.OllitraultJ.-Y.PoskanzerA. M.Eccentricity distributions in nucleus-nucleus collisionsYanL.OllitraultJ.Universal fluctuation-driven eccentricities in proton-proton, proton-nucleus, and nucleus-nucleus collisionsYanL.OllitraultJ.-Y.PoskanzerA. M.Azimuthal anisotropy distributions in high-energy collisionsBravinaL. V.FotinaE. S.KorotkikhV. L.Anisotropic flow fluctuations in hydro-inspired freeze-out model for relativistic heavy ion collisionsJiaJ.Event-shape fluctuations and flow correlations in ultra-relativistic heavy-ion collisionsPetersenH.QinG. Y.BassS. A.MüllerB.Triangular flow in event-by-event ideal hydrodynamics in Au + Au collisions at sNN=200A GeVQiuZ.HeinzU. W.Event-by-event shape and flow fluctuations of relativistic heavy-ion collision fireballsNiemiH.DenicolG. S.HolopainenH.HuovinenP.Event-by-event distributions of azimuthal asymmetries in ultrarelativistic heavy-ion collisionsHeinzU.QiuZ.ShenC.Fluctuating flow angles and anisotropic flow measurementsGardimF. G.GrassiF.LuzumM.OllitraultJ.-Y.Breaking of factorization of two-particle correlations in hydrodynamicsZhouY.Searches for pT dependent fluctuations of flow angle and magnitude in Pb-Pb and p-Pb collisionsKozlovI.LuzumM.DenicolG. S.JeonS.GaleC.Signatures of collective behavior in small systemsAndronicA.StoiceaG.PetroviciM.Transition from in-plane to out-of-plane azimuthal enhancement in Au + Au collisionsChungP.AjitanandN. N.AlexanderJ. M.Centrality and momentum-selected elliptic flow: tighter constraints for the nuclear equation of stateAdamsJ.AdlerC.AggarwalM. M.Azimuthal anisotropy at the relativistic heavy ion collider: the first and fourth harmonicsTeaneyD.YanL.Event-plane correlations and hydrodynamic simulations of heavy ion collisionsBhaleraoR. S.OllitraultJ. Y.PalS.Event-plane correlatorsBilandzicA.ChristensenC. H.GulbrandsenK.HansenA.ZhouY.Generic framework for anisotropic flow analyses with multiparticle azimuthal correlationsBhaleraoR. S.LuzumM.OllitraultJ.-Y.Understanding anisotropy generated by fluctuations in heavy-ion collisionsTeaneyD.YanL.Nonlinearities in the harmonic spectrum of heavy ion collisions with ideal and viscous hydrodynamicsSchukraftJ.TimminsA.VoloshinS. A.Ultra-relativistic nuclear collisions: event shape engineeringHuoP.JiaJ.MohapatraS.Elucidating the event-by-event flow fluctuations in heavy-ion collisions via the event-shape selection techniqueWangX.-N.GyulassyM.Hijing: a Monte Carlo model for multiple jet production in pp, pA, and AA collisionsGyulassyM.WangX.-N.HIJING 1.0: a Monte Carlo program for parton and particle production in high energy hadronic and nuclear collisionsZhouY.ShiS. S.XiaoK.WuK. J.LiuF.Higher moments of net baryon distribution as probes of the QCD critical pointLinZ.-W.KoC. M.LiB.-A.ZhangB.PalS.Multiphase transport model for relativistic heavy ion collisionsZhouY.Measurements of correlations of anisotropic flow harmonics in Pb-Pb Collisions with ALICEhttps://arxiv.org/abs/1512.05397