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Initial Vorticities of Quark-Gluon Matter in Heavy-ion Collisions

Anke Lei Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China.
Department of Physics, Wuhan University of Technology, Wuhan 430070, China.
Dujuan Wang Department of Physics, Wuhan University of Technology, Wuhan 430070, China.
wangdj@whut.edu.cn Dai-Mei Zhou Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China.
Ben-Hao Sa Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China.
China Institute of Atomic Energy, P. O. Box 275 (10), Beijing, 102413 China. Laszlo Pal Csernai Department of Physics and Technology, University of Bergen, Allegaten 55, 5007 Bergen, Norway.
Frankfurt Institute for Advanced Studies, Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany. Larissa V. Bravina Department of Physics, University of Oslo, POB 1048 Blindern, N-0316 Oslo, Norway.

(Day Month Year; Day Month Year)

Abstract

We calculate four types of initial vorticities in Au+Au collisions at energies $\sqrt{S_{NN}}$ = 5–200 GeV using a microscopic transport model PACIAE. Our simulation shows the non-monotonic dependence of the initial vorticities on the collision energies. The energy turning point is around 10-15 GeV for different vorticities but not sensitive to impact parameter.

keywords:

Vorticity; Heavy-ion Collisions; PACIAE.

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PACS numbers: 21.65.Qr, 25.70.-z, 25.75.-q

1 Introduction

In non-central heavy-ion collisions, two colliding nuclei penetrate each other and produce a huge initial orbital angular momentum (OAM). A fraction of the OAM could be retained in the reaction area then leads to rotation and nonzero vorticity. Theoretical studies show that the final hyperon polarization could serve as a good probe of vorticity[1, 2, 3, 4]. STAR and ALICE collaborations have reported the measurements for global and local $\Lambda$ polarization in Au+Au and Pb+Pb collisions [5, 6, 7, 8, 9, 10, 11]. The vorticity and polarization have been well understood and studied in many numerical models[12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 29]. It is worth mentioning that the choice of different types of vorticities gives the same azimuthal angle dependence of polarization as the experimental measurements, while the previous studies predicted the opposite trend [23, 31]. The vorticity and polarization provide a new way to understand Quark Matter (QM) in heavy-ion collisions.

On the other hand, some studies showed a non-monotonic dependence of the initial vorticities on the collision energy and predicted the maximum polarization would occur at $\sqrt{S_{NN}}\approx$ 3 GeV or 7.7 GeV [32, 30, 29]. However, recent HADES and STAR measurements of global $\Lambda$ polarization at $\sqrt{S_{NN}}$ = 2.4 GeV and 3 GeV (fixed-target) follow the increasing trend towards the lower collision energies observed before [33, 34, 9, 5]. The disagreement implies that there may be other effects to take into account in the low-energy region, such as the freeze-out time, the equilibrium/non-equilibrium treatments and the different hadronization scenarios [32, 28]. It still needs more studies. In this work, we focus on the behavior of the initial vorticities and extend to different types of vorticities, which could provide complementary information of vorticity and polarization.

The paper is organized as follows: the model setups are described in Sec. 2. In Sec. 3 we present and discuss our numerical results of the initial vorticities. Finally, we give a summary in Sec. 4.

2 Model Setups

A microscopic parton and hadron cascade model PACIAE [35] is employed in this paper to simulate Au+Au collisions at $\sqrt{s_{NN}}$ = 5–200 GeV. PACIAE model includes four stages: the partonic initialization, the partonic rescattering, the hadronization, and the hadronic rescattering. The coordinate system is set up as follows. The projectile and target nucleus move along the $+Z$ and $-Z$ direction, respectively. The impact parameter $\bm{b}$ points from the center of the target nucleus to the center of the projectile nucleus, along the $+X$ direction.

Same as Ref. 28, a generalized coarse-graining method is used to structure the continuous fluid cell. We firstly divide particles into grid-cells according to their space-time coordinates. The cell size is set to be $0.5\times 0.5\times 0.5~{}\mathrm{fm^{3}}$ , and the time slice is 0.5 $\mathrm{fm/c}$ . The energy density $\epsilon$ and momentum density $p$ of a cell will be obtained by summing over the corresponding quantities of the particles inside the cell. Then, as shown in Fig. 1 (taking a two-dimensional case as an example), we add $\epsilon$ and $p$ of each nearest four side cell and four corner cells into the central one and average the quantities over the number of these cells. After that, the corresponding average coarse-graining energy density $\overline{\epsilon}$ and momentum density $\overline{p}$ are obtained. The flow velocity field is defined as $\overline{p}/\overline{\epsilon}$ .

