# Review of anisotropic flow correlations in ultrarelativistic heavy-ion collisions

###### Abstract

Anisotropic flow phenomena is a key probe of the existence of Quark-Gluon Plasma. Several new observable 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.

###### pacs:

25.75.Ld, 25.75.GzGB

## I 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) Lee:1978mf (); Shuryak:1980tp (). 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 Ollitrault:1992bk (); Voloshin:1994mz (). It can be characterized by an expansion of the single-particle azimuthal distribution :

(1) |

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 () in terms of and as:

(2) |

In the last decade, the experimental measurements of anisotropic flow Alt:2003ab (); Ackermann:2000tr (); Adler:2001nb (); Adler:2002pu (); Adams:2004bi (); Adams:2005zg (); Abelev:2008ae (); Adamczyk:2013gw (); Adcox:2002ms (); Adler:2003kt (); Adare:2006ti (); Afanasiev:2009wq (); Adare:2010ux (); Adare:2011tg (); Back:2002gz (); Back:2004mh (); Back:2004zg (); Manly:2005zy (); Back:2005pc (); Alver:2006wh (); Alver:2007qw (); Alver:2010rt (); Aamodt:2010pa (); ALICE:2011ab (); Abelev:2012di (); Abelev:2013cva (); Abelev:2014mda (); Abelev:2014pua (); Adam:2015eta (); Adam:2015vje (); Adam:2016izf (); ALICE:2016kpq (); ATLAS:2011ah (); ATLAS:2012at (); Aad:2013fja (); Aad:2013xma (); Aad:2014fla (); Aad:2014eoa (); Aad:2014vba (); Aad:2014lta (); Aad:2015lwa (); Chatrchyan:2012wg (); Chatrchyan:2012xq (); Chatrchyan:2012ta (); Chatrchyan:2012vqa (); Chatrchyan:2013nka (); Chatrchyan:2013kba (); CMS:2013bza (); Khachatryan:2014jra (); Khachatryan:2015waa (); Khachatryan:2015oea (), combined with theoretical advances from calculations made in a variety of frameworks Huovinen:2001cy (); Kolb:2000fha (); Luzum:2008cw (); Song:2007ux (); Song:2010mg (); Niemi:2015qia (), have led to a 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 Heinz:2013th (); Luzum:2013yya (); Huovinen:2013wma (); Shuryak:2014zxa (); Song:2013gia (); Dusling:2015gta (), which is close to a quantum limit 1/4 Kovtun:2004de ().

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 Collaboration:2011yba (); Aad:2014vba (); Chatrchyan:2012ta (); Chatrchyan:2013kba (); Zhou:2016fvj () and the event-by-event distributions Aad:2013xma (), it was realized that the newly proposed Elliptic-Power function Yan:2014afa (); Yan:2013laa (); Yan:2014nsa () gives the best description of underlying of single harmonic distributions Bravina:2015sda (); Zhou:2015eya (); Jia:2014jca (). 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 Petersen:2010cw (); Qiu:2011iv (); Niemi:2012aj (), but only recently the and dependent flow angle () and magnitude () were predicted by hydrodynamic calculations Heinz:2013bua (); Gardim:2012im (). Many indications were quickly obtained in experiments by looking at the deviations from unity of Zhou:2014bba () and factorization ratio Zhou:2014bba (); CMS:2013bza (); Khachatryan:2015oea (). These measurements were nicely predicted or reproduced by hydrodynamic calculations, and are found to be sensitive to either the initial-state density fluctuations and/or the shear viscosity of the expanding fireball medium Heinz:2013bua (); Gardim:2012im (); Kozlov:2014hya (). Most of these above mentioned studies are focused on the fluctuations of single flow harmonics and its corresponding symmetry planes, as a function of collisions centrality, transverse momentum and pseudorapidity . Results of correlations between symmetry planes Aad:2014fla (); ALICE:2011ab () reveal a new type of correlations between different order flow-vectors, which was investigated in the observable of before Andronic:2000cx (); Chung:2001qr (); Adams:2003zg (). In particular, some of the symmetry planes correlations show quite different centrality dependence from the initial- and final-state, and this characteristic sign change during system evolution is correctly reproduced by theoretical calculations Qiu:2011iv (); Teaney:2013dta (); Niemi:2015qia (), thus confirms the validity of hydrodynamic framework in heavy-ion collisions and further yields valuable additional insights into the fluctuating initial conditions and hydrodynamic response Qiu:2011iv (); Bhalerao:2013ina (); Niemi:2015qia ().

