Systematic study of elliptic flow parameter in the relativistic nuclear collisions at RHIC and LHC energies

We employed the new issue of a parton and hadron cascade model PACIAE 2.1 to systematically investigate the charged particle elliptic flow parameter $v_2$ in the relativistic nuclear collisions at RHIC and LHC energies. With randomly sampling the transverse momentum $x$ and $y$ components of the particles generated in string fragmentation on the circumference of an ellipse instead of circle originally, the calculated charged particle $v_2(\eta)$ and $v_2(p_T)$ fairly reproduce the corresponding experimental data in the Au+Au/Pb+Pb collisions at $\sqrt{s_{NN}}$=0.2/2.76 TeV. In addition, the charged particle $v_2(\eta)$ and $v_2(p_T)$ in the p+p collisions at $\sqrt s$=7 TeV as well as in the p+Au/p+Pb collisions at $\sqrt{s_{NN}}$=0.2/5.02 TeV are predicted.


I. INTRODUCTION
To explore the phase transition from the hadronic matter (HM) to quark-gluon matter (QGM) is one of the fundamental aims of relativistic nuclear collisions. A couple years ago, four international collaborations of BRAHMS, PHOBOS, STAR, and PHENIX at RHIC have published white papers [1][2][3][4] to declare their evidences for the discovery of strongly coupled quark-gluon plasma (sQGP). One of the most important signals is the large elliptic flow parameter of produced particle in the Au+Au collisions at √ s N N =200 GeV.
The measurement of particle elliptic flow parameter v 2 is not trivial. Several methods have been proposed, such as the event plane method [5], Lee-Yang zero point method [6], and the cumulant method [7] etc. The cumulant method is even distinguished with two-, four-, and six-particle cumulants. The discrepancy among the v 2 values measured with the event plane method, Lee-Yang zero point method, and the cumulant method may reach a few ten percent as shown in Fig. 4 and 5 of [8] and in Fig. 11 of [9]. Recently, one even argued that the event plane method is obsolete [10].
On the other hand, the particle elliptic flow parameter v 2 is also not easy to investigate theoretically. The conventional (hadronic) transport (cascade) models always underestimated the v 2 experimental data in the nucleusnucleus collisions at RHIC and/or LHC energies. In [11] it was mentioned that the charged particle v 2 experimental data is around 0.05 in the Au+Au collisions at highest RHIC energy (estimated from v 2 (η) in [12]), while the UrQMD model provides only half of this value. They have pointed out that a lack of pressure in the model at this energy may be the reason and that the partonic rescattering has to be taken into account in order to describe the data.
Similarly, the default AMPT model (AMPT def ) also underestimated the v 2 experimental data in the nucleusnucleus collisions at RHIC energies [13]. In order to meet with experimental data they updated AMPT def to the AMPT sm with string melting. In the AMPT sm model the hadrons (strings) from HIJING [14] are all melted to the partons. Relying on the rescattering among huge number of partons AMPT sm is able to account for the v 2 experimental data, provided the parton-parton cross section is enlarged to ten mb. Of course, the AMPT sm model has to hadronize the partons after rescattering by the coalescence model rather than the string fragmentation in AMPT def .
In the non-center nucleus-nucleus collisions the geometric overlap zone leads to the initial particle spatial asymmetry distribution. It is then dynamically developed to the final hadronic state transverse momentum asymmetry due to the partonic rescattering [11] and the strong electromagnetic field [15] etc.. We have pointed out that the transverse momentum p ′ x and p ′ y of produced particle from string fragmentation are randomly arranged on a circle with radius of p ′ T in the PACIAE 2.0 model [16] (in the PYTHIA model [17] originally). Here the observable with superscript ( ′ ) refers to the string fragmentation local frame distinguished from the without superscript one referred to the nucleusnucleus cms frame. This symmetric arrangement strongly cancels the final hadronic state transverse momentum asymmetry developed from the initial spatial asymmetry. In the new issue of a PACIAE model (PACIAE 2.1 [18]) we randomly distribute the p ′ x and p ′ y of produced particle from the string fragmentation on the circumference of an ellipse instead of circle. PACIAE 2.1 is then able to describe the v 2 experimental data.
