An experimental review on heavy flavor $v_{2}$ in heavy-ion collision

For over a decade now, the primary purpose of relativistic heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) has been to study the properties of QCD matter under extreme conditions -high temperature and high density. The heavy-ion experiments at both RHIC and LHC have recorded a wealth of data in p+p, p+Pb, d+Au, Cu+Cu, Cu+Au, Au+Au, Pb+Pb and U+U collisions at energies ranging from $\sqrt{s_{NN}}$ = 7.7 GeV to 7 TeV. Heavy quarks are considered good probe to study the QCD matter created in relativistic collisions due to their very large mass and other unique properties. A precise measurement of various properties of heavy flavor hadrons provides an insight into the fundamental properties of the hot and dense medium created in these nuclei-nuclei collisions, such as transport coefficient and thermalization and hadronization mechanisms. The main focus of this paper is to present a review on the measurements of azimuthal anisotropy of heavy flavor hadrons and to outline the scientific opportunities in this sector due to future detector upgrade. We will mainly discuss the elliptic flow of open charmed meson ($D$-meson), $J/\psi$ and leptons from heavy flavor decay at RHIC and LHC energy.

preserving the information from the system at early stage [6][7][8][9][10][11][12][13]. While traversing the hot and dense matter produced in nucleus-nucleus collisions, hard partons (partons with high transverse momentum p T ) produced in the early stages of the collision lose energy dominantly due to multiple scatterings and radiative energy loss. Hence, become quenched. Theoretical models predict that the mechanism as well as average energy loss will be different for heavy quarks compared to light quarks [14][15][16]. Therefore, high p T charmed mesons (D 0 , D ± , D * , D ± S etc.) will show different suppression with respect to light mesons (π, K, K 0 S , etc.). In contrast, measurements of heavy flavor decay electrons at RHIC and charm hadrons at the LHC have shown significant suppression at high transverse momentum, p T , similar to that of light hadrons for central collisions. Therefore, a complete understanding of the energy loss mechanisms in the QGP medium requires a systematic and precise measurements of the properties of various hadrons carrying different quark flavors at RHIC and the LHC. The dependence of the partonic energy loss on the in-medium path length is expected to be different for different energy loss mechanism. It is suggested that low-momentum heavy quarks could undergo hadronization both via fragmentation in the vacuum and recombination with other quarks from the medium [17]. Azimuthal anisotropy measurements of the production of heavy-flavor hadron with respect to the reaction plane can be very useful in addressing these questions.
For a given rapidity window the second coefficient is given by where p x and p y are the x and y components of the particle momenta. At small transverse momentum, p T , a large v 2 is considered to be an evidence for the collective hydrodynamical expansion of the medium. Positive v 2 , if observed at very high p T is expected to be due to path-length dependent energy loss by hard partons. Unlike light quarks and gluons, which can be produced or annihilated during the entire evolution of the medium, heavy quarks are expected to be produced mainly in initial hard scattering processes and their annihilation rate is small. Therefore, for all p T , the final state heavy-flavor hadrons originate from heavy quarks that have experienced each stage of the system evolution.
The paper is organized in the following way. Section II describes sensitivity of heavy flavor hadron as probe of QCD medium. In Sec. III, elliptic flow of heavy-flavor decay electron are briefly discussed. Section IV and V describe elliptic flow of open charmed meson and J/ψ, respectively, measured at RHIC and LHC. Comparisons between model and data are also presented in Sec. IV and V. Finally, we summarize in Sec. VI .

