Polarization phenomenon in heavy-ion collisions
重离子碰撞中的极化现象

Abstract 摘要

The strongly interacting system created in ultrarelativistic nuclear collisions behaves almost as an ideal fluid with rich patterns of the velocity field exhibiting strong vortical structure. Vorticity of the fluid, via spin-orbit coupling, leads to particle spin polarization. Due to the finite orbital momentum of the system, the polarization on average is not zero; it depends on the particle momenta reflecting the spatial variation of the local vorticity.
在超高能核碰撞中产生的强相互作用系统几乎表现为具有丰富速度场模式的理想流体，展现出强旋涡结构。流体的涡度通过自旋轨道耦合导致粒子自旋极化。由于系统的有限轨道动量，平均极化不为零；它取决于粒子动量，反映了局部涡度的空间变化。

In the last few years, this field experienced a rapid growth due to experimental discoveries of the global and local polarizations. Recent measurements triggered further development of the theoretical description of the spin dynamics and suggestions of several new mechanisms for particle polarization. In this review, we focus mostly on the experimental results. We compare the measurements with the existing theoretical calculations but try to keep the discussion of possible underlying physics at the qualitative level. Future measurements and how they can help to answer open theoretical questions are also discussed. We pay a special attention to the employed experimental methods, as well as to the detector effects and associated corrections to the measurements.
在过去几年中，由于全球和本地极化的实验发现，这一领域经历了快速增长。最近的测量触发了对自旋动力学的理论描述的进一步发展，并提出了几种新的粒子极化机制。在本综述中，我们主要关注实验结果。我们将测量结果与现有的理论计算进行比较，但尽量保持对可能的基础物理学讨论在定性水平上。未来的测量以及它们如何帮助回答未解的理论问题也在讨论中。我们特别关注所采用的实验方法，以及探测器效应和与测量相关的修正。

polarization; vorticity; nuclear collisions.
极化；涡度；核碰撞。

Contents 内容

• 1 Introduction: Polarization as a collective phenomenon
  1 引言：极化作为一种集体现象
• 2 Global and local polarizations
  2 全球和本地极化
  • 2.1 Nonrelativistic vorticity and the global polarization, $⟨P_{-\mathrm{y}}⟩$
    2.1 非相对论涡度和全局极化， $⟨P_{-\mathrm{y}}⟩$
  • 2.2 Role of the magnetic field
    2.2 磁场的作用
  • 2.3 Anisotropic flow and polarization along the beam direction
    2.3 沿着束流方向的各向异性流动和极化
  • 2.4 Circular polarization, $P_{\varphi }$; polarization along $x$-direction, $P_{\mathrm{x}}$
    2.4 循环极化， $P_{\varphi }$ ；沿 $x$ 方向的极化， $P_{\mathrm{x}}$
• 3 Spin and polarization in hydrodynamic description
  3 旋转和极化在流体力学描述中
  • 3.1 Kinematic vorticity, thermal gradients, acceleration
    3.1 运动涡度、热梯度、加速度

1 Introduction: Polarization as a collective phenomenon
1 引言：极化作为一种集体现象

The discovery of the global polarization in heavy-ion collisions, the hyperon polarization along the system orbital momentum [1, 2], followed by the measurements of the polarization along the beam direction [3], opened totally new opportunities for study of the nuclear collision dynamics and the properties of the quark-gluon plasma (QGP), as well as for deeper understanding of the spin and its transport in QGP medium. These polarization measurements are among the most significant discoveries made in heavy-ion collision program along with observations of the strong elliptic flow and jet quenching [4, 5, 6, 7], and have generated intense theoretical discussions as well as experimental activities.
在重离子碰撞中发现了全局极化，系统轨道角动量沿着超子的极化[1, 2]，随后测量了沿着束流方向的极化[3]，为研究核碰撞动力学和夸克胶子等离子体（QGP）的性质，以及更深入理解 QGP 介质中的自旋及其传输开辟了全新的机会。这些极化测量是重离子碰撞计划中最重要的发现之一，与观察到的强椭圆流和喷注熄灭[4, 5, 6, 7]一起，引发了激烈的理论讨论和实验活动。

The phenomenon of the global polarization in heavy-ion collisions arises from the partial conversion of the orbital angular momentum of the colliding nuclei into the spin angular momentum of produced particles [8, 9, 10]. As a result, the particles on average become polarized along the direction of the initial orbital momentum of the two colliding nuclei. The term "global" in the name of the phenomenon indicates that the component of the particle polarization along the system orbital momentum is not zero when averaged over all produced particles. The origin of the polarizationin heavy ion collisions is in the collective motion of the strongly interacting fluid, and it is unlikely to be related to the hyperon polarization (with respect to the production plane) observed in $pp$ and $pA$ collisions.[11]
全球极化现象在重离子碰撞中的现象是由于碰撞核的轨道角动量部分转化为产生粒子的自旋角动量[8, 9, 10]。因此，平均而言，粒子沿着两个碰撞核的初始轨道角动量方向极化。现象名称中的“全球”一词表明，当对所有产生的粒子进行平均时，粒子极化沿着系统轨道角动量的分量不为零。重离子碰撞中极化的起源在于强相互作用流体的集体运动，与在 $pp$ 和 $pA$ 碰撞中观察到的超子极化（相对于产生平面）不太可能有关[11]。

In a non-central nuclear collision, the most prominent pattern in the initial collective velocity distribution is a shear of the velocity field, $d{v}_z/dx\ne 0$, where the $z$ direction is chosen along the beam, and the $x$ direction is along the impact parameter vector defined as a vector connecting the centers of the two nuclei (pointing from the "target" to the "projectile" nucleus, the latter defined as the one moving in the positive $z$ direction, see Fig. 1. Such a shear in the velocity field leads to nonzero vorticity characterizing the local orbital angular momentum density. Particle binary interactions in the system would have on average non-zero orbital angular momentum, which will be partially converted into spin of the final-state particles. For example, in a system with non-zero vorticity consisting of pions, the colliding pions have a preferential direction for their orbital angular momentum, and the spin of $\rho$ mesons produced in such collisions (${\pi }^{+}{\pi }^{-}\to {\rho }^0$) would point in that very direction.[9]
在非中心核碰撞中，初始集体速度分布中最突出的模式是速度场的剪切，其中 $d{v}_z/dx\ne 0$ ，其中 $z$ 方向沿着束流选择， $x$ 方向沿着作为连接两个核心中心的冲击参数矢量定义的矢量（从“目标”指向“弹丸”核，后者定义为正 $z$ 方向移动的核心，参见图 1。速度场中的这种剪切导致表征局部轨道角动量密度的非零涡度。系统中的粒子二进制相互作用平均具有非零轨道角动量，这部分将被部分转化为最终粒子的自旋。例如，在由介子组成的非零涡度系统中，碰撞的介子具有其轨道角动量的优选方向，而在这种碰撞中产生的 $\rho$ 介子的自旋（ ${\pi }^{+}{\pi }^{-}\to {\rho }^0$ ）将指向该方向。[9]

The idea of the global polarization is almost 20 years old with the initial predictions for the quark and final particles polarization as high as "in the order of tens of a percent".[8] As pointed in Ref.[9], the global polarization phenomenon, if that strong, could affect the interpretation of different measurements. In particular, the polarization of the vector mesons would have a significant contribution to the elliptic flow measurements.[9, 12] The decay products of the vector resonances with spin pointing perpendicular to the reaction plane have the angular distribution enhancing the in-plane particle production and thus contributing to the elliptic flow measurements. The first measurements[13] of the global polarization of $\mathrm{\Lambda }$ hyperons in Au+Au collisions at 200 GeV by the STAR Collaboration put an upper limit on hyperon polarization of $|P_{\mathrm{\Lambda }}|\le 0.02$. Subsequently, the theoretical predictions have
全球极化的概念已经有近 20 年的历史，最初对夸克和最终粒子极化的预测高达“几十个百分点”。正如参考文献[9]所指出的，如果全球极化现象如此强烈，可能会影响对不同测量结果的解释。特别是，矢量介子的极化将对椭圆流测量产生重要贡献。具有垂直于反应平面的自旋的矢量共振的衰变产物具有增强平面内粒子产生的角分布，从而对椭圆流测量产生贡献。STAR 合作组织在 200 GeV 的 Au+Au 碰撞中首次测量了 $\mathrm{\Lambda }$ 超子的全球极化[13]，并对超子极化设定了一个上限 $|P_{\mathrm{\Lambda }}|\le 0.02$ 。随后，理论预测已经。

Figure 1: Schematic view of a nuclear collision with $x$ and $z$ being the impact parameter and beam directions, respectively. The system orbital angular momentum as well as the magnetic field points into the page, opposite to the direction of the $y$ axis. Solid yellow arrows indicate collective velocity field at $z=0$. Open arrow indicates vorticity of the system.
图 1：核碰撞的示意图，其中 $x$ 和 $z$ 分别为撞击参数和束流方向。系统的轨道角动量以及磁场指向页面内部，与 $y$ 轴的方向相反。实心黄色箭头表示 $z=0$ 处的集体速度场。开放箭头表示系统的涡度。

been improved [14] to be consistent with these experimental results.
已经改进[14]以与这些实验结果保持一致。

Particle polarization is determined by the vorticity of the fluid element where the particle has been produced. Due to the strong space-momentum correlation present in the system, the polarization of the particles in a certain momentum range would reflect the vorticity in the regions where those particles are predominantly emitted from. Thus, while averaged over the entire system only the "global" polarization component survives, the polarization in general depends on the particle momentum. Such polarization is often referred to as "local". The non-trivial local vorticity can originate, for example, due to propagation of highly energetic jets produced by the partons hard scattering [15, 16], or due to collective expansion of the system [17, 18, 19, 20]. An important example of that is the polarization along the beam direction due to (transverse) anisotropic flow, discussed first in Ref. [18] on a basis of a simple Blast-Wave model, as well as observed in full hydrodynamical calculations [19].
粒子极化由粒子产生的流体元素的涡度决定。由于系统中存在强烈的空间动量相关性，某一动量范围内的粒子极化将反映出这些粒子主要从哪些区域发射出来的涡度。因此，尽管在整个系统上平均只有“全局”极化分量存留，但极化通常取决于粒子动量。这种极化通常被称为“局部”。非平凡的局部涡度可以源自于，例如，由部分子硬散射产生的高能喷流的传播，或者由系统的集体膨胀引起。其中一个重要的例子是由于（横向）各向异性流引起的沿着束流方向的极化，首次在基于简单爆发波模型的文献[18]中讨论，同时也在完整的流体力学计算中观察到[19]。

A short review format does not allow to describe and discuss in detail all the results and the questions under discussion. Our goal is to provide a more general picture of the field, to emphasize the major developments in our understanding of the phenomena and formulate outstanding questions, and to outline the relation of the current and future measurements to the underlying physics. In the first section of the review, we focus the discussion on the nature of the phenomenon. We use a very simple picture based on Glauber and Blast-Wave models for illustrations and rough estimates. Then we discuss the experimental side of the measurements, emphasizing the details important for the interpretation of the results and their uncertainties, as well as what is needed to accomplish this or other measurements. We proceed with an overview of available results and their current theoretical interpretations. The overview of the experimental results is followed by a summary of what we have learned so far, open questions, and future perspectives.
短评格式无法详细描述和讨论所有结果和讨论中的问题。我们的目标是提供该领域的更一般图景，强调我们对现象理解的主要发展，并提出突出问题，概述当前和未来测量与基础物理之间的关系。在评论的第一部分中，我们将讨论重点放在现象的本质上。我们使用基于 Glauber 和 Blast-Wave 模型的简单图像进行说明和粗略估计。然后我们讨论测量的实验方面，强调对结果解释和其不确定性重要的细节，以及完成这些或其他测量所需的条件。我们继续概述可用结果及其当前的理论解释。实验结果的概述后跟着我们迄今为止学到的内容总结，未解问题和未来展望。

In the following discussion we refer to $x$, $y$, and $z$ components of the polarization according to the coordinate system depicted in Figs. 1 and 2. In such a coordinate system, the global polarization would be given by the average of $-P_{\mathrm{y}}=P_{-y}$, and the polarization along the beam direction is given by $P_{\mathrm{z}}$. The global averages (over all particles and momenta) of $P_{\mathrm{z}}$ and $P_{\mathrm{x}}$ components (more exactly, the components of the total spin angular momentum) are expected to be zero.
在以下讨论中，我们根据图 1 和图 2 中所示的坐标系，提到极化的 $x$ 、 $y$ 和 $z$ 组成部分。在这样一个坐标系中，全局极化将由 $-P_{\mathrm{y}}=P_{-y}$ 的平均值给出，沿着光束方向的极化由 $P_{\mathrm{z}}$ 给出。全局平均值（对所有粒子和动量）的 $P_{\mathrm{z}}$ 和 $P_{\mathrm{x}}$ 组成部分（更确切地说，总自旋角动量的组成部分）预计将为零。

2 Global and local polarizations
2 全球和本地极化

Nonrelativistic vorticity and the global polarization, $⟨P_{-\mathrm{y}}⟩$
非相对论涡度和全局极化， $⟨P_{-\mathrm{y}}⟩$

A very significant development leading to a fast progress in this field was an application of the statistical methods to vortical fluid with non-zero spin particles [21], and development of the hydrodynamical calculations based on the assumption of the local angular momentum equilibrium [22]. A rough estimate of the polarization can be obtained with the help of a simple nonrelativistic a formula describing the particle distribution in a fluid with nonzero vorticity:[23]
这一领域取得快速进展的一个非常重要的发展是将统计方法应用于具有非零自旋粒子的涡旋流体[21]，并基于局部角动量平衡假设开展了流体动力学计算的发展[22]。可以通过一个简单的非相对论性公式来粗略估计极化，该公式描述了具有非零涡度的流体中的粒子分布：[23]

Footnote a: Note that for hyperons used in polarization measurements $m_H\gg T$, where $T$ is the temperature and $m_H$ is a hyperon mass.
脚注 a：请注意，用于极化测量的超子 $m_H\gg T$ ，其中 $T$ 是温度， $m_H$ 是超子质量。

$$\begin{array}{}\text{(1)}& w\propto \exp [-(E+\mathit{\omega }(\mathbf{s}+\mathbf{l})-\frac{\mu }{s}\mathbf{s}\mathbf{B})/T],\end{array}$$

where $\mathit{\omega }=\frac{1}{2}\mathbf{\nabla }\times \mathbf{v}$ is the local vorticity of the fluid velocity ($\mathbf{v}$) field, $\mathbf{B}$ is an external magnetic field, $\mu$ is the particle magnetic moment, and $T$ is the temperature of the system at equilibrium. Then the polarization of particles with spin $s$ is given by:
其中 $\mathit{\omega }=\frac{1}{2}\mathbf{\nabla }\times \mathbf{v}$ 是流体速度（ $\mathbf{v}$ ）场的局部涡度， $\mathbf{B}$ 是外部磁场， $\mu$ 是粒子磁矩， $T$ 是系统在平衡时的温度。然后，具有自旋 $s$ 的粒子的极化由以下公式给出：

$$\begin{array}{}\text{(2)}& \mathbf{P}=\frac{⟨\mathbf{s}⟩}{s}\approx \frac{(s+1)}{3}\frac{(\mathit{\omega }+\mu \mathbf{B}/s)}{T}.\end{array}$$

For spin $1/2$ particles, the vorticity contribution to the polarization is $P=\omega /2T$b. Averaged over the entire system volume, the vorticity direction coincides with the direction of the system orbital angular momentum. Note that the magnetic field created by fast positively charged nuclei is also pointing in the same direction.
对于自旋 $1/2$ 粒子，涡度对极化的贡献为 $P=\omega /2T$ b。在整个系统体积上平均，涡度方向与系统轨道角动量方向一致。请注意，由快速带正电核创建的磁场也指向同一方向。

Footnote b: The nonrelativistic estimate can be also obtained by noting that the entropy $\sigma (E)$ of the rotating gas can be approximated as $\sigma (E-L^2/(2I))$, where $L$ is the orbital momentum, and $I$ is the system inertia. Under condition of angular momentum conservation, $S+L=\mathrm{const}$, this leads to $\partial \sigma /\partial S=L/(IT)=\omega /T$.
脚注 b：非相对论估计也可以通过注意到旋转气体的熵 $\sigma (E)$ 可以近似为 $\sigma (E-L^2/(2I))$ 来获得，其中 $L$ 是轨道动量， $I$ 是系统惯性。在角动量守恒条件下， $S+L=\mathrm{const}$ ，这导致 $\partial \sigma /\partial S=L/(IT)=\omega /T$ 。

Figure 1 shows a cartoon of a non-central nuclear collision with solid arrows indicating the velocity field of the matter at the plane $z=0$. One can estimate the vorticity as ${\omega }_y\approx -\frac{1}{2}\partial v_z/\partial x$ where $v_z$ is the net-velocity along the beam direction that depends on the number of participants coming from the target vs. projectile nuclei. The magnitude of $v_z$ is reflected in the length of the solid arrows in Fig. 1. For a rough estimate of the vorticity, we present the velocity ($v_z$) distribution in the transverse plane in Fig. 2(a). In these calculations, the velocity was estimated as $v_z=(n_{\mathrm{P}}-n_{\mathrm{T}})/(n_{\mathrm{P}}+n_{\mathrm{T}})$ where $n_{\mathrm{P}}$ and $n_{\mathrm{T}}$ are the densities of the projectile and target nucleon participants (nucleons that experienced inelastic collisions) obtained by a simple Glauber model. The middle and right plots in Fig. 2 show the derivatives $d{v}_z^{\ast }/dx$ and $d{v}_z^{\ast }/dy$ (with the asterisks denoting the quantities in the rest frame of the fluid) weighted with the density of participating nucleons (roughly proportional to the produced particle density). From these estimates, one concludes that the vorticity might be as large as a few percent of fm${}^{-1}$. Then the nonrelativistic formula (1) yields for the spin $1/2$ particle polarization, $P\approx \omega /(2T)$, in the range of a few percent (assuming $T\sim 100$ MeV). Note that this simple estimate ignores the effect of nuclear transparency at high energies where the vorticity values could be significantly lower.
图 1 显示了一个非中心核碰撞的卡通图，实心箭头表示平面 $z=0$ 上物质的速度场。人们可以估计涡度为 ${\omega }_y\approx -\frac{1}{2}\partial v_z/\partial x$ ，其中 $v_z$ 是沿着束流方向的净速度，取决于来自靶核与弹道核的参与者数量。图 1 中实心箭头的长度反映了 $v_z$ 的大小。为了粗略估计涡度，我们在图 2(a)中展示了横截面平面上的速度（ $v_z$ ）分布。在这些计算中，速度被估计为 $v_z=(n_{\mathrm{P}}-n_{\mathrm{T}})/(n_{\mathrm{P}}+n_{\mathrm{T}})$ ，其中 $n_{\mathrm{P}}$ 和 $n_{\mathrm{T}}$ 是通过简单的 Glauber 模型获得的弹道核和靶核核子参与者（经历非弹性碰撞的核子）的密度。图 2 中的中间和右侧图显示了加权参与核子密度的导数 $d{v}_z^{\ast }/dx$ 和 $d{v}_z^{\ast }/dy$ （星号表示流体静止参考系中的量），大致与产生的粒子密度成正比。从这些估计中，人们可以得出结论，涡度可能相当于几个百分之几的 fm ${}^{-1}$ 。 然后，非相对论公式（1）得出自旋 $1/2$ 粒子极化 $P\approx \omega /(2T)$ ，在几个百分点范围内（假设 $T\sim 100$ MeV）。请注意，这个简单的估计忽略了高能量处核透明度的影响，那里的涡度值可能会显著降低。

The vorticity of the system, especially its component along the system's orbital momentum, is directly related to the asymmetries in the initial velocity fields and thus it is intimately related to the directed flow $v_1$.[24, 25, 26] The $v_1$ is defined by the first Fourier moment $v_1=⟨\cos (\phi -{\mathrm{\Psi }}_{\mathrm{RP}})⟩$ of the produced particles' azimuthal distribution relative to the collision reaction plane angle ${\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}$. Hydrodynamic simulations show that the orbital angular momentum stored in the system and the directed flow of charged particles are almost directly proportional to each other [26]. This allows for an empirical estimate of the collision energy dependence of the global polarization [18]. The STAR results for the directed flow [27, 28] and the hyperon global polarization [1, 2] from the RHIC Beam Energy Scan program show that the slopes of $v_1$ at midrapidity ($\mathrm{d}{v}_1/\mathrm{d}\eta$) for charged hadrons and the hyperon polarization are indeed strongly correlated. The charged-particle directed flow in Pb-Pb collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=2.76$ TeV [29] is about three times smaller than at the top RHIC energy of 200 GeV [30]. This suggest that the global polarization at the LHC energies should be also about three times smaller than at RHIC and decreasing from $\sqrt{s_{\mathrm{N}\mathrm{N}}}=2.76$ TeV to 5.02 TeV by about $\sim 20\mathrm{\%}$[31]. Even smaller polarization values at the LHC are expected when the directed flow is considered as a combination of the two effects - the tilt of the source in the longitudinal direction and the dipole flow originating from the asymmetry in the initial energy density distributions [32]. Taking into account that only the contribution to the directed flow from the tilted source is related to the vorticity and that its contribution relative to the dipole flow decreases with the collision energy [32], one arrives to an estimate for the global polarization at the LHC energies of the order of 0.15-0.2 of that at the top RHIC energy.
系统的涡度，特别是沿着系统轨道动量的分量，与初始速度场中的不对称性直接相关，因此与定向流 $v_1$ 密切相关。[24, 25, 26] $v_1$ 由产生粒子的方位分布相对于碰撞反应平面角度 ${\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}$ 的第一傅里叶矩定义。流体动力学模拟表明，系统中储存的轨道角动量和带电粒子的定向流几乎是直接成比例的[26]。这允许对全局极化的碰撞能量依赖性进行经验估计[18]。来自 RHIC 束能量扫描计划的定向流[27, 28]和超子全局极化[1, 2]的 STAR 结果表明，中快度（ $\mathrm{d}{v}_1/\mathrm{d}\eta$ ）处的带电强子和超子极化的斜率确实强相关。在 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=2.76$ TeV 的 Pb-Pb 碰撞中，带电粒子的定向流[29]约为顶部 RHIC 能量 200GeV[30]的三分之一。 这表明，在 LHC 能量下的全球极化应该也比 RHIC 小约三倍，并且从 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=2.76$ TeV 到 5.02 TeV 减少约 $\sim 20\mathrm{\%}$ [31]。在考虑定向流时，LHC 上的极化值甚至更小，定向流被认为是源在纵向方向的倾斜和源自初始能量密度分布不对称性的偶极流的组合[32]。考虑到只有来自倾斜源的定向流对涡度有关，而其相对于偶极流的贡献随碰撞能量的增加而减少[32]，人们得出 LHC 能量下全球极化的估计约为顶部 RHIC 能量的 0.15-0.2。

