Local spin polarization in high energy heavy ion collisions

TL;DR

This paper investigates how the azimuthal angle affects hyperon spin polarization in high-energy heavy-ion collisions using viscous hydrodynamics, showing results consistent with experimental data and highlighting the importance of the spin chemical potential's choice.

Contribution

It demonstrates the impact of different initial conditions and the choice of spin chemical potential on hyperon polarization, aligning simulations with experimental observations.

Findings

Azimuthal dependence of hyperon polarization varies with initial conditions.

Choosing spin chemical potential proportional to temperature vorticity reproduces experimental results.

Temperature vorticity remains approximately conserved in quark-gluon plasma.

Abstract

We revisit the azimuthal angle dependence of the local spin polarization of hyperons in heavy-ion collisions at 200 GeV in the framework of the (3+1)D viscous hydrodynamic model CLVisc. Two different initial conditions are considered in our simulation: the optical Glauber initial condition without initial orbital angular momentum and the AMPT initial condition with an initial orbital angular momentum. We find that the azimuthal angle dependence of the hyperon polarization strongly depends on the choice of the so-called "spin chemical potential" $\Omega_{\mu\nu}$. With $\Omega_{\mu\nu}$ chosen to be proportional to the temperature vorticity, our simulation shows qualitatively coincidental results with the recent measurements at RHIC for both the longitudinal and transverse polarization. We argue that such a coincidence may be related to the fact that the temperature vorticity is approximately conserved in the hot quark-gluon matter.

Equations

$$n_{\mu }\mathrm{Tr}(\hat{\rho }{\hat{T}}^{\mu \nu })$$

$$n_{\mu }\mathrm{Tr}(\hat{\rho }{\hat{J}}^{\mu ,\alpha \beta })$$

$$n_{\mu }\mathrm{Tr}(\hat{\rho }{\hat{N}}^{\mu })$$

$${\hat{J}}^{\mu ,\alpha \beta }=x^{\alpha }{\hat{T}}^{\mu \beta }-x^{\beta }{\hat{T}}^{\mu \alpha }+{\hat{S}}^{\mu ,\alpha \beta }.$$

$$n_{\mu }\mathrm{Tr}(\hat{\rho }{\hat{S}}^{\mu ,\alpha \beta })=n_{\mu }{S}^{\mu ,\alpha \beta }.$$

$${\hat{\rho }}_{\mathrm{LE}}=\frac{1}{Z_{\mathrm{LE}}}\exp \left[-\int d{\Sigma }_{\mu }\left({\hat{T}}^{\mu \nu }{\beta }_{\nu }-\frac{1}{2}{\Omega }_{\alpha \beta }{\hat{S}}^{\mu ,\alpha \beta }-\zeta {\hat{N}}^{\mu }\right)\right],$$

$${\omega }_{\mu \nu }^{(K)}=-\frac{1}{2}({\partial }_{\mu }{u}_{\nu }-{\partial }_{\nu }{u}_{\mu }),$$

$${\omega }_{\mu \nu }^{(K)}={\epsilon }_{\nu }{u}_{\mu }-{\epsilon }_{\mu }{u}_{\nu }+{ϵ}_{\nu \mu \rho \eta }{u}^{\rho }{\omega }^{\eta },$$

$${\omega }_{\mu \nu }^{(\mathrm{NR})}={ϵ}_{\nu \mu \rho \eta }{u}^{\rho }{\omega }^{\eta }.$$

$${\omega }_{\mu \nu }^{(T)}$$

$$=T{\omega }_{\mu \nu }^{(K)}+\frac{1}{2}(u_{\mu }{\partial }_{\nu }T-u_{\nu }{\partial }_{\mu }T)$$

$$\equiv T{\omega }_{\mu \nu }^{(K)}+{\omega }_{\mu \nu }^{(T)}(T),$$

$${\int }_{\Xi }{\omega }_{\mu \nu }^{(T)}d{x}^{\mu }\wedge d{x}^{\nu }=-{\oint }_{C}T{u}_{\mu }d{x}^{\mu }.$$

$$(\epsilon +P)\frac{d}{d\tau }{u}^{\mu }={\nabla }^{\mu }P,$$

$$\frac{d}{d\tau }(T{u}^{\mu })={\partial }^{\mu }T,$$

$$\frac{d}{d\tau }\oint T{u}_{\mu }d{x}^{\mu }=\oint {\partial }_{\mu }Td{x}^{\mu }=0.$$

$${\omega }_{\mu \nu }^{(\mathrm{th})}$$

$$=\frac{1}{T}{\omega }_{\mu \nu }^{(K)}-\frac{1}{2T^2}(u_{\mu }{\partial }_{\nu }T-u_{\nu }{\partial }_{\mu }T)$$

$$=\frac{1}{T}{\omega }_{\mu \nu }^{(K)}+{\omega }_{\mu \nu }^{(\mathrm{th})}(T),$$

$${\hat{T}}^{\mu \nu }{\partial }_{\mu }{\beta }_{\nu }-\frac{1}{2}{\hat{S}}^{\mu ,\alpha \beta }{\partial }_{\mu }{\Omega }_{\alpha \beta }+\frac{1}{2}({\hat{T}}^{\alpha \beta }-{\hat{T}}^{\beta \alpha }){\Omega }_{\alpha \beta }=0,$$

$${\partial }_{\mu }{\beta }_{\nu }+{\partial }_{\nu }{\beta }_{\mu }$$

$${\partial }_{\mu }{\Omega }_{\alpha \beta }$$

$${\Omega }_{\alpha \beta }$$

$$S^{\mu }(p)=-\frac{1}{8m}{ϵ}^{\mu \rho \sigma \tau }{p}_{\tau }\frac{\int d{\Sigma }_{\lambda }{p}^{\lambda }{\omega }_{\rho \sigma }^{(\mathrm{th})}{f}_{FD}(1-f_{FD})}{\int d{\Sigma }_{\lambda }{p}^{\lambda }{f}_{FD}}+O(({\omega }_{\mu \nu }^{(\mathrm{th})}{)}^2),$$

