On Particle Production in Lead-Gold Collision and Azimuthal Anisotropy at Top SPS Energy

Abstract

In a multisource thermal model, we analyze the dependence of elliptic flow 𝑣₂ on the transverse momentum 𝑃_𝑇. The model results are compared with the data of 𝜋⁻, 𝐾⁰_𝑆, 𝑝, and Λ measured in Pb + Au collisions at top SPS energy, 17.3 GeV. It is found that the azimuthal anisotropy in the evolution process of high-energy collisions is correlated highly to the number of participant nucleons.

1. Introduction

Collective flows are a common phenomenon of heavy ion collisions at high energy [1–5]. In a noncentral collision of two spherical nuclei, the overlapping zone between the two nuclei is not circular but it rather takes an almond shape, where pressure gradients are formed due to the azimuthal anisotropy. Then, the initial spatial anisotropy of the overlapping zone is converted into momentum space anisotropy of the particle distribution. The momentum distribution of emitted particles is quantified in terms of the Fourier expansion for the particle invariant azimuthal distribution [6]. These are defined by 𝐸𝑑𝑁𝑑³𝑝=𝑑𝑁2𝜋𝑝_𝑇𝑑𝑝_𝑇𝑑𝑦1+2^∞_𝑛=1𝑣_𝑛cos𝑛𝜙−Ψ_𝑛, (1) where 𝜙_𝑛 and Ψ_𝑛 represent the azimuthal angle of an emitted particle in laboratory reference frame and the azimuthal angle of the reaction plane in the same frame, respectively. The 𝑣_𝑛 represents the magnitude of the 𝑛th-order anisotropy. The second-order coefficient of the anisotropy flow 𝑣₂ is called the elliptic flow, which is related to the almond shape mentioned above. The elliptic flow is one of the key observables measured at the RHIC, where large values of 𝑣₂ are considered as a sign of collectivity in the system created in the high-energy collisions. Recently, an event-by-event ideal hydrodynamic model is used to study the elliptic flow of thermal photons in Au + Au collisions at RHIC and Pb + Pb collisions at the LHC, and the results show that there is a persistent tension between experimental data and photons of hydrodynamics [7].

In this paper, we will discuss the elliptic flow in Pb + Au collision at √𝑠_NN=17.3 GeV, which is top SPS energy. To extract properties of the matter created in the asymmetry reaction, we improve a multisource thermal model, which was used to describe the particle production in our previous work [8].

2. Azimuthal Anisotropy and Number of Participant Nucleons

The initial transverse coordinate-space anisotropy of the participant region is converted into an azimuthal momentum-space anisotropy [9]. But we obtain the evolution information of such matter by final-state particles measured in detectors [10]. The transverse momentum 𝑝^′_𝑇 of the particles has an invariant form in different reference fames: 𝑓_𝑝′𝑇𝑝^′_𝑇=1𝜎²−𝑝exp^𝑇′22𝜎², (2) which is called Rayleigh distribution.

When two ultrarelativistic nuclei collide at a nonzero impact parameter, their overlap area in the transverse plane has a short axis which is parallel to the impact parameter and a long axis which is perpendicular to the impact parameter. Considering the mechanics effect, these sources depart from the isotropic emission. The distribution functions of 𝑝^∗_𝑥 and 𝑝^∗_𝑦 are 𝑝^∗_𝑥=𝐴_𝑥𝑝^′_𝑥+𝐵_𝑥,𝑝^∗_𝑦=𝐴_𝑦𝑝^′_𝑦+𝐵_𝑦, (3) where 𝐴_𝑥,𝑦 and 𝐵_𝑥,𝑦 denote the expansion coefficient of the source and the movement coefficient of the source, respectively. For Maxwell's ideal gas, 𝐴_𝑥,𝑦=1 and 𝐵_𝑥,𝑦=0, and the nonexpanding source has no contribution to the collective flow. The particle distribution is 𝑓_{𝑝∗𝑇,𝜙}^∗(𝑝^∗_𝑇,𝜙^∗). In the final state, the normalized distribution of total particles can be expressed as 𝑓_{𝑝𝑇,𝜙}𝑝_𝑇,𝜙=𝑢𝑓_𝑝′𝑇𝑝^′_𝑇+(1−𝑢)𝑓_{𝑝∗𝑇,𝜙}𝑝^∗_𝑇,𝜙, (4) where 𝑢 and 1−𝑢 denote the contribution of isotropic sources and anisotropic sources, respectively. The 1−𝑢 is proportional to the number of nucleons participating 𝑁_part: 𝑁𝑢=1−𝑘^part𝐴_Pb+𝐴_Au, (5) where 𝐴_Pb and 𝐴_Au are the mass numbers of the Pb and Au, respectively.

