# Revisit local spin polarization beyond global equilibrium
in relativistic heavy ion collisions

Cong Yi<EMAIL_ADDRESS>Department of Modern Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China  Shi Pu
<EMAIL_ADDRESS>Department of Modern Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China  Di-Lun Yang
<EMAIL_ADDRESS>Institute of Physics, Academia Sinica, Taipei,
11529, Taiwan

###### Abstract

We have studied local spin polarization in the relativistic hydrodynamic
model. Generalizing the Wigner functions previously obtained from chiral
kinetic theory in Ref. (Hidaka:2017auj, ) to the massive case, we present the
possible contributions up to the order of $\hbar$ from thermal vorticity,
shear viscous tensor, other terms associated with the temperature and
chemical-potential gradients, and electromagnetic fields to the local spin
polarization. We then implement the (3+1) dimensional viscous hydrodynamic
model to study the spin polarizations from these sources with a small chemical
potential and ignorance of electromagnetic fields by adopting an equation of
state different from those in other recent studies. Although the shear
correction alone upon local polarization results in the sign and azimuthal-
angle dependence more consistent with experimental observations, as also
discovered in other recent studies, it is mostly suppressed by the
contributions from thermal vorticity and other terms that yield an opposite
trend. It is found that the total local spin polarization could be very
sensitive to the equation of states, the ratio of shear viscosity over entropy
density, and freezeout temperature.

##  I Introduction

In non-central heavy-ion collisions, large orbital angular momenta (OAM) can
be generated and transferred to the quark gluon plasma (QGP) in the form of
vorticity fields. Proposed by Liang and Wang (ZTL_XNW_2005PRL, ;
ZTL_XNW_2005PLB, ) in early pioneer works, the global OAM could trigger the
spin polarization of interacting partons through spin-orbit coupling. The
magnitude of vorticity can be accordingly extracted through the spin
polarization of hadrons created from the experiments. Later on, relativistic
fermions with spin in thermal equilibrium have been systematically studied in
the statistical model and a more precise relation to the possible experimental
observation has been established (Becattini:2007nd, ; Becattini:2007sr, ;
Becattini:2013fla, ). On the other hand, the properties and dynamical
evolution of vorticity are further analyzed via numerical simulations in AMPT
and HIJING (Jiang:2016woz, ; Deng_2016PRC, ; LiHui_prc2017_lpwx, ). Assuming
the global equilibrium condition and employing the modified Cooper-Frye
formula for spin polarization Becattini:2013fla ; Fang_2016PRC_polar , the
spin polarization of $\Lambda$ hyperons in heavy ion collisions has been
estimated based on UrQMD (WeiDexian_prc2019_wdh, ) and hydrodynamic models
(Csernai:2013bqa, ; Becattini:2013vja, ; Becattini:2015ska, ; Pang:2016igs, ;
Wu:2020yiz, ). See also Refs. (Betz:2007kg, ; Ivanov_prc2017_is, ;
Ivanov_prc2018_is, ; Ivanov_prc2019_its, ) for other related studies. The
theoretic predictions and follow-up studies (Karpenko_epjc2017_kb, ;
XieYilong_prc2017_xwc, ; LiHui_prc2017_lpwx, ; Sun:2017xhx, ;
WeiDexian_prc2019_wdh, ; Shi_plb2019_sll, ) remarkably agree with the later
measurement of the global polarization for $\Lambda$ and $\bar{\Lambda}$
hyperons by STAR collaboration (STAR:2017ckg, ). It is also shown an average
angular velocity or vorticity of QGP is as large as $\omega\sim 10^{22}s^{-1}$
(STAR:2017ckg, ), which reveals that the QGP could be the most vortical fluid
so far.

Despite the agreement between theory and experiment in global polarization,
STAR collaboration has also measured the azimuthal angle dependence of the
spin polarization along the beam and the out-plane directions in Au+Au
collisions at $200$ GeV (Adam:2019srw, ; Niida:2018hfw, ), known as the
longitudinal and transverse local spin polarization, respectively. Most of
theoretical simulations, e.g. relativistic hydrodynamics
(Becattini_prl2018_bk, ; Fu:2020oxj, ) and transport models
(XieYilong_prc2017_xwc, ; Xia:2018tes, ; WeiDexian_prc2019_wdh, ) failed to
describe the measurements of local polarization with a few exception such as
the numerical simulation from the kinetic theory of massless fermions in Ref.
(Liu:2019krs, ) and results from some phenomenological models in Ref.
(Voloshin_2018epjWeb, ; Wu:2019eyi, ; Wu:2020yiz, ). In general, these
theoretical estimations have found the local polarization with an opposite
sign against the experimental result. Also, the feed-down effects are found to
be negligible for recoiling the tension (Xia:2019fjf, ;
Becattini_epjc2019_bcs, ). This disagreement between theory and the experiment
for local spin polarization is dubbed as the “sign” problem.