$$S^{\mu }(p)=-\frac{1}{8m}{ϵ}^{\mu \rho \sigma \tau }{p}_{\tau }\frac{\int d{\Sigma }_{\lambda }{p}^{\lambda }{\Omega }_{\rho \sigma }{f}_{FD}(1-f_{FD})}{\int d{\Sigma }_{\lambda }{p}^{\lambda }{f}_{FD}}+O({\Omega }_{\mu \nu }^2),$$

$${\nabla }_{\mu }{T}^{\mu \nu }=0,$$

$${\pi }^{\mu \nu }={\eta }_{\upsilon }{\sigma }^{\mu \nu }-{\tau }_{\pi }\left[{△}_{\alpha }^{\mu }{△}_{\beta }^{\nu }{u}^{\lambda }{\nabla }_{\lambda }{\pi }^{\alpha \beta }+\frac{4}{3}{\pi }^{\mu \nu }\theta \right],$$

$${\nabla }_{\mu }{\lambda }^{\nu }={\partial }_{\mu }{\lambda }^{\nu }+{\Gamma }_{\mu \rho }^{\nu }{\lambda }^{\rho },$$

$${\nabla }_{\mu }{\lambda }^{\rho \sigma }={\partial }_{\mu }{\lambda }^{\rho \sigma }+{\Gamma }_{\mu \lambda }^{\rho }{\lambda }^{\lambda \sigma }+{\Gamma }_{\mu \lambda }^{\sigma }{\lambda }^{\rho \lambda },$$

$${\Gamma }_{\rho \sigma }^{\mu }=\frac{1}{2}{g}^{\mu \lambda }({\partial }_{\sigma }{g}_{\lambda \rho }+{\partial }_{\rho }{g}_{\lambda \sigma }-{\partial }_{\lambda }{g}_{\rho \sigma }).$$

$${\mathcal{P}}_x(p)$$

$${\mathcal{P}}_y(p)$$

$${\mathcal{P}}_z(p)$$

$$p^{\mu }=(m_T\cosh Y,p_x,p_y,m_T\sinh Y),$$

$$\overrightarrow{\mathcal{P}}({\varphi }_p)=\frac{1}{\Delta p_T}{\int }_{p_T^{\min }}^{p_T^{\max }}d{p}_T\overrightarrow{\mathcal{P}}(p),$$

Full text

I Introduction

It is well known that the rotation and spin polarization are correlated and can be converted to each other in materials Einstein and de Haas (1915); Barnett (1935). Recently, the polarization of electrons in a vortical fluid has been observed Takahashi et al. (2016). Similar phenomena also exist in high-energy heavy-ion collisions in which huge orbital angular momenta (OAM) are produced in peripheral collisions Liang and Wang (2005a); Becattini et al. (2008); Gao et al. (2008); Huang et al. (2011); Wang (2017) (for a recent review, see, e.g. Wang (2017)). The huge OAM are distributed into the quark gluon plasma created in heavy-ion collisions in the form of local vorticity Becattini et al. (2015a); Pang et al. (2016a); Deng and Huang (2016); Jiang et al. (2016), which result in the local polarization of hadrons along the vorticity direction Becattini et al. (2013); Fang et al. (2016) due to the spin-orbit coupling Liang and Wang (2005a); Gao et al. (2008). The net effect of the local polarization at all space-time points on the freeze-out hyper-surface gives the global polarization in the direction of the reaction plane or the OAM of two colliding nuclei Liang and Wang (2005a, b); Voloshin (2004); Betz et al. (2007); Becattini et al. (2008); Gao et al. (2008); Wang (2017).

The global polarization of $\Lambda$ and $\bar{\Lambda}$ has been measured by the STAR collaboration in Au+Au collisions at $\sqrt{s_{NN}}=7.7-200$ GeV Adamczyk et al. (2017); Adam et al. (2018). The data show a decreasing trend in collision energies from about 2% at 7.7 GeV to about 0.3% at 200 GeV.

There are several theoretical approaches which have been developed to study the global and local polarization in heavy ion collisions. The statistic-hydro model is based on the spin-vorticity coupling in the thermal distribution function Becattini et al. (2013); Becattini and Grossi (2015); Becattini et al. (2017); Florkowski et al. (2018a, b). So the average spin polarization is proportional to the so-called thermal vorticity (see the definition in the next section) if the thermal vorticity is small. Another theoretical approach is the Wigner function (WF) formalism Heinz (1983); Elze et al. (1986); Vasak et al. (1987); Zhuang and Heinz (1996); Florkowski et al. (1996); Blaizot and Iancu (2002); Wang et al. (2002), which has been revived Gao et al. (2012); Chen et al. (2013); Gao and Wang (2015); Hidaka et al. (2017); Gao et al. (2017, 2018a); Huang et al. (2018); Gao et al. (2018b); Liu et al. (2019) to study the chiral magnetic effect (CME) Vilenkin (1980); Kharzeev et al. (2008); Fukushima et al. (2008) (for reviews, see, e.g., Ref. Kharzeev et al. (2013, 2016); Huang (2016); Hattori and Huang (2017)) and chiral vortical effect (CVE) Vilenkin (1978); Erdmenger et al. (2009); Banerjee et al. (2011); Son and Surowka (2009); Gao et al. (2012); Hou et al. (2012) for massless fermions. Recently, the kinetic theory for spin-1/2 massive fermions has been formulated in the WF framework Fang et al. (2016); Weickgenannt et al. (2019); Gao and Liang (2019); Hattori et al. (2019a); Wang et al. (2019), which is useful in describing the evolution of the spin polarization. This is because the axial vector component gives the spin phase space distribution of fermions. At equilibrium, when the thermal vorticity is small, the spin polarization of fermions from the WF formalism is proportional to the thermal vorticity, consistent with the statistic-hydro model.