Role of the magnetic field
磁场的作用

In addition to possessing of the large orbital angular momentum, the system also experiences a strong magnetic field, of the order of $B\sim e/m_{\pi }^2\sim {10}^{14}$ T, generated in the initial state of the collision [33, 34, 35] by the fast moving electrically charged nuclei and by the spectator protons after the collision. The direction of the magnetic field coincides with that of the orbital angular momentum. Therefore, the measured global polarization would include a contribution from the magnetic field, see Eq. (2), especially if the magnetic field is sustained for longer time by the presence of the QGP [36]. Unlike to the case for vorticity contribution, the contribution from the
除了具有大轨道角动量外，该系统还经历了一个强磁场，约为 $B\sim e/m_{\pi }^2\sim {10}^{14}$ T，由快速移动的带电核和碰撞后的旁观质子在碰撞初态中产生[33, 34, 35]。磁场的方向与轨道角动量的方向一致。因此，测量的全局极化将包括磁场的贡献，参见方程（2），特别是如果磁场通过 QGP 的存在延续更长时间。与涡度贡献不同，来自

Figure 2: (a) A transverse plane distribution of the $z$-component of the velocity of participant nucleons in the center-of-mass frame, (b) $d{v}_z/dx$ distribution weighted with participant nucleon density, (c) the same for $d{v}_z/dy$, based on the Glauber model.
图 2：(a) 质心系中参与核子速度的 $z$ -分量的横截面分布，(b) 与参与核子密度加权的 $d{v}_z/dx$ 分布，(c) 基于格劳伯模型的 $d{v}_z/dy$ 相同。

magnetic field is opposite for particles and antiparticles because of the difference in signs of the magnetic moments $\mu$ in Eq. (2). The lifetime of the initial magnetic field depends on the electric conductivity of the QGP, which is poorly known. Precise measurements of the polarization difference between particles and antiparticles can provide an important constraint on the magnitude of the magnetic field at the hadronization time [23, 37], as well as on medium conductivity. Such information is of particular importance for study of the chiral magnetic effect [38].
磁场对粒子和反粒子是相反的，因为在方程（2）中磁矩的符号差异 $\mu$ 。初始磁场的寿命取决于 QGP 的电导率，这是很少知道的。粒子和反粒子之间极化差异的精确测量可以为强子化时的磁场大小提供重要约束[23, 37]，以及对介质电导率的约束。这些信息对于研究手征磁效应[38]具有特殊重要性。

At lower collision energies, the passing time of two nuclei becomes larger and therefore the lifetime of the magnetic field is extended. Also, the medium created in the collision has positive net-charge due to baryon stopping. If the system with non-zero charge rotates, a magnetic field might be created at relatively later stage. Such late-stage magnetic fields may also contribute to the observed polarization [39].
在较低的碰撞能量下，两个核子的相互作用时间变长，因此磁场的寿命延长。此外，碰撞中产生的介质由于重子阻止而具有正净电荷。如果带有非零电荷的系统旋转，磁场可能在相对较晚的阶段产生。这种晚期磁场也可能对观察到的极化有贡献。

In the global polarization picture based on the vorticity, one expects different particles to be polarized depending only on the particle spin in accordance with Eq. (2). A deviation could arise from effects of the initial magnetic field mentioned above, and from the fact that different particles are produced at different times or regions as the system freezes out [40], or through meson-baryon interactions [41]. Therefore, to understand the nature of the global polarization, it is crucial to measure the polarization of different particles, and if possible, particles with different spins. The polarization measurement with particles of different magnetic moments would provide additional information on the magnitude of the magnetic field. For example, the magnetic moment of $\mathrm{\Omega }$ hyperon is three times larger than that of $\mathrm{\Lambda }$ hyperon (${\mu }_{\mathrm{\Omega }-}=-2.02{\mu }_N$ and ${\mu }_{\mathrm{\Lambda }}=-0.613{\mu }_N$ where ${\mu }_N$ is the nuclear magneton). Thus $\mathrm{\Omega }$ hyperons are more sensitive to the magnetic field contribution to the polarization.
基于涡度的全球极化图像中，人们预期不同粒子的极化仅取决于粒子自旋，符合方程（2）。偏差可能来自上述初始磁场效应，以及不同粒子在系统冷冻时在不同时间或区域产生，或通过介子-重子相互作用。因此，要理解全球极化的性质，关键是测量不同粒子的极化，如果可能的话，还要测量具有不同自旋的粒子。具有不同磁矩的粒子的极化测量将提供有关磁场强度的额外信息。例如，Λ^0 超子的磁矩是Λ^1 超子的三倍（其中μ_N 是核磁子）。因此，Λ^0 超子对磁场对极化的贡献更为敏感。

Comparing the polarization between particles and antiparticles, it might be important to account for the effects of non-zero baryon chemical potential [42], especially at lower collisions energies. Besides directly affecting the quark distributions, the non-zero chemical potential could lead to different freeze-out conditions, and thus to different effective vorticities responsible for the particle polarization. But overall such effects are expected to be relatively small.
比较粒子和反粒子之间的极化，可能需要考虑非零重子化学势[42]的影响，特别是在较低的碰撞能量下。除了直接影响夸克分布外，非零化学势还可能导致不同的冻结条件，从而导致不同的负责粒子极化的有效涡度。但总体而言，这些影响预计会相对较小。

Anisotropic flow and polarization along the beam direction
沿着束流方向的各向异性流动和极化

Anisotropic flow leads to non-trivial collective velocity fields in the transverse direction, which in its turn would manifest itself via particle polarization along the beam direction [18]. The polarization $P_{\mathrm{z}}$ component dependence on the azimuthal angle would in general follow the anisotropic flow pattern of the same harmonic. We use a simple Blast Wave model to illustrate this phenomenon below.
各向异性流导致横向方向上的非平凡集体速度场，这反过来会通过粒子在束流方向上的极化来表现出来[18]。极化 $P_{\mathrm{z}}$ 分量对方位角的依赖通常会遵循相同谐波的各向异性流模式。我们使用简单的爆发波模型来说明这一现象。

In a simplest version of the Blast-Wave model including anisotropic flow [18, 43, 44], the particle production source at freeze-out is parameterized with 5 parameters: temperature $T$, maximum transverse radial flow velocity (rapidity) ${\rho }_{t,max}$ and amplitude of the azimuthal modulation in expansion velocity denoted as $b_n$, parameter $R$ characterizing the size, and the spatial anisotropy parameter $a_n$. The source, see Fig. 3(left), is then described by the following equations:
在包括各向异性流动的 Blast-Wave 模型的最简单版本中[18, 43, 44]，冻结时的粒子产生源被参数化为 5 个参数：温度 $T$ ，最大横向径向流速度（快度） ${\rho }_{t,max}$ 和扩张速度中的方位调制振幅标记为 $b_n$ ，表征大小的参数 $R$ ，以及空间各向异性参数 $a_n$ 。然后，源，见图 3（左），由以下方程描述：

$$\begin{array}{}\text{(3)}& r_{max}=R[1-a_n\cos (n{\varphi }_s)],{\rho }_t={\rho }_{t,max}\left[\frac{r}{r_{max}({\varphi }_s)}\right][1+b_n\cos (n{\varphi }_s)].\end{array}$$

It is assumed that the source element located at azimuthal angle ${\varphi }_s$ is boosted with velocity ${\rho }_t$ perpendicular to the surface of the source (${\varphi }_b$). Note that the parameters $a_n$ and $b_n$ are usually small, $a_n$, $b_n<0.1$, as follows from comparison of the model to the anisotropic flow and azimuthally sensitive femtoscopic measurements [45]. In the limit of $a_n\ll 1,b_n\ll 1$, the longitudinal component of the vorticity can be written as:
假设位于方位角 ${\varphi }_s$ 处的源元素以垂直于源表面（ ${\varphi }_b$ ）的速度 ${\rho }_t$ 进行增强。请注意，参数 $a_n$ 和 $b_n$ 通常很小， $a_n$ ， $b_n<0.1$ ，这是通过将模型与各向异性流和方位敏感的飞秒尺度测量结果进行比较得出的[45]。在 $a_n\ll 1,b_n\ll 1$ 的极限情况下，涡度的纵向分量可以写成：

$$\begin{array}{}\text{(4)}& {\omega }_z=\frac{1}{2}(\mathrm{\nabla }\times \mathbf{v}{)}_z\approx (\frac{{\rho }_{t,nmax}}{R})\sin (n{\varphi }_s)[b_n-a_n].\end{array}$$

It results in the following estimate for the hyperon polarization:
它导致超子极化的以下估计：

$$\begin{array}{}\text{(5)}& P_z\approx \frac{{\omega }_z}{2T}\approx 0.1\sin (n{\varphi }_s)[b_n-a_n],\end{array}$$

where it is assumed that ${\rho }_{t,nmax}\sim 1$, $R\sim 10$ fm, and $T\sim 100$ MeV. In practice, the coefficients $b_n$ and $a_n$ are both of the order of a few percent [45], often close to each other. That results in the magnitudes of $z$-polarization not greater than a few per-mill, almost an order of magnitude lower than what was obtained in original hydrodynamics calculations [19]. Note that the estimate above is valid for anisotropic flow of any harmonics $n$.
在这里假设 ${\rho }_{t,nmax}\sim 1$ ， $R\sim 10$ fm 和 $T\sim 100$ MeV。在实践中，系数 $b_n$ 和 $a_n$ 都在几个百分点的数量级上 [45]，通常彼此接近。这导致 $z$ 的极化幅度不大于几千分之几，几乎比原始流体力学计算中得到的结果低一个数量级 [19]。请注意，上述估计对于任何谐波的各向异性流都是有效的 $n$ 。

Figure 3: Transverse plane schematic view of the system leading to elliptic (left) and triangular flow (right). Red solid arrows indicate the expansion velocity, the largest along the greatest density gradients defining the flow angles, ${\mathrm{\Psi }}_2$ (coincides with $x$-axis) and ${\mathrm{\Psi }}_3$. In the left sketch, ${\varphi }_s$ denotes azimuthal angle of the source element which is boosted to ${\varphi }_b$ direction. Blue open arrows indicate local vorticities induced by the anisotropic flow.
图 3：系统导致椭圆（左）和三角形流动（右）的横截面示意图。红色实箭头表示膨胀速度，沿着定义流动角度的最大密度梯度最大（与 $x$ -轴重合）和 ${\mathrm{\Psi }}_3$ 。在左侧草图中， ${\varphi }_s$ 表示源元素的方位角，该角被提升到 ${\varphi }_b$ 方向。蓝色空箭头表示由各向异性流动引起的局部涡度。

Circular polarization, $P_{\varphi }$; polarization along $x$-direction, $P_{\mathrm{x}}$
循环极化， $P_{\varphi }$ ；沿 $x$ 方向的极化， $P_{\mathrm{x}}$

Non-uniform stopping in the transverse plane, and dependence of the expansion velocity on rapidity leads to toroidal structure of the velocity field [20, 46]. Theoretical calculations suggest that a vortex ring could be created at very forward/backward regions, most prominent in the central collisions. Even a better pronounced vortex ring could be created when smaller object passes through larger object such as central asymmetric collisions of Cu+Au, $d$+Au, and $p$+Au as first proposed in Ref. [18], see Fig. 4. The calculations [47] show that the polarization due to such vortex rings could reach values as high as a few percent. The smaller object could be replaced with a jet instead of nuclei [16]. According to the simulation of jet interacting with medium [15, 48], vortex rings can be created around the path which jet passes through in the medium.
在横向平面上的非均匀停止，以及扩张速度对快速度的依赖导致速度场的环形结构[20, 46]。理论计算表明，在非常前/后区域可能会形成涡环，最明显的是在中心碰撞中。甚至在较小物体穿过较大物体时，如 Cu+Au、 $d$ +Au 和 $p$ +Au 的中心不对称碰撞中，可能会形成更明显的涡环，正如在参考文献[18]中首次提出的，见图 4。计算[47]表明，由于这种涡环引起的极化可能达到几个百分点。较小的物体可以用喷流代替核子[16]。根据与介质相互作用的喷流模拟[15, 48]，涡环可以在喷流穿过介质的路径周围形成。

The axis of such a vortex ring is along the azimuthal direction relative to the smaller-nucleus-going or jet-going direction. The expected polarization in case of central A+B collisions can be expressed as $P_{\varphi }\propto {\hat{p}}_{\mathrm{T}}\times \hat{z}$ where ${\hat{p}}_T$ and $\hat{z}$ are the unit vectors along the particle transverse momentum and the beam direction, respectively (replace $\hat{z}$ with a unit vector pointing along the jet axis in case for the jet-induced polarization). Such measurement would require a careful treatment of the detector acceptance effects, excluding left-right asymmetry in particle reconstruction.
这样一个涡环的轴沿着方位角方向，相对于较小核前进或射流前进方向。在中心 A+B 碰撞的情况下，预期的极化可以表示为 $P_{\varphi }\propto {\hat{p}}_{\mathrm{T}}\times \hat{z}$ ，其中 ${\hat{p}}_T$ 和 $\hat{z}$ 分别是沿着粒子横向动量和束流方向的单位矢量（在射流诱导极化的情况下，用一个沿着射流轴的单位矢量替换 $\hat{z}$ ）。这样的测量需要对探测器接受效应进行仔细处理，排除粒子重建中的左右不对称性。

Finally we note the importance of the polarization measurements along the impact parameter direction, $P_{\mathrm{x}}$. Theoretical models [20, 49] suggest that $P_{\mathrm{x}}$ has an azimuthal dependence following $\sin (2\varphi )$ curve where $\varphi$ is the hyperon's azimuthal angle relative to the reaction plane. Such a dependence on the azimuthal angle is also expected from simple calculations based on the Glauber model, see right panel in Fig. 2 where ${\omega }_x\propto \frac{1}{2}\partial v_z/\partial y$. This component could also have a contribution from the so-called shear induced polarization, see the discussion in Sec. 3.2. Similarly to
最后，我们注意到沿着冲击参数方向的极化测量的重要性。理论模型[20, 49]表明， $P_{\mathrm{x}}$ 具有一个沿着 $\sin (2\varphi )$ 曲线的方位角依赖性，其中 $\varphi$ 是超子相对于反应平面的方位角。从简单的基于 Glauber 模型的计算中也预期在方位角上有这样的依赖性，参见图 2 中右侧面板，其中 ${\omega }_x\propto \frac{1}{2}\partial v_z/\partial y$ 。这个分量也可能来自所谓的剪切诱导极化，参见第 3.2 节中的讨论。类似地

Figure 4: A schematic view of a central asymmetric collision, such as Cu+Au, before (left) and after (right) the collision. The blue open arrows indicate the vorticity at the outer edges of the collision zone.
图 4：中心不对称碰撞（例如 Cu+Au）的示意图，在碰撞之前（左）和之后（右）。蓝色的开放箭头表示碰撞区域外缘的涡度。

the $P_{\varphi }$ measurements, these measurements could be technically difficult accounting for the acceptance effect.
这些 $P_{\varphi }$ 测量值，这些测量值在技术上可能会很难考虑到接受效应。

3 Spin and polarization in hydrodynamic description
3 旋转和极化在流体力学描述中

Kinematic vorticity, thermal gradients, acceleration
动力学涡度，热梯度，加速度

The Blast Wave model described in Sec. 2.3 is likely a gross oversimplification of the reality. It accounts, though still approximately, only for the contribution form so-called kinematic vorticity neglecting several other potentially important contributions. At the same time, as we discuss below, it describes surprisingly well the main features of the data. In relativistic hydrodynamics, the mean spin vector of $s=1/2$ particles with mass $m$ and four-momentum $p$ is given by the following equation:[22]
在第 2.3 节中描述的爆炸波模型很可能是对现实的粗略简化。尽管仍然大致上，它仅考虑了所谓的运动涡度的贡献，忽略了其他几个潜在重要的贡献。与此同时，正如我们在下文讨论的那样，它出奇地很好地描述了数据的主要特征。在相对论流体力学中，具有质量 $m$ 和四动量 $p$ 的 $s=1/2$ 粒子的平均自旋矢量由以下方程给出：[22]

$$\begin{array}{}\text{(6)}& S^{\mu }(x,p)=-\frac{1}{8m}(1-n_F)e^{\mu \tau \rho \sigma }{p}_{\tau }{\varpi }_{\rho \sigma },\end{array}$$

where $n_F$ is the Fermi-Dirac distribution and $\varpi$ is the thermal vorticity defined as:
其中 $n_F$ 是费米-狄拉克分布， $\varpi$ 是定义为热涡度的热量

$$\begin{array}{}\text{(7)}& {\varpi }_{\mu \nu }=\frac{1}{2}[{\partial }_{\nu }(u_{\mu }/T)-{\partial }_{\mu }(u_{\nu }/T)];\end{array}$$

$u_{\mu }=(\gamma ,\gamma \mathbf{v})$ is the fluid 4-velocity ($\gamma$ is the gamma factor), and $T$ is the proper temperature. Thermal vorticity can be subdivided into two parts - the part including the temperature gradients ${\partial }_{\mu }(1/T)$ and the part proportional to the kinematic vorticity ${\omega }_{\mu \nu }^{\mathrm{K}}=\frac{1}{2}({\partial }_{\nu }{u}_{\mu }-{\partial }_{\mu }{u}_{\nu })$. The latter in its turn can be separated into the part proportional to the acceleration and the spatial (transverse) part:
$u_{\mu }=(\gamma ,\gamma \mathbf{v})$ 是流体 4-速度（ $\gamma$ 是伽玛因子）， $T$ 是适当的温度。 热涡度可以分为两部分 - 包括温度梯度 ${\partial }_{\mu }(1/T)$ 的部分和与动力涡度 ${\omega }_{\mu \nu }^{\mathrm{K}}=\frac{1}{2}({\partial }_{\nu }{u}_{\mu }-{\partial }_{\mu }{u}_{\nu })$ 成比例的部分。 后者又可以分为与加速度成比例的部分和空间（横向）部分：

$$\begin{array}{}\text{(8)}& {\omega }_{\mu \nu }^{\mathrm{K}}=\frac{1}{2}(A_{\mu }{u}_{\nu }-A_{\nu }{u}_{\mu })+\frac{1}{2}({\mathrm{\nabla }}_{\nu }{u}_{\mu }-{\mathrm{\nabla }}_{\mu }{u}_{\nu }),\end{array}$$

where the acceleration vector $A^{\mu }=D{u}^{\mu }$ ($D=u^{\nu }{\partial }_{\nu }$ is the co-moving time derivative) and the ${\mathrm{\nabla }}_{\mu }={\partial }_{\mu }-u_{\mu }D$ is the so-called "orthogonal" (to $u^{\mu }$) derivative. The transverse part of the kinematic vorticity can be also expressed via the vorticity (angular velocity) vector
加速度矢量 $A^{\mu }=D{u}^{\mu }$ （ $D=u^{\nu }{\partial }_{\nu }$ 是共动时间导数）， ${\mathrm{\nabla }}_{\mu }={\partial }_{\mu }-u_{\mu }D$ 是所谓的“正交”（对 $u^{\mu }$ ）导数。运动涡度的横向部分也可以通过涡度（角速度）矢量来表示。

$$\begin{array}{}\text{(9)}& {\omega }^{\mu }=\frac{1}{2}{ϵ}^{\mu \nu \rho \sigma }{u}_{\nu }{\partial }_{\rho }{u}_{\sigma },\end{array}$$

as 作为

$$\begin{array}{}\text{(10)}& \frac{1}{2}({\mathrm{\nabla }}_{\nu }{u}_{\mu }-{\mathrm{\nabla }}_{\mu }{u}_{\nu })={ϵ}_{\mu \nu \rho \sigma }{\omega }^{\rho }{u}^{\sigma }.\end{array}$$

Using these notations and combining everything together, the spin vector can be written as
使用这些符号并将所有内容结合在一起，自旋矢量可以写成

$$S^{\mu }(x,p)=\frac{1}{8m}(1-n_F)$$ $$\begin{array}{}\text{(11)}& \times [-{ϵ}^{\mu \nu \rho \sigma }{p}_{\sigma }\frac{1}{T^2}({\partial }_{\nu }T)u_{\rho }+2\frac{{\omega }^{\mu }(u^{\nu }{p}_{\nu })-u^{\mu }({\omega }^{\nu }{p}_{\nu })}{T}-\frac{1}{T}{ϵ}^{\mu \nu \rho \sigma }{p}_{\sigma }{A}_{\nu }{u}_{\rho }],\end{array}$$with three terms describing contributions of the temperature gradient, the vorticity, and the acceleration, respectively. Note that in an ideal uncharged fluid the temperature gradient term and the acceleration contribution are related by the equation of motion ${\mathrm{\nabla }}_{\mu }T=T{A}_{\mu }$.
$$S^{\mu }(x,p)=\frac{1}{8m}(1-n_F)$$ $$\begin{array}{}\text{(11)}& \times [-{ϵ}^{\mu \nu \rho \sigma }{p}_{\sigma }\frac{1}{T^2}({\partial }_{\nu }T)u_{\rho }+2\frac{{\omega }^{\mu }(u^{\nu }{p}_{\nu })-u^{\mu }({\omega }^{\nu }{p}_{\nu })}{T}-\frac{1}{T}{ϵ}^{\mu \nu \rho \sigma }{p}_{\sigma }{A}_{\nu }{u}_{\rho }],\end{array}$$ 分别描述了温度梯度、涡度和加速度的贡献。请注意，在理想的无电荷流体中，温度梯度项和加速度贡献通过运动方程 ${\mathrm{\nabla }}_{\mu }T=T{A}_{\mu }$ 相关。

It is instructive to rewrite the expression Eq. 12 in the rest frame of the fluid, where $u^{\mu }=(1,0,0,0)$, $D={\partial }^t$, ${\mathrm{\nabla }}_{\mu }=(0,\mathrm{\nabla })$, and ${\omega }^{\mu }=(0,\omega )$:
在流体的静止参考系中重新表达表达式方程 12 是有益的，其中 $u^{\mu }=(1,0,0,0)$ ， $D={\partial }^t$ ， ${\mathrm{\nabla }}_{\mu }=(0,\mathrm{\nabla })$ 和 ${\omega }^{\mu }=(0,\omega )$ ：

$$\begin{array}{}\text{(12)}& S^0(x,p)=\frac{1}{8m}(1-n_F)\frac{\omega \cdot \mathbf{p}}{T},\end{array}$$ $$\begin{array}{}\text{(13)}& \mathbf{S}(x,p)=\frac{1}{8m}(1-n_F)(-\frac{\mathbf{p}\times \mathrm{\nabla }T}{T^2}+2\frac{E\omega }{T}+\frac{\mathbf{p}\times \mathbf{A}}{T}).\end{array}$$