II. ELLIPTIC FLOW OF HEAVY FLAVOR AS A SENSITIVE PROBE
We have recently studied the elliptic flow of open charm mesons, using quark coalescence as a mechanism of hadronization within the framework of a multi-phase transport model (AMPT) [6]. This study includes effect of partonic interaction cross-section, QCD coupling constant and specific viscosity on elliptic flow of open charm mesons within the transport model approach. The AMPT model is a hybrid transport model [31]. It uses the same initial conditions as in HIJING. In the AMPT model, the value of parton parton scattering cross-section, σ P P , is calculated by where α s and µ are the QCD coupling constant and screening mass respectively. Using the framework of AMPT model one can study the effect of specific viscosity on elliptic flow of hadrons. For a system of massless quarks and gluons at temperature T (T = 378 MeV at RHIC energy in AMPT [32]), the specific viscosity is given by [32] η s s ≈ 3π 40α 2 Hadronization of heavy quarks are not implemented in AMPT model. It only gives phase space information heavy quarks at freeze-out. We have implemented quark coalescence mechanism to form open charm mesons using phase space information of quarks available from AMPT model. Within the framework of coalescence mechanism [33], the probability of producing a hadron from a soup of partons is determined by the overlap of the phase space distribution of partons at freeze-out with the parton Wigner phase space function inside the hadron. The Wigner phase space function for quarks inside a meson is obtained from its constituent quark wave function [34] where the relative momentum between the two quarks is k = (k 1 − k 2 )/2 and the quark wave function is given by spherical harmonic oscillator described as with r = (r 1 − r 2 ) being the relative coordinate and σ is the size parameter related to the root mean square radius as r 2 = (3/8) 1/2 σ. We have taken σ = 0.47 fm 2 from Ref. [34].  This is consistent with the interpretation that increased sheer viscosity reduces transverse expansion due to increased interactions and hence, reduces v 2 . We can also see that the change in v 2 for charged hadrons is ∼ 15%, whereas for GeV, v 2 is consistent with zero as shown in Fig. 2(b). A very high precision measurement is required at 39 and 62.4 GeV, to understand NPE v 2 at these energies.
The nuclear modification factors (R AA ) and elliptic flow of NPE in Pb+Pb collisions at 2.76 TeV is shown in The results from the models calculations [40][41][42], that include parton energy loss in the hot and dense QCD medium, are shown as lines for both v 2 and R AA in Fig. 3. The simultaneous description of the measured v 2 and R AA is challenging for models. BAMPS [40] gives a good description of NPE v 2 but predicts a larger inmedium suppression than measured. In BAMPS approach, heavy quarks are transported through the medium while undergoing collisional and radiative energy loss. The prediction from POWLANG [42] describes the NPE R AA but their calculation underestimates NPE v 2 . In POWLANG, heavy quarks are transported following a Langevin approach and consider collisional energy loss only. The prediction from Rapp et al [41] (TAMU) and MC@sHQ+EPOS, Coll+Rad(LPM) [43]  results obtained with the scalar product and two-particle cumulant methods respectively. The event plane is estimated from TPC tracks within |η| < 0.8. For the other methods, TPC tracks in |η| < 0.8 were used as reference particle. The elliptic flow of D 0 , D + and D * + mesons are consistent within uncertainties. At very high p T (p T > 12 GeV/c), v 2 is consistent with zero within the large statistical uncertainties. In the p T range between 2 < p T < 6 GeV/c, the measured v 2 is found to be larger than zero with 5.7 σ significance. It suggests that low p T charm quarks possibly participate in the collective expansion of the medium. However, the possibility that the observed D-meson v 2 is completely due to the contribution from light-quark in a scenario with hadronization via recombination cannot be ruled out. We need high precision data at low p T and more theoretical understanding about the charm quark hadronization to understand the origin of collectivity of measured D-mesons v 2 .
The p T dependence of D 0 v 2 in the three centrality classes 0-10%, 10-30% and 30-50% are presented in Fig. 5 [44]. v 2 of charged hadrons are also shown for comparison [48]. Both the measurements are done with the event plane method.
For these three centrality classes, the D 0 meson v 2 is comparable in magnitude to that of inclusive charged hadrons.
These results indicate that the interactions with the medium constituents transfer information of the azimuthal anisotropy of the system to the charmed particles.
The STAR experiment at RHIC has also reported the first preliminary results of D 0 (p T ) v 2 at mid-rapidity (|y|  [45]. Measurements are done at mid-rapidity (|y| <1.0). The blue and black data points are the D 0 v 2 measured using two-particle correlation and event plane method respectively. Results from both the method are consistent within statistical uncertainty. The D ± v 2 is shown by red symbol and calculated using event plane method. D 0 azimuthal anisotropy is non-zero for 2 < p T < 5 GeV/c. The right panel of Fig. 6 shows the comparison of D 0 v 2 with other mesons species (K 0 S and φ). It seems that D 0 v 2 for p T < 4 GeV/c is systematically lower than K 0 S [49] and φ [50], but one should be very careful while comparing different particle species for a wide centrality bin e.g.

Coalescence Based Models
The model by Cao et al. is based on the Langevin approach where the space-time evolution of the medium is modeled using viscous hydrodynamic. In this model, hadronization is done using quark coalescence mechanism. This model describes R AA in central collisions very well, but tend to underestimate the v 2 at low p T . The model [57], labeled as MC@sHQ+EPOS, Coll+Rad(LPM), is a perturbative QCD (pQCD) model that includes collisional and radiative energy loss mechanisms for heavy quarks. Hadronization is performed via quark recombination in this model. It underestimates the low-p T suppression, but yields a substantial anisotropy (∼10%) which slightly underestimates observed data. It correctly describes high-p T suppression. In the TAMU model [58], heavy-quark transport coefficient is calculated within a non-perturbative T-matrix approach. This model includes hydrodynamic medium evolution and quark coalescence as a mechanism of hadronization. This model provides a good description of the observed suppression of D mesons over the entire p T range. However, it fails to reproduces observed anisotropy for p T > 4 GeV/c. The Ultra relativistic Quantum Molecular Dynamics (UrQMD) [59] model is based on a microscopic transport theory where the phase space description of the reactions are important. The hybrid UrQMD model includes a realistic description of the medium evolution by combining hadronic transport and ideal hydrodynamics. Hadroniztion via quark recombination is implemented. The model describes the measured anisotropy and suppression in the interval 4 < p T < 8 GeV/c, but fails to explain the data for very low and high p T region.