Since most of the previous simulations rely on the assumption that spin
polarization for $\Lambda$ hyperons at the freezeout hypersurface is mainly
induced by the thermal vorticity in light of the statistical model
(Becattini:2013fla, ) and Wigner-function approach (Fang:2016uds, ) in global
equilibrium, it is generally believed that the non-equilibrium effects beyond
the assumption for global equilibrium is essential to delineate the local spin
polarization. Consequently, many theoretical efforts have been inputted to
study dynamical spin polarization. One microscopic theory for tracking the
dynamical evolution of spin transport for relativistic fermion is the quantum
kinetic theory (QKT). The QKT was developed for the massless fermions
(Stephanov:2012ki, ; Son:2012zy, ; Chen:2012ca, ; Manuel:2013zaa, ;
Manuel:2014dza, ; Chen:2014cla, ; Chen:2015gta, ; Hidaka:2016yjf, ;
Hidaka:2017auj, ; Mueller:2017lzw, ; Hidaka:2018mel, ; Huang:2018wdl, ;
Gao:2018wmr, ; Liu:2018xip, ; Lin:2019ytz, ; Lin:2019fqo, ; Yamamoto:2020zrs,
) at the very beginning, known as the chiral kinetic theory (CKT). To describe
the spin dynamics in particular for strange (s) quarks or $\Lambda$ hyperons,
QKT has been extended to the case for massive fermions (Gao:2019znl, ;
Weickgenannt:2019dks, ; Hattori:2019ahi, ; Wang:2019moi, ; Yang:2020hri, ;
Weickgenannt:2020aaf, ; Weickgenannt:2020sit, ; Li:2019qkf, ; Liu:2020flb, ;
Wang:2020pej, ; Weickgenannt:2021cuo, ; Sheng:2021kfc, ; Wang:2021qnt, ). In
Ref. (Zhang:2019xya, ), another microscopic model for spin polarization
through particle collisions is proposed. As for the macroscopic descriptions,
one may need to add the spin effects to the relativistic hydrodynamics, i.e.
relativistic spin hydrodynamics (Florkowski_prc2018_ffjs, ;
Florkowski:2018myy, ; Yang:2018lew, ; Becattini_plb2019_bfs, ;
Florkowski:2018fap, ; Hattori:2019lfp, ; Bhadury:2020puc, ; Shi:2020htn, ;
Montenegro:2017lvf, ; Montenegro:2017rbu, ; Li:2020eon, ). It has been derived
from many approaches, e.g. the entropy principle (Hattori:2019lfp, ;
Fukushima:2020qta, ; Fukushima:2020ucl, ), the Lagrangian effective theory
(Montenegro:2017lvf, ; Montenegro:2017rbu, ), kinetic approaches
(Florkowski_prc2018_ffjs, ; Florkowski:2018myy, ; Becattini_plb2019_bfs, ;
Florkowski:2018fap, ; Bhadury:2020puc, ; Shi:2020htn, ), and general
discussion from field theory (Gallegos:2021bzp, ). Also, see recent reviews
(Wang:2017jpl, ; Becattini:2020ngo, ; Becattini:2020sww, ; Gao:2020vbh, ;
Liu:2020ymh, ) and the references therein.

Nevertheless, to simulate the spin polarization of relativistic fermions far
from equilibrium is technically difficult and computationally expensive. It is
instead natural to explore the spin polarization near local equilibrium. From
the detailed balance of massless fermions with 2-to-2 scattering in CKT, it
has been shown that the Wigner function pertinent to spin polarization
contains several corrections besides the thermal vorticity in local
equilibrium as derived by some of the authors of this paper Hidaka:2017auj  in
2017. Recently, the authors in Refs. (Liu:2020dxg, ; Liu:2021uhn, ) have found
similar contributions for massive fermions in local equilibrium, which have
smooth connection to part of the massless result. Based on their findings with
peculiarly the shear correction, the hydrodynamic simulations in Ref.
(Fu:2021pok, ) show qualitative agreement with experimental data by computing
the polarization of $s$ quarks in $\Lambda$ hyperons. Simultaneously, the
authors in Ref. (Becattini:2021iol, ) have added the contribution from shear
viscous tensor to the local spin polarization near isothermal equilibrium, in
which the numerical results also agree with the experimental data
qualitatively.

A natural question then arises: whether the numerical finding above with the
shear corrections is sensitive to the parameters chosen in the hydrodynamics
simulations? If the answer is positive, it implies that the correct ”sign” for
local polarization may not be solely attributed to the shear correction in
local equilibrium and further non-equilibrium corrections depending on
interaction should be considered. Therefore, we follow the early work
(Hidaka:2017auj, ) done by some of the authors in the present paper and list
all possible local-equilibrium corrections to the local spin polarization up
to the order of $\hbar$. We then implement $(3+1)$-dimensional viscous
hydrodynamics to investigate the polarization induced by these effects. In
order to examine the dependence of the numerical parameters in simulations, we
use a different equation of states (EoS) as opposed to the one adopted in Ref.
(Fu:2021pok, ) and discuss the dependence of freezeout temperature and the
ratio of shear viscosity to entropy density.