In the above expressions, $E$ and $\mathbf{p}$ are the energy and momentum of the particle in the fluid rest frame. In the nonrelativistic limit, the contribution related to the angular velocity (coinciding with nonrelativistic vorticity) is the largest, with the contribution from temperature gradients and acceleration being suppressed by $v/c$ powers.
在上述表达中， $E$ 和 $\mathbf{p}$ 是粒子在流体静止参考系中的能量和动量。在非相对论极限下，与角速度相关的贡献（与非相对论涡度重合）最大，温度梯度和加速度的贡献被 $v/c$ 次幂抑制。

For completeness we also present an equation for the average spin vector transformation
为了完整性，我们还提供了平均自旋矢量变换的方程

$$\begin{array}{}\text{(14)}& {\mathbf{S}}^{\ast }=\mathbf{S}-\frac{\mathbf{p}\cdot \mathbf{S}}{E(E+m)}\mathbf{p},\end{array}$$

which should be used for calculation of the spin vector in the particle rest frame.
应该用于在粒子静止参考系中计算自旋矢量。

Shear-induced polarization and the spin Hall effect
剪切诱导极化和自旋霍尔效应

Very recently two groups [50, 51] independently reported a new mechanism for the spin polarization - so called "shear induced polarization" (SIP) originated in symmetric part of the velocity gradients ${\xi }^{\mu \nu }=1/2({\partial }^{\mu }{u}^{\nu }+{\partial }^{\nu }{u}^{\mu })$. Note that the expression for the polarization due to symmetric part of the velocity gradient tensor obtained by two groups are similar but not exactly the same with one qualitative difference as that the expressions obtained in Ref. [51] explicitly depends on the freeze-out hypersurface shape, while the expression in Ref. [50] allows "local" interpretation. For our qualitative discussion of the effect below we will use the definition in Ref. [50]
最近，两个独立的团体[50, 51]最近报告了一个新的自旋极化机制 - 所谓的“剪切诱导极化”（SIP），起源于速度梯度的对称部分 ${\xi }^{\mu \nu }=1/2({\partial }^{\mu }{u}^{\nu }+{\partial }^{\nu }{u}^{\mu })$ 。请注意，由两个团体得到的速度梯度张量对称部分引起的极化表达式相似，但并非完全相同，其中一个定性差异在于 Ref. [51]中得到的表达式明确取决于冻结超曲面的形状，而 Ref. [50]中的表达式允许“局部”解释。对于我们下面对效应的定性讨论，我们将使用 Ref. [50]中的定义。

The origin of SIP is the motion of a particle in anisotropic fluid. It is zero if the particle is moving with the fluid velocity, which is in contrast to the polarization due to vorticity. It is clearly seen if the corresponding expressions are written in the fluid rest frame $u^{\mu }=(1,0,0,0)$:
SIP 的起源是颗粒在各向异性流体中的运动。如果颗粒与流体速度相同，则为零，这与涡度引起的极化形成对比。如果相应的表达式在流体静止参考系中写出，这一点就很明显。

$$\begin{array}{}\text{(15)}& S_i^{(\mathrm{v}\mathrm{o}\mathrm{r}\mathrm{t})}\approx \frac{E}{8mT}{ϵ}_{ikj}\frac{1}{2}({\partial }_k{v}_j-{\partial }_j{v}_k),\end{array}$$

$$\begin{array}{}\text{(16)}& S_i^{(\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{r})}\approx \frac{1}{4mTE}{ϵ}_{ikj}{p}_k{p}_m\frac{1}{2}({\partial }_j{v}_m+{\partial }_m{v}_j).\end{array}$$

One can see that the SIP contribution to the polarization is suppressed by the order of $(p/E{)}^2$ compared to vorticity contribution and become zero for particles moving with the fluid velocity. It was also pointed out recently that the chemical potential gradients could also contribute to the polarization. This contribution identified as the "spin Hall effect" (SHE) [52]. In the fluid rest frame :
一个可以看到的事实是，与涡度贡献相比，SIP 对极化的贡献被抑制了 $(p/E{)}^2$ 阶，并且对于与流体速度移动的粒子来说，这种贡献变为零。最近还指出化学势梯度也可能对极化产生贡献。这种贡献被称为“自旋霍尔效应”（SHE）[52]。在流体静止参考系中：

$$\begin{array}{}\text{(17)}& S_i^{(\mathrm{S}\mathrm{H}\mathrm{E})}\approx \frac{1}{4mE}{ϵ}_{ikj}{p}_k{\partial }_j(\mu /T).\end{array}$$

Similarly to SIP, the polarization due to the gradients in baryon chemical potential (SHE - spin Hall effect) is also suppressed by a power of $p/E$. The SHE might be important in particular for the interpretation of the difference in polarization of particles and antiparticles. Note that the role of chemical potential was also studied earlier in a different content with the conclusion that for nonrelativistic hyperons the effect is almost negligible [42]. The new effects, both SIP and SHE, are related to the motion of the particle in anisotropic fluid, and thus expected to be small for nonrelativistic particles compared to the polarization due to vorticity. This is the main reason why the calculations that involve quark degree of freedom result in stronger effects compared to those where SIP and SHE contributions are calculated directly for (nonrelativistic) hyperons. We discuss this in more detail in relation to the experimental measurements of the polarization along the beam direction ($P_{\mathrm{z}}$).
类似于 SIP，由于重子化学势梯度而产生的极化（SHE - 自旋霍尔效应）也被 $p/E$ 的幂次抑制。SHE 可能对解释粒子和反粒子极化差异特别重要。请注意，化学势的作用在不同的内容中也曾进行过研究，得出结论称对于非相对论超子，该效应几乎可以忽略不计[42]。新效应，即 SIP 和 SHE，与粒子在各向异性流体中的运动有关，因此预计对于非相对论粒子而言，与由涡度引起的极化相比将会较小。这也是为什么涉及夸克自由度的计算结果会比那些直接计算（非相对论）超子的 SIP 和 SHE 贡献的计算结果产生更强烈的影响的主要原因。我们将在与沿着束流方向的极化的实验测量相关的更多细节中讨论这一点（ $P_{\mathrm{z}}$ ）。

Additional comments 额外评论

While several model calculations do show a significant contribution to the hyperon polarization from temperature gradients and acceleration, in our more qualitative discussion we mostly argue on the basis of the contribution from kinematic vorticity. The freeze-out temperature of the system is about $\sim 100$ MeV, and all the hyperons are nonrelativistic in the local fluid frame. For that reason we also often estimate the polarization in the fluid frame, although all the experimental measurements are performed in the particle rest frame. We do treat the hyperons as relativistic in the laboratory frame though, as the fluid collective motion is relativistic. The nonrelativistic treatment might fail if the final particle polarization is due to the coalescence of initially (during the system evolution before the hadronization) polarized (constituent) quarks, with masses that are only factor of 2-3 higher than the temperature.
尽管几种模型计算确实显示出温度梯度和加速度对超子极化的显著贡献，但在我们更为定性的讨论中，我们主要是基于动力学涡量的贡献进行论证。系统的冻结温度约为 $\sim 100$ MeV，所有超子在局部流体参考系中都是非相对论的。因此，我们经常在流体参考系中估计极化，尽管所有实验测量都是在粒子静止参考系中进行的。然而，在实验室参考系中，我们将超子视为相对论的，因为流体的集体运动是相对论的。如果最终粒子极化是由于最初（在强子化之前系统演化期间）极化（成分）夸克的聚合而导致，而这些夸克的质量仅比温度高 2-3 倍，那么非相对论处理可能会失败。

All hydrodynamic calculations use the Cooper-Frye prescription [53] for the fluid freeze-out. This prescription has several known problems (see e.g., Refs. [54, 55]), which might be not very important for calculations of the particle spectra, but it is not known how good the Cooper-Frye prescription is for calculation of the polarization. In particular, the contributions from the temperature gradients and acceleration might be questionable, as the very concept of freeze-out assumes insignificance of those effect. Then. their contributions would be related to the corresponding relaxation times of the system. Note, that if found significant, the measurement of those effects might provide unique information about the velocity and temperature gradients at freeze-out, for which the particle spectra are mostly insensitive.
所有流体动力学计算都使用 Cooper-Frye prescription [53] 进行流体冷冻过程的计算。这种 prescription 存在一些已知问题（参见例如，Refs. [54, 55]），这些问题对于粒子谱的计算可能并不是非常重要，但目前尚不清楚 Cooper-Frye prescription 在计算极化方面的表现如何。特别是，温度梯度和加速度的贡献可能是有问题的，因为冷冻过程的概念本身意味着这些效应的微不足道。然后，它们的贡献将与系统的相应弛豫时间相关。请注意，如果发现这些效应显著，那么对这些效应的测量可能提供关于冷冻过程中速度和温度梯度的独特信息，而粒子谱对此大多不敏感。

Vortical effects may also strongly affect the baryon dynamics of the system, leading to a separation of baryon and antibaryons along the vorticity direction (perpendicular to the reaction plane) - the so-called Chiral Vortical Effect (CVE) [38]. The CVE is similar in many respect to the more familiar Chiral Magnetic Effect (CME) - the electric charge separation along the magnetic field. For reviews on those and similar effects, as well as the status of the experimental search for those phenomena, see Refs. [38, 56]. For a reliable theoretical calculation of both effects, one has to know the vorticity of the created system as well as the evolution of (electro)magnetic field.
涡旋效应也可能强烈影响系统的重子动力学，导致重子和反重子沿涡旋方向（垂直于反应平面）分离 - 所谓的手征涡旋效应（CVE）[38]。 CVE 在许多方面类似于更为熟悉的手征磁效应（CME）-沿磁场的电荷分离。有关这些和类似效应的评论，以及对这些现象的实验搜索状况，请参阅参考文献[38, 56]。要可靠地计算这两种效应，必须了解所创建系统的涡旋以及（电磁）场的演变。

In view of the recent polarization measurements in ultra-relativistic heavy-ion collisions, note the discussion [57] of a physical meaning of the spin angular momentum in quantum field theory and relativistic hydrodynamics.
鉴于超相对论重离子碰撞中最近的极化测量，注意量子场论和相对论流体力学中自旋角动量的物理意义的讨论[57]。

4 How is it measured
4 如何测量

Self-analyzing weak decays of hyperons
超子的弱衰变自我分析

The hyperon weak decays provide a most straightforward way to experimentally measure polarization of particles produced in heavy-ion collisions. Because of its parity-violating weak decay, the angular distribution of the decay products at the hyperon rest frame obeys the following relation:
超子弱衰变提供了一种最直接的实验方法，用于实验测量在重离子碰撞中产生的粒子的极化。由于其破坏宇称的弱衰变，超子静止参考系中的衰变产物的角分布遵循以下关系：

$$\begin{array}{}\text{(18)}& \frac{dN}{d{\mathrm{\Omega }}^{\ast }}=\frac{1}{4\pi }(1+{\alpha }_{\mathrm{H}}{\mathbf{P}}_{\mathrm{H}}^{\ast }\cdot {\hat{\mathbf{p}}}_{\mathrm{B}}^{\ast }),\end{array}$$

where ${\alpha }_{\mathrm{H}}$ is the hyperon decay parameter, ${\mathbf{P}}_{\mathrm{H}}^{\ast }$ is the hyperon polarization vector, and ${\hat{\mathbf{p}}}_{\mathrm{B}}^{\ast }$ is the unit vector in the direction of the daughter baryon momentum, and the $\mathrm{\Omega }$ is its solid angle. The asterisk is used to denote quantities in the hyperon rest frame. The decay parameter ${\alpha }_{\mathrm{H}}$ reflects analyzing power in the measurement, it is different for different hyperons. In case of $\mathrm{\Lambda }$ hyperon decay of $\mathrm{\Lambda }\to p+{\pi }^{-}$, the decay parameter is ${\alpha }_{\mathrm{\Lambda }}=0.732\pm 0.014$[58]. The decay parameter for $\overline{\mathrm{\Lambda }}\to \overline{p}+{\pi }^{+}$ is usually treated same as $\mathrm{\Lambda }$, i.e., ${\alpha }_{\mathrm{\Lambda }}=-{\alpha }_{\overline{\mathrm{\Lambda }}}$, assuming that charge conjugation parity (CP) symmetry is conserved, although the world average data shows slightly higher value ${\alpha }_{\overline{\mathrm{\Lambda }}}=-0.758\pm 0.014$[58]. In this paper, we follow this convention unless it is specified in the figure. We also note that the recent studies [59, 60] indicate that ${\alpha }_{\mathrm{\Lambda }}$ could be be larger by a few percent compared to the aforementioned value (and be closer to ${\alpha }_{\mathrm{\Lambda }}\approx {\alpha }_{\overline{\mathrm{\Lambda }}}$), leading to a few % reduction of the measured polarization.
其中 ${\alpha }_{\mathrm{H}}$ 是超子衰变参数， ${\mathbf{P}}_{\mathrm{H}}^{\ast }$ 是超子极化矢量， ${\hat{\mathbf{p}}}_{\mathrm{B}}^{\ast }$ 是指向子重子动量的单位矢量， $\mathrm{\Omega }$ 是其固体角度。星号用于表示超子静止参考系中的量。衰变参数 ${\alpha }_{\mathrm{H}}$ 反映了测量中的分析能力，对于不同的超子是不同的。在 $\mathrm{\Lambda }$ 超子衰变为 $\mathrm{\Lambda }\to p+{\pi }^{-}$ 的情况下，衰变参数为 ${\alpha }_{\mathrm{\Lambda }}=0.732\pm 0.014$ [58]。 $\overline{\mathrm{\Lambda }}\to \overline{p}+{\pi }^{+}$ 的衰变参数通常被视为与 $\mathrm{\Lambda }$ 相同，即 ${\alpha }_{\mathrm{\Lambda }}=-{\alpha }_{\overline{\mathrm{\Lambda }}}$ ，假设电荷共轭宇称（CP）对称性是守恒的，尽管世界平均数据显示略高的值 ${\alpha }_{\overline{\mathrm{\Lambda }}}=-0.758\pm 0.014$ [58]。在本文中，除非在图中另有说明，我们将遵循这一惯例。我们还注意到最近的研究[59, 60]表明 ${\alpha }_{\mathrm{\Lambda }}$ 可能比前述值大几个百分点（接近 ${\alpha }_{\mathrm{\Lambda }}\approx {\alpha }_{\overline{\mathrm{\Lambda }}}$ ），导致测量极化减少几个百分点。

4.1.1 Multistrange hyperons and two-step decays
4.1.1 多奇异超子和两步衰变

Multistrange hyperons such as $\mathrm{\Xi }$ and $\mathrm{\Omega }$ decay in two steps. For example, in case of ${\mathrm{\Xi }}^{-}$ hyperon (spin-1/2), ${\mathrm{\Xi }}^{-}\to \mathrm{\Lambda }+{\pi }^{-}$ with subsequent decay $\mathrm{\Lambda }\to p+{\pi }^{-}$. If ${\mathrm{\Xi }}^{-}$ is polarized, its polarization is partially transferred to the daughter $\mathrm{\Lambda }$. Both steps in such a cascade decay are parity-violating and thus can be used for an independent measurement of the parent hyperon polarization. The decay parameter for ${\mathrm{\Xi }}^{-}\to \mathrm{\Lambda }+{\pi }^{-}$ is ${\alpha }_{{\mathrm{\Xi }}^{-}}=-0.401\pm 0.010$[58]. This value of ${\alpha }_{\mathrm{\Xi }}$ was constrained by the measurement of the product of ${\alpha }_{\mathrm{\Xi }}{\alpha }_{\mathrm{\Lambda }}$ with the ${\alpha }_{\mathrm{\Lambda }}$ measured separately, therefore the change of ${\alpha }_{\mathrm{\Lambda }}$ would affect the value of ${\alpha }_{\mathrm{\Xi }}$ as well. We also note that the recent direct measurement of ${\alpha }_{\mathrm{\Xi }}$[59] suggests a slightly different value (${\alpha }_{\mathrm{\Xi }}=-0.376\pm 0.007$).
多奇异超子，如Λ和Ξ，在两个步骤中衰变。例如，在Ξ超子（自旋-1/2）的情况下，首先发生 $\mathrm{\Lambda }\to p+{\pi }^{-}$ 的 ${\mathrm{\Xi }}^{-}\to \mathrm{\Lambda }+{\pi }^{-}$ 。如果 ${\mathrm{\Xi }}^{-}$ 极化，其极化部分转移到子 $\mathrm{\Lambda }$ 。这种级联衰变中的两个步骤都是破坏宇称的，因此可以用于独立测量父超子的极化。 ${\mathrm{\Xi }}^{-}\to \mathrm{\Lambda }+{\pi }^{-}$ 的衰变参数为 ${\alpha }_{{\mathrm{\Xi }}^{-}}=-0.401\pm 0.010$ [58]。 ${\alpha }_{\mathrm{\Xi }}$ 的这个值受到了 ${\alpha }_{\mathrm{\Xi }}{\alpha }_{\mathrm{\Lambda }}$ 与 ${\alpha }_{\mathrm{\Lambda }}$ 分别测量的乘积的约束，因此 ${\alpha }_{\mathrm{\Lambda }}$ 的变化也会影响 ${\alpha }_{\mathrm{\Xi }}$ 的值。我们还注意到最近对 ${\alpha }_{\mathrm{\Xi }}$ 的直接测量[59]表明一个略有不同的值（ ${\alpha }_{\mathrm{\Xi }}=-0.376\pm 0.007$ ）。

The polarization of the daughter baryon in the weak decay of a spin-1/2 hyperon is described by the Lee-Yang formula[61, 62, 63] with three decay parameters; $\alpha$, $\beta$, and $\gamma$, where $\alpha$ is a parity-violating part reflecting decay asymmetry as mentioned above, $\beta$ accounts for the violation of the time reversal symmetry, and $\gamma$ satisfies ${\alpha }^2+{\beta }^2+{\gamma }^2=1$. For a particular case of $\mathrm{\Xi }\to \mathrm{\Lambda }+\pi$ decay, the daughter $\mathrm{\Lambda }$ polarization in its rest frame can be written as:
女儿重子在自旋 1/2 超子弱衰变中的极化由 Lee-Yang 公式[61, 62, 63]描述，其中包括三个衰变参数; $\alpha$ ， $\beta$ 和 $\gamma$ ，其中 $\alpha$ 是一个反映衰变不对称性的破坏宇称的部分， $\beta$ 解释了时间反演对称性的破坏， $\gamma$ 满足 ${\alpha }^2+{\beta }^2+{\gamma }^2=1$ 。对于 $\mathrm{\Xi }\to \mathrm{\Lambda }+\pi$ 衰变的特定情况，女儿 $\mathrm{\Lambda }$ 在其静止参考系中的极化可以写成：

$$\begin{array}{}\text{(19)}& {\mathbf{P}}_{\mathrm{\Lambda }}^{\ast }=\frac{({\alpha }_{\mathrm{\Xi }}+{\mathbf{P}}_{\mathrm{\Xi }}^{\ast }\cdot {\hat{\mathit{p}}}_{\mathrm{\Lambda }}^{\ast }){\hat{\mathit{p}}}_{\mathrm{\Lambda }}^{\ast }+{\beta }_{\mathrm{\Xi }}{\mathbf{P}}_{\mathrm{\Xi }}^{\ast }\times {\hat{\mathit{p}}}_{\mathrm{\Lambda }}^{\ast }+{\gamma }_{\mathrm{\Xi }}{\hat{\mathit{p}}}_{\mathrm{\Lambda }}^{\ast }\times ({\mathbf{P}}_{\mathrm{\Xi }}^{\ast }\times {\hat{\mathit{p}}}_{\mathrm{\Lambda }}^{\ast })}{1+{\alpha }_{\mathrm{\Xi }}{\mathbf{P}}_{\mathrm{\Xi }}^{\ast }\cdot {\hat{\mathit{p}}}_{\mathrm{\Lambda }}^{\ast }},\end{array}$$

where ${\hat{\mathit{p}}}_{\mathrm{\Lambda }}^{\ast }$ is the unit vector of the $\mathrm{\Lambda }$ momentum, and $\mathrm{\Xi }$ polarization ${\mathit{P}}_{\mathrm{\Xi }}^{\ast }$ is given in the $\mathrm{\Xi }$ rest frame. Averaging over the angular distribution of the daughter $\mathrm{\Lambda }$ in the rest frame of the $\mathrm{\Xi }$ given by Eq. (18) leads to
其中 ${\hat{\mathit{p}}}_{\mathrm{\Lambda }}^{\ast }$ 是 $\mathrm{\Lambda }$ 动量的单位矢量， $\mathrm{\Xi }$ 极化 ${\mathit{P}}_{\mathrm{\Xi }}^{\ast }$ 在 $\mathrm{\Xi }$ 静止参考系中给出。在由方程（18）给出的 $\mathrm{\Xi }$ 静止参考系中对子 $\mathrm{\Lambda }$ 的角分布进行平均可导致

$$\begin{array}{}\text{(20)}& {\mathbf{P}}_{\mathrm{\Lambda }}^{\ast }=C_{{\mathrm{\Xi }}^{-}\mathrm{\Lambda }}{\mathbf{P}}_{\mathrm{\Xi }}^{\ast }=\frac{1}{3}(1+2{\gamma }_{\mathrm{\Xi }}){\mathbf{P}}_{\mathrm{\Xi }}^{\ast }.\end{array}$$

Using the measured value for the ${\gamma }_{\mathrm{\Xi }}$ parameter,[58, 63] the polarization transfer coefficient for ${\mathrm{\Xi }}^{-}$ to $\mathrm{\Lambda }$ decay leads to:
使用测得的 ${\gamma }_{\mathrm{\Xi }}$ 参数数值，[58, 63]， ${\mathrm{\Xi }}^{-}$ 到 $\mathrm{\Lambda }$ 衰变的极化转移系数为：

$$\begin{array}{}\text{(21)}& C_{{\mathrm{\Xi }}^{-}\mathrm{\Lambda }}=\frac{1}{3}(1+2\times 0.916)=+0.944.\end{array}$$

This shows that the polarization of ${\mathrm{\Xi }}^{-}$ is transferred to daughter $\mathrm{\Lambda }$ almost at its full value. We also would like to point out that the value of the $\gamma$ parameter is constrained by the measured $\alpha$ and $\varphi$ as $\gamma =(1-{\alpha }^2){\cos }^2\varphi$ where $\varphi ={\tan }^{-1}\beta /\gamma$, therefore the change in $\alpha$ parameter as well as $\varphi$ value would also lead to a change in the $\gamma$ parameter.
这表明 ${\mathrm{\Xi }}^{-}$ 的极化几乎完全转移到了女儿 $\mathrm{\Lambda }$ 。我们还想指出 $\gamma$ 参数的值受到测得的 $\alpha$ 和 $\varphi$ 的约束，如 $\gamma =(1-{\alpha }^2){\cos }^2\varphi$ 所示，其中 $\varphi ={\tan }^{-1}\beta /\gamma$ ，因此 $\alpha$ 参数的变化以及 $\varphi$ 值也会导致 $\gamma$ 参数的变化。