In the next section, section II, a parton and hadron cascade model PACIAE, its new issue of PACIAE 2.1, and the definition of elliptic flow parameter are briefly introduced. The calculated charged particle v 2 (η) and v 2 (p T ) are compared with the corresponding experimental data of the Au+Au/Pb+Pb collisions at √ s N N = 0.2/2.76 TeV in the section III. Additionally, the predictions for charged particle v 2 (η) and v 2 (p T ) in the p+p collisions at √ s=7 TeV and in the p+Au/p+Pb collisions at √ s N N = 0.2/5.02 TeV are also given in the section III. The last section is devoted to the conclusions.

II. MODELS
The PACIAE model is based on PYTHIA [17]. However, the PYTHIA model is for high energy hadron-hadron (hh) collisions but the PACIAE model is mainly for nucleus-nucleus collisions. In the PYTHIA model a hh collision is decomposed into parton-parton collisions. The hard parton-parton collision is described by the lowest leading order perturbative QCD (LO-pQCD) parton-parton interactions with the modification of parton distribution function in a hadron. The soft parton-parton collision, a non-perturbative process, is considered empirically. The initial-and final-state QCD radiations and the multiparton interactions are also taken into account. So the consequence of a hh collision is a partonic multijet state composed of the diquarks (anti-diquarks), quarks (antiquarks), and the gluons, besides a few hadronic remnants. It is followed by the string construction and fragmentation, thus a final hadronic state is obtained for a hh (pp) collision eventually.
In the PACIAE model [16], the nucleons in a nucleus-nucleus collision are first randomly distributed in the spatial phase space according to the Woods-Saxon distribution. The participant nucleons, resulted from Glauber model calculation, are required to be inside the overlap zone, formed when two colliding nuclei path through each other at a given impact parameter. The spectator nucleons are required to be outside the overlap zone but inside the nucleus-nucleus collision system. Then we decompose a nucleus-nucleus collision into nucleon-nucleon (N N ) collisions according to nucleon straight-line trajectories and the N N total cross section. Each N N collision is then dealt by PYTHIA with the string fragmentation switched-off and the diquarks (anti-diquarks) broken into quark pairs (anti-quark pairs). A partonic initial state (composed of the quarks, antiquarks, and the gluons) is obtained for a nucleus-nucleus collision after all of the N N collision pairs were exhausted. This partonic initial stage is followed by a parton evolution stage, where parton rescattering is performed by the Monte Carlo method with 2 → 2 LO-pQCD cross sections [19]. The hadronization stage follows the parton evolution stage. The Lund string fragmentation model and a phenomenological coalescence model are provided for the hadronization. However, the string fragmentation model is selected in this calculations. Then the rescattering among produced hadrons is dealt with the usual two body collision model [16]. In this hadronic evolution stage, only the rescatterings among π, K, p, n, ρ(ω), ∆, Λ, Σ, Ξ, Ω, and their antiparticles are considered for simplicity.  [24], here NSD refers to the non-single diffractive.
The PACIAE 2.0 model [16] is mainly different from AMPT sm as follows: 1. The partonic initial state is obtained by breaking the strings from PYTHIA in PACIAE 2.0, but by breaking hadrons from HIJING in AMPT sm.
2. The gg → gg elastic scattering cross section is utilized in the parton rescattering in AMPT sm but specific scattering cross section is used for individual qq (gg) scattering processes in PACIAE 2.0 .
3. In the AMPT sm model the partons after rescattering are hadronized by the coalescent model but by string fragmentation in the present PACIAE calculations. Because of the first difference, the number of initial partons in PACIAE 2.0 is much less than the one in AMPT sm. Hence the strength of partonic rescattering effect in the former is not as strong as that in the later. Therefore relying on partonic rescattering only the PACIAE model is hard to describe v 2 experimental data, unlike AMPT sm. The rearrangement for the transverse momentum x and y components of the particles from string fragmentation, mentioned above, is then required.