To describe the STAR data on the global polarization of hyperons which is along the direction of the reaction plane, the hydrodynamic and transport models have been used to calculate the vorticity field Baznat et al. (2013); Csernai et al. (2013, 2014); Teryaev and Usubov (2015); Jiang et al. (2016); Deng and Huang (2016); Ivanov and Soldatov (2017); Li et al. (2017); Wei et al. (2019). In the hydrodynamic framework, the velocity and in turn the vorticity fields in the fireball can be obtained naturally. The transport models describe the phase space evolution of a particle system through collisions among particles, so the position and momentum of each particle in the system at any time is given. To obtain the fluid velocity and then the vorticity at one space-time point, the suitable coarse graining procedure has to be used. Once the vorticity field is obtained, the global polarization of hyperons can be calculated from an integral over the freeze-out hyper-surface which agrees well with the data Karpenko and Becattini (2017); Xie et al. (2017); Li et al. (2017); Sun and Ko (2017); Wei et al. (2019).

The polarization of hyperons as a function of the azimuthal angle in the transverse plane has been recently measured in the STAR experiment Adam et al. (2018, 2019). However the data for the polarization along both the longitudinal and the transverse directions cannot be described by the hydrodynamic models (including A Multi-phase Transport (AMPT) model from which the vorticity field is extracted by the coarse graining method) Becattini and Karpenko (2018); Xia et al. (2018); Wei et al. (2019) based on the coupling of the thermal vorticity and the spin at equilibrium. There is a sign difference between the data and these model calculations. Although one model based on the chiral kinetic theory can explain the sign of the data Sun and Ko (2019), it cannot reproduce the magnitude of the data. Recent studies showed that the feed-down effects cannot resolve the sign difference Xia et al. (2019); Becattini et al. (2019).

The disagreement between theories and experiments indicates that the spin degree of freedom may not reach equilibrium in the fireball and thus the spin polarization may not be determined by the thermal vorticity. The relation between spin and thermal vorticity is dictated by the condition of local thermodynamic equilibrium if the spin tensor does not play a physical role Becattini et al. (2018). This calls for new approaches, for examples, the spin can be treated as an independent dynamical variable in the spin kinetic theory and spin hydrodynamics, or dissipative terms should be considered which are possibly larger than believed. Recently, the framework of spin hydrodynamics was developed Florkowski et al. (2018c); Florkowski and Ryblewski (2018); Hattori et al. (2019b). The spin evolution based on particle collisions was derived Zhang et al. (2019). The purpose of the present paper is not to make a numerical study based on these new approaches, instead, our purpose is not that ambitious: we will explore different choices of the so-called “spin chemical potential” $\Omega_{\mu\nu}$ and calculate the corresponding local hyperon polarization. The underlying reason is that, beyond global equilibrium, the thermal vorticity is not guaranteed to be the spin chemical potential, and thus the latter becomes a free parameter Florkowski and Ryblewski (2018); Hattori et al. (2019b); Becattini et al. (2018). In the (3+1)D hydrodynamic model CLVisc Pang et al. (2012, 2018), we will assume that the spin chemical potential $\Omega_{\mu\nu}$ is still determined by the fluid velocity and temperature (or equivalently the energy density). This means that $\Omega_{\mu\nu}$, being an anti-symmetric tensor, can be regarded as a type of vorticity (with appropriate normalization to make the dimension correct). We will thus explore four different definitions for $\Omega_{\mu\nu}$ or vorticity and calculate the local hyperon polarization and compare with the data. In our hydrodynamic simulation, we will examine two different initial conditions: the optical Glauber initial condition without initial OAM and the AMPT initial condition with an initial OAM.

The paper is organized as follows. In Section II we give a brief discussion about our motivation. In Section III we introduce our hydrodynamic model which we use for the simulation. We present our numerical results in Section IV. We give some discussions in Section V. Finally, we give a summary of our results in Section VI.

II Spin polarization and vorticity

The thermodynamic equilibrium in quantum field theory can be described by the density operator $\hat{\rho}$. Its form at local equilibrium can be obtained by maximizing the entropy $S=-\mathrm{Tr}(\hat{\rho}\ln\hat{\rho})$ with fixed densities of the energy-momentum, the angular momentum, and the conserved charge current on a space-like hyper-surface $\Sigma^{\mu}=n^{\mu}\Sigma$ pointing to a time-like direction $n^{\mu}$ Zubarev et al. (1979); van Weert (1982); Becattini et al. (2015b); Hayata et al. (2015),

[TABLE]

where $\hat{T}^{\mu\nu}$, $\hat{J}^{\mu,\alpha\beta}$ and $\hat{N}^{\mu}$ are the density operators of the energy-momentum tensor, the angular momentum tensor, and the conserved charge current, respectively. Note that $\hat{T}^{\mu\nu}$ is not necessarily symmetric. The quantities $T^{\mu\nu}$, $J^{\mu,\alpha\beta}$ and $N^{\mu}$ are their expectation values. For simplicity we will call $\hat{T}^{\mu\nu}$ ($T^{\mu\nu}$) and $\hat{J}^{\mu,\alpha\beta}$ ($J^{\mu,\alpha\beta}$) the energy-momentum and angular momentum tensor respectively though they are actually tensor densities. The angular momentum density operator includes the orbital and spin parts

[TABLE]

Thus, the second constraint in Eq. (1) can be equivalently expressed as

[TABLE]

The form of the density operator under the constraints (1), or with the second constraint in Eq. (1) being replaced by the constraint (3), that maximizes the entropy reads

[TABLE]

where $\beta_{\nu}$, $\Omega_{\alpha\beta}$ and $\zeta$ are Lagrangian multipliers which have physical meanings: $\beta_{\nu}=u_{\nu}/T$ with $u_{\nu}$ being the four-velocity and $T$ being the temperature, $\zeta=\mu/T$ with $\mu$ being the chemical potential, and $\Omega_{\alpha\beta}$ plays the role of the chemical potential for the angular momentum 111More precisely, it is $T\Omega_{\alpha\beta}$ that plays the role of a chemical potential for the angular momentum.. In the following, we will simply call $\Omega_{\alpha\beta}$ the spin chemical potential as it determines the spin polarization at local equilibrium. The density operator $\hat{\rho}_{{\rm LE}}$ in (4) defines the local thermal equilibrium and in general depends on the time.