The polarization of the daughter baryon in a two-particle decay of spin-3/2 hyperon, i.e., $\mathrm{\Omega }\to \mathrm{\Lambda }+K$, can be also described by three parameters ${\alpha }_{\mathrm{\Omega }}$, ${\beta }_{\mathrm{\Omega }}$, and ${\gamma }_{\mathrm{\Omega }}$.[64] The decay parameter ${\alpha }_{\mathrm{\Omega }}$ determines the angular distribution of $\mathrm{\Lambda }$ in the $\mathrm{\Omega }$ rest frame and is measured to be small:[58]${\alpha }_{\mathrm{\Omega }}=0.0157\pm 0.0021$; this means that the measurement of $\mathrm{\Omega }$ polarization via analysis of the daughter $\mathrm{\Lambda }$ angular distribution is practically impossible. The polarization transfer in this decay is determined by the ${\gamma }_{\mathrm{\Omega }}$ parameter as:[64, 65, 66]
子重子的极化在自旋-3/2 超子的两粒子衰变中也可以用三个参数 ${\alpha }_{\mathrm{\Omega }}$ ， ${\beta }_{\mathrm{\Omega }}$ 和 ${\gamma }_{\mathrm{\Omega }}$ 来描述。[64]衰变参数 ${\alpha }_{\mathrm{\Omega }}$ 确定了 $\mathrm{\Omega }$ 静止系中 $\mathrm{\Lambda }$ 的角分布，并测量结果很小：[58] ${\alpha }_{\mathrm{\Omega }}=0.0157\pm 0.0021$ ；这意味着通过分析子 $\mathrm{\Lambda }$ 角分布来测量 $\mathrm{\Omega }$ 极化几乎是不可能的。这种衰变中的极化转移由 ${\gamma }_{\mathrm{\Omega }}$ 参数确定：[64, 65, 66]

$$\begin{array}{}\text{(22)}& {\mathbf{P}}_{\mathrm{\Lambda }}^{\ast }=C_{{\mathrm{\Omega }}^{-}\mathrm{\Lambda }}{\mathbf{P}}_{\mathrm{\Omega }}^{\ast }=\frac{1}{5}(1+4{\gamma }_{\mathrm{\Omega }}){\mathbf{P}}_{\mathrm{\Omega }}^{\ast }.\end{array}$$

The parameter ${\gamma }_{\mathrm{\Omega }}$ is unknown but considering the time-reversal violation parameter ${\beta }_{\mathrm{\Omega }}$ to be small, one can expect that the unmeasured parameter ${\gamma }_{\mathrm{\Omega }}$ is ${\gamma }_{\mathrm{\Omega }}\approx \pm 1$, based on the constraint ${\alpha }^2+{\beta }^2+{\gamma }^2=1$. This results in a polarization transfer to be $C_{{\mathrm{\Omega }}^{-}\mathrm{\Lambda }}\approx 1$ or $C_{{\mathrm{\Omega }}^{-}\mathrm{\Lambda }}\approx -0.6$. As discussed later, the measurement of $\mathrm{\Omega }$ global polarization can help to resolve the ambiguity of ${\gamma }_{\mathrm{\Omega }}$ under assumption of the vorticity picture in heavy-ion collisions.
参数 ${\gamma }_{\mathrm{\Omega }}$ 是未知的，但考虑到时间反演破坏参数 ${\beta }_{\mathrm{\Omega }}$ 很小，可以预期未测量的参数 ${\gamma }_{\mathrm{\Omega }}$ 是 ${\gamma }_{\mathrm{\Omega }}\approx \pm 1$ ，基于约束 ${\alpha }^2+{\beta }^2+{\gamma }^2=1$ 。这导致极化转移为 $C_{{\mathrm{\Omega }}^{-}\mathrm{\Lambda }}\approx 1$ 或 $C_{{\mathrm{\Omega }}^{-}\mathrm{\Lambda }}\approx -0.6$ 。正如后面讨论的那样，测量 $\mathrm{\Omega }$ 的全局极化可以帮助解决在重离子碰撞中涡度图像假设下 ${\gamma }_{\mathrm{\Omega }}$ 的歧义。

Global polarization measurement
全球极化测量

Polarization component along the initial orbital angular momentum $\mathbf{L}$ for hyperons, referred to as global polarization when averaged over all produced particles, can be obtained by integrating Eq. (18) over the polar angle of daughter baryon ${\theta }_{\mathrm{B}}^{\ast }$ and the reaction plane angle ${\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}$, considering the projection of the polarization onto the direction $\mathbf{L}$. In Eq. (18), ${\mathbf{P}}_{\mathrm{H}}^{\ast }\cdot {\hat{\mathbf{p}}}_{\mathrm{B}}^{\ast }$ can be substituted with $P_{\mathrm{H}}\cos {\theta }^{\ast }=P_{\mathrm{H}}\sin {\theta }_{\mathrm{B}}^{\ast }\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }_{\mathrm{B}}^{\ast })$ where ${\theta }^{\ast }$ is the angle between the polarization vector and momentum of daughter baryon in the hyperon rest frame, and ${\theta }_{\mathrm{B}}^{\ast }$ and ${\varphi }_{\mathrm{B}}^{\ast }$ are azimuthal and polar angles of daughter baryon in the hyperon rest frame. Then the average of $\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }_{\mathrm{B}}^{\ast })$ is calculated as
沿着超子的初始轨道角动量 $\mathbf{L}$ 的极化分量，当对所有产生的粒子进行平均时被称为全局极化，可以通过在子核子 ${\theta }_{\mathrm{B}}^{\ast }$ 的极角和反应平面角 ${\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}$ 上积分 Eq.（18）来获得，考虑极化在方向 $\mathbf{L}$ 上的投影。在 Eq.（18）中， ${\mathbf{P}}_{\mathrm{H}}^{\ast }\cdot {\hat{\mathbf{p}}}_{\mathrm{B}}^{\ast }$ 可以用 $P_{\mathrm{H}}\cos {\theta }^{\ast }=P_{\mathrm{H}}\sin {\theta }_{\mathrm{B}}^{\ast }\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }_{\mathrm{B}}^{\ast })$ 替换，其中 ${\theta }^{\ast }$ 是超子静止系中极化矢量和子核子动量之间的角度， ${\theta }_{\mathrm{B}}^{\ast }$ 和 ${\varphi }_{\mathrm{B}}^{\ast }$ 是超子静止系中子核子的方位角和极角。然后计算 $\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }_{\mathrm{B}}^{\ast })$ 的平均值。

$$⟨\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }_{\mathrm{B}}^{\ast })⟩$$ $$\begin{array}{}\text{(23)}& =\int d{\mathrm{\Omega }}^{\ast }\int \frac{d{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}}{2\pi }\frac{1}{4\pi }[1+{\alpha }_{\mathrm{H}}{P}_{\mathrm{H}}\sin {\theta }_{\mathrm{B}}^{\ast }\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }_B^{\ast })]\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }_B^{\ast })\end{array}$$ $$\begin{array}{}\text{(24)}& =\frac{{\alpha }_{\mathrm{H}}{P}_{\mathrm{H}}}{2}\int d{\mathrm{\Omega }}^{\ast }\sin {\theta }_{\mathrm{B}}^{\ast }.\end{array}$$

Hence, this leads to
因此，这导致

$$P_{\mathrm{H}}=\frac{8}{\pi {\alpha }_{\mathrm{H}}}\frac{1}{(4/\pi )\int d{\mathrm{\Omega }}^{\ast }\sin {\theta }_{\mathrm{B}}^{\ast }}⟨\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }_{\mathrm{B}}^{\ast })⟩$$ $$\begin{array}{}\text{(25)}& =\frac{8}{\pi {\alpha }_{\mathrm{H}}}\frac{1}{A_0}⟨\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }_{\mathrm{B}}^{\ast })⟩,\end{array}$$

where $A_0=(4/\pi )\int d{\mathrm{\Omega }}^{\ast }\sin {\theta }_{\mathrm{B}}^{\ast }$ is an acceptance correction factor and usually estimated in a data-driven way. If the detector can measure all produced hyperons of interest, the factor $A_0$ becomes unity. In practice, $A_0$ slightly deviates from unity and depends on the event multiplicity and the hyperon transverse momentum $p_T$. Equation (25) does not account for the polarization dependence on azimuthal angle, but such a dependence as well as the presence of elliptic flow might affect the measurements. For a more detailed discussion of the acceptance and tracking efficiency corrections including $P_{\mathrm{H}}$ azimuthal dependence, see Sec. 4.6.
其中 $A_0=(4/\pi )\int d{\mathrm{\Omega }}^{\ast }\sin {\theta }_{\mathrm{B}}^{\ast }$ 是一个接受校正因子，通常以数据驱动的方式估计。如果探测器能够测量所有感兴趣的产生的超子，因子 $A_0$ 将变为单位。在实践中， $A_0$ 略微偏离单位，并取决于事件多重性和超子横向动量 $p_T$ 。方程（25）不考虑极化对方位角的依赖性，但这种依赖性以及椭圆流的存在可能会影响测量结果。有关接受和跟踪效率校正的更详细讨论，包括 $P_{\mathrm{H}}$ 方位角依赖性，请参见第 4.6 节。

Experimentally the first-order event plane angle ${\mathrm{\Psi }}_1$ is used as a proxy of ${\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}$. Then Eq. (25) can be rewritten to take into account for the event plane resolution as follows[13]
实验上，一阶事件平面角 ${\mathrm{\Psi }}_1$ 被用作 ${\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}$ 的代理。然后，方程（25）可以重写以考虑事件平面分辨率，如下所示[13]。

$$\begin{array}{}\text{(26)}& P_{\mathrm{H}}=\frac{8}{\pi {\alpha }_{\mathrm{H}}}\frac{1}{A_0}\frac{⟨\sin ({\mathrm{\Psi }}_1-{\varphi }_{\mathrm{B}}^{\ast })⟩}{\text{Res}({\mathrm{\Psi }}_1)},\end{array}$$

where $\text{Res}({\mathrm{\Psi }}_1)$ is the event plane resolution defined as $⟨\cos ({\mathrm{\Psi }}_1-{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}})⟩$. The azimuthal angle ${\mathrm{\Psi }}_1$ can be determined by measuring spectator fragments and provides the direction of the initial orbital angular momentum.[67]
其中 $\text{Res}({\mathrm{\Psi }}_1)$ 是定义为 $⟨\cos ({\mathrm{\Psi }}_1-{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}})⟩$ 的事件平面分辨率。方位角 ${\mathrm{\Psi }}_1$ 可以通过测量旁观者碎片来确定，并提供初始轨道角动量的方向。[67]

Equation 26 provides a possibility to measure global polarization of hyperons by measuring only azimuthal distributions of the daughter baryon. While this approach based on well established anisotropic flow techniques, a slightly better statistical accuracy could be achieved by measuring the full angular distribution, includingpolar angle. In this case,
方程 26 提供了一种通过仅测量子重粒子的方位分布来测量超子全局极化的可能性。虽然这种方法基于成熟的各向异性流技术，但通过测量完整的角分布，包括极角，可能实现稍微更好的统计精度。在这种情况下，

$$\begin{array}{}\text{(27)}& P_{\mathrm{H}}=\frac{3}{{\alpha }_{\mathrm{H}}}⟨\cos {\theta }^{\ast }⟩=\frac{3}{{\alpha }_{\mathrm{H}}}⟨\sin {\theta }_{\mathrm{B}}^{\ast }\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }_{\mathrm{B}}^{\ast })⟩.\end{array}$$

For a discussion of the acceptance and tracking efficiency corrections including the azimuthal dependence, see Sec. 4.6.
有关接受和跟踪效率校正的讨论，包括方位角依赖性，请参阅第 4.6 节。

4.2.1 Global polarization in which frame?
4.2.1 全球极化在哪个框架中？

As defined in Eq. (18), the polarization is measured in the hyperon rest frame and the global polarization is the polarization component along the orbital angular momentum $\mathbf{L}$ direction in the center-of-mass frame of heavy-ion collisions as shown in Eq. (26). Strictly speaking, the $\mathbf{L}$ direction in $\mathrm{\Lambda }$ rest frame is different from the $\mathbf{L}$ direction in the center-of-mass frame of heavy-ion collisions, and therefore the proper treatment of the reference frame and measured polarization is needed. Reference[68] studies the effect of the frame difference in the measurement of global polarization and found that the effect is small and reaches about 10% for high transverse momentum ($p_{\mathrm{T}}\sim$ 4-5 GeV/$c$). Note that the mean $p_T$ for $\mathrm{\Lambda }$ is $\sim$ 1 GeV/$c$[69] depending on the centrality and collision energy, and could be slightly higher ($p_{\mathrm{T}}\sim$ 1.5 GeV/$c$) with the kinematic cut for $\mathrm{\Lambda }$ used in the polarization measurement.
如方程（18）中定义的，极化是在超子静止参考系中测量的，全局极化是沿着重离子碰撞质心参考系中轨道角动量 $\mathbf{L}$ 方向的极化分量，如方程（26）所示。严格来说， $\mathrm{\Lambda }$ 静止参考系中的 $\mathbf{L}$ 方向与重离子碰撞质心参考系中的 $\mathbf{L}$ 方向不同，因此需要正确处理参考系和测量的极化。参考文献[68]研究了在测量全局极化时参考系差异的影响，并发现该影响很小，在高横向动量（ $p_{\mathrm{T}}\sim$ 4-5 GeV/ $c$ ）时达到约 10%。请注意，对于 $\mathrm{\Lambda }$ 的平均 $p_T$ 是 $\sim$ 1 GeV/ $c$ [69]，取决于中心度和碰撞能量，并且在极化测量中使用的动力学截断可能略高（ $p_{\mathrm{T}}\sim$ 1.5 GeV/ $c$ ）。

Measuring polarization induced by anisotropic flow
通过各向异性流引起的极化测量

As already mentioned in Sec. 2.3, one can expect that azimuthal anisotropic flow would lead to a vorticity pointing along the beam direction. The orientation, along or opposite to the beam, in this case depends on the azimuthal angle of the particle. Similarly to the case of the global polarization, the longitudinal component of the polarization $P_{\mathrm{z}}$ can be obtained by integrating Eq. (18) as $\int dN/d{\mathrm{\Omega }}^{\ast }\cos {\theta }^{\ast }d{\mathrm{\Omega }}^{\ast }$ in which ${\mathbf{P}}_{\mathrm{H}}\cdot {\hat{p}}_{\mathrm{B}}^{\ast }$ is replaced by $P_{\mathrm{z}}\cos {\theta }_{\mathrm{B}}^{\ast }$, where ${\theta }_{\mathrm{B}}^{\ast }$ is the polar angle of daughter baryon in the parent hyperon rest frame. This leads to
如第 2.3 节中已经提到的，人们可以期望方位各向异性流会导致一个沿着束流方向指向的涡度。在这种情况下，沿着或相反于束流的方向取决于粒子的方位角。类似于全局极化的情况，可以通过将方程（18）积分得到极化的纵向分量 $P_{\mathrm{z}}$ ，其中 ${\mathbf{P}}_{\mathrm{H}}\cdot {\hat{p}}_{\mathrm{B}}^{\ast }$ 被 $P_{\mathrm{z}}\cos {\theta }_{\mathrm{B}}^{\ast }$ 替换，其中 ${\theta }_{\mathrm{B}}^{\ast }$ 是父超子静止系中子弹子的极角。这导致

$$\begin{array}{}\text{(28)}& P_{\mathrm{z}}=\frac{1}{A_{\mathrm{z}}}\frac{3⟨\cos {\theta }_{\mathrm{B}}^{\ast }⟩}{{\alpha }_{\mathrm{H}}},\end{array}$$

where $A_{\mathrm{z}}$ is the acceptance correction factor (see Sec. 4.6.2). For the case of perfect detector acceptance, $A_{\mathrm{z}}=1$. The factor $A_{\mathrm{z}}$ can be calculated using the data and is known to weekly depend on the transverse momentum and collision centrality. For more details on the correction factors, see Sec. 4.6.
其中 $A_{\mathrm{z}}$ 是接受度校正因子（见第 4.6.2 节）。对于完美探测器接受度的情况， $A_{\mathrm{z}}=1$ 。因子 $A_{\mathrm{z}}$ 可以使用数据计算，并且已知其弱依赖于横向动量和碰撞中心度。有关校正因子的更多详细信息，请参见第 4.6 节。

As follows from considerations in Sec. 2.3, see Eq. 4, the polarization induced by $n$-th harmonic anisotropic flow, if any, is expected to depend on the azimuthal angle of hyperons as $P_z(\varphi )\propto \sin [n(\varphi -{\mathrm{\Psi }}_n)]$. Then such a polarization can be quantified by the corresponding Fourier coefficient
根据 2.3 节的考虑，参见方程 4，由 $n$ -th 谐波各向异性流引起的极化，如果有的话，预计将取决于超子的方位角为 $P_z(\varphi )\propto \sin [n(\varphi -{\mathrm{\Psi }}_n)]$ 。然后，这种极化可以通过相应的傅里叶系数来量化。

$$\begin{array}{}\text{(29)}& P_{z,sn}=⟨P_z\sin [n({\varphi }_H-{\mathrm{\Psi }}_n)]⟩,\end{array}$$where ${\mathrm{\Psi }}_n$ is n-th harmonic event plane. Then, similarly to the case for global polarization as in Eq. (26), Eq. (29) can be rewritten accounting for the event plane resolution
$$\begin{array}{}\text{(29)}& P_{z,sn}=⟨P_z\sin [n({\varphi }_H-{\mathrm{\Psi }}_n)]⟩,\end{array}$$ 其中 ${\mathrm{\Psi }}_n$ 是第 n 个谐波事件平面。然后，类似于全局极化的情况，如方程（26）中所示，方程（29）可以重新编写，考虑到事件平面分辨率

$$\begin{array}{}\text{(30)}& P_{z,sn}=\frac{⟨P_z\sin [n({\varphi }_H-{\mathrm{\Psi }}_n)]⟩}{\text{Res}(n{\mathrm{\Psi }}_n)},\end{array}$$

where $\text{Res}(\text{n}{\mathrm{\Psi }}_n)$ is the resolution of n-th harmonic event plane defined as $⟨\cos [n({\mathrm{\Psi }}_n^{\mathrm{o}\mathrm{b}\mathrm{s}}-{\mathrm{\Psi }}_n)]⟩$ ("obs" denotes an observed angle).
其中 $\text{Res}(\text{n}{\mathrm{\Psi }}_n)$ 是定义为 $⟨\cos [n({\mathrm{\Psi }}_n^{\mathrm{o}\mathrm{b}\mathrm{s}}-{\mathrm{\Psi }}_n)]⟩$ 的 n-th 谐波事件平面的分辨率（“obs”表示观察角度）。

Feed-down effect 传递效应

It is known that a significant amount of $\mathrm{\Lambda }$ and $\mathrm{\Xi }$ hyperons comes from decays of heavier particles, such as ${\mathrm{\Sigma }}^0$, ${\mathrm{\Sigma }}^{\ast }$, and $\mathrm{\Xi }$ baryons for $\mathrm{\Lambda }$, and $\mathrm{\Xi }(1530)$ baryons for $\mathrm{\Xi }$. While the secondary particles from weak decays can be reduced, though not completely, using information on the decay topology, particles decayed via strong interaction cannot be separated from primary particles experimentally due to their short lifetimes. If the parent particles are polarized, the polarization is transferred to the daughter hyperons with certain polarization transfer factor, as discussed with Eqs. 20 and 22. The transfer factor $C$ depends on the type of decays and could be negative, for instance, $C_{{\mathrm{\Sigma }}^0\mathrm{\Lambda }}=-1/3$ for the electromagnetic decay of ${\mathrm{\Sigma }}^0\to \mathrm{\Lambda }+\gamma$.[23] Based on model studies,[23, 49, 70, 71, 72] such feed-down contribution is found to suppress the polarization of inclusively measured $\mathrm{\Lambda }$ compared to primary $\mathrm{\Lambda }$ by 10-20% depending on the model used. In case for $\mathrm{\Xi }$ hyperons, $\mathrm{\Xi }(1530)$ has spin 3/2 and the polarization transfer factor in the decay of $\mathrm{\Xi }(1530)\to {\mathrm{\Xi }}^{-}+\pi$ is equal to unity. Therefore the feed-down contribution for $\mathrm{\Xi }$ leads to the enhancement of the polarization of inclusive $\mathrm{\Xi }$ by $\sim$25%.[72]
已知，大量 $\mathrm{\Lambda }$ 和 $\mathrm{\Xi }$ 超子来自于更重粒子的衰变，例如 ${\mathrm{\Sigma }}^0$ ， ${\mathrm{\Sigma }}^{\ast }$ 和 $\mathrm{\Xi }$ 重子对 $\mathrm{\Lambda }$ ，以及 $\mathrm{\Xi }(1530)$ 重子对 $\mathrm{\Xi }$ 。虽然通过衰变拓扑信息可以减少弱衰变产生的次级粒子，但无法从实验上将通过强相互作用衰变的粒子与初级粒子分离，因为它们的寿命很短。如果母粒子极化，极化将以一定的极化转移因子转移到子超子上，如方程 20 和 22 所讨论的。转移因子 $C$ 取决于衰变类型，可能为负，例如， $C_{{\mathrm{\Sigma }}^0\mathrm{\Lambda }}=-1/3$ 为 ${\mathrm{\Sigma }}^0\to \mathrm{\Lambda }+\gamma$ 的电磁衰变。[23]基于模型研究，[23, 49, 70, 71, 72]发现这种传递贡献会抑制包括测量的 $\mathrm{\Lambda }$ 的极化，使其比初级 $\mathrm{\Lambda }$ 低 10-20%，具体取决于所使用的模型。对于 $\mathrm{\Xi }$ 超子， $\mathrm{\Xi }(1530)$ 的自旋为 3/2， $\mathrm{\Xi }(1530)\to {\mathrm{\Xi }}^{-}+\pi$ 的衰变中的极化转移因子等于 1。 因此， $\mathrm{\Xi }$ 的 feed-down 贡献导致了包容性 $\mathrm{\Xi }$ 的极化增强了 $\sim$ 25%。[72]

Although the effect of feed-down is not so significant, it is important to assess the effect, especially when extracting physical quantities such as the vorticity and magnetic field at the freeze-out. Note that the feed-down correction relies on the assumption of local thermodynamic equilibrium for spin degree of freedom as formulated in Ref.[23]. But it is not clear if the relaxation time is similar for the vorticity and magnetic field. Furthermore, actual situation may be more complicated since some of the particles have a shorter lifetime than the system lifetime (10-15 fm/$c$).
尽管下行馈送效应并不那么显著，但评估效应仍然很重要，特别是在提取像涡度和磁场这样的物理量时。请注意，下行馈送校正依赖于在文献[23]中制定的自旋自由度的局部热力学平衡假设。但目前尚不清楚涡度和磁场的弛豫时间是否相似。此外，实际情况可能更加复杂，因为一些粒子的寿命比系统寿命更短（10-15 fm/ $c$ ）。