In the Monte Carlo calculation, the transverse momenta are given by 𝑝^∗_𝑇=𝐴_𝑥𝜎−2ln𝑟₁cos2𝜋𝑟₂+𝐵_𝑥²+𝐴_𝑦𝜎−2ln𝑟₃cos2𝜋𝑟₄+𝐵_𝑦^21/2, (6) where 𝑟₁, 𝑟₂, 𝑟₃, and 𝑟₄ denote random numbers in [0,1]. According to the measurement in experiments, the elliptic flow in the model is defined as 𝑣₂=𝑝²_𝑥−𝑝²_𝑦𝑝²_𝑥+𝑝²_𝑦. (7) It is interesting to note that several different definitions of centralities in 𝐴+𝐴 collision were given through other quantities because they are not a directly measurable quantity [11]. The centrality can usually be calculated by multiplicity distribution or fractional cross-section or impact parameter 𝑏, 𝑏²/(𝑅_Pb+𝑅_Au)² [12]. The 𝑅_Pb and 𝑅_Au are radii of Pb and Au, respectively. Theoretically, the number of participant nucleons in a collision of Pb + Au at the impact parameter 𝑏 is 𝑁_part(𝑏)=𝑁^Pb_part(𝑏)+𝑁^Au_part(𝑏). (8) They can be estimated easily in the geometry method [13]: 𝑁^Pb_part(𝑏)=𝑑𝑉𝜌_Pb𝑥²+𝑏−𝑦²+𝑧²×𝜃𝑅_Pb−𝑥²+𝑏−𝑦²+𝑧²𝜃𝑅_Au−𝑥²+𝑦²,𝑁^Au_part(𝑏)=𝑑𝑉𝜌_Au𝑥²+𝑦²+𝑧²×𝜃𝑅_Au−𝑥²+𝑦²+𝑧²𝜃𝑅_Pb−𝑥²+𝑏−𝑦². (9) The nuclear density function 𝜌_Pb/Au is the Woods-Saxon density profile [14]: 𝜌_Pb/Au(𝑟)=𝜌₀1+exp𝑟−𝑅_Pb/Au,/𝑐 (10) where 𝜌₀ refers to normal nuclear density and 𝑐≈0.54 fm stands for the nuclear diffusion edge from low energy electron-nucleus scattering experiments [15].

First, in order to check the reliability of the model, Figure 1 presents the transverse momentum spectra of charged pions, charged kaons and protons at midrapidity for Pb-Pb collisions at √𝑠_NN=17.3 GeV. The symbols denote the data measured by NA49 Collaboration [16]. The curves denote our results with 𝐴_𝑥=3.10±0.45 and 𝜎=0.44±0.14. The calculated results agree with the experimental data except for the region of 𝑝_𝑇<0.6 GeV/𝑐.

Figure 1 

Invariant 𝑝𝑇-differential yields of 𝜋+, 𝐾+, 𝐾, and protons. The experimental data [16] are denoted by the scattered symbols. The theoretical results are denoted by the solid line.