In relativistic hydrodynamics, in order to obtain the spin vector, we need to first obtain $T$, $u^{\mu}$, and $\Omega_{\mu\nu}$ by solving the hydrodynamic equations in which the spin degree of freedom (or equivalently $\Omega_{\mu\nu}$) is treated on the same footing as $T$ and $u^{\mu}$. Such a framework is the spin hydrodynamics Florkowski et al. (2018c); Hattori et al. (2019b). However, the numerical spin hydrodynamics has not been established yet. Therefore we will adopt an usual (3+1)D hydrodynamic model, CLVisc Pang et al. (2012, 2018), which can give the space-time evolution of $T$ and $u^{\mu}$. Since $\Omega_{\mu\nu}$ is antisymmetric, we then assume that $\Omega_{\mu\nu}$ can be constructed from $T$ and $u^{\mu}$ as $\Omega_{\mu\nu}=-(1/2)\lambda(T)[\partial_{\mu}(g(T)u_{\nu})-\partial_{\nu}(g(T)u_{\mu})]\equiv\lambda(T)\omega_{\mu\nu}$ or its projections where $\lambda$ and $g$ are scalar functions of $T$ and $\omega_{\mu\nu}$ is the vorticity tensor 222In principle, it is also allowed to use the Hodge dual of the vorticity tensor to construct $\Omega_{\mu\nu}$. However, when the global equilibrium is approached, it is known that $\Omega_{\mu\nu}$ should approach the thermal vorticity up to a constant (depending on the symmetry properties of $T^{\mu\nu}$). We therefore do not consider such a possibility here.. In our numerical simulation, four types of vorticity will be considered, namely, the kinematic vorticity, the relativistic extension of the non-relativistic vorticity (NR vorticity), the thermal vorticity, and the temperature vorticity (T-vorticity). All these four types of vorticities have definite physical meaning and have been widely studied for quite some time in literature. For example, in Ref. Becattini et al. (2015a) the properties of the kinematic, thermal and T-vorticity have been studied in heavy ion collisions, in Ref. Karpenko and Becattini (2019) the NR vorticity is defined in a study of Lambda polarization in heavy ion collisions.

The kinematic vorticity is defined by

[TABLE]

where $u^{\mu}=\gamma(1,\boldsymbol{v})$ is the four-velocity and $\gamma$ is the Lorentz factor. The mass dimension of the kinematic vorticity is 1. The kinematic vorticity tensor can be decomposed into the part parallel and the part orthogonal to the fluid velocity

[TABLE]

where $\varepsilon_{\mu}=-(1/2)u^{\nu}\partial_{\nu}u_{\mu}$ and $\omega^{\mu}=(1/2)\epsilon^{\mu\nu\rho\sigma}u_{\nu}(\partial_{\rho}u_{\sigma})$. In comparison with the decomposition of the electromagnetic field strength $F^{\mu\nu}$, the vector $\varepsilon^{\mu}$ is like an ‘electric’ field while the vorticity vector $\omega^{\mu}$ is like a ‘magnetic’ field. It is clear that $\omega^{\mu}$ is a direct extension of the vorticity defined in non-relativistic hydrodynamics, $\boldsymbol{\omega}=(1/2)\nabla\times\boldsymbol{v}$. We thus define the last term in Eq. (6) as the NR vorticity tensor Karpenko and Becattini (2019)

[TABLE]

Similar to $\omega_{\mu\nu}^{(\mathrm{NR})}$, there has been an attempt to use the spatial components of the thermal vorticity as the spin chemical potential to study the longitudinal spin polarization of hyperons Florkowski et al. (2019).

The temperature vorticity or T-vorticity is defined by

[TABLE]

where the temperature enters the space-time derivative. The mass dimension of the temperature vorticity is 2. We have decomposed $\omega_{\mu\nu}^{(T)}$ into the part involving the space-time gradient of the temperature and the part without it which is proportional to the kinematic vorticity.

An important property of the T-vorticity is that it obeys a conservation law Gao et al. (2014); Becattini et al. (2015a); Deng and Huang (2016). Suppose $\Xi$ is a two-dimensional hyper-surface and $C$ is its boundary, thus the flux of the temperature vorticity on $\Xi$ is equal to the corresponding circulation of $Tu^{\mu}$ along the boundary $C$

[TABLE]

Since the viscosity of the hot matter in the fireball is small, we can approximately apply the Euler equation for an ideal fluid

[TABLE]

where $\varepsilon$ and $P$ are the energy density and pressure respectively, $d/d\tau=u^{\mu}\partial_{\mu}$ is co-moving time derivative, and $\nabla_{\mu}=\partial_{\mu}-u_{\mu}(d/d\tau)$. Rewriting the Euler equation in the following form

[TABLE]

one easily finds

[TABLE]

This is the relativistic Helmholtz-Kelvin theorem: the flux of the T-vorticity tensor is conserved with the fluid cell along $u^{\mu}$. We will see that this imposes a strong influence on the spin polarization.

The thermal vorticity $\omega_{\mu\nu}^{(\mathrm{th})}$ is defined by

[TABLE]

where $\beta=1/T$. The thermal vorticity is dimensionless. Similar to $\omega_{\mu\nu}^{(T)}$ we have also decomposed $\omega_{\mu\nu}^{(\mathrm{th})}$ into the part involving the space-time gradient of the temperature and the part without it which is proportional to $\omega_{\mu\nu}^{(K)}$. We see in Eq. (8) and (13) that $\omega_{\mu\nu}^{(T)}(T)$ and $\omega_{\mu\nu}^{(\mathrm{th})}(T)$ have the opposite sign.

