#### Abstract

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.

#### 1. 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 :where is the azimuthal angle of emitted particles, is the th order flow vector defined as , its magnitude is the th order anisotropic flow harmonic, and its orientation is symmetry plane (participant plane) angle . Alternatively, this anisotropy can be generally given by the joint probability density function (PDF) in terms of and as In the last decade, the experimental measurements of anisotropic flow [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 [63–68], which is close to a quantum limit 1/4 [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 distributions [40], it was realized that the newly proposed Elliptic-Power function [76–78] gives the best description of underlying PDF of single harmonic distributions [72, 79, 80]. On the other hand, it has been known for a while that both the flow harmonic (magnitude) and its symmetry plane (orientation) of the flow vector fluctuate event-by-event [81–83], but only recently and dependent flow angle () and magnitude () were predicted by hydrodynamic calculations [84, 85]. Many indications were quickly obtained in experiments by looking at the deviations from unity of [86] and factorization ratio [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 , and pseudorapidity . 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 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 and 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.

#### 2. Correlations of and Fluctuations

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 distribution, it is critical to investigate the relationship between and . Considering the naive ellipsoidal shape of the overlap region in noncentral heavy-ion collisions generating nonvanishing even flow harmonics , 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 was proposed to study the relationship between and [83]. It is defined aswhere is the standard deviation of the quantity ; is 1 (or −1) if and are linearly (antilinearly) correlated and is 0 if they are not correlated. It was shown in Figure 1 that there is an anticorrelation between and , while a correlation was observed between and . In addition, it was demonstrated that depends on both the initial conditions and , while is only sensitive to [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 and , are discussed in the following section.

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##### 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 . Figure 2 shows the performance of event shape selection on (a) and (b) in ATLAS detector. For each centrality the data sample is divided into several event classes according to or distributions. Then and relationship was investigated by measurements of and in each event class from ESE selection. Without using ESE selection, a boomerang-like pattern was observed for the centrality dependence of - correlation. This is mainly due to the fact that has weaker centrality dependence than . By using ESE, it was observed in Figure 3(b) that, for event class with the same centrality (shown as the same color), decreases as increases. It suggests that is anticorrelated with . Considering the linear hydrodynamic response of and from eccentricity and triangularity , the anticorrelation between and might reveal the anticorrelation between and of the initial geometry. This indication of initial anticorrelations between and is observed in model calculations [96, 97].

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Figure 4 shows the investigation of relationship between and . A boomerang-like pattern, although weaker than that for - 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 increases with increasing . 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 from is further investigated by fitting the correlation pattern using , where and 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].

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These (anti)correlation patterns between and 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 . If , the isotropic part of the corresponding four-particle cumulant is given by For a detector with uniform acceptance in azimuthal direction, the asymmetric terms, , , are averaged to zero. The single event 4-particle correlation could be calculated asAnd the single event 2-particle correlation could be obtained asThen, the weights of and 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 is independent of the symmetry planes and and is expected to be less sensitive to nonflow correlations, which should be strongly suppressed in 4-particle cumulants. This was confirmed by 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 and are nonzero, while are compatible with zero in HIJING simulations [36]. This confirms that measurements are nearly insensitive to nonflow correlations. Therefore, it is believed that is nonzero if there is (anti)correlations of and . The investigation of will allow us to know whether finding larger than in an event will enhance or reduce the probability of finding larger than in that event, which provides unique information for the event-by-event simulations of anisotropic flow harmonics.

Figure 5 shows the first calculation of (solid markers) and (open markers) as a function of centrality from AMPT model [93]. Nonzero values for both and are observed. Positive suggests a correlation between the event-by-event fluctuations of and , which indicates that finding larger than in an event enhances the probability of finding larger than in that event. On the other hand, the negative results of imply that finding larger than enhances the probability of finding smaller than [93].

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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 and 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 decreases. Additionally, the “10 mb, no rescattering” setup seems to give slightly smaller magnitudes of and .

Further studies have been performed in AMPT initial conditions, based on the observable of which is defined as [72]. The centrality dependence of and 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 . In contrast, negative and decreasing trend was observed for in the AMPT initial conditions. This shows that finding larger than in an event enhances the probability of finding larger than , while in parallel enhancing the probability of finding smaller than in that event. Same conclusions were obtained using MC-Glauber initial conditions [75].

Based on AMPT calculations, it seems that the signs of (for ) in the final state are determined by the correlations of in the initial state, while its magnitude also depends on the properties of the created system. This clearly suggests that 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 (red squares) and (blue circles) are presented in Figure 6(a). Positive values of are observed for all cases of centrality. This confirms a correlation between the event-by-event fluctuations of and . On the other hand, the measured negative results of show the anticorrelation between and 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 itself. This further proves that nonzero values of measured in experiments cannot be explained by nonflow effects solely.

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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 , , and harmonics in central and mid-central collisions [62]. However, it can only qualitatively but not quantitatively predict measurements by ALICE. For given parameterization tuned by individual flow harmonic, the calculation cannot describe and 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 0.08 and 0.20. It is noticed that the one with MC-Glb IC and 0.08 is compatible with measurement and the calculation with MC-AMPT IC and 0.08 can describe measurement [70]. However, just like EKRT calculations, none of these combinations is able to describe and simultaneously. Thus, it is concluded that the new observables provide better handle on the initial conditions and than each of the individual harmonic measurements alone.

After being presented for the first time at Quark Matter 2015 conference, preliminary results of and gained a lot of attention [102]. One of the key suggestions was to normalize by dividing with the products in order to get rid of influences from individual flow harmonics. The results are shown in Figure 6(b), with normalized and observables by dividing with the products and , respectively [36]. The 2-particle correlations and are obtained with a pseudorapidity gap of to suppress contributions from nonflow effects. It was shown in Figure 8(a) that the normalized observable exhibits clear sensitivity to different parameterizations and the initial conditions, which provides a unique opportunity to discriminate between various possibilities of the detailed setting of of the produced QGP and the initial conditions used in hydrodynamic calculations. On the other hand, normalized is independent of the setting of . In addition, it was demonstrated in Figure 9 that the normalized , also named in the following text, is compatible with its corresponding observable in the initial state. Thus, 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.

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Predictions of relationship between other harmonics are provided in [70] and shown in Figure 8. Besides different sensitivities to IC and 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 is observed in central collision using , while the maximum value is seen in more peripheral collision if is used.

Compared to the previous measurements of relationship between flow harmonics investigated using the ESE technique, 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 and 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 or 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 of the created QGP in heavy-ion collisions.

#### 3. Summary

In the past two decades, the underlying PDF of each single harmonic 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 . 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 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).