Fragmentation Based Models
In WHDG model, the observed anisotropy results from path-length dependent energy loss and hadronization is performed using vacuum fragmentation function. This model describes R AA in central collisions reasonably well, but tend to underestimate the v 2 at low p T . BAMPS model is a partonic transport model which includes multiparton scattering based on Boltzmann approach. Like WHDG model, in BAMPS, hadronization is performed using vacuum fragmentation functions. BAMPS model describes both R AA and v 2 reasonably well. POWLANG [56] is also a transport model which is based on collisional processes treated within the framework of Langevin dynamics.
Hadronization, in this model, is done using vacuum fragmentation functions. This model overestimates the high-p T suppression, and significantly underestimates observed v 2 at low p T .
In summary, models including hadronization of charm quarks from recombination with light quarks from the medium (e.g. TAMU) provide a better description of the data at low transverse momentum.  [45,46] and RAA (0-10%) [47] in Au+Au collisions at √ sNN = 200 GeV and comparison with selected theoretical models [54,58,60].

C. Model Comparisons at RHIC Energy
anisotropy. The TAMU and SUBATECH describe D 0 -meson v 2 data reasonably well. Fig. 9 (left panel) shows comparison between our AMPT model calculations [6] for D 0 -meson v 2 and measured D 0 -meson v 2 at 2.76 TeV for 30-50% central collisions by the ALICE experiment. Here σ P P is taken to be 1.5 mb and 10mb with other parameters tuned for LHC data (charged hadron v 2 and multiplicity). Previous study shows that 1.5 mb parton parton scattering cross-section is sufficient to described charged hadron v 2 at mid-rapidity for p T < 2.0 GeV/c. However, we find that cross-sections of both both 1.5 and 10 mb underestimate LHC D 0 -meson v 2 data.
It would be very interesting to see how the data and model behave at low p T (below 2 GeV/c). Therefore, the results from future ALICE upgrade [61] will be very useful to study both heavy flavor and charged hadrons v 2 at low p T .
The right panel of Fig. 9 shows the comparison between D 0 v 2 with AMPT model predictions at top RHIC energy.
AMPT model calculation roughly explain data within large statistical errors.

V. ELLIPTIC FLOW OF J/ψ
The J/ψ meson is bound state of charm (c) and anti-charm (c) quark. It was discovered independently by two research groups on 11 November 1974 [62,63]. The importance of this discovery is highlighted by the fact that the subsequent rapid changes in high-energy physics around that time came to be collectively known as the "November Revolution". In relativistic heavy-ion collisions, J/ψ can be produced mainly by recombination of charm (c) and anticharm and/or direct pQCD processes. By measuring anisotropic flow of J/ψ, one may infer the relative contribution of J/ψ particles from recombination and from direct pQCD processes. J/ψ produced from quark recombination will inherit the flow of charm quarks. On the other-hand, if J/ψ is produced from direct pQCD processes, it should have very little v 2 . A detailed comparison between experimental measurements and models on J/ψ v 2 will be helpful to understand the production mechanism of J/ψ. Figure 10 show elliptic flow of for J/ψ at mid-rapidity in 0-80% min-bias Au+Au events at √ s N N =200 GeV [64] compared with charged hadrons [65] and φ meson [66] in the upper panel and with theoretical calculations in the lower panel. J/ψ v 2 is found to be very small in comparison to that of charged hadrons and φ-meson. Models which include J/ψ production from coalescence of thermalized cc [68], gives the maximum of J/ψ v 2 to be almost the same in magnitude as light hadrons. v 2 of J/ψ produced from initial pQCD processes [67] is predicted to be very small compared to light hadrons. Models that include J/ψ production from both initial pQCD process and coalescence mechanism [69,70] also give much smaller J/ψ v 2 in comparison to light hadrons.
In summary, models thats include J/ψ production from both initial pQCD process and coalescence production or entirely from initial pQCD process describe the data better at top RHIC energy. At this point, it is still unclear and we would need very high precision measurements to estimate the fraction of the total J/ψ yield that comes from   TeV within the rapidity range 2.5<y<4.0 [71]. J/ψ v 2 for non-central (20-60%) Pb-Pb collisions at √ s N N = 2.76 TeV is shown in Fig. 11. Unlike RHIC, an indication of non-zero J/ψ v 2 is observed with a maximum value of v 2 = 0.090 ± 0.041 (stat) ± 0.019 (syst) for non-central (20-60%) Pb-Pb collisions. Calculations from two transport models [72,73] are also shown for comparison. Transport model calculations that include a J/ψ regeneration component (30%) from deconfined charm quarks in the medium describes data very well. A precise measurement of heavy flavor hadrons will provide information on fundamental properties of the medium, such as the transport coefficient, and hadronization mechanisms. As we discussed, current heavy flavor measurements are limited by statistics. In order to circumvent this, recent upgrades have been made in STAR with the introduction of the Heavy Flavor Tracker (HFT) and Muon Telescope Detector (MTD) and dedicated high statistics run in 2016.
We hope to see the results from these soon. The ALICE experiment is also upgrading its detectors to pursue high precision measurements in the heavy flavor sector [61]. and v 2 could be interesting to shed light on heavy quark dynamics. The new ITS is expected to allow for a precise measurement of the all D-mesons v 2 down to very low momentum [61].