The importance of the thermal vorticity relies on the fact that at global equilibrium, $\Omega_{\mu\nu}$ equals to $\omega_{\mu\nu}^{(\mathrm{th})}$ provided that the energy-momentum tensor $\hat{T}^{\mu\nu}$ has a non-vanishing anti-symmetric component Becattini et al. (2018); Hattori et al. (2019b); Liu et al. . This can be seen from the following procedure (the analysis based on the dissipative spin hydrodynamics or the kinetic theory gives the same conclusion). The global equilibrium is the state that the density operator (4) becomes independent of the choice of the hyper-surface $\Sigma_{\mu}$, so that

[TABLE]

where we used $\partial_{\mu}\hat{S}^{\mu,\alpha\beta}=\hat{T}^{\beta\alpha}-\hat{T}^{\alpha\beta}$. The above condition is fulfilled when 333We also note that these are sufficient but not necessary conditions for global equilibrium. For example, for conformal fluid, the right-hand side of the first condition can be relaxed to $\phi(x)g_{\mu\nu}$ with $\phi$ a scalar Liu et al. (2019).

[TABLE]

Note that if $\hat{T}^{\mu\nu}$ is symmetric, the spin tensor is conserved $\partial_{\mu}\hat{S}^{\mu,\alpha\beta}=0$ and the third condition in (15) does not hold which means that $\Omega_{\mu\nu}$ remains an independent variable even at global equilibrium. Note that the first line of Eq. (15) is called the Killing equation Becattini (2012) whose solution is $\beta_{\mu}=b_{\mu}+\omega_{\mu\alpha}^{\mathrm{(th)}}x^{\alpha}$, where $b_{\mu}$ and $\omega_{\mu\alpha}^{\mathrm{(th)}}$ are constants.

For spin-1/2 fermions at local equilibrium, when $\omega_{\mu\nu}^{(\mathrm{th})}$ is small, the average spin vector (defined as the Pauli-Lubanski vector) over the hyper-surface $\Sigma_{\mu}$ can be expressed as Becattini et al. (2013); Fang et al. (2016); Liu et al.

[TABLE]

where $f_{FD}=1/[\exp(p_{\mu}\beta^{\mu}-\zeta)+1]$ is the Fermi-Dirac distribution, normally $\Sigma_{\mu}$ is chosen as the freeze-out hyper-surface for hyperon polarization at the freeze-out. In the calculation we will set $\zeta=0$ as the net baryon density is almost zero in the hot fireball created in heavy ion collisions at high energies. In this paper, we assume that Eq. (16) can be generalized by replacing $\omega_{\mu\nu}^{(\mathrm{th})}$ with the spin chemical potential $\Omega_{\mu\nu}$ as

[TABLE]

where we will consider four types of vorticities as the spin chemical potentials, namely, $\Omega_{\rho\sigma}=\frac{1}{T}\omega_{\rho\sigma}^{(K)}$, $\frac{1}{T^{2}}\omega_{\rho\sigma}^{(T)}$, $\omega_{\rho\sigma}^{(\mathrm{th})}$, $\frac{1}{T}\omega_{\rho\sigma}^{(\mathrm{NR})}$. Here we have chosen suitable factors, $\lambda(T)=1/T,1/T^{2},1,1/T$, respectively, to make the spin chemical potential dimensionless. Note that Eq. (17) is the main assumption of this paper.

In the following, we will use the (3+1)D hydrodynamic model CLVisc to calculate the four types of vorticities and then use Eq. (17) to obtain the spin vector and then the corresponding spin polarization.

III Hydrodynamic model

The space-time evolution of the hot quark gluon plasma and dense hadronic matter is described by second order relativistic hydrodynamic equations,

[TABLE]

where $T^{\mu\nu}=(\varepsilon+P)u^{\mu}u^{\nu}-Pg^{\mu\nu}+\pi^{\mu\nu}$ is the energy-momentum tensor which is symmetric, $\varepsilon$ is the local energy density in the co-moving frame of the fluid, $P=P(\varepsilon)$ is the pressure determined by the QCD equation of state, $u^{\mu}$ is the fluid four-velocity obeying $u_{\mu}u^{\mu}=1$, $g^{\mu\nu}=\mathrm{diag}(1,-1,-1,-1/\tau^{2})$ is the metric tensor, $\pi^{\mu\nu}$ is the shear-stress tensor whose evolution is solved using a separate group of equations,

[TABLE]

where $\eta_{\upsilon}$ is the shear viscous coefficient, $\sigma^{\mu\nu}\equiv 2\nabla^{<\mu}u^{\nu>}\equiv 2\triangle^{\mu\nu\alpha\beta}\nabla_{\alpha}u_{\beta}$ is the symmetric shear tensor, $\tau_{\pi}=5\eta_{\upsilon}/(Ts)$ is the relaxation time for the shear viscosity, $\triangle^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}$ is the projection operator that makes the resulting contracted vector orthogonal to $u^{\mu}$, $\triangle^{\mu\nu\alpha\beta}=\frac{1}{2}(\triangle^{\mu\alpha}\triangle^{\nu\beta}+\triangle^{\mu\beta}\triangle^{\nu\alpha})-\frac{1}{3}\triangle^{\mu\nu}\triangle^{\alpha\beta}$ is the double projection operator that makes the resulting contracted tensor symmetric, traceless and orthogonal to $u^{\mu}$, $\theta\equiv\nabla_{\mu}u^{\mu}$ is the expansion rate. The operator $\nabla_{\mu}$ is the covariant derivative operator defined as

[TABLE]

[TABLE]

for vectors $\lambda^{\mu}$ and tensors $\lambda^{\mu\nu}$. The $\Gamma$’s are Christoffel symbols solved as a function of $g^{\mu\nu}$,

[TABLE]

The above (3+1)D viscous hydrodynamic equations are solved numerically using CLVisc Pang et al. (2012, 2018) with s95p-pce lattice QCD equation-of-state Borsanyi et al. (2012), and two different initial conditions: optical Glauber initial condition without initial OAM and AMPT initial condition with initial OAM are applied to check the dependences of the results on initial conditions.