Vector mesons spin alignment
矢量介子自旋对齐

Vector mesons, $s=1$ particles, can be also utilized to study the particle polarization in heavy-ion collisions. Unlike in the case of hyperons' weak decay, vector mesons predominantly decay via parity conserved (strong or electromagnetic) interaction. Therefore, one cannot determine the direction of the polarization of vector mesons, and their polarization state is usually reported via so-called spin-alignment measurements. The spin state of a vector meson is described by the spin density matrix ${\rho }_{mn}$. The diagonal elements of this matrix have a meaning of the probabilities for spin projections onto a quantization axis to have values $0,\pm 1$; ${\rho }_{00}$ represents the probability for the spin projection to be zero. As $s_z=\pm 1$ projections can not be distinguished, and the sum of the probabilities has to be unity, only one independent diagonal element, usually ${\rho }_{00}$, can be measured. In the case of vector meson decay into two (pseudo)-scalar mesons, ${\rho }_{00}$ can be determined directly from the angular distributions of the vector mesons decay products (given by the squares of the corresponding spherical harmonics):
矢量介子， $s=1$ 粒子，也可用于研究重离子碰撞中的粒子极化。与超子弱衰变不同，矢量介子主要通过守恒偶极（强或电磁）相互作用衰变。因此，无法确定矢量介子的极化方向，它们的极化状态通常通过所谓的自旋对齐测量来报告。矢量介子的自旋状态由自旋密度矩阵 ${\rho }_{mn}$ 描述。该矩阵的对角元素表示自旋在一个量子化轴上的投影取值为 $0,\pm 1$ 的概率； ${\rho }_{00}$ 表示自旋投影为零的概率。由于 $s_z=\pm 1$ 个投影无法区分，并且概率之和必须为单位，因此只能测量一个独立的对角元素，通常为 ${\rho }_{00}$ 。在矢量介子衰变为两个（伪）标量介子的情况下， ${\rho }_{00}$ 可以直接从矢量介子衰变产物的角分布（由相应球谐函数的平方给出）中确定。

$$\begin{array}{}\text{(31)}& \frac{dN}{d{\mathrm{\Omega }}^{\ast }}=\frac{3}{8\pi }[1-{\rho }_{00}+(3{\rho }_{00}-1){\cos }^2{\theta }^{\ast }],\end{array}$$

where ${\theta }^{\ast }$ is the angle of one of the daughter particles with respect to the polarization direction in the rest frame of the vector meson.For the global spin alignment measurement, the polarization direction is given by the orbital angular momentum direction of the system, perpendicular to the reaction plane. In the case of unpolarized particles, ${\rho }_{00}$ equals $1/3$. The deviation of ${\rho }_{00}$ from $1/3$ would indicate spin alignment of vector mesons.
其中 ${\theta }^{\ast }$ 是其中一个子粒子相对于矢量介子静止参考系中的极化方向的角度。对于全局自旋对齐测量，极化方向由系统的轨道角动量方向给出，垂直于反应平面。在无极化粒子的情况下， ${\rho }_{00}$ 等于 $1/3$ 。 ${\rho }_{00}$ 与 $1/3$ 的偏差将表明矢量介子的自旋对齐。

The spin alignment, $\mathrm{\Delta }\rho ={\rho }_{00}-1/3$, can be measured by directly analyzing $\cos {\theta }^{\ast }$ distribution given in Eq. 31, or considering $⟨{\cos }^2{\theta }^{\ast }⟩$ as follows
旋转对齐， $\mathrm{\Delta }\rho ={\rho }_{00}-1/3$ ，可以通过直接分析等式 31 中给定的 $\cos {\theta }^{\ast }$ 分布来测量，或者考虑如下 $⟨{\cos }^2{\theta }^{\ast }⟩$ 。

$$\begin{array}{}\text{(32)}& ⟨{\cos }^2{\theta }^{\ast }⟩=\int d{\mathrm{\Omega }}^{\ast }\frac{3}{8\pi }[1-{\rho }_{00}+(3{\rho }_{00}-1){\cos }^2{\theta }^{\ast }]{\cos }^2{\theta }^{\ast }=\frac{1}{3}+\frac{2}{5}\mathrm{\Delta }\rho .\end{array}$$

It results in 它导致

$$\begin{array}{}\text{(33)}& \mathrm{\Delta }\rho =\frac{5}{2}(⟨{\cos }^2{\theta }^{\ast }⟩-\frac{1}{3}).\end{array}$$

Taking also into account the event plane resolution[73] one arrives to the equation:
考虑到事件平面分辨率[73]，我们得到以下方程式：

$$\begin{array}{}\text{(34)}& {\rho }_{00}=\frac{1}{3}+\frac{4}{1+3\text{Res}(2\mathrm{\Psi })}({\rho }_{00}^{\text{ obs}}-\frac{1}{3}),\end{array}$$

where ${\rho }_{00}^{\text{scriptsize obs}}$ is the measured ("observed") signal and $\text{Res}(2\mathrm{\Psi })$ is the event plane resolution defined as $⟨\cos [2(\mathrm{\Psi }-{\mathrm{\Psi }}_{\text{scriptsize RP}})]⟩$ where $\mathrm{\Psi }$ can be either the first-order or second-order event plane. One could also analyze the daughter product distribution relative to the reaction plane[18, 73] similarly to that performed in the global polarization measurement:
其中 ${\rho }_{00}^{\text{scriptsize obs}}$ 是测量（“观察到的”）信号， $\text{Res}(2\mathrm{\Psi })$ 是事件平面分辨率，定义为 $⟨\cos [2(\mathrm{\Psi }-{\mathrm{\Psi }}_{\text{scriptsize RP}})]⟩$ ，其中 $\mathrm{\Psi }$ 可以是一阶或二阶事件平面。人们也可以分析与反应平面相关的子产物分布[18, 73]，类似于全局极化测量中所进行的操作：

$$\begin{array}{}\text{(35)}& {\rho }_{00}=\frac{1}{3}-\frac{4}{3}\frac{⟨\cos [2({\varphi }^{\ast }-\mathrm{\Psi })]⟩}{\text{Res}(2\mathrm{\Psi })},\end{array}$$

where ${\varphi }^{\ast }$ is the azimuthal angle of the daughter product in the parent rest frame.
其中 ${\varphi }^{\ast }$ 是父亲静止参考系中子产品的方位角。

In the case of vector meson decaying into two fermions, e.g. $J/\psi \to e^{+}{e}^{-}$, the interpretation of the final angular distribution in terms of the vector meson polarization is less straightforward, as it involves the spin wave functions of the daughter fermions. In this case the angular distribution of the daughter particles is often parameterized with a set of lambda parameters. For the distribution integrated over azimuthal angle, it reduces to
在矢量介子衰变成两个费米子的情况下，例如 $J/\psi \to e^{+}{e}^{-}$ ，用矢量介子极化来解释最终角分布并不那么直接，因为它涉及到子费米子的自旋波函数。在这种情况下，子粒子的角分布通常用一组 lambda 参数来参数化。对于沿方位角角度积分的分布，它简化为

$$\begin{array}{}\text{(36)}& \frac{dN}{d\cos {\theta }^{\ast }}\propto \frac{1}{3+{\lambda }_{\theta }}[1+{\lambda }_{\theta }{\cos }^2{\theta }^{\ast }].\end{array}$$The ${\lambda }_{\theta }$ parameter can be then determined from
$$\begin{array}{}\text{(36)}& \frac{dN}{d\cos {\theta }^{\ast }}\propto \frac{1}{3+{\lambda }_{\theta }}[1+{\lambda }_{\theta }{\cos }^2{\theta }^{\ast }].\end{array}$$ ${\lambda }_{\theta }$ 参数可以从中确定

$$\begin{array}{}\text{(37)}& ⟨{\cos }^2{\theta }^{\ast }⟩=\frac{1+3{\lambda }_{\theta }/5}{3+{\lambda }_{\theta }}.\end{array}$$

If the masses of the fermions are small, the helicity conservation tells that they should be in the spin state with projection on their momentum $\pm 1$. In this case, ${\lambda }_{\theta }$ parameter is related to the probability for a vector meson to have spin projection zero via equation[74]
如果费米子的质量很小，螺旋度守恒告诉我们它们应该处于自旋状态，其在动量上的投影为 $\pm 1$ 。在这种情况下， ${\lambda }_{\theta }$ 参数与矢量介子具有自旋投影零的概率相关，通过方程[74]。

$$\begin{array}{}\text{(38)}& {\lambda }_{\theta }=\frac{1-3{\rho }_{00}}{1+{\rho }_{00}}.\end{array}$$

Detector acceptance effects
探测器接受效应

4.6.1 Polarization along the initial angular momentum
4.6.1 沿着初始角动量的极化

We start with deriving the correction for polarization measurements based on Eq. 25. For the case of an imperfect detector, one has to take into account that in the calculation of the average $⟨\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }^{\ast })⟩$, the integral over solid angle $d{\mathrm{\Omega }}^{\ast }=d{\varphi }^{\ast }\sin {\theta }^{\ast }d{\theta }^{\ast }$ of the hyperon decay baryon's 3-momentum ${\mathbf{p}}^{\ast }$ in the hyperon rest frame, is affected by detector acceptance:
我们从基于方程 25 的极化测量校正开始。对于不完美探测器的情况，必须考虑到在计算平均值 $⟨\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }^{\ast })⟩$ 时，固体角 $d{\mathrm{\Omega }}^{\ast }=d{\varphi }^{\ast }\sin {\theta }^{\ast }d{\theta }^{\ast }$ 上的积分在超子静止参考系中受到探测器接受度的影响：

$$⟨\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }^{\ast })⟩=\int \frac{d{\mathrm{\Omega }}^{\ast }}{4\pi }\frac{d{\varphi }_H}{2\pi }A({\mathbf{p}}_H,{\mathbf{p}}^{\ast })\underset{0}{\overset{2\pi }{\int }}\frac{d{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}}{2\pi }\{1+2v_{2,H}\cos [2({\varphi }_H-{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}})]\}$$ $$\begin{array}{}\text{(39)}& \times \sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }^{\ast })[1+{\alpha }_H{P}_H({\mathbf{p}}_H;{\varphi }_H-{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}})\sin {\theta }^{\ast }\cdot \sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }^{\ast })].\end{array}$$

Here ${\mathbf{p}}_H$ is the hyperon 3-momentum, and $A({\mathbf{p}}_H,{\mathbf{p}}_p^{\ast })$ is a function to account for detector acceptance. The integral of this function over $(d{\mathrm{\Omega }}_p^{\ast }/4\pi )(d{\varphi }_H/2\pi )$ is normalized to unity. The polarization component along the system orbital angular momentum could depend on the relative azimuthal angle $({\varphi }_H-{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}})$. Taking into account the symmetry of the system, one can expand the polarization as a function of $({\varphi }_H-{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}})$ in a sum over even harmonics. Keeping below only the first two terms:
这里 ${\mathbf{p}}_H$ 是超子的三动量， $A({\mathbf{p}}_H,{\mathbf{p}}_p^{\ast })$ 是一个用来考虑探测器接受度的函数。该函数在 $(d{\mathrm{\Omega }}_p^{\ast }/4\pi )(d{\varphi }_H/2\pi )$ 上的积分被归一化为单位。沿着系统轨道角动量的极化分量可能取决于相对方位角 $({\varphi }_H-{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}})$ 。考虑到系统的对称性，可以将极化展开为 $({\varphi }_H-{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}})$ 的函数，作为偶次谐波的总和。仅保留前两项如下：

$$\begin{array}{}\text{(40)}& P_H({\varphi }_H-{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}},p_t^H,{\eta }^H)=P_0(p_t^H,{\eta }^H)+2P_2(p_t^H,{\eta }^H)\cos [2({\varphi }_H-{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}})].\end{array}$$

Substituting it into Eq. 39 and integrating over ${\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}$ one gets
将其代入方程 39 并在 ${\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}$ 上积分得到

$$\begin{array}{}\text{(41)}& ⟨\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }^{\ast })⟩=\frac{{\alpha }_H}{2}\int \frac{d{\mathrm{\Omega }}^{\ast }}{4\pi }\frac{d{\varphi }_H}{2\pi }A({\mathbf{p}}_H,{\mathbf{p}}^{\ast })\sin {\theta }^{\ast }\end{array}$$ $$\times [(P_0+2P_2{v}_2)-(P_2+P_0{v}_2)\cos [2({\varphi }_H-{\varphi }^{\ast })]]$$ $$=\frac{{\alpha }_H\pi }{8}[A_0(P_0+2P_2{v}_2)-A_2(P_2+P_0{v}_2)],$$

where the "acceptance" functions $A_0(p_t^H,{\eta }^H)$ and $A_2(p_t^H,{\eta }^H)$ are defined by:
其中，“接受”函数 $A_0(p_t^H,{\eta }^H)$ 和 $A_2(p_t^H,{\eta }^H)$ 的定义如下：

$$A_0(p_t^H,{\eta }^H)=\frac{4}{\pi }\int \frac{d{\mathrm{\Omega }}^{\ast }}{4\pi }\frac{d{\varphi }_H}{2\pi }A({\mathbf{p}}_H,{\mathbf{p}}^{\ast })\sin {\theta }^{\ast }.$$ (42) $$A_2(p_t^H,{\eta }^H)=\frac{4}{\pi }\int \frac{d{\mathrm{\Omega }}^{\ast }}{4\pi }\frac{d{\varphi }_H}{2\pi }A({\mathbf{p}}_H,{\mathbf{p}}^{\ast })\sin \text{For}theperfectacceptance\text{(}{A}_0=1\text{)}and\text{(}{A}_2=0\text{)}.Similarlyoneobtains:\text{[}⟨\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }^{\ast })\cos [2({\varphi }_H-{\varphi }^{\ast })]⟩$$ $$\begin{array}{}\text{(44)}& =\frac{{\alpha }_H\pi }{8}[A_0(P_2+P_0{v}_2)-\frac{1}{2}{A}_2(P_0+3P_2{v}_2)].\end{array}$$

Another set of equations can be derived for the method based on calculation of the $⟨\cos {\theta }^{\ast }⟩$, Eq. 27. In this case
基于计算 $⟨\cos {\theta }^{\ast }⟩$ ，方程 27，可以推导出另一组方程。

$$\begin{array}{}\text{(45)}& ⟨\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }^{\ast })\sin {\theta }^{\ast }⟩=\frac{{\alpha }_H}{3}[{\tilde{A}}_0(P_0+2P_2{v}_2)-{\tilde{A}}_2(P_2+P_0{v}_2)],\end{array}$$

$$⟨\sin ({\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}-{\varphi }^{\ast })\sin {\theta }^{\ast }\cos [2({\varphi }_H-{\varphi }^{\ast })]⟩$$ $$\begin{array}{}\text{(46)}& =\frac{{\alpha }_H}{3}[{\tilde{A}}_0(P_2+P_0{v}_2)-\frac{1}{2}{\tilde{A}}_2(P_0+3P_2{v}_2)],\end{array}$$

where 哪里

$$\begin{array}{}\text{(47)}& {\tilde{A}}_0(p_t^H,{\eta }^H)=\frac{3}{2}\int \frac{d{\mathrm{\Omega }}^{\ast }}{4\pi }\frac{d{\varphi }_H}{2\pi }A({\mathbf{p}}_H,{\mathbf{p}}^{\ast }){\sin }^2{\theta }^{\ast },\end{array}$$ $$\begin{array}{}\text{(48)}& {\tilde{A}}_2(p_t^H,{\eta }^H)=\frac{3}{2}\int \frac{d{\mathrm{\Omega }}^{\ast }}{4\pi }\frac{d{\varphi }_H}{2\pi }A({\mathbf{p}}_H,{\mathbf{p}}^{\ast }){\sin }^2{\theta }^{\ast }\cos [2({\varphi }_H-{\varphi }^{\ast })].\end{array}$$

4.6.2 Polarization along the beam direction
4.6.2 沿着光束方向的极化

For $P_{\mathrm{z}}$ measurement, we consider the average of $⟨\cos {\theta }_B^{\ast }⟩$ using Eq. 18, where ${\theta }_B^{\ast }$ is the polar angle of the daughter baryon in its parent hyperon rest frame, relative to the beam direction.
对于 $P_{\mathrm{z}}$ 测量，我们考虑使用方程式 18 计算 $⟨\cos {\theta }_B^{\ast }⟩$ 的平均值，其中 ${\theta }_B^{\ast }$ 是子重子在其父超子静止参考系中的极角，相对于束流方向。

$$\begin{array}{}\text{(49)}& ⟨\cos {\theta }_{\mathrm{B}}^{\ast }⟩=\int \frac{d{\mathrm{\Omega }}^{\ast }}{4\pi }A({\mathbf{p}}_H,{\mathbf{p}}^{\ast })(1+{\alpha }_{\mathrm{H}}{P}_{\mathrm{z}}\cos {\theta }_{\mathrm{B}}^{\ast })\cos {\theta }_{\mathrm{B}}^{\ast }\end{array}$$ $$\begin{array}{}\text{(50)}& ={\alpha }_H{P}_{\mathrm{z}}\int \frac{d{\mathrm{\Omega }}^{\ast }}{4\pi }A({\mathbf{p}}_H,{\mathbf{p}}^{\ast }){\cos }^2{\theta }_{\mathrm{B}}^{\ast }.\end{array}$$

Thus: 因此：

$$\begin{array}{}\text{(51)}& P_z=\frac{1}{A_{\mathrm{z}}}\frac{3⟨\cos {\theta }_{\mathrm{B}}^{\ast }⟩}{{\alpha }_{\mathrm{H}}},\end{array}$$

where $A_{\mathrm{z}}$ is acceptance correction factor defined as
其中 $A_{\mathrm{z}}$ 是接受修正因子，定义为

$$\begin{array}{}\text{(52)}& A_{\mathrm{z}}(p_t^H,{\eta }^H)=3\int \frac{d{\mathrm{\Omega }}^{\ast }}{4\pi }A({\mathbf{p}}_H,{\mathbf{p}}^{\ast }){\cos }^2{\theta }_{\mathrm{B}}^{\ast }.\end{array}$$

The factor $A_z$ can be determined in a data driven way, similar to the acceptance correction factors in the global polarization measurement, and is typically close to unity [3
因子 $A_z$ 可以以数据驱动的方式确定，类似于全球极化测量中的接受校正因子，并且通常接近于单位

4.6.3 Acceptance effects in spin alignment measurements
4.6.3 自旋定向测量中的接受效应

Spin alignment measurements are significantly more difficult compared to the measurements of the hyperon polarization. The difficulty comes from the fact that while the acceptance effects in the polarization measurements can only change the magnitude of the effect, in the spin alignment measurement the acceptance effects could lead to false spurious signal. We demonstrate this below providing equations for the acceptance correction to the signal for the case of vector mesons experiencing elliptic flow.
自旋对齐测量与超子极化测量相比要困难得多。困难在于，极化测量中的接受效应只能改变效应的幅度，而在自旋对齐测量中，接受效应可能导致虚假信号。我们在下面提供了方程式，用于描述椭圆流经历的矢量介子情况下信号的接受校正。

One of the main tracking efficiency effects is due to different probabilities if vector meson reconstruction when the daughter particles are emitted along the momentum of the parent particles or perpendicular to that. A toy model study on decay daughters can show that such efficiency effect can be well parameterized by parameter $a_2$ in the equation
主要跟踪效率影响之一是由于矢量介子重建时，当子粒子沿着父粒子的动量发射或垂直于父粒子的动量发射时，存在不同的概率。对衰变子进行的玩具模型研究表明，这种效率影响可以通过方程中的参数 $a_2$ 很好地参数化。

$$\begin{array}{}\text{(53)}& A({\varphi }^{\ast })=A_0(1+2a_2(p_{\mathrm{T}})\cos [2({\varphi }^{\ast }-\varphi )],\end{array}$$

where $\varphi$ denotes the vector meson azimuthal angle in the laboratory frame.
其中 $\varphi$ 表示实验室参考系中的矢量介子方位角。

Then following Eq. 34 and accounting for elliptic flow and efficiency effects, one finds:
然后根据方程 34，并考虑椭圆流和效率影响，我们可以得到：

$$\begin{array}{}\text{(54)}& \mathrm{\Delta }{\rho }^{\mathrm{o}\mathrm{b}\mathrm{s}}\equiv {\rho }_{00}^{\mathrm{o}\mathrm{b}\mathrm{s}}-\frac{1}{3}=\int \frac{d{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}}}{2\pi }\frac{d{\varphi }^{\ast }}{2\pi }\frac{d\varphi }{2\pi }(-4/3)\cos [2({\varphi }^{\ast }-{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}})]\end{array}$$ $$\times (1+2v_2(p_{\mathrm{T}})\cos [2(\varphi -{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}})])(1+2a_2(p_{\mathrm{T}})\cos [2({\varphi }^{\ast }-\varphi )])$$ $$\times (1-\frac{3}{2}\mathrm{\Delta }\rho (p_{\mathrm{T}})\cos [2({\varphi }^{\ast }-{\mathrm{\Psi }}_{\mathrm{R}\mathrm{P}})])=\mathrm{\Delta }\rho -\frac{4}{3}{a}_2{v}_2,$$

where the superscript "obs" (observed) denotes the value obtained by the direct application of Eq. 34 to the data. One can see that even in the case of "real" $\mathrm{\Delta }\rho$ being zero, the observed signal is not zero due to interplay of the elliptic flow and tracking efficiency effects.
上标“obs”（观察到的）表示通过直接应用方程式 34 到数据获得的值。可以看到，即使在“真实” $\mathrm{\Delta }\rho$ 为零的情况下，由于椭圆流和跟踪效率效应的相互作用，观察到的信号并不为零。

Similarly this effect biases the flow measurement as:
同样，这种效应会使流量测量产生偏差：

$$\begin{array}{}\text{(55)}& v_2^{\mathrm{o}\mathrm{b}\mathrm{s}}=v_2-\frac{3}{4}{a}_2\mathrm{\Delta }\rho ,\end{array}$$

and the determination of parameter $a_2$ from data can be done with:
参数 $a_2$ 的确定可以通过数据完成：

$$\begin{array}{}\text{(56)}& a_2^{\mathrm{o}\mathrm{b}\mathrm{s}}=a_2-\frac{4}{3}{v}_2\mathrm{\Delta }\rho .\end{array}$$

The actual values of the spin alignment and elliptic flow can be obtained by solving the above equations Eqs. 54-56 with respect to $v_2$, $a_2$, and $\mathrm{\Delta }\rho$.
通过解决上述方程式 Eqs. 54-56 关于 $v_2$ ， $a_2$ 和 $\mathrm{\Delta }\rho$ 的值，可以获得自旋对齐和椭圆流的实际值。