Figure 2 shows elliptic flows 𝑣₂ of 𝐾⁰_𝑆 produced in Pb-Au collisions at √𝑠_NN=17.3 GeV. The symbols are the data measured by the CERES Collaboration [17] and NA49 Collaboration [18]. Collision centralities are 5.3–13% of 𝜎_geo for CERES and top 13% for NA49 data. The curves are our results with 𝐴_𝑥=3.10±0.45, which depends on collision energies and the centrality. The other parameters used in the calculations are 𝑘=0.24±0.04 and 𝜎=0.45±0.18. The observed 𝑣₂ increases with increasing 𝑝_𝑇 at the low transverse momentum and saturates in 𝑝_𝑇>2 GeV/𝑐. The modeling results are in approximate agreement with the experimental data. Figures 3 and 4 show elliptic flows 𝑣₂ of 𝑝_𝑇 for 𝜋⁻ and 𝜋⁺ produced in Pb-Au collisions at √𝑠_NN=17.3 GeV. The symbols denote the data measured by the CERES Collaboration [17]. The curves denote our results. One sees again that the model can approximately describe the elliptic flows in Pb-Au collisions at the top SPS energy.

Figure 2 

Differential elliptic flow 𝑣2(𝑝𝑇) of 𝐾0𝑆. The CERES data [17] and the NA49 data [18] are denoted by the scattered symbols. The theoretical results are denoted by the solid line.

Figure 3 

Differential elliptic flow 𝑣2(𝑝𝑇) of negative pion candidates after the HBT correction. The CERES data [17] are denoted by the scattered symbols. The theoretical results are denoted by the solid line.

Figure 4 

The same as in Figure 4, but for 𝜋− and 𝜋+ candidates.

In Figure 5, we show differential elliptic flows 𝑣₂(𝑝_𝑇) of protons in Pb-Au collisions at √𝑠_NN=17.3 GeV. In the given centrality, the dependence of 𝑣₂ on 𝑝_𝑇 is similar to that for the pion. In Figure 6, we present differential elliptic flows 𝑣₂(𝑝_𝑇) of 𝐾⁰_𝑆 mesons and Λ hyperons in Pb-Au collisions at √𝑠_NN=17.3 GeV. We also see the same behavior for the same centrality like the above figures. In order to understand the azimuthal anisotropy of final-state particles in the asymmetric collision, Figure 7 gives a comparison to the results of NA49 [19] at the same energy √𝑠_NN=17.3 GeV and gives a comparison to STAR results [20] at √𝑠_NN=200 GeV. The NA49 and CERES data are in reasonably good agreement. In order to compare STAR to CERES results, the former have been rescaled to the centrality used in the CERES experiment. The 𝑣₂ values measured at the RHIC energy are 15–20% higher due to the higher beam energy. Our results are in approximate agreement with the data measured by the three collaborations.

Figure 5 

The same as in Figure 4, but for the proton.

Figure 6 

The same as in Figure 4, but for 𝐾0𝑆 mesons and Λ hyperons.

Figure 7 

Comparison of Λ and 𝐾0𝑆 elliptic flow between CERES and NA49 [19] and STAR [20].

3. Conclusions

In the above discussions, we have investigated the azimuthal anisotropy by using the elliptic flow 𝑣₂ versus the transverse momentum 𝑝_𝑇 in Pb + Au collisions at the top SPS energy √𝑠_NN=17.3 GeV. The amplitude of the azimuthal anisotropy can be measured by the elliptic flow, which is a sensitive probe of the matter created in the collisions. We improve the multisource statistical model to discuss the transverse momentum dependence of elliptic flows for the particles produced in the asymmetric system, Pb-Au collision at the top SPS energy. Considering the collision geometry, the modelling results are obtained in the framework of the multisource statistical model. The results agree approximately with the data of the CERN-CERES/NA45 Collaboration. More importantly, this model has partly described the particle production in high-energy collisions [8]. In the descriptions of elliptic flows 𝑣₂(𝑝_𝑇) in the asymmetric system (Pb + Au collision) at SPS energies, the present work is a new successful attempt.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grants no. 11247250, no. 11005071, and no. 10975095, the National Fundamental Fund of Personnel Training under Grant no. J1103210, and the Shanxi Provincial Natural Science Foundation under Grants no. 2013021006 and no. 2011011001.