IV Numerical results for hyperon polarization

In this section we will present our numerical results for the polarization of $\Lambda$ hyperons through vorticity fields by Eq. (17). We choose the coordinate system for collisions of two gold nuclei at 200 GeV in 20-50% centrality, see Fig. 1. The spatial indices $\mu=1,2,3$ in $S^{\mu}(p)$ in (17) correspond to the $x$, $y$ and $z$ direction respectively, so sometimes we write $\mu=1,2,3$ as $\mu=x,y,z$. To test effect of different choices for the spin chemical potential coupled to the spin tensor, we choose four types of vorticities: the kinematic vorticity, the T-vorticity, the thermal vorticity and the NR vorticity. We use the hydrodynamic model CLVisc to compute the vorticity field on the freeze-out hyper-surface.

We use two types of the initial condition: the optical Glauber initial condition without initial OAM and AMPT initial condition with an initial OAM. In the optical Glauber initial condition, the initial energy density distribution is boost invariant at mid-rapidity and is symmetric about the y-axis. As a result, it does not provide any initial OAM. On the other hand, the AMPT initial condition uses HIJING strings. These strings are attached to forward and backward going participants whose distributions are not symmetric about the y-axis in non-central collisions. On one side of the y-z plane, the midpoints of those strings are shifted to forward rapidity in the projectile-going direction. On the other side of the y-z plane, the midpoints of those strings are shifted to backward rapidity in the target-going direction. This forward-backward asymmetry in the AMPT model introduces non-zero initial OAM along the negative y-axis. One should keep in mind that there is no forward-backward asymmetry if the length of strings is infinity (before string fragmentation). In that case, all strings cover mid-rapidity. The system would be perfectly boost invariant along the space-time rapidity and symmetric about the y-axis. This corresponds to extreme high energy collisions where the initial OAM disappears at mid-rapidity. It is consistent with experimental observation that the global polarization is stronger in lower energy collisions. For collisions at low beam energies, the strings from the AMPT model have finite fluctuating lengths Pang et al. (2016b). The lengths of strings are determined by the longitudinal-momentum differences between their two end points which are quarks and diquarks from the projectile and the target, whose longitudinal momenta are sampled from parton distribution functions. In this way, the lengths of strings are finite and fluctuating. This helps to propagate the forward-backward asymmetry to left-right asymmetry at mid-rapidity, which is responsible for the initial OAM.

We calculate the transverse and longitudinal polarization of $\Lambda$ hyperons in the rapidity range $Y\in[-\Delta Y/2,\Delta Y/2]$

[TABLE]

where $S^{\mu}(p)$ is given by Eq. (17) and $p$ denotes the four-momentum of the $\Lambda$ hyperon,

[TABLE]

with $m_{T}=\sqrt{m_{\Lambda}^{2}+p_{x}^{2}+p_{y}^{2}}$. We choose the rapidity range $Y\in[-1,1]$ or $\Delta Y=2$ in the calculation.

IV.1 Results with optical Glauber initial condition

With the optical Glauber initial condition, we present the results for the longitudinal polarization for $\Lambda$. Figure 2 shows $\mathcal{P}_{z}^{(i)}(p)$ for four types of spin chemical potentials or vorticities $i=K,T,\mathrm{th},\mathrm{NR}$. The transverse momentum $p_{x}$ and $p_{y}$ all range from $-3$ to $3$ GeV. We see that the T-vorticity has the sign $(+,-,+,-)$ from the first to fourth quadrants consistent with the data. The kinematic, thermal and NR vorticity have the sign $(-,+,-,+)$ opposite in comparison with the data. Note that there is no contribution from the space-time gradient of the temperature in the kinematic vorticity, the temperature gradient part in the T-vorticity has the opposite sign to that in the thermal vorticity. In Fig. 2 we see that the magnitude of $\mathcal{P}_{z}^{(K)}(p)$ is smaller than that of $\mathcal{P}_{z}^{(\mathrm{th})}(p)$ and $\mathcal{P}_{z}^{(T)}(p)$, indicating the dominance of the temperature gradient parts in the thermal vorticity and T-vorticity.

The longitudinal polarizations from four types of vorticities as functions of azimuthal angles in transverse momenta are shown in Fig. 3. The azimuthal angle relative to the reaction plane (zx-plane) is defined as $\tan\phi_{p}=p_{y}/p_{x}$. The azimuthal angle distribution of the polarization $\overrightarrow{\mathcal{P}}(\phi_{p})$ is obtained by taking an average over $p_{T}=\sqrt{p_{x}^{2}+p_{y}^{2}}$ for $\overrightarrow{\mathcal{P}}(p)$

[TABLE]

where $\Delta p_{T}=p_{T}^{\mathrm{max}}-p_{T}^{\mathrm{min}}$ denotes the range of the transverse momentum. In Fig. 3 we see that $\mathcal{P}_{z}^{(T)}(\phi_{p})\sim\sin(2\phi_{p})$ which is consistent with the data, while all $\mathcal{P}_{z}^{(i)}(\phi_{p})\sim-\sin(2\phi_{p})$ with $i=K,\mathrm{th},\mathrm{NR}$ which have the wrong sign in comparison with the data. The magnitude of $\mathcal{P}_{z}^{(\mathrm{th})}(\phi_{p})$ is the largest since it is the sum of the kinematic vorticity contribution and the temperature gradient contribution which have the same sign. But in $\mathcal{P}_{z}^{(T)}(\phi_{p})$ the kinematic vorticity and temperature gradient contribution have the opposite sign and the latter is dominant over the former. This is the reason that $\omega_{\mu\nu}^{(T)}(T)$ and $\omega_{\mu\nu}^{(\mathrm{th})}(T)$ have the opposite sign as shown in Eqs. (8,13).

Since there is no initial OAM in the optical Glauber initial condition, the polarizations in the $y$ direction $\mathcal{P}_{y}(p)$ are vanishing for all four types of vorticities.

IV.2 Results with AMPT initial condition

In this subsection, we present the results for the AMPT initial condition which encodes the initial OAM of two nuclei in $-y$ direction.

The results of $\mathcal{P}_{z}(p)$ and $\mathcal{P}_{y}(p)$ for four types of vorticities are shown in Fig. 4 and Fig. 5 respectively. We see in Fig. 4 that the signs of $\mathcal{P}_{z}(p)$ with the AMPT initial condition are the same as those with the Glauber initial condition but the magnitudes of $\mathcal{P}_{z}(p)$ with the AMPT initial condition are smaller than those with the Glauber initial condition except $\mathcal{P}_{z}(p)$ for the NR vorticity that has almost the same magnitude with both initial conditions.