The spatial overlap zone formed in non-center nucleus-nucleus collision is almond-like, which is always assumed to be an ellipse with a half minor axis of a r = R A (1 − δ r ) along the x axis (axis of impact parameter) and a half major axis of b r = R A (1 + δ r ) along the y axis (here R A refers to the radius of nucleus provided a symmetry nucleus-nucleus collisions is considered). Originally this initial spatial asymmetry may develop dynamically into a final hadronic state momentum asymmetry due to the parton rescattering and the strong electromagnetic field etc.. Unfortunately, in the PYTHIA (PACIAE 2.0) model once the transverse momentum p ′ T of the produced particle from string fragmentation is randomly sampled according to the exponential and/or Gaussian distribution, its p where φ ′ refers to the azimuthal angle of particle transverse momentum. This symmetry arrangement strongly cancels the final hadronic state transverse momentum asymmetry developed dynamically from the initial spatial asymmetry. As a prescription to minimize this cancelation, in PACIAE 2.1 [18] we randomly distributed p We know from ideal hydrodynamic calculation [20] that the integrated elliptic flow parameter of final hadronic state is approximately proportional to the initial spatial eccentricity of nuclear overlap zone. Therefore we assume that the introduced deformation parameter of δ p here can be related to the deformation parameter of δ r in the initial spatial phase space, i. e.
where C is an extra model parameter instead of δ p . We also know that the spatial eccentricity of nucleon distribution in the initial overlap zone, reaction plane eccentricity for instance [21], can be expressed as where < ... > denotes the average over the nucleon spatial distribution. This spatial eccentricity should be identical with the geometrical eccentricity [22] of the ellipse of initial spatial overlap zone. Using ǫ rp instead of ǫ g on the left hand side of Eq. (5) and inserting b r = R A (1 + δ r ) as well as a r = R A (1 − δ r ) on the right hand side of Eq. (5), we obtain an algebraic equation of degree 2 in the unknown δ r This equation has two analytical roots: The one less than unity is a physical root Another one larger than unity is an unphysical root because δ r must be ≤ 1. The approximation of introduced in PACIAE 2.1 [18] is just a specifically approximated root of Eq. (6). For the p+p and p+A collisions, the weak initial spatial fluctuation (asymmetry) is also possible to be dynamically developed to the final hadronic state transverse momentum asymmetry and the Eq. (3) steamed from hydrodynamic calculation [20] may also be reliable. Just because of the lack of a proper definition for the initial spatial fluctuation (eccentricity ?), we regard δ p itself as an extra model parameter temporarily.
The Fourier expansion of particle transverse momentum azimuthal distribution reads [5,23] where φ refers to the azimuthal angle of particle transverse momentum, Ψ r stands for the azimuthal angle of reaction plane. In the theoretical study, if the beam direction and impact parameter vector are fixed, respectively, on the p z and p x axes in the nucleus-nucleus cms frame, then the reaction plane is just the p x − p z plane [23]. Therefore the reaction plane angle, Ψ r , between the reaction plane and the p x axis [23] introduced for extracting the elliptic flow experimentally [5] is zero. The equation (9) and the harmonic coefficients there reduce to where ... p denotes the particle-wise average, i.e. the average over all particles in all events [5].

III. RESULTS AND DISCUSSIONS
In the PACIAE 2.1 model simulations, the model parameters are all fixed as the same as default values given in PYTHIA, except the K factor, β, and ∆t. They are, respectively, the higher order term corrections for the LO-pQCD parton-parton cross section [17], a factor in the Lund string fragmentation function [17], and the least time interval of two distinguishably consecutive collisions in the partonic initial and evolution stages [16]. These model parameters are first fitted to the experimental data of charged particle pseudorapidity density and are given in Tab. I. Later on, these fitted parameters are used in all of the simulations. Additionally, in this study the participant eccentricity [21] of is used instead of reaction plane eccentricity ǫ rp . In the above equation σ xy is equal to < xy > − < x >< y >.