In addition to all these observables, the (anti-)correlations between anisotropic flow harmonics and are found to be extremely interesting Bilandzic:2013kga (); Bhalerao:2014xra (); Niemi:2015qia (); Aad:2015lwa (); SC:hydro (). A completely new set of information on the joint is carried by the rich pattern observed in experiments. On the other hand, no existed theoretical calculations Bilandzic:2013kga (); Bhalerao:2014xra (); Niemi:2015qia (); SC:hydro () could provide quantitative descriptions of data ALICE:2016kpq (). Thus, it’s crucial to investigate in depth of the relationship between different flow harmonics: whether they are correlated, anti-correlated or not correlated, from both experimental and theoretical point of view.

## Ii 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 Bilandzic:2013kga (); Bhalerao:2011yg (); Teaney:2012ke (); Bhalerao:2014xra (); ALICE:2016kpq (). In order to better understand the event-by-event distribution, it’s critical to investigate the relationship between and . Considering the naive ellipsoidal shape of the overlap region in non-central heavy ion collisions generates non-vanishing 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 odd flow harmonics for central and non-central collisions Jia:2014jca (). A linear correlation function was proposed to study the relationship between and Niemi:2012aj (). It is defined as:

(3) |

where is the standard deviation of the quantity , is 1 (or -1) if and are linearly (anti-linearly) correlated, and is 0 if not correlated.

It was shown in Fig 1 that there is an anti-correlationsbetween 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 Niemi:2012aj (). Nevertheless, it cannot be accessible easily in experimental measurements, which rely on two- and multi-particle 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.

### ii.1 Event Shape Engineering (ESE)

The first experimental attempt was made by ATLAS Collaboration Aad:2015lwa (), using the Event-Shape Engineering (ESE) Schukraft:2012ah (). This is a technique to select events according to the magnitude of reduced flow vector . Fig. 2 shows the performance of event shape selection on (left) and (right) in ATLAS detector. For each centrality the data sample is divided into several event classes according to the or distributions. Then the and relationship was investigated by measurements of and in each event class from ESE selection. Without using ESE selection, a boomerang-like patten was observed for the centrality dependence of - correlation. This is mainly due to the fact that has a weaker centrality dependence than . By using ESE, it was observed in Fig. 3 (right) that for event class with the same centrality (shown as the same color), decreases as increasing. It suggests that is anti-correlated with . Considering the linear hydrodynamic response of and from eccentricity and triangularity , the anti-correlation between and might reveal the anti-correlation between and of the initial geometry. This indication of initial anti-correlations between and is observed in model calculations Schukraft:2012ah (); Huo:2013qma ().

Figure 4 shows the investigation of relationship between and . A boomerang-like patten, although weaker than that for the - relationship shown in Fig. 3 (left), is observed in Fig. 4 (left), prior to the ESE selection. After the ESE selection, it is found in Fig. 4 (right) 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 Aad:2015lwa (). This non-linear contribution of from is further investigated by fitting the correlation pattern using , where and denote the linear and non-linear components. It is found that the linear component has a weak centrality dependence, while the non-linear component, increasing dramatically with collision centrality, becomes the dominant contribution in the most peripheral collisions Aad:2015lwa ().

These (anti)correlation patten 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 non-flow effects, and they require sub-dividing such calculations and modeling resolutions associated with ESE due to finite event-wise multiplicities. Considering the computational constraints, this approach can not be performed easily in hydrodynamic calculations which usually are based on limited statistics compared to experimental data.

### ii.2 Symmetric Cumulants ()

A new type of observable for the analyses of flow harmonic correlations, Symmetric Cumulants (originally named Standard Candles (SC) in Bilandzic:2013kga ()), was proposed as . If , the isotropic part of the corresponding four-particle cumulant is given by:

(4) | |||||

For a detector with uniform acceptance in azimuthal direction, the asymmetric terms, e.g. , are averaged to zero. The single event 4-particle correlation could be calculated as:

(5) | |||||

And the single event 2-particle correlation could be obtained as:

(6) |

Then, the weights of and are used to get the event-averaged 2- and 4-particle correlations, as introduced in Bilandzic:2013kga (). Due to the definition, this new type of 4-particle cumulant is independent of the symmetry planes and , and is expected to be less sensitive to non-flow correlations, which should be strongly suppressed in 4-particle cumulants. This was confirmed by the calculation using HIJING model Wang:1991hta (); Gyulassy:1994ew () which does not include anisotropic collectivity but e.g. azimuthal correlations due to jet production. It is observed that both and are non-zero, while are compatible with zero in HIJING simulations ALICE:2016kpq (). This confirms that the measurements are nearly insensitive to non-flow 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 a 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 Bilandzic:2013kga (). Non-zero values for both and are observed. The 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 implies that finding larger than enhances the probability of finding smaller than Bilandzic:2013kga ().