The equations above demonstrate only one example of the tracking efficiency effects leading to a spurious spin alignment signal. Another example was discussed in Ref. [75], where the authors investigated (and found to be significant) the effect of the finite rapidity acceptance on $\mathrm{\Delta }\rho$ measurements.
上述方程只展示了一个跟踪效率效应导致虚假自旋对齐信号的例子。另一个例子在参考文献[75]中讨论过，作者们调查了（并发现是显著的）有限快度接受度对 $\mathrm{\Delta }\rho$ 测量的影响。

5 Overview of experimental results
5 实验结果概述

Global polarization of $\mathrm{\Lambda }$ hyperons
$\mathrm{\Lambda }$ 超子的全球极化

5.1.1 Energy dependence 5.1.1 能源依赖

Global polarization of $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ hyperons has been measured in a wide range of collision energies. The first observation of non-zero global polarization was reported in Au+Au collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=7.7$-39 GeV in the first phase of the beam energy scan program (BES-I) at RHIC by the STAR Collaboration;[1] later it was also confirmed with a better significance at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=200$ GeV.[2] At the LHC energies the measurements were performed by the ALICE Collaboration.[76] Figure 5 presents a compilation of the results of the global polarization measurements for $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ hyperons at mid-rapidity for mid-central collisions as a function of collision energy. The polarization increases as the collision energy decreases. One would naively expect that the initial orbital angular momentum becomes larger at higher energy,[49] therefore the polarization would have the same trend, but this argument does not take into account that the initial angular momentum has to be spread over much larger rapidity region and more produced particles, and that the particle production at the midrapidity is almost boost invariant. Another reason for the observed energy dependence might be a dilution effect of the vorticity due to longer lifetime of the system at higher collision energies.
$\mathrm{\Lambda }$ 和 $\overline{\mathrm{\Lambda }}$ 超子的全局极化已在广泛的碰撞能量范围内进行了测量。在 RHIC 的 STAR 合作组织的束流能量扫描计划（BES-I）的第一阶段中，首次观察到了 Au+Au 碰撞中的非零全局极化，能量为 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=7.7$ -39 GeV；后来在 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=200$ GeV 时也以更高的显著性得到了证实。在 LHC 能量下，测量是由 ALICE 合作组织进行的。图 5 展示了 $\mathrm{\Lambda }$ 和 $\overline{\mathrm{\Lambda }}$ 超子在中心快度的中心碰撞中的全局极化测量结果的汇编，作为碰撞能量的函数。随着碰撞能量的降低，极化增加。人们可能会天真地期望初始轨道角动量在能量较高时变得更大，因此极化也会有相同的趋势，但这个论点没有考虑到初始角动量必须分布在更大的快度区域和更多产生的粒子上，并且在中心快度处的粒子产生几乎是布局不变的。 观察到的能量依赖性的另一个原因可能是由于在更高的碰撞能量下系统寿命更长导致的涡度稀释效应。

Most of the theoretical calculations rely on the assumptions that (a) the system is in a local thermal equilibrium and (b) that the spin polarization is not modified at later non-equilibrium stages, e.g., by hadronic rescattering.[77, 78] Neither of these assumptions is obvious. Nevertheless, most of the calculations, based on different approaches, such as hydrodynamic models,[49, 79, 80, 81] chiral kinetic approach,[82] and a transport model,[70] surprisingly well reproduce the observed energy dependence of the global polarization at the quantitative level, as seen in Fig. 5. Note that there still exists a disagreement between the data and models in differential measurements, which we discuss in the following sections. Based on Eq. (2), the vorticity can be estimated as $\omega \approx k_{B}T(P_{\mathrm{\Lambda }}+P_{\overline{\mathrm{\Lambda }}})/\hslash$ with $T$ being the system temperature at the time of particle emission. The polarization averaged over $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ in the BES-I results in $\omega \approx (9\pm 1)\times {10}^{21}$ s${}^{-1}$, leading to the finding of the most vortical fluid ever observed.[1]
大多数理论计算依赖于以下假设：(a) 系统处于局部热平衡状态，(b) 自旋极化在后续非平衡阶段不会被强子再散射等修改。这些假设都不是显而易见的。然而，大多数基于不同方法的计算，如流体动力学模型、手征动力学方法和传输模型，出奇地很好地重现了全局极化的观测能量依赖性，如图 5 所示。需要注意的是，在微分测量方面，数据与模型之间仍存在分歧，我们将在接下来的章节中讨论。根据方程(2)，涡度可以估计为 0，其中 t 是粒子发射时系统温度。在 BES-I 中，平均极化结果为 s，导致发现有史以来观测到的最具涡动性的流体。

From empirical estimates[18] based on the directed flow measurements, see Sec. 2.1, the global polarization signal at the LHC energies is expected to be an order of a few per mill. The results from the ALICE Collaboration are consistent with zero with statistical uncertainties of the order of the expected signal. At lower energies, it is expected that the kinematic vorticity becomes maximum around $\sqrt{s_{\mathrm{N}\mathrm{N}}}=3$ GeV and vanishes at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=2m_N$ ($m_N$ is the nucleon mass) near the threshold of nucleon pair production because the total angular momentum of the system at such energies becomes close to zero.[85, 86, 87] In such high baryon density region, the system would no longer experience a partonic phase but be in a hadronic phase during theentire system evolution. Therefore, it would be interesting to check whether the polarization changes smoothly with the beam energy. Recently the STAR Collaboration has reported $\mathrm{\Lambda }$ global polarization in Au+Au collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=3$ GeV [83], followed by results on $\mathrm{\Lambda }$ global polarization in Au+Au collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=2.4$ GeV and Ag+Ag collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=2.55$ GeV by the HADES Collaboration [88]. The results indicate that the global polarization still increase at these energies, although the current uncertainties may be too large to see the expected trend.
根据基于定向流测量的经验估计[18]，参见第 2.1 节，预计 LHC 能量下的全球极化信号将达到几千分之几的数量级。ALICE 合作组的结果与零一致，统计误差与预期信号的数量级相近。在较低能量下，预计运动涡度在 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=3$ GeV 左右达到最大值，并在 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=2m_N$ （ $m_N$ 为核子质量）附近消失，接近核子对产生的阈值，因为在这种能量下，系统的总角动量接近于零[85, 86, 87]。在这种高重子密度区域，系统将不再经历部分子相，而是在整个系统演化过程中处于强子相。因此，检查极化是否随着束流能量平稳变化将是有趣的。最近，STAR 合作组报告了 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=3$ GeV 的 Au+Au 碰撞中的 $\mathrm{\Lambda }$ 全球极化[83]，随后 HADES 合作组在 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=2.4$ GeV 的 Au+Au 碰撞和 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=2.55$ GeV 的 Ag+Ag 碰撞中报告了 $\mathrm{\Lambda }$ 全球极化结果[88]。 结果表明，尽管当前的不确定性可能太大以至于看不到预期的趋势，但全球极化仍在这些能量上增加。

Calculation from the three-fluid dynamics (3FD) [81] incorporating the equation of state (EoS) for the first-order phase transition (1PT) captures the trend of the experimental data. The 3FD model also shows sensitivity of the global polarization to EoS as seen in some difference in the calculations for the first-order phase transition and hadronic (HG) EoS.
从三流体动力学（3FD）[81]的计算中融入了一级相变（1PT）的状态方程（EoS），捕捉了实验数据的趋势。 3FD 模型还显示了对 EoS 的全局极化的敏感性，如在一级相变和强子（HG）EoS 的计算中的一些差异中所见。

5.1.2 Particle-antiparticle difference
5.1.2 粒子-反粒子差异

As discussed in Sec. 2.2, the initial and/or later-stage magnetic field created in heavy-ion collisions could lead to a difference in the global polarizations of particles and antiparticles. The experimental results, presented in Fig 5, do not show any significant difference in polarizations of $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$, already indicating that the thermal vorticity, rather than the magnetic field contribution, is the dominant source of the observed global polarization. Figure 6 presents directly the differences in the global polarizations of $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ as a function of $\sqrt{s_{\mathrm{N}\mathrm{N}}}$[89]. The new RHIC BES-II results from Au+Au collisions at 19.6 GeV and 27 GeV greatly improve the statis
如第 2.2 节所讨论的，重离子碰撞中产生的初始和/或后期磁场可能导致粒子和反粒子的全局极化差异。图 5 中呈现的实验结果并未显示 $\mathrm{\Lambda }$ 和 $\overline{\mathrm{\Lambda }}$ 的极化有显著差异，已经表明热涡度而非磁场贡献是观察到的全局极化的主要来源。图 6 直接展示了 $\mathrm{\Lambda }$ 和 $\overline{\mathrm{\Lambda }}$ 的全局极化差异作为 $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ 的函数[89]。新的 RHIC BES-II 结果来自 19.6 GeV 和 27 GeV 的 Au+Au 碰撞，极大地改善了统计数据。

Figure 5: Collision energy dependence of $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ global polarization for mid-central heavy-ion collisions [83] compared to various model calculations [49, 70, 79, 81, 82]. The experimental data from the original publications are rescaled accounting for the recent update of the $\mathrm{\Lambda }$ decay parameters [84] indicated in the figure.
图 5：与各种模型计算[49, 70, 79, 81, 82]相比，中央重离子碰撞的 $\mathrm{\Lambda }$ 和 $\overline{\mathrm{\Lambda }}$ 全局极化对碰撞能量的依赖性[83]。根据图中所示最近更新的 $\mathrm{\Lambda }$ 衰变参数[84]，对原始出版物中的实验数据进行重新调整。

tical uncertainty in the measurements, and show no significant difference between particle-antiparticle polarizations. Following Eq. 2, one could put an upper limit on the magnetic field effect assuming the local thermodynamic equilibrium for the spin degrees of freedom:
在测量中存在技术上的不确定性，并且显示出粒子-反粒子极化之间没有显著差异。根据方程式 2，可以假设自旋自由度的局部热力学平衡，对磁场效应设定一个上限。

$$\begin{array}{}\text{(57)}& \mathrm{\Delta }{P}_H=P_{\overline{\mathrm{\Lambda }}}-P_{\mathrm{\Lambda }}=\frac{2|{\mu }_{\mathrm{\Lambda }}|B}{T},\end{array}$$

where ${\mu }_{\mathrm{\Lambda }}=-{\mu }_{\overline{\mathrm{\Lambda }}}=-0.613{\mu }_N$ with ${\mu }_N$ being the nuclear magneton. Thus, one arrives at the upper limit on the magnitude of the magnetic field $B\lesssim {10}^{13}$ T assuming the temperature $T=150$ MeV and ignoring the feed-down contributions (see Sec. 4.4). The estimated magnitude of the magnetic field is still considerably large and presents an important input for the dynamical modeling e.g., magneto-hydrodynamics, constraining the electric conductivity of the plasma.
其中 ${\mu }_{\mathrm{\Lambda }}=-{\mu }_{\overline{\mathrm{\Lambda }}}=-0.613{\mu }_N$ 与 ${\mu }_N$ 为核磁子。因此，假设温度为 $T=150$ MeV，忽略下行贡献（见第 4.4 节），可得到磁场 $B\lesssim {10}^{13}$ T 的幅度上限。估计的磁场幅度仍然相当大，并为动力学建模（例如磁流体力学）提供了重要输入，限制了等离子体的电导率。

It should be noted that several other sources could contribute to the polarization difference. Ref. [40] suggests that the different space-time distributions and emission times of $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ hyperons lead to the polarization difference. $\overline{\mathrm{\Lambda }}$ hyperons, emitted earlier in time, are less affected by the dilution of the vorticity with the system expansion, leading to larger polarization of $\overline{\mathrm{\Lambda }}$. On the other hand, Ref. [90] argues that the formation time of $\mathrm{\Lambda }$ is smaller than that of $\overline{\mathrm{\Lambda }}$, leading to larger polarization of $\mathrm{\Lambda }$. The actual situation might be even more complicated since the spin-orbit coupling may take place at quark level. Ref. [41] reported that the strong interaction
应该注意到，还有其他几个来源可能会对极化差异做出贡献。参考文献[40]指出， $\mathrm{\Lambda }$ 和 $\overline{\mathrm{\Lambda }}$ 超子的不同时空分布和发射时间导致了极化差异。较早发射的 $\overline{\mathrm{\Lambda }}$ 超子受到系统膨胀时涡度稀释影响较小，导致 $\overline{\mathrm{\Lambda }}$ 的极化较大。另一方面，参考文献[90]认为 $\mathrm{\Lambda }$ 的形成时间小于 $\overline{\mathrm{\Lambda }}$ ，导致 $\mathrm{\Lambda }$ 的极化较大。实际情况可能更加复杂，因为自旋轨道耦合可能发生在夸克水平。参考文献[41]报告称强相互作用。

Figure 6: Collision energy dependence of the difference in global polarizations of $\overline{\mathrm{\Lambda }}$ and $\mathrm{\Lambda }$ hyperons, $P_{\overline{\mathrm{\Lambda }}}-P_{\mathrm{\Lambda }}$. The figure is taken from Ref. [89].
图 6： $\overline{\mathrm{\Lambda }}$ 和 $\mathrm{\Lambda }$ 超子全局极化差异的碰撞能量依赖性， $P_{\overline{\mathrm{\Lambda }}}-P_{\mathrm{\Lambda }}$ 。该图摘自参考文献[89]。

with meson field could make $\overline{\mathrm{\Lambda }}$ polarization larger. The effect of chemical potential becomes important at lower energies as it appears in the Fermi-Dirac distribution (see Eq. (6)). Non-zero baryon chemical potential is expected to lead to larger polarization of $\overline{\mathrm{\Lambda }}$,[42] though the effect may be rather small. The feed-down effects with non-zero baryon chemical potential might lead to the opposite relation[23] but it would depend on the relative abundance at different phase space. Having these complications in mind, non-significant difference in the observed global polarization of $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$, $P_{\overline{\mathrm{\Lambda }}}-P_{\mathrm{\Lambda }}$, does not exclude a limited contribution from the magnetic field.
使用介子场可以使 $\overline{\mathrm{\Lambda }}$ 极化更大。化学势的影响在较低能量时变得重要，因为它出现在费米-狄拉克分布中（见方程（6））。预计非零重子化学势会导致 $\overline{\mathrm{\Lambda }}$ 的极化更大，尽管影响可能相当小。具有非零重子化学势的下行效应可能导致相反的关系，但这将取决于不同相空间中的相对丰度。考虑到这些复杂性，观察到的 $\mathrm{\Lambda }$ 和 $\overline{\mathrm{\Lambda }}$ ， $P_{\overline{\mathrm{\Lambda }}}-P_{\mathrm{\Lambda }}$ 的全局极化没有显著差异，不排除磁场的有限贡献。

5.1.3 Differential measurements
5.1.3 差分测量

Recently available high statistics data permit to study global polarization differentially, as a function of centrality, transverse momentum, and rapidity. Model calculations show that the initial angular momentum of the system increases from central to mid-central collisions and then decreases in peripheral collisions since the energy density decreases,[10] but the vorticity, hence the global polarization, are expected to increase in more peripheral collisions.[91] Figure 7(left) shows centrality dependence of $\mathrm{\Lambda }$ ($\overline{\mathrm{\Lambda }}$) global polarization in Au+Au collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=200$ and 3 GeV,[2, 83] where the increasing trend towards peripheral collisions can be clearly seen. Viscous hydrodynamics models[92, 92] qualitatively describe the centrality dependence of global polarization as shown in the figure.
最近可用的高统计数据允许根据中心性、横向动量和快度的函数差异性地研究全局极化。模型计算显示，系统的初始角动量从中心碰撞到中心碰撞增加，然后在外围碰撞中减少，因为能量密度减小，但涡度，因此全局极化，预计在更外围的碰撞中会增加。图 7（左）显示 Au+Au 碰撞中 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=200$ 和 3 GeV 的中心性依赖性 $\mathrm{\Lambda }$ （ $\overline{\mathrm{\Lambda }}$ ）全局极化，可以清楚地看到朝外围碰撞的增长趋势。粘性流体动力学模型定性地描述了全局极化的中心性依赖性，如图所示。

As already mentioned, the "global" polarization refers to the polarization component along the system orbital angular momentum averaged over all particles and all momenta. The same component (denoted as $P_{-y}$), but measured for a particular kinematics, can deviate from the global average; in this case the term "local" polarization is more appropriate. For example, the initial velocity shear resulting in the global vorticity would change with rapidity, i.e., the shear might be larger in forward/backward rapidity, also depending on the collision energy [91, 93]. Theoretical models such as hydrodynamics and transport models predict the rapidity dependence differently,[94, 95, 96, 97, 87] some models predict that the polarization goes up in forward (backward) rapidity while the others predict decreasing trend in larger rapidities. The hydrodynamic models using different initial conditions and frameworks also predict different trends (see Fig. 7(right)). The first study was performed at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=200$ GeV${}^2$ as shown in Fig. 7(right) and no significant rapidity dependence was observed, which may be expected at high collision energy as the shear should be weaker at midrapidity because of longitudinally boost invariance. Recent measurement at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=3$ GeV from STAR[83] also found no strong rapidity dependence within $-0.2<y<1$, even at the rapidity close to the beam rapidity ($y_{\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{m}}=1.02$ at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=3$ GeV). Similarly, no dependence on rapidity is observed at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=2.55$ GeV by the HADES experiment.[88] The uncertainties of the data are still large and this question should be further studied in future analyses with better statistics and upgraded/new detectors.
如前所述，“全局”极化是指沿系统轨道角动量的极化分量，平均分布在所有粒子和所有动量上。同一分量（表示为 $P_{-y}$ ），但针对特定动力学测量时，可能会偏离全局平均值；在这种情况下，“局部”极化这个术语更为恰当。例如，导致全局涡度的初始速度剪切会随着快度变化，即，剪切在前/后快度可能更大，也取决于碰撞能量[91, 93]。理论模型如流体力学和输运模型对快度依赖性的预测不同，[94, 95, 96, 97, 87]一些模型预测极化在前（后）快度上升，而其他模型预测在更大快度下降趋势。使用不同初始条件和框架的流体力学模型也预测不同的趋势（见图 7（右））。第一项研究在 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=200$ GeV ${}^2$ 进行，如图所示。 7（右）并且没有观察到显著的快度依赖性，这可能在高碰撞能量时会预期，因为由于纵向动量不变性，中快度处的剪切应该较弱。来自 STAR[83]的 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=3$ GeV 的最近测量也发现在 $-0.2<y<1$ 内没有强烈的快度依赖性，即使在接近束流快度（ $y_{\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{m}}=1.02$ 在 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=3$ GeV）的快度处也是如此。同样，HADES 实验在 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=2.55$ GeV 处也没有观察到快度依赖性。数据的不确定性仍然很大，这个问题应该在未来的分析中进一步研究，以获得更好的统计数据和升级/新探测器。

It should be noted that the polarization $P_{-\mathrm{y}}$ component seems to have little dependence on the hyperon transverse momentum $p_T$[83, 88, 89, 2], which qualitatively agrees with theoretical models that predict a mild $p_T$ dependence. Figure 8(left) shows hyperons' transverse momentum dependence of the polarization along the system angular momentum in Au+Au collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=200$ GeV, compared to hydrodynamic model calculations with two different initial conditions [23]: Monte Carlo Glauber with the initial source tilt and UrQMD initial state. The UrQMD initial condition includes the initial flow from a preequilibrium phase that would affect the initial velocity field. Similar trend was also seen at lower collision energies [88, 89, 83].
应该注意到，极化 $P_{-\mathrm{y}}$ 分量似乎与超子横向动量 $p_T$ 几乎没有关联[83, 88, 89, 2]，这与预测轻微 $p_T$ 依赖性的理论模型定性一致。图 8（左）显示了 Au+Au 碰撞中系统角动量方向上超子横向动量依赖性的极化，与两种不同初始条件的流体动力学模型计算进行了比较[23]：具有初始源倾斜和 UrQMD 初始状态的蒙特卡洛 Glauber。UrQMD 初始条件包括来自预平衡相的初始流，这会影响初始速度场。类似的趋势也在较低的碰撞能量中观察到[88, 89, 83]。

The STAR Collaboration also studied charge asymmetry ($A_{\mathrm{c}\mathrm{h}}$) dependence of the global polarization for a possible relation to anomalous chiral effects [38]. According to Ref. [98], the global polarization could be explained by axial charge separation due to the chiral vortical effect. In addition, the axial current ${\mathbf{J}}_5$ can be generated in the system with nonzero vector chemical potential ${\mu }_{\mathrm{v}}$ under a strong magnetic field $\mathbf{B}$ (${\mathbf{J}}_5\propto Qe{\mu }_{\mathrm{v}}\mathbf{B}$), aka chiral separation effect, where $Qe$ represents net electric charge of particles. For massless quarks, their momentum direction is aligned (anti-aligned) with spin direction for right-handed (left-handed) quarks. Thus the ${\mathbf{J}}_5$, if generated, might contribute to the hyperon global polarization. The event charge asymmetry defined as $A_{\mathrm{c}\mathrm{h}}=(N_{+}-N_{-})/(N_{+}+N_{-})$ where $N_{+}(N_{-})$ is the number of positively (negatively) charged particles was used to study the possible relation with the polarization assuming $A_{\mathrm{c}\mathrm{h}}\propto {\mu }_{\mathrm{v}}$. Figure 8(right) shows $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$
STAR 协作组还研究了全局极化的电荷不对称性（ $A_{\mathrm{c}\mathrm{h}}$ ）对异常手征效应[38]可能关系的依赖性。根据参考文献[98]，全局极化可以通过由手征涡旋效应引起的轴向电荷分离来解释。此外，在强磁场 $\mathbf{B}$ （ ${\mathbf{J}}_5\propto Qe{\mu }_{\mathrm{v}}\mathbf{B}$ ）下，带有非零矢量化学势 ${\mu }_{\mathrm{v}}$ 的系统中可以产生轴向电流 ${\mathbf{J}}_5$ ，即手征分离效应，其中 $Qe$ 代表粒子的净电荷。对于无质量夸克，他们的动量方向与右手（左手）夸克的自旋方向对齐（反对齐）。因此，如果生成了 ${\mathbf{J}}_5$ ，可能会对超子的全局极化产生贡献。事件电荷不对称性定义为 $A_{\mathrm{c}\mathrm{h}}=(N_{+}-N_{-})/(N_{+}+N_{-})$ ，其中 $N_{+}(N_{-})$ 是正（负）电荷粒子的数量，用于研究与假设 $A_{\mathrm{c}\mathrm{h}}\propto {\mu }_{\mathrm{v}}$ 的极化可能关系。图 8（右）显示 $\mathrm{\Lambda }$ 和 $\overline{\mathrm{\Lambda }}$ 。