Refer to caption — Figure 1: Schematic picture of the generalizaed coarse-graining.

In addition, the local cell temperature $T$ will be extracted from the local cell energy density $\overline{\epsilon}$ via the relation [36]

\overline{\epsilon}=\pi^{2}(16+10.5N_{f})T^{4}/30,

(1)

where $N_{f}=3$ is number of considered quark flavors. Thus the vorticity could be calculated from velocity and temperature. Four types of vorticities (non-relativistic vorticity $\omega^{NR}_{ij}$ , kinematic vorticity $\omega_{\mu\nu}^{K}$ , temperature vorticity $\omega_{\mu\nu}^{T}$ and thermal vorticity $\omega_{\mu\nu}^{th}$ ) are considered here. They are defined as

$\displaystyle\omega^{NR}_{ij}$	$\displaystyle=\frac{1}{2}\left(\partial_{i}v_{j}-\partial_{j}v_{i}\right),$	(2)
$\displaystyle\omega_{\mu\nu}^{K}$	$\displaystyle=-\frac{1}{2}\left(\partial_{\mu}u_{\nu}-\partial_{\nu}u_{\mu}\right),$	(3)
$\displaystyle\omega_{\mu\nu}^{T}$	$\displaystyle=-\frac{1}{2}\left[\partial_{\mu}\left(Tu_{\nu}\right)-\partial_{\nu}\left(Tu_{\mu}\right)\right],$	(4)
$\displaystyle\omega_{\mu\nu}^{th}$	$\displaystyle=-\frac{1}{2}\left[\partial_{\mu}\left(u_{\nu}/T\right)-\partial_{\nu}\left(u_{\mu}/T\right)\right],$	(5)

where $v_{i}(i=1,2,3)$ denotes the components of flow four-velocity $u^{\mu}=$ $\gamma(1,\bm{v})$ with $\gamma=1/\sqrt{1-\bm{v}^{2}}$ being the Lorentz factor, and $T$ is the local temperature. We calculate vorticities of the partonic stage where QGP starts to form and evolve.

3 Numerical Results

In this section, we present our numerical results of initial vorticities at $\sqrt{S_{NN}}=$ 5–200 GeV. We focus on the vorticities along with the OAM, in the $-Y$ direction. For characterizing the overall vorticity of the system, an energy-density-weighted vorticity is used here:

\left<-\omega_{zx}\right>\equiv\left<-\omega_{Y}\right>=\frac{\sum_{i}^{N_{cell}}\overline{\epsilon_{i}}\omega_{i}}{\sum_{i}^{N_{cell}}\overline{\epsilon_{i}}},

(6)

where $N_{cell}$ is the total number of the cells with non-zero energy density, $\overline{\epsilon_{i}}$ and $\omega_{i}$ are the energy density and the vorticity of the $i$ -th cell.

In Fig. LABEL:fig:iniParVor, we show the initial vorticities among four types of vorticities at time 0.5 fm/c which could be compared with work in Ref. 32. One can see that they show similar behaviors. For larger impact parameters, the magnitude of initial vorticities is larger. This trend is consistent with experimental measurements of polarization, where the global $\Lambda$ polarization is found to be larger in more peripheral collisions [6, 9, 10]. We note here that at zero impact parameter, the initial vorticities are nearly vanishing. This is to be expected.

Our results show that the initial vorticities first increase then decrease as $\sqrt{S_{NN}}$ grows and the turning point is at around 10 GeV in non-relativistic, kinematic, and thermal vorticities. The turning point of temperature vorticity moves back to around 15 GeV under the direct influence of temperature. However, the turning point is not sensitive to impact parameters. As impact parameters increase, the turning point does not change significantly. Although our turning energy is different from the previous study in UrQMD and IQMD [32], our results do show that the initial vorticities have a non-monotonic dependence on the energy.

4 Summary

In this work, we calculated four types of initial vorticities and study their energy dependence using the microscopic transport model PACIAE. It was reconfirmed that the initial vorticities have a non-monotonous dependence on increasing collision energies. The energy turning point was at 10-15 GeV for different types of vorticities but was not sensitive to impact parameter. It requires further verification and studies in low- and moderate-energy regions.

Acknowledgments

The authors thank Yilong Xie for the helpful discussions. This work was supported by National Natural Science Foundation of China (Grant No. 11905163, 11775094).

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