We can take an average over $p_{T}$ for $\mathcal{P}_{z}(p)$ in a transverse momentum range to obtain $\mathcal{P}_{z}(\phi_{p})$. The results for $\mathcal{P}_{z}(\phi_{p})$ are shown in Fig. 6 for two transverse momentum ranges. We see that the magnitudes of $\mathcal{P}_{z}(\phi_{p})$ become smaller in the $p_{T}$ range with smaller transverse momenta. For the range $p_{T}\in[0,1.2]$ GeV, the magnitude of $\mathcal{P}_{z}(\phi_{p})$ matches the data. But if we choose $p_{T}\in[0,3]$ GeV, the magnitude of $\mathcal{P}_{z}(\phi_{p})$ is one order of magnitude larger than the data.

In contrast to the vanishing $\mathcal{P}_{y}(p)$ with the Glauber initial condition, we obtain finite values of $\mathcal{P}_{y}(p)$ in the AMPT initial condition as shown in Fig. 5. The results for $\mathcal{P}_{y}(\phi_{p})$ are displayed in Fig. 7. All four types of vorticities give the correct sign of the initial OAM in $-y$ direction. Note that only $\mathcal{P}_{y}^{(T)}(\phi_{p})$ for the T-vorticity gives the falling trend in $\phi_{p}$ consistent with the data. Although we have the correct trend in $\phi_{p}$ in $\mathcal{P}_{y}^{(T)}(\phi_{p})$, our results fall slower than the data as $\phi_{p}$ increases. Our results for $\mathcal{P}_{y}^{(T)}(\phi_{p})$ match the data at $\phi_{p}=0$, but at $\phi_{p}=\pi/2$ our results are $\mathcal{P}_{y}^{(T)}(\phi_{p})\approx 0.25$ while the data approach zero.

We also calculated $\mathcal{P}_{y}(Y)$ in $-y$ direction as functions of the rapidity $Y$, the results are shown in Fig. 8. We see that $\mathcal{P}_{y}(Y)$ is an even function of $Y$ and increases slowly with $|Y|$. The values of $\mathcal{P}_{y}(Y)$ are very close for the kinematic and NR-vorticity, and $\mathcal{P}_{y}(Y)$ for the T-vorticity is the largest while that for thermal vorticity is the smallest.

IV.3 With different average method in momentum

We can also choose a different method of the average over transverse momenta and rapidity to replace Eqs. (23,25). From Eq. (17) we can take an average of the denominator and numerator separately to obtain the $i$-th component of the polarization vector,

[TABLE]

Note that we have introduced an additional $p_{T}$ factor into the $p_{T}$ integrals in both the denominator and numerator since it corresponds to the Lorentz invariant integral $d^{3}p/E_{p}$. The numerical results for $\mathcal{P}_{z}(\phi_{p})$ are presented in Fig. 9. We see that with the same cutoffs for $p_{T}$, the results for $\mathcal{P}_{z}(\phi_{p})$ from Eq. (26) are a little larger than from Eqs. (23,25). The same behavior also occurs in the results for $\mathcal{P}_{y}(\phi_{p})$ with two different average methods.

V Discussions

We make some remarks about the results. We have checked the first two conditions in (15) and found that they are not fulfilled, so the use of the thermal vorticity as the spin chemical potential is not justified in the hydro-simulation.

For $\mathcal{P}_{y}(\phi_{p})$, we find that only the T-vorticity gives the right trend in $\phi_{p}$ comparing to the data, although it decreases slower than the data. Except the trend in $\phi_{p}$, all vorticities can give the global polarization consistent with the data. The reason why the T-vorticity can give the right trend in $\phi_{p}$ may be understood as follows. The T-vorticity is conserved so that the T-vorticity flux is frozen in the fluid and move with the fluid cell. In this sense, we can regard the T-vorticity flux as a kind of conserved charge. At the early stage of a non-central collision, the T-vorticity in the out-of-plane direction may be induced by the global OAM, then as the pressure gradient is stronger in the in-plane direction than the out-of-plane direction, the T-vorticity will have a positive elliptic flow which results in the unique $\phi_{p}$ dependence as shown in Fig. 7. This suggests that if the spin is (quasi-)conserved, after polarized in the early stage by the OAM, the pressure gradient would lead to a similar $\phi_{p}$ dependence as that for the T-vorticity. This may be verified by the simulation using spin hydrodynamics Florkowski et al. (2018c); Hattori et al. (2019b).

We see very different and even opposite behaviors of $\mathcal{P}_{z}(\phi_{p})$ from different vorticities. This might be related to the fact that $\mathcal{P}_{z}(\phi_{p})$ is one order magnitude smaller than $\mathcal{P}_{y}(\phi_{p})$ since there is no initial OAM in the $z$ direction. Also $\mathcal{P}_{z}(\phi_{p})$ is almost independent of $\mathcal{P}_{y}(\phi_{p})$. This can be seen from the observation that the results of $\mathcal{P}_{z}(\phi_{p})$ from all types of vorticities in the Glauber initial condition (without initial OAM) have the same behaviors as in the AMPT initial condition (with initial OAM). In the optical Glauber initial condition, we found that $\mathcal{P}_{y}(\phi_{p})$ from all types of vorticities are vanishing since there is no orbital angular momentum encoded in the initial state.

Only the T-vorticity in our simulation can describe the data of $\mathcal{P}_{z}(\phi_{p})$ which is the main finding of the paper. The temperature part $\omega_{\mu\nu}^{(T)}(T)$ in the T-vorticity (8) plays an essential role in producing the right sign of $\mathcal{P}_{z}(\phi_{p})$: the sign of $\omega_{\mu\nu}^{(T)}(T)$ is different from $\omega_{\mu\nu}^{(K)}$ but with larger magnitude, so the T-vorticity takes the sign of $\omega_{\mu\nu}^{(T)}(T)$. It is just the opposite way for the thermal vorticity (13) to make its sign: the temperature part $\omega_{\mu\nu}^{(\mathrm{th})}(T)$ has the same sign as $\omega_{\mu\nu}^{(K)}$.