Figure 7: (Left) Centrality dependence of $\mathrm{\Lambda }(\overline{\mathrm{\Lambda }})$ of $P_{-\mathrm{y}}$ polarization component in Au+Au collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=3$, and 200 GeV compared to viscous hydrodynamic model calculation [92]. (Right) Rapidity dependence of $\mathrm{\Lambda }(\overline{\mathrm{\Lambda }})$$P_{-\mathrm{y}}$ compared to Particle-in-Cell Relativistic (PICR) hydrodynamics model [96] and viscous hydrodynamic model CLVisc [95]. Note that the data for 3 GeV in the left (right) plot are scaled by 0.1 (0.2), and the average pseudorapidity for 200 GeV is converted to the rapidity in the right panel.
图 7：（左）Au+Au 碰撞中 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=3$ 的 $P_{-\mathrm{y}}$ 极化分量的中心性依赖性，与粘性流体动力学模型计算[92]相比，以及 200 GeV。 （右）与粒子在 PICR 相对论（PICR）流体动力学模型[96]和粘性流体动力学模型 CLVisc[95]相比的快度依赖性 $\mathrm{\Lambda }(\overline{\mathrm{\Lambda }})$ $P_{-\mathrm{y}}$ 。请注意，左图（右图）中 3 GeV 的数据按 0.1（0.2）缩放，并且 200 GeV 的平均赝快度转换为右图中的快度。

global polarization as a function of $A_{\mathrm{c}\mathrm{h}}$ for mid-central Au+Au collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ = 200 GeV. There seems a slight dependence on $A_{\mathrm{c}\mathrm{h}}$ and the slopes look different for $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$, although the effect is only at $\sim 2\sigma$ level. The effect of the chemical potential may be an alternative explanation of the difference if the charge asymmetry is correlated with the baryon number asymmetry [42, 99].
全球极化作为中央 Au+Au 碰撞的函数 $A_{\mathrm{c}\mathrm{h}}$ 在 $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ = 200 GeV 时。似乎对 $A_{\mathrm{c}\mathrm{h}}$ 有轻微依赖，对 $\mathrm{\Lambda }$ 和 $\overline{\mathrm{\Lambda }}$ 斜率看起来不同，尽管效应仅在 $\sim 2\sigma$ 水平。如果电荷不对称性与重子数不对称性相关，化学势的影响可能是差异的另一种解释[42, 99]。

Azimuthal angle dependence of the polarization is also of great interest and has been the subject of debate. The experimental preliminary result from STAR [100] shows larger polarization for hyperons emitted in the in-plane direction than those in out-of-plane direction as shown in Fig. 9, while hydrodynamic and transport models predict it oppositely, i.e., larger polarization in out-of-plane direction [26, 79, 94, 101]. Based on Glauber simulation shown in Fig. 2(b), one expects ${\omega }_J(\propto d{v}_z/dx)$ to be larger in in-plane direction ($x$-direction in the plot), which is consistent with experimental results. As shown in Fig. 9, the calculation including only the contribution from the kinematic vorticity leads to the opposite sign, while the inclusion of the shear term leads to the correct sign. We discuss this question further in Sec. 5.4 together with the results on polarization along the beam direction in relation to the so-called "spin sign crisis".
方位角依赖性的极化也备受关注，并成为争论的焦点。来自 STAR 的实验初步结果显示，与图 9 中所示的平面方向发射的超子相比，垂直平面方向发射的超子极化更大，而流体动力学和输运模型预测相反，即垂直平面方向的极化更大。根据图 2(b)中显示的 Glauber 模拟，人们预期 ${\omega }_J(\propto d{v}_z/dx)$ 在平面方向（图中的 $x$ -方向）更大，这与实验结果一致。如图 9 所示，仅考虑动力学涡度贡献的计算导致相反的符号，而考虑剪切项则导致正确的符号。我们将在第 5.4 节进一步讨论这个问题，同时结合沿着束流方向的极化结果，以及与所谓的“自旋符号危机”相关的结果。

Global polarization of multi-strange hyperons
多奇异超子的全球极化

Based on the picture of the rotating system, any non-zero spin particles should be polarized in a similar way, along the direction of the initial orbital angular momentum. According to Eq. (2), the magnitude of the polarization depends on the spin of particles. Thus, it is of great interest to study the polarization of different particles with different spin. The STAR Collaboration reported global polarization
根据旋转系统的图像，任何非零自旋粒子都应该以类似的方式极化，沿着初始轨道角动量的方向。根据方程（2），极化的大小取决于粒子的自旋。因此，研究具有不同自旋的不同粒子的极化是非常有趣的。STAR 合作组织报告了全局极化。

Figure 8: $P_{-y}$ polarization component of $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ as a function of (left) transverse momentum dependence and (right) charge asymmetry $A_{\mathrm{c}\mathrm{h}}$ normalized with its RMS in Au+Au collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ = 200 GeV. The figures are adapted from Ref. [2].
图 8: $P_{-y}$ 的极化分量作为（左）横向动量依赖性和（右）电荷不对称性 $A_{\mathrm{c}\mathrm{h}}$ 的函数，与 Au+Au 碰撞中 $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ = 200 GeV 的 RMS 归一化。这些图表改编自参考文献[2]。

of ${\mathrm{\Xi }}^{-}$ (${\overline{\mathrm{\Xi }}}^{+}$) and ${\mathrm{\Omega }}^{-}$ (${\overline{\mathrm{\Omega }}}^{+}$) hyperons in 200 GeV Au+Au collisions, see Fig. 10. Two independent methods (see Sec. 4.1.1) were used to measure ${\mathrm{\Xi }}^{-}$ (${\overline{\mathrm{\Xi }}}^{+}$) polarization and the results combining ${\mathrm{\Xi }}^{-}$ and ${\overline{\mathrm{\Xi }}}^{+}$, and averaging over the two methods is found to be positive at the $2\sigma$ level ($⟨P_{\mathrm{\Xi }}⟩=0.47\pm 0.10(\text{stat})\pm 0.23(\text{syst})\mathrm{\%}$ for 20-80% centrality), supporting the global vorticity picture. The cascade polarization is measured to be slightly larger than that of inclusive $\mathrm{\Lambda }$, but the significance of that is below $1\sigma$. The results on ${\mathrm{\Omega }}^{-}$ global polarization hint even larger polarization indicating a possible hierarchy of $P_{\mathrm{\Omega }}>P_{\mathrm{\Xi }}>P_{\mathrm{\Lambda }}$ but with large uncertainties. Based on Eq. (2), the following relation: $P_{\mathrm{\Lambda }}=P_{{\overline{\mathrm{\Xi }}}^{-}}=\frac{3}{5}{P}_{{\mathrm{\Omega }}^{-}}$ is expected. Recent model study shows that this relation is valid only for primary particles, while it leads to $P_{\mathrm{\Lambda }}<P_{{\mathrm{\Xi }}^{-}}<P_{{\mathrm{\Omega }}^{-}}$ after taking into account the feed-down contribution [72], which seems to be consistent with the data. More precise measurements are needed to clarify the particle/spin dependence of the global polarization.
在 200 GeV Au+Au 碰撞中测量 ${\mathrm{\Xi }}^{-}$ （ ${\overline{\mathrm{\Xi }}}^{+}$ ）和 ${\mathrm{\Omega }}^{-}$ （ ${\overline{\mathrm{\Omega }}}^{+}$ ）超子的极化，见图 10。使用了两种独立方法（见 4.1.1 节）来测量 ${\mathrm{\Xi }}^{-}$ （ ${\overline{\mathrm{\Xi }}}^{+}$ ）的极化，将 ${\mathrm{\Xi }}^{-}$ 和 ${\overline{\mathrm{\Xi }}}^{+}$ 的结果结合起来，并在两种方法上取平均值，发现在 $2\sigma$ 水平上为正（20-80%中心度为 $⟨P_{\mathrm{\Xi }}⟩=0.47\pm 0.10(\text{stat})\pm 0.23(\text{syst})\mathrm{\%}$ ），支持全局涡度图像。级联极化被测量为略大于包容 $\mathrm{\Lambda }$ 的极化，但其显著性低于 $1\sigma$ 。关于 ${\mathrm{\Omega }}^{-}$ 全局极化的结果暗示着更大的极化，表明可能存在 $P_{\mathrm{\Omega }}>P_{\mathrm{\Xi }}>P_{\mathrm{\Lambda }}$ 的层次结构，但存在较大的不确定性。根据方程（2），预期以下关系： $P_{\mathrm{\Lambda }}=P_{{\overline{\mathrm{\Xi }}}^{-}}=\frac{3}{5}{P}_{{\mathrm{\Omega }}^{-}}$ 。最近的模型研究表明，这种关系仅适用于初级粒子，而在考虑到衰变贡献后，会导致 $P_{\mathrm{\Lambda }}<P_{{\mathrm{\Xi }}^{-}}<P_{{\mathrm{\Omega }}^{-}}$ [72]，这似乎与数据一致。需要更精确的测量来澄清全局极化的粒子/自旋依赖性。

It is worth mentioning that ${\mathrm{\Omega }}^{-}$ hyperon has larger magnetic moment (${\mu }_{{\mathrm{\Omega }}^{-}}=-2.02{\mu }_{\mathrm{N}}$) compared to those for $\mathrm{\Lambda }$ (${\mu }_{\mathrm{\Lambda }}=-0.613{\mu }_{\mathrm{N}}$) and ${\mathrm{\Xi }}^{-}$ (${\mu }_{{\mathrm{\Xi }}^{-}}=-0.65{\mu }_{\mathrm{N}}$). Therefore, the polarization difference between ${\mathrm{\Omega }}^{-}$ and ${\overline{\mathrm{\Omega }}}^{+}$, if any, should be more sensitive to the magnetic field created in the collisions. Another thing to be mentioned is that one of the decay parameter ${\gamma }_{\mathrm{\Omega }}$ is unknown, but expected to be close to either $+1$ or $-1$ (see Sec. 4.1.1). Assuming the vorticity picture, one can determine the sign of ${\gamma }_{\mathrm{\Omega }}$. Currently the experimental result on $\mathrm{\Omega }$ global polarization has large uncertainty but future high statistics data will allow to resolve the ambiguity.
值得一提的是， ${\mathrm{\Omega }}^{-}$ 超子的磁矩较 $\mathrm{\Lambda }$ （ ${\mu }_{\mathrm{\Lambda }}=-0.613{\mu }_{\mathrm{N}}$ ）和 ${\mathrm{\Xi }}^{-}$ （ ${\mu }_{{\mathrm{\Xi }}^{-}}=-0.65{\mu }_{\mathrm{N}}$ ）更大。因此，如果有的话， ${\mathrm{\Omega }}^{-}$ 和 ${\overline{\mathrm{\Omega }}}^{+}$ 之间的极化差异应更加敏感于碰撞中产生的磁场。另一件值得一提的事情是，其中一个衰变参数 ${\gamma }_{\mathrm{\Omega }}$ 是未知的，但预计接近于 $+1$ 或 $-1$ （见第 4.1.1 节）。假设涡度图像，可以确定 ${\gamma }_{\mathrm{\Omega }}$ 的符号。目前关于 $\mathrm{\Omega }$ 全局极化的实验结果存在较大的不确定性，但未来高统计数据将有助于解决这种模棱两可。

Figure 9: Polarization of $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ hyperons along the initial angular momentum $P_J=P_{-y}$ as a function of hyperons’ azimuthal angle relative to the second-order event plane ${\mathrm{\Psi }}_2$ in 20-50% Au+Au collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=200$ GeV (preliminary result from STAR[100]), comparing to the hydrodynamic model (vHLLE for 20-60% Au+Au collisions)[102] where $T_{\mathrm{d}\mathrm{e}\mathrm{c}}$ is a decoupling temperature assuming the isothermal freeze-out. This figure is taken from Ref.[102].
图 9：在 20-50% Au+Au 碰撞中， $\sqrt{s_{\mathrm{N}\mathrm{N}}}=200$ GeV 时， $\mathrm{\Lambda }$ 和 $\overline{\mathrm{\Lambda }}$ 超子沿着初始角动量 $P_J=P_{-y}$ 的极化作为超子相对于第二阶事件平面 ${\mathrm{\Psi }}_2$ 的方位角的函数（来自 STAR[100]的初步结果），与水动力模型（20-60% Au+Au 碰撞的 vHLLE）[102]进行比较，其中 $T_{\mathrm{d}\mathrm{e}\mathrm{c}}$ 是假定等温冻结的解耦温度。此图摘自 Ref.[102]。

Global spin alignment of vector mesons
矢量介子的全局自旋对齐

The vorticity should also lead to the global polarization of the vector mesons, such as $K^{\ast 0}$ and $\varphi$, revealing itself via global spin alignment [9, 12]. The first measurement of the spin alignment was made by the STAR Collaboration at RHIC using 200 GeV Au+Au collisions in 2008 [104] but there was no clear signal taking into account the uncertainties of the measurement. More recently, the ALICE and STAR Collaborations reported finite signals [105, 106], i.e., deviation of ${\rho }_{00}$ from 1/3. Figure 11 shows ${\rho }_{00}$ of $K^{\ast 0}$ and $\varphi$ mesons as a function of collision centrality in a form of the number of participants from MC Glauber simulation, in Pb+Pb collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=2.76$ TeV. At lower $p_{\mathrm{T}}$, the results for both $K^{\ast 0}$ and $\varphi$ mesons indicate ${\rho }_{00}<1/3$. The STAR results on $\varphi$-meson ${\rho }_{00}$ show large positive deviation from 1/3 (${\rho }_{00}>1/3$) for $p_{\mathrm{T}}>1.2$ GeV/$c$ at lower collision energies, while results on $K^{\ast 0}$ are consistent with zero as shown in Fig. 12. The dependence of $\varphi$-meson spin alignment signal on transverse momentum and centrality is not systematic; the signal seems to change sign and become negative at higher transverse momenta as well as in more central collisions, At present the dependence on transverse momentum and centrality can not be explained in any scenario.
涡度还应导致矢量介子的全局极化，例如 $K^{\ast 0}$ 和 $\varphi$ ，通过全局自旋对齐来展现自己[9, 12]。自旋对齐的第一次测量是由 STAR 合作组织在 2008 年使用 200 GeV Au+Au 碰撞在 RHIC 进行的[104]，但考虑到测量的不确定性，没有明确的信号。最近，ALICE 和 STAR 合作组织报告了有限信号[105, 106]，即 ${\rho }_{00}$ 偏离 1/3。图 11 显示了在 Pb+Pb 碰撞中，作为 MC Glauber 模拟中参与者数量形式的碰撞中心性函数的 ${\rho }_{00}$ 和 $K^{\ast 0}$ 介子，能量为 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=2.76$ TeV。在较低 $p_{\mathrm{T}}$ ，对 $K^{\ast 0}$ 和 $\varphi$ 介子的结果表明 ${\rho }_{00}<1/3$ 。在较低碰撞能量下，STAR 关于 $\varphi$ -介子 ${\rho }_{00}$ 的结果显示出明显的正偏离 1/3（ ${\rho }_{00}>1/3$ ），而关于 $K^{\ast 0}$ 的结果与图 12 中所示的零一致。 $\varphi$ -介子自旋对齐信号与横向动量和中心度的依赖关系并不系统；信号似乎在更高的横向动量以及更中心的碰撞中改变符号并变为负值，目前无法用任何情景解释横向动量和中心度的依赖关系。

Note that in the vorticity scenario, the spin alignments signal is expected to be very small $\mathrm{\Delta }\rho \approx (\omega /T{)}^2/3\approx 4P_H^2/3$. Taking into account the hyperon global polarization measurements presented in Fig. 5, the spin alignment signal should be of the order of ${10}^{-5}$ at the top RHIC energy and of the order of ${10}^{-3}$ at lowest BES energy, which is too small to explain the reported large deviation. If the vector mesons are produced via quark coalescence, ${\rho }_{00}$ of vector mesons can be expressed
请注意，在涡度场景中，预计自旋对齐信号非常小 $\mathrm{\Delta }\rho \approx (\omega /T{)}^2/3\approx 4P_H^2/3$ 。考虑到图 5 中呈现的超子全局极化测量结果，自旋对齐信号应该在顶部 RHIC 能量处为 ${10}^{-5}$ 数量级，在最低 BES 能量处为 ${10}^{-3}$ 数量级，这太小了，无法解释报告的大偏差。如果矢量介子是通过夸克凝聚产生的， ${\rho }_{00}$ 个矢量介子可以被表达。

Figure 10: Global polarization of $\mathrm{\Xi }$ and $\mathrm{\Omega }$ hyperons compared to that of $\mathrm{\Lambda }$ ($\overline{\mathrm{\Lambda }}$) as well as transport model calculations. The figure is taken from Ref. [103].
图 10：与 $\mathrm{\Lambda }$ （ $\overline{\mathrm{\Lambda }}$ ）以及输运模型计算相比， $\mathrm{\Xi }$ 和 $\mathrm{\Omega }$ 的全球极化。该图取自参考文献[103]。

via the quark (antiquark) polarization $P_q(P_{\overline{q}})$ as ${\rho }_{00}=(1-P_q{P}_{\overline{q}})/(3+P_q{P}_{\overline{q}})\approx (1-4P_q{P}_{\overline{q}}/3)/3$.[12] If $P_q=P_{\overline{q}}=\omega /(2T)$, ${\rho }_{00}\approx [1-(\omega /T{)}^2/3]/3$ which is consistent with the thermal approach.[23] If the particle production for the $p_{\mathrm{T}}$ of interest is dominated by fragmentation process, the ${\rho }_{00}$ approximates ${\rho }_{00}\approx (1+4P_q^2/3)/3$ leading to ${\rho }_{00}>1/3$, but the deviation is again expected to be very small.[12] The only possibility to have large signal in the vorticity based scenario could arise if the vorticity fluctuations are much larger than its average. Note that spin alignment signal is proportional to the (mean root) square of vorticity, while the hyperon polarization is proportional to its average.
通过夸克（反夸克）极化 $P_q(P_{\overline{q}})$ 为 ${\rho }_{00}=(1-P_q{P}_{\overline{q}})/(3+P_q{P}_{\overline{q}})\approx (1-4P_q{P}_{\overline{q}}/3)/3$ 。[12] 如果 $P_q=P_{\overline{q}}=\omega /(2T)$ ， ${\rho }_{00}\approx [1-(\omega /T{)}^2/3]/3$ 与热力学方法一致。[23] 如果感兴趣的 $p_{\mathrm{T}}$ 的粒子产生主要由碎裂过程主导， ${\rho }_{00}$ 近似于 ${\rho }_{00}\approx (1+4P_q^2/3)/3$ 导致 ${\rho }_{00}>1/3$ ，但偏差预计再次非常小。[12] 在基于涡度的情景中获得大信号的唯一可能性是，如果涡度波动远大于其平均值。请注意，自旋对齐信号与涡度的（均方根）平方成正比，而超子极化与其平均值成正比。

References[108, 109] suggest that the mean field of $\varphi$-mesons could play a role in $\varphi$${\rho }_{00}$ but not in $K^{\ast 0}$${\rho }_{00}$ because of mixing of different flavors. The model involving the strong force seems to explain the energy dependence of $\varphi$${\rho }_{00}$ as shown with the solid line in Fig. 12. Note that in the given $p_{\mathrm{T}}$ range the $\varphi$${\rho }_{00}$ at the LHC is consistent with zero.
参考文献[108, 109]表明 $\varphi$ -介子的平均场可能在 $\varphi$ ${\rho }_{00}$ 中发挥作用，但在 $K^{\ast 0}$ ${\rho }_{00}$ 中却不起作用，因为不同风味的混合。涉及强力的模型似乎能够解释 $\varphi$ ${\rho }_{00}$ 的能量依赖性，如图 12 中实线所示。请注意，在给定的 $p_{\mathrm{T}}$ 范围内，LHC 上的 $\varphi$ ${\rho }_{00}$ 与零一致。

Charm quarks are produced via hard scattering of partons at the collision, with a time scale of $\tau \sim 1/(2m_{\mathrm{H}\mathrm{Q}})\sim 0.1$ fm/c. Therefore, one may expect larger effect of the initial magnetic field as well as vorticity on the polarization of $J/\psi$ which consists of charm and anti-charm quarks. The ALICE Collaboration reported inclusive $J/\psi$ polarization relative to the event plane in Pb+Pb collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ = 5.02 TeV at forward rapidity ($2.5<y<4$).[110] Figure 13 shows $J/\psi$ polarization parameter ${\lambda }_{\theta }$ (see Eq. 36) as a function of centrality. Non-zero ${\lambda }_{\theta }$ means finite polarization of $J/\psi$. The observed signal of $0.1<{\lambda }_{\theta }<0.2$ corresponds to $0.29>{\rho }_{00}>0.25$, a large negative deviation from 1/3 and opposite to that of $\varphi$-mesons. This measurement was performed at forward rapidity ($2.5<y<4$) and the measurements at mid-rapidity both at the LHC and RHIC[111] energies are needed for a better understanding of the phenomena. The regeneration mechanism of the $J/\psi$ production that becomes significant at the LHC energies, especially at low $p_{\mathrm{T}}$, might also complicate the interpretation of the data.
魅力夸克是通过部分子在碰撞中的硬散射产生的，时间尺度为 $\tau \sim 1/(2m_{\mathrm{H}\mathrm{Q}})\sim 0.1$ fm/c。因此，人们可能期望初始磁场以及涡度对由魅力夸克和反魅力夸克组成的 $J/\psi$ 的极化产生更大影响。ALICE 合作组在 $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ = 5.02 TeV 的 Pb+Pb 碰撞中报告了相对于事件平面的包容性 $J/\psi$ 极化，位于前向快度（ $2.5<y<4$ ）。图 13 显示了 $J/\psi$ 极化参数 ${\lambda }_{\theta }$ （见方程 36）随着中心度的变化。非零 ${\lambda }_{\theta }$ 表示 $J/\psi$ 的有限极化。观测到的 $0.1<{\lambda }_{\theta }<0.2$ 信号对应于 $0.29>{\rho }_{00}>0.25$ ，与 1/3 的大负偏差相反，与 $\varphi$ -介子的相反。这项测量是在前向快度（ $2.5<y<4$ ）进行的，需要在 LHC 和 RHIC[111]能量下的中快度进行测量，以更好地理解这些现象。在 LHC 能量下，尤其是在低 $p_{\mathrm{T}}$ 时， $J/\psi$ 产生的再生机制可能也会使数据的解释变得更加复杂。

Figure 11: Global spin alignment of $K^{\ast 0}$ and $\varphi$ mesons shown as the spin density matrix element ${\rho }_{00}$ with quantization axis chosen along the system orbital angular momentum measured in Pb+Pb collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ =2.76 TeV by the ALICE Collaboration. The figure is taken from Ref.[105].
图 11：全球 $K^{\ast 0}$ 和 $\varphi$ 介子的自旋对齐，显示为自旋密度矩阵元素 ${\rho }_{00}$ ，量子化轴选择沿着在 ALICE 协作组在 $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ =2.76 TeV 的 Pb+Pb 碰撞中测量的系统轨道角动量。该图取自参考文献[105]。