Meanwhile, the physical root of Eq. (7) is employed instead of the specifically approximated root of Eq. (8).
We compare the calculated charged particle v 2 (η) and v 2 (p T ) in the 20-40% and 40-50% central Au+Au collisions at √ s N N =0.2 TeV with the corresponding experimental data in the left and right panels of Fig. 1, respectively. The PHENIX data were taken from [29] (using the results of event-plane method). One sees in the left panel that the PACIAE 2.1 results calculated by C=2 well agree with the PHENIX data. The right panel shows that the model results calculated by C=1 reproduce PHENIX data quite well in the p T < 3 GeV/c region. However, the theoretical result decreases with p T increasing is faster than experimental data in the p T >3 GeV/c region. As most of particles are generated below p T ∼2 TeV/c (about 95 percent of the total multiplicity), one always satisfies the agreement between model calculations and experimental data within p T ≤2 GeV/c, cf. Fig. 7 in the first quotation of Ref. [13] for instance. As for the best model parameter C ∼2 in the left panel but 1 in the right panel, which may be attributed to the difference in the studied centrality bin, 20-40% in former but 40-50% in the later. Thus the centrality dependence of parameter C should be studied later.
Similarly, the calculated charged particle v 2 (η) and v 2 (p T ) in the 40-50% central Pb+Pb collisions at √ s N N =2.76 TeV are compared with the corresponding CMS data [9] (using the results of Lee-Yang zero point method for v 2 (η) and event-plane method for v 2 (p T )) in Fig. 2. We see in this figure that the PACIAE 2.1 model is also able to describe the CMS data by adjusting the extra parameter C.
In the Figures 3, 4, and 5 we give the PACIAE 2.1 model predictions for the charged particle v 2 (η) and v 2 (p T ) in the minimum bias (MB) p+Au and p+Pb, as well as in the non-single diffractive (NSD) p+p collisions at √ s N N =0.2, 5.02, and 7 TeV, respectively. We see in these figures that the elliptic flow parameter may reach a amount of 0.04, 0.07, and 0.016 (estimated from v 2 (η)) in the p+Au, p+Pb, and p+p collisions at √ s N N =0.2, 5.02, and 7 TeV, respectively. This amount of the elliptic flow parameter may be measurable experimentally . One sees in Fig. 5 that v 2 seems to be proportional to the value of deformation parameter δ p in the p+p collisions. However, the behavior of v 2 (η) and v 2 (p T ) changing with δ p is needed to be further investigated in detail.

IV. CONCLUSIONS
In summary, We have employed the new issue of a parton and hadron cascade model PACIAE 2.1 investigating systematically the charged particle elliptic flow parameter v 2 in the relativistic nuclear collisions at RHIC and LHC energies. Because of the new introduced mechanism of random arrangement of the particles from string fragmentation on the circumference of an ellipse instead of circle originally, the calculated charge particle v 2 (η) and v 2 (p T ) in the Au+Au/Pb+Pb collisions at √ s N N =0.2/2.76 TeV describe the corresponding experimental data fairly well. Meanwhile, the charged particle v 2 (η) and v 2 (p T ) in the p+Au/p+Pb collisions at √ s N N =0.2/5.02 TeV and in the p+p collisions at √ s=7 TeV are predicted. The elliptic flow parameter in these reactions reaches a measurable amount.
As mentioned in the first section that the elliptic flow parameter is important observable relevant to the exploring of sQGP. However, the measurement of v 2 is not trivial. The discrepancy among v 2 values measured by the event plane method [5], Lee-Yang zero point method [6], and the cumulant method [7] may reach a few ten percent as shown in Fig. 4 and 5 of [8] and Fig. 11 of [9]. On the other hand, the obscures also exist among the various v 2 model calculations as mentioned in the first section. So the further studies for v 2 asymmetry etc. are still required both experimentally and theoretically.
This work is just a first step along the novel approach. Further investigations, such as the cross section effect, energy and centrality dependence of C parameter, as well as the detail study for the dependence of v 2 (p T ) and v 2 (η) on δ p (C) etc., are really required.