The implication of the T-vorticity by the data may possibly indicate: (1) The time behavior of the temperature at the freeze-out is essential for the T-vorticity to reproduce the correct sign of $\mathcal{P}_{z}(\phi_{p})$. (2) The T-vorticity might be coupled with the spin in a similar way that a magnetic moment is coupled to a magnetic field. Considering an ideal fluid without a conserved charge density (such as the baryon number density) which is the case in the current hydro-simulation for high energy heavy ion collisions, $Tu^{\mu}$ can be regarded as a vector potential and the T-vorticity tensor is then the corresponding field strength tensor, so the conservation of T-vorticity flux is similar to the conservation of the magnetic flux in an ideally conducting fluid, see Eqs. (11,12). However, such a picture is not yet rigorously established and it is also unclear how the roles of T-vorticity and thermal vorticity change when the system approaches global equilibrium. Nevertheless, for collisions at lower energies in which the baryon number density is finite, the conservation of the T-vorticity flux does not hold anymore Gao et al. (2014). Thus, the behavior of $\mathcal{P}_{z}(\phi_{p})$ in low energy collisions might provide a test of this point of view. (3) The assumption that the spin chemical potential can be constructed using $T$ and $u^{\mu}$ might not be correct, so the fact that the T-vorticity can qualitatively reproduce the experimental data for $\mathcal{P}_{z}(\phi_{p})$ and $\mathcal{P}_{y}(\phi_{p})$ is just accidental. This may be tested by using the spin hydrodynamics which is, however, beyond the scope of this work and we leave it for future. (4) It is also possible that it is a coincidence from the main assumption that the spin vector is given by the T-vorticity in the same way as the thermal vorticity. The true relationship between the spin vector on the freeze-out hyper-surface and all these vorticities is unclear and has to be figured out.

All our results depend on a set of parameters and assumptions. For $\mathcal{P}_{z}(\phi_{p})$ and $\mathcal{P}_{y}(\phi_{p})$, one of the most sensitive parameter is the cutoffs in $p_{T}$ in Eq. (25). For example, as shown in Fig. 6, if we choose the range $p_{T}\in[0,1.2]$ GeV, the theoretical results match the data of $\mathcal{P}_{z}(\phi_{p})$. But if we choose a larger range $p_{T}\in[0,3]$ GeV, our theoretical results are much larger the data of $\mathcal{P}_{z}(\phi_{p})$. The aim of this paper is a qualitative study instead of a quantitative one. We will carry out a detailed and quantitative study of the effects in the future.

VI Summary

There is a disagreement between theoretical model calculations and recent experimental data about the azimuthal angle dependence of both the longitudinal and transverse polarization of hyperons. These theoretical models are mainly based on the hydrodynamic or kinetic descriptions of the fluid vorticity and express the spin polarization in terms of the thermal vorticity. However, away from global equilibrium, the linear relationship between the spin polarization and thermal vorticity may not be valid (higher order contribution might be relevant). Instead, the spin polarization (or equivalently the spin chemical potential) itself should be regarded as a dynamical variable. Recently there have been attempts in formulating the theory of relativistic hydrodynamics with the spin chemical potential as a (quasi-)hydrodynamic variable, but so far there has been no reliable numerical implementation of the spin hydrodynamics in the market yet.

In this paper, we assume that the spin vector is determined from the spin chemical potential $\Omega_{\mu\nu}$ in the same way as from the thermal vorticity when the thermal vorticity is small, see Eq. (16) and (17). We also assume that the spin chemical potential $\Omega_{\mu\nu}$ is still determined by the fluid velocity and temperature, which means that $\Omega_{\mu\nu}$ can be regarded as being proportional to a type of vorticity. In relativistic hydrodynamics there are various types of vorticities such as the kinematic, temperature and thermal vorticity. There is also a relativistic extension of the non-relativistic vorticity. We thus explore the possibility that the spin chemical potential is proportional to these four vorticities and the spin vector is given by Eq. (17).

We use CLVisc, a (3+1)D viscous hydrodynamic model, to compute the vorticity field. We choose two different initial conditions for the hydro-simulation: the optical Glauber one without initial orbital angular momentum and the AMPT one with an initial orbital angular momentum. We calculated $\mathcal{P}_{z}(\phi_{p})$ and $\mathcal{P}_{y}(\phi_{p})$ as functions of $\phi_{p}$, the azimuthal angle in transverse momentum, for four types of vorticities: the kinematic, temperature, thermal and relativistic extension of the non-relativistic vorticity. Our results show: (1) All types of vorticities have the correct sign of $\mathcal{P}_{y}$ for the AMPT initial condition. With the optical Glauber initial condition, they all give vanishing results for $\mathcal{P}_{y}(\phi_{p})$ since there is no orbital angular momentum encoded in the initial state. For $\mathcal{P}_{y}(\phi_{p})$ with the AMPT initial condition, only the temperature vorticity has the same trend as the data, although its magnitude does not agree with the data. (2) For the azimuthal angle distribution in the longitudinal polarization, $\mathcal{P}_{z}(\phi_{p})$, only the temperature vorticity reproduces the sign of the oscillation in the azimuthal angle in data. Other three types of vorticities have a sign difference from the data. (3) The oscillation behavior of $\mathcal{P}_{z}(\phi_{p})$ (not the magnitude) is insensitive to the initial conditions with or without the orbital angular momentum encoded.

Acknowledgements.

The authors thanks F. Becattini, X.L. Sheng and X.L. Xia for insightful discussions. HZW and QW are supported in part by the National Natural Science Foundation of China (NSFC) under Grant No. 11535012 and No. 11890713, and the Key Research Program of the Chinese Academy of Sciences under the Grant No. XDPB09. XGH is supported by NSFC under Grants No. 11535012 and No. 11675041.