Obviously the spin alignment measurements still need further investigations, both theoretically and experimentally. These are difficult measurements, and as discussed in Sec. 4.5 and 4.6.3 strongly dependent on complete understanding of the tracking and acceptance effects. Future analyses, in particular based on new high statistics data will allow us to study the spin alignment in much more detail including other particles such as charged $D^{\ast }$[112, 113] and $\mathrm{{\rm Y}}$[114].
显然，自旋对齐测量仍需要进一步的理论和实验研究。这些是困难的测量，正如在第 4.5 节和 4.6.3 节中讨论的那样，它们在很大程度上依赖于对跟踪和接受效应的完全理解。未来的分析，特别是基于新的高统计数据，将使我们能够更详细地研究自旋对齐，包括其他粒子，如带电 $D^{\ast }$ [112, 113]和 $\mathrm{{\rm Y}}$ [114]。

Polarization along the beam direction
沿着光束方向的极化

As discussed in Sec. 2.3, anisotropic transverse flow leads to nonzero vorticity component along the beam direction, with the direction of vorticity changing with the azimuthal angle[18, 19], as depicted by open arrows in Fig. 3(left). The polarization along the beam direction $P_{\mathrm{z}}$ was first measured with $\mathrm{\Lambda }$ hyperons by the STAR Collaboration at RHIC[3]. Later, it was also observed by the ALICE Collaboration
如第 2.3 节所讨论的，各向异性横向流导致光束方向上存在非零涡度分量，涡度方向随方位角变化[18, 19]，如图 3（左）中的开放箭头所示。光束方向上的极化 $P_{\mathrm{z}}$ 是由 STAR 协作组在 RHIC 上首次通过 $\mathrm{\Lambda }$ 超子测量的[3]。后来，ALICE 协作组也观察到了这一现象。

Figure 12: $K^{\ast 0}$ and $\varphi$ mesons spin density matrix element with quantization axis along the system orbital angular momentum in Au+Au collisions by the STAR experiment. The results are shown as a function of collision energy $\sqrt{s_{\mathrm{NN}}}$, transverse momentum, and centrality. The figures are taken from Ref.[107].
图 12：在 Au+Au 碰撞中，由 STAR 实验测得的系统轨道角动量沿量子化轴的 $K^{\ast 0}$ 和 $\varphi$ 介子自旋密度矩阵元。结果显示为碰撞能量 $\sqrt{s_{\mathrm{NN}}}$ 、横向动量和中心度的函数。这些图表摘自参考文献[107]。

at the LHC energy [115]. As expected from the elliptic flow picture, $P_{\mathrm{z}}$ exhibits a quadrupole or $\sin (2\varphi )$ pattern as shown in Fig. 14(left). The polarization is quantified by a second-order Fourier sine coefficient and studied as a function of centrality, see Fig. 15. The results show a clear centrality dependence similar to that of elliptic flow except in most peripheral collisions. The polarization magnitudes at RHIC and the LHC are rather similar, indicating weak collision energy dependence unlike in the global polarization case.
在 LHC 能量[115]。从椭圆流图像中可以看到， $P_{\mathrm{z}}$ 呈现出四极或 $\sin (2\varphi )$ 模式，如图 14（左）所示。极化通过二阶傅立叶正弦系数量化，并随着中心度的变化进行研究，参见图 15。结果显示出明显的中心度依赖性，类似于椭圆流，除了在大多数外围碰撞中。在 RHIC 和 LHC 的极化幅度相当相似，表明与全局极化情况不同，碰撞能量依赖性较弱。

It was found that hydrodynamic and transport models that successfully reproduce the energy dependence of the global polarization fail badly in predictions of the magnitude and the sign (phase) of the azimuthal angle modulation differently [101, 117, 3, 19, 20, 95, 96, 100]. This was true for several hydrodynamics models using different approaches and initial conditions. The chiral kinetic approach accounting for the nonequilibrium effects of the spin degrees of freedom gives the correct sign of the $P_{\mathrm{z}}$ modulation [118]. Interestingly, the Blast-Wave model which is a simplified model of hydrodynamics with a few freeze-out parameters [45] (taken from STAR publication in 2005!) describes the data very well [3].
发现，成功重现全球极化能量依赖性的流体动力学和输运模型在预测方位角角度调制的幅度和符号（相位）方面表现不佳[101, 117, 3, 19, 20, 95, 96, 100]。这对于使用不同方法和初始条件的几种流体动力学模型都是如此。考虑自旋自由度的非平衡效应的手征动力学方法给出了 $P_{\mathrm{z}}$ 调制的正确符号[118]。有趣的是，爆震波模型是一个简化的流体动力学模型，具有少量冻结参数[45]（取自 2005 年 STAR 出版物！）很好地描述了数据[3]。

More recently, two independent groups pointed out that accounting for contribution from the fluid velocity shear (see Sec. 3.2) might help to explain the disagreement between the data and theoretical calculations. As shown in Fig. 14(left), the contribution from the kinematic shear, as that in the hydrodynamic model [102], exhibits an opposite sign in $P_{\mathrm{z}}$ modulation to that of the kinematic vorticity, and as a consequence, combining the two effects leads to a trend similar to the data if additionally the model assumes the isothermal freeze-out. Hydrodynamic model (MUSIC with AMPT initial conditions) including the shear contribution [119], and assuming that $\mathrm{\Lambda }$ inherits the polarization from the strange quark, can also qualitatively describe the measurements including the centrality dependence as shown
最近，两个独立的团体指出，考虑来自流体速度剪切的贡献（见第 3.2 节）可能有助于解释数据与理论计算之间的分歧。如图 14（左）所示，来自运动剪切的贡献，如在流体动力学模型[102]中一样，与运动涡度的贡献在 $P_{\mathrm{z}}$ 调制中呈相反符号，因此，结合这两种效应会导致一个与数据类似的趋势，如果模型另外假设等温冻结。包括剪切贡献的流体动力学模型（具有 AMPT 初始条件的 MUSIC），并假设 $\mathrm{\Lambda }$ 继承了奇异夸克的极化，也可以在包括中心度依赖性的测量中定性描述。

Figure 13: Spin alignment of inclusive $J/\psi$ along the system orbital angular momentum in Pb+Pb collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=5.02$ TeV from ALICE Collaboration [110]. The measurement was performed within $2.5<y<4$ and $2<P_{\mathrm{T}}<6$ GeV/$c$.
图 13：ALICE 协作组在 $\sqrt{s_{\mathrm{N}\mathrm{N}}}=5.02$ TeV 的 Pb+Pb 碰撞中沿系统轨道角动量的包容性 $J/\psi$ 自旋对齐。该测量在 $2.5<y<4$ 和 $2<P_{\mathrm{T}}<6$ GeV/ $c$ 范围内进行。

Figure 14: (Left) Raw signal of polarization along the beam direction, $⟨\cos {\theta }_{\mathrm{F}}^{\mathrm{e}}⟩$, of $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ hyperons as a function of azimuthal angle relative to the second-order event plane in Au+Au collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ = 200 GeV.[3] Hydrodynamic model calculations including kinematic vorticity and kinematic shear separately, as well as the sum of the two are shown by lines.[102] (Right) Same as left figure for polarization relative to the third-order event plane in R+Ru and Zr+Zr collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ = 200 GeV.[116]
图 14：（左）极化信号沿着光束方向， $⟨\cos {\theta }_{\mathrm{F}}^{\mathrm{e}}⟩$ ， $\mathrm{\Lambda }$ 和 $\overline{\mathrm{\Lambda }}$ 超子的极化信号作为方位角相对于 Au+Au 碰撞中第二阶事件平面的函数在 $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ = 200 GeV 时显示。[3] 包括运动涡度和运动剪切的流体力学模型计算分别显示在线上，以及两者的总和。[102]（右）相对于 R+Ru 和 Zr+Zr 碰撞中第三阶事件平面的极化信号的左图相同在 $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ = 200 GeV 时显示。[116]

Figure 15: The second-order sine modulation of $\mathrm{\Lambda }$ polarization along the beam direction as a function of centrality at RHIC[3] and the LHC[115] compared to various model calculations.[3, 119] The experimental data are rescaled with $\mathrm{\Lambda }$ decay parameter ${\alpha }_{\mathrm{\Lambda }}=0.732$.[84]
图 15：与各种模型计算[3, 119]相比，RHIC[3]和 LHC[115]上的中心度函数沿着光束方向的 $\mathrm{\Lambda }$ 极化的二阶正弦调制。实验数据使用 $\mathrm{\Lambda }$ 衰变参数 ${\alpha }_{\mathrm{\Lambda }}=0.732$ 重新调整。

in Fig. 15. But the predictions change to the opposite sign if the polarization is calculated using $\mathrm{\Lambda }$ mass. It should be noted that the thermal vorticity and shear contributions are largely canceled out [120, 121, 123] and the final result depends strongly on the detailed implementation of those contributions. Thus the spin sign puzzle still needs more investigations.
在图 15 中。但是，如果使用 $\mathrm{\Lambda }$ 质量计算极化，预测将变为相反的符号。值得注意的是，热涡度和剪切贡献在很大程度上被抵消[120, 121, 123]，最终结果强烈取决于这些贡献的详细实施。因此，自旋符号之谜仍需要更多的调查。

As predicted in Ref. [18], higher harmonic anisotropic flow should also lead to a similar vorticity structure and polarization along the beam direction. Recently, the STAR Collaboration has reported $\mathrm{\Lambda }$ polarization along the beam direction relative to the third harmonic event plane in isobar Ru+Ru and Zr+Zr collisions $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ = 200 GeV [116]; these results are shown in Fig. 14(right). The sine modulation of $P_{\mathrm{z}}$ relative to the third-order event plane was observed similarly to the second-order case, indicating a sextupole pattern of vorticity induced by triangular flow as depicted in Fig. 3(right).
根据参考文献[18]的预测，更高谐波各向异性流也应导致类似的涡度结构和沿着束流方向的极化。最近，STAR 协作组报告了在同位素 Ru+Ru 和 Zr+Zr 碰撞中，相对于第三谐波事件平面沿着束流方向的 $\mathrm{\Lambda }$ 极化，能量为 200 GeV[116]；这些结果显示在图 14（右）中。与第三阶事件平面相对的正弦调制 $P_{\mathrm{z}}$ 也被观察到，类似于二阶情况，表明由三角流引起的涡度的六极模式，如图 3（右）所示。

Figure 16(left) shows $P_z$ sine coefficients relative to the second and third order harmonic event planes as a function of centrality in the isobar collisions. The third-order result seems to increase towards peripheral collisions as the second-order does. Calculations from hydrodynamic model with two different implementations of the shear induced polarization (SIP), based on Ref. [51] by Becattini-Buzzegoli-Palermo (BBP) and on Ref. [50] by Liu-Yin (LY), are also compared. The calculations with "SIP${}_{\mathrm{B}\mathrm{B}\mathrm{P}}$" reasonably well describe the data for both the second and third-orders except peripheral collisions. The calculation with "SIP${}_{\mathrm{L}\mathrm{Y}}$" leads to the opposite sign to the data but note that it provides the correct sign if the mass of strange
图 16（左）显示 $P_z$ 正弦系数相对于同位素碰撞中心度的第二和第三阶谐波事件平面的变化情况。第三阶结果似乎朝向外围碰撞增加，就像第二阶一样。基于 Becattini-Buzzegoli-Palermo（BBP）的参考文献[51]和基于 Liu-Yin（LY）的参考文献[50]的两种不同实现的剪切诱导极化（SIP）的流体动力学模型计算也进行了比较。使用“SIP ${}_{\mathrm{B}\mathrm{B}\mathrm{P}}$ ”进行的计算相当好地描述了第二和第三阶的数据，除了外围碰撞。使用“SIP ${}_{\mathrm{L}\mathrm{Y}}$ ”进行的计算导致与数据相反的符号，但请注意，如果奇怪的质量是正确的符号。

Figure 16: The second and third-order sine modulation of $\mathrm{\Lambda }$ polarization along the beam direction as a function of centrality (left) and hyperons’ transverse momentum (right) in Ru+Ru and Zr+Zr collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ = 200 GeV [116]. Solid and dashed lines are the calculations from hydrodynamic model with particular implementation of the shear induced polarization (SIP) [123]. See texts for the detail.
图 16：在 Ru+Ru 和 Zr+Zr 碰撞中，沿着光束方向的 $\mathrm{\Lambda }$ 极化的二阶和三阶正弦调制作为中心度（左）和超子的横向动量（右）的函数在 $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ = 200 GeV [116]。实线和虚线是具有剪切诱导极化（SIP）特定实现的流体力学模型的计算[123]。有关详细信息，请参阅文本。

quark is used instead of $\mathrm{\Lambda }$ mass as shown in Fig. 15. It is also worth to mention that the calculation with a nearly zero specific shear viscosity (denoted as "ideal hydro") leads to almost zero $P_z$ sine coefficient, which indicates that the $P_z$ measurement could provide an additional constraint on the shear viscosity of the medium. Figure 16(right) shows $p_{\mathrm{T}}$ dependence of the second and third-order $P_{\mathrm{z}}$ sine coefficients. The third-order result is found to be comparable in magnitude to the second-order result, slightly smaller at low $p_{\mathrm{T}}$ and showing a hint of overpassing the second-order at high $p_{\mathrm{T}}$. This trend is similar to what was observed in $p_{\mathrm{T}}$ dependence of the elliptic and triangular flow [124], which further supports the picture of anisotropic-flow-driven polarization. The model incorporating the shear induced polarization of SIP${}_{\mathrm{B}\mathrm{B}\mathrm{P}}$ is comparable to the data at low $p_{\mathrm{T}}$ but not the $p_{\mathrm{T}}$ dependence in detail.
夸克被用来代替 $\mathrm{\Lambda }$ 质量，如图 15 所示。值得一提的是，几乎为零的特定剪切粘度（标记为“理想流体力学”）的计算导致几乎为零的 $P_z$ 正弦系数，这表明 $P_z$ 测量可以为介质的剪切粘度提供额外的约束。图 16（右）显示了第二和第三阶 $P_{\mathrm{z}}$ 正弦系数的依赖关系。第三阶结果在低 $p_{\mathrm{T}}$ 时与第二阶结果相当，稍微小一些，并且在高 $p_{\mathrm{T}}$ 时显示出超过第二阶的迹象。这种趋势类似于所观察到的椭圆和三角流的 $p_{\mathrm{T}}$ 依赖关系[124]，进一步支持各向异性流驱动的极化图像。包含 SIP 的剪切诱导极化的模型在低 $p_{\mathrm{T}}$ 处与数据相当，但在细节上不符合 $p_{\mathrm{T}}$ 依赖关系。

6 Open questions and future perspective
6 个开放问题和未来展望

Summarizing the discussion in Sec. 5, one tend to conclude that while the theoretical description of the global polarization, including its energy dependence, is rather good, our understanding of the local polarization measurements, in particular the azimuthal angle dependence of the polarization along the beam direction is far from satisfactory. Surprisingly, the data is much better described by "naive" BlastWave model including only nonrelativistic vorticity, than by more sophisticated hydrodynamical calculations (including contribution from temperature gradients and acceleration), which very often differ from the data at the qualitative level. Recent calculations including the shear induced polarization make the comparison somewhat better, but still unsatisfactory. The disagreements with theoretical models definitely need further investigation in future, including the role of different freeze-out scenarios, validity of Cooper-Frye prescription, relative contributions of kinematic vorticity, acceleration, SIP, SHE, and temperature gradients. Comparison of more advanced calculations with new measurements should be also able to provide information of vorticity evolution and spin equilibration relaxation times.
总结第 5 节中的讨论，人们倾向于得出这样的结论：虽然对于全局极化的理论描述，包括其能量依赖性，相当不错，但我们对于局部极化测量的理解，特别是沿着束流方向的方位角依赖性，远非令人满意。令人惊讶的是，数据更适合由仅包括非相对论涡度的“天真”BlastWave 模型描述，而不是更复杂的流体力学计算（包括来自温度梯度和加速度的贡献），后者往往在定性水平上与数据不同。最近的计算包括剪切诱导的极化，使比较略微好一些，但仍然不尽人意。与理论模型的分歧明显需要未来进一步调查，包括不同冻结场景的作用、Cooper-Frye 配方的有效性、运动学涡度、加速度、SIP、SHE 和温度梯度的相对贡献。 更先进的计算与新的测量结果的比较也应该能够提供涡度演变和自旋平衡弛豫时间的信息。

From experimental point of view, in the next few years several new precise measurements will be performed to shed more light on the topics of interest, such as particle-antiparticle polarization splitting, rapidity and azimuthal angle dependencies, and particle species dependence. Below we list possible near-future measurements intended to provide more information on the vorticity and polarization phenomena in heavy-ion collisions.
从实验的角度来看，在接下来的几年里，将进行几项新的精密测量，以更好地阐明感兴趣的主题，如粒子-反粒子极化分裂、快度和方位角依赖性，以及粒子种类依赖性。以下我们列出可能的近期测量，旨在提供有关重离子碰撞中涡度和极化现象的更多信息。

• Polarization splitting between particles and antiparticles, including particles with larger magnitude of the magnetic moment such as $\mathrm{\Omega }$. It will further constrain the magnetic field time evolution and its strength at freeze-out, and the electric conductivity of quark-gluon plasma.
  粒子和反粒子之间的极化分裂，包括具有较大磁矩幅度的粒子，如 $\mathrm{\Omega }$ 。这将进一步限制磁场的时间演化及其在冻结时的强度，以及夸克胶子等离子体的电导率。
• Precise measurements of multistrange hyperon polarization to study particle species dependence and confirm the vorticity-based picture of polarization. Measurement with $\mathrm{\Omega }$ will also constrain unknown decay parameter ${\gamma }_{\mathrm{\Omega }}$.
  多奇异超子极化的精确测量，以研究粒子种类依赖性并确认基于涡度的极化图像。使用 $\mathrm{\Omega }$ 进行测量还将限制未知的衰变参数 ${\gamma }_{\mathrm{\Omega }}$ 。
• Precise differential measurements of the azimuthal angle and rapidity dependence of $P_J$ ($P_{-y}$).
  精确测量方位角和快度依赖性的 $P_J$ （ $P_{-y}$ ）。
• Detailed measurement of $P_{\mathrm{z}}$ induced by elliptic and higher harmonic flow. In particular this study could help to identify the contribution from SIP, which is expected to be different for different harmonics.
  由椭圆和更高谐波流引起的 $P_{\mathrm{z}}$ 的详细测量。特别是这项研究可以帮助识别来自 SIP 的贡献，预计对不同谐波有所不同。
• Application of the event-shape-engineering technique[125] testing the relationship between anisotropic flow and polarization.
  事件形状工程技术的应用[125]，测试各向异性流和极化之间的关系。
• Measuring $P_{\mathrm{x}}$ to complete all the components of polarization and compare the data to the Glauber estimates and full hydrodynamical calculations.
  测量 $P_{\mathrm{x}}$ 以完成所有极化组件，并将数据与 Glauber 估计和完整的流体力学计算进行比较。
• Circular polarization $P_{\varphi }$ to search for toroidal vortex structures
  循环极化 $P_{\varphi }$ 以搜索环形涡旋结构
• The particle-antiparticle difference in the polarization dependence on azimuthal angle at lower collision energies testing the Spin-Hall Effect.
  在较低的碰撞能量下测试自旋霍尔效应时，极化依赖于方位角的粒子-反粒子差异。
• Understanding of the vector meson spin alignment measurements including new results with corrections of different detector effects.
  理解矢量介子自旋对齐测量，包括纠正不同探测器效应的新结果。
• Measurement of the hyperon polarization correlations to access the scale of vorticity fluctuations.
  测量超子极化相关性以访问涡度波动的尺度。
• Measurement of the hyperon polarization in $pp$ collisions to establish/disprove possible relation to the single spin asymmetry effect.
  在 $pp$ 碰撞中测量超子极化，以建立/证伪可能与单旋转不对称效应的关系。

7 Summary 7 总结

The polarization phenomena in heavy-ion collisions appeared to be an extremely interesting and important subject overarching such questions as the nature of the spin and spin structure of the hadrons, evolution of the QGP and its hadronization, and finally the freeze-out of the system. While many, or better to say, most of the details of the entire picture is far from being even well formulated, it is clear that following this direction we might expect many important discoveries.
重离子碰撞中的极化现象似乎是一个非常有趣和重要的主题，涵盖了诸如强子的自旋和自旋结构的性质、QGP 的演化及其强子化，以及系统的冻结等问题。虽然整个画面的许多细节远未得到很好的阐明，但很明显，沿着这个方向前进，我们可能会期待许多重要的发现。

The observed global polarization of hyperons in heavy-ion collisions is found to be well described by hydrodynamic and microscopic transport models based on the local vorticity of the fluid averaged over the freeze-out hypersurface under assumption of the local thermal equilibrium of the spin degrees of freedom. Furthermore, the measurement of hyperon polarization along the beam direction confirmed the local vorticity induced by anisotropic collective flow, adding to the evidences of ideal fluid dynamics of the quark-gluon plasma. These measurements opened new direction to study the dynamics of quark-gluon plasma and spin transport in the hot and dense medium, triggering a lot of theoretical interest on spin dynamics in general. Despite the successful description of the average global polarization, when looking into the detailed comparison between the data and models in differential measurements, there are still many open questions to be solved. The spin alignment measurements of vector mesons are very intriguing, but far from satisfactory understanding. More precise measurements with different particle species and a wider detector acceptance that will be available in near future at RHIC and the LHCand future experiments at new facilities will be extremely helpful to shed light on existing issues.
在重离子碰撞中观察到的超子全局极化被发现可以很好地由基于流体局部涡度的流体力学和微观输运模型描述，这些模型是在冻结超曲面上对流体的局部热平衡自旋自由度的假设下建立的。此外，沿着束流方向测量的超子极化证实了由各向异性集体流引起的局部涡度，进一步证明了夸克胶子等离子体的理想流体动力学。这些测量开辟了研究夸克胶子等离子体动力学和热密介质中自旋输运的新方向，引发了许多关于自旋动力学的理论兴趣。尽管成功描述了平均全局极化，但在详细比较数据和模型在微分测量中的情况时，仍然有许多待解决的问题。矢量介子的自旋对齐测量非常有趣，但远未达到令人满意的理解。 在不久的将来，在 RHIC 和 LHC 以及新设施的未来实验中，使用不同的粒子种类和更广泛的探测器接受度进行更精确的测量将极大地有助于阐明现有问题。

Acknowledgments 致谢

The discussions with F. Becattini, and C. Shen are gratefully acknowledged.
感谢与 F. Becattini 和 C. Shen 的讨论。

S. Voloshin is supported by the U.S. Department of Energy Office of Science, Office of Nuclear Physics under Award No. DE-FG02-92ER40713. T. Niida is supported by JSPS KAKENHI Grant Number JP22K03648.
S. Voloshin 得到美国能源部科学办公室核物理办公室的支持，奖励编号为 DE-FG02-92ER40713。T. Niida 得到 JSPS KAKENHI 资助，资助编号为 JP22K03648。