Several configurations of the AMPT model have been investigated to better understand the results based on AMPT simulations Bilandzic:2013kga (). 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 Zhou:2010us (); Lin:2004en (). 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 Zhou:2010us (); Lin:2004en (). The and calculations for three different scenarios: (a) 3 mb; (b) 10 mb; (c)10 mb, no rescattering are presented in Fig. 5 (left). 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 ‘10mb, 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 Zhou:2015eya (). The centrality dependence of and are presented as red circles and blue diamonds in Fig. 5 (left bottom). 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 Zhou:2016fvj ().

Based on AMPT calculations, it seems that the signs of (for 2, 3, 4) 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 Fig. 6 (left). Positive values of are observed for all centralities. This confirms a correlation between the event-by-event fluctuations of and . On the other hand, the measured negative results of show the anti-correlation between and magnitudes. The same measurements are performed using the like-sign technique, which is another powerful approach to estimate non-flow effects Aamodt:2010pa (). It was found that the difference between correlations for like-sign and all charged combinations, which might be mainly due to non-flow effects, are much smaller compared to the magnitudes of itself. This further proves that non-zero values of measured in experiments cannot be explained by non-flow effects solely.

In addition, the comparison between experimental data and the event-by-event perturbative-QCDsaturationhydro (“EKRT”) calculations Niemi:2015qia (), which incorporate both initial conditions and hydrodynamic evolution, is shown in Fig. 6. It was shown that this model can capture quantitatively the centrality dependence of individual , and harmonics in central and mid-central collisions Niemi:2015qia (). However, it can only qualitatively, but not quantitatively predict the measurements by ALICE. For a given parameterization tuned by individual flow harmonic, the calculation can not describe and simultaneously for any single centrality. Experimental measurements are also compared to the VISH21 model calculations (see Fig. 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 the measurement SC:hydro (). However, just like EKRT calculations, none of these combinations is able to describe and simultaneously. Thus, it is concluded that the new SC observables provide a better handle on the initial conditions and than each of the individual harmonic measurement alone.

After being presented for the first time at Quark Matter 2015 conference, preliminary results of and gained a lot of attention Zhou:2015slf (). 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 Fig. 6 (right), with normalized and observables by dividing with the products and , respectively ALICE:2016kpq (). The 2-particle correlations and are obtained with a pseudorapidity gap of to suppress contributions from non-flow effects. It was shown in Fig. 8 (top left) that the normalized observable exhibits a 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 Fig. 5 (right) that the normalized , also named in the following text, is compatible with its corresponding observable in the initial state. Thus, the could be taken as golden observable to directly constrain initial conditions without demands for precise knowledge of transport properties of the system SC:hydro (). Furthermore, none of existing theoretical calculations can reproduce the data, there is still a long way to go for the development of hydrodynamic calculations.

Observables | Equations | number of particles | Exp. | Th. |
---|---|---|---|---|

4 | ALICE:2016kpq () | SC:hydro (), Bhalerao:2014xra (), Zhou:2015eya () | ||

4 | ALICE:2016kpq () | SC:hydro (), Bhalerao:2014xra (), Zhou:2015eya (), Giacalone:2016afq () | ||

4 | SC:hydro (), Bhalerao:2014xra (), Giacalone:2016afq () | |||

4 | ||||

4 | SC:hydro () | |||

4 | SC:hydro (), Bhalerao:2014xra (), Giacalone:2016afq () | |||

4 | ||||

4 | ||||

4 | ||||

4 | ||||

… | … | 6 |

Predictions of relationship between other harmonics are provided in SC:hydro () and shown in Fig. 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 0.20, while the maximum value is seen in more peripheral collision if 0.08 is used.

Compared to the previous measurements of relationship between flow harmonics investigated using the ESE technique, the 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, , , which is clearly non trivial. Although one did not use the information of symmetry planes in both and studies, recent study just reveals that flow harmonic correlations might be not completely independent on symmetry plane correlations Giacalone:2016afq (). The proportionality relations between symmetric cumulants involving higher harmonics or and symmetry plane correlations is derived, which seems 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 into the nature of fluctuating initial conditions and of the created QGP in heavy ion collisions.

## Iii Summary

In the past two decades, the underlying of each single harmonic was investigated in great details. However, it is an open question at the moment how the joint underlying , including different order symmetry planes and harmonics, is described, 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 , 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.

## Iv Acknowledgments

The author thanks J.J. Gaardhøje, K. Gajdošová, L. Yan, J.Y. Ollitrault and H. Song for the comments on the manuscripts and fruitful discussions. The author is supported by the Danish Council for Independent Research, Natural Sciences, and the Danish National Research Foundation (Danmarks Grundforskningsfond).

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