The structure of the paper is as follows. In Sec. II, we review the main
results in our early work (Hidaka:2017auj, ) and present all possible
corrections in local equilibrium to polarization up to the order of $\hbar$
explicitly. After that, we implement the viscous hydrodynamical simulations
with AMPT initial condition to study the polarization and discuss the
dependence of freezeout temperature and the ratio of shear viscosity to
entropy density in Sec. III. We summarize in Sec. IV.

Through out this paper, we use the Minkowski metric
$g_{\mu\nu}=\text{diag}\\{+,-,-,-\\}$ and define the Levi-Civita tensor
$\epsilon^{\mu\nu\alpha\beta}$ with the convention
$\epsilon^{0123}=-\epsilon_{0123}=1$. We also introduce the notations
$A_{(\rho}B_{\sigma)}\equiv(A_{\rho}B_{\sigma}+A_{\sigma}B_{\rho})/2$ and
$A_{[\rho}B_{\sigma]}\equiv(A_{\rho}B_{\sigma}-A_{\sigma}B_{\rho})/2$.

##  II Theoretical analysis from quantum kinetic theories

We are interested in the polarization (pseudo) vector characterized by the
axial-charge current density in phase space,

$\mathcal{J}_{5}^{\mu}(p,X)\equiv 2\int_{p\cdot
n}[\mathcal{J}^{\mu}_{+}(p,X)-\mathcal{J}^{\mu}_{-}(p,X)],$ (1)

where $\int_{p\cdot n}\equiv\int dp\cdot n\theta(p\cdot n)/(2\pi)$ with
$\theta(p\cdot n)$ an unit-step function 111For the computation of spin
polarization, we usually apply the on-shell Wigner functions. Here we thus
integrate over the energy defined as $p\cdot n$, where $n^{\mu}$ may be chosen
as the fluid four velocity in thermal equilibrium. Also, we further introduce
a unit-step function to omit the contribution from anti-fermions.. Here
$\mathcal{J}^{\mu}_{+}(p,X)$ and $\mathcal{J}^{\mu}_{-}(p,X)$ denote the
Wigner functions for right and left-handed fermions, respectively. Given
$\mathcal{J}_{5}^{\mu}(p,X)$, we can calculate the spin polarization of
$\Lambda$ hyperons via the modified Cooper-Frye formula Becattini:2013fla ,

$\mathcal{S}^{\mu}({\bf p})=\frac{\int d\Sigma\cdot
p\mathcal{J}_{5}^{\mu}(p,X)}{2m_{\Lambda}\int d\Sigma\cdot\mathcal{N}(p,X)},$
(2)

where $\mathcal{N}^{\mu}(p,X)\equiv 2\int_{p\cdot
n}[\mathcal{J}^{\mu}_{+}(p,X)+\mathcal{J}^{\mu}_{-}(p,X)]$ is the number
density in phase space and $m_{\Lambda}$ is the mass of $\Lambda$ and
$\Sigma_{\mu}$ is the normal vector of the freezeout surface.

As derived in the early work (Hidaka:2017auj, ), the Wigner functions for
right or left-handed fermions near local equilibrium are given by 222In
general, there exist off-equilibrium corrections pertinent to collisions,
which are however higher-order in gradient expansion starting from
$\mathcal{O}(\partial)$ and $\mathcal{O}(\partial^{2})$ for parity even and
odd terms, respectively. These corrections are neglected here.

$\displaystyle\mathcal{J}_{\lambda}^{\mu}(p,X)$ $\displaystyle=$
$\displaystyle 2\pi\textrm{sign}(u\cdot
p)\left\\{p^{\mu}+\lambda\frac{\hbar}{2}\delta(p^{2})\left[u^{\mu}(p\cdot\omega)-\omega^{\mu}(u\cdot
p)\right.\right.$ (3)
$\displaystyle\left.\left.-2S_{(u)}^{\mu\nu}\tilde{E}_{\nu}\right]\partial_{u\cdot
p}+\lambda\frac{\hbar}{4}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}\partial_{\nu}^{p}\delta(p^{2})\right\\}f_{\lambda}^{(0)},$

where $\lambda=\pm$ for right and left handed fermions, $u^{\mu}$ the fluid
four velocity,

$\displaystyle S_{(u)}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\epsilon^{\mu\nu\alpha\beta}p_{\alpha}u_{\beta}/(2u\cdot p),$
$\displaystyle\tilde{E}_{\nu}$ $\displaystyle=$ $\displaystyle
E_{\nu}+T\partial_{\nu}\frac{\mu_{\lambda}}{T}+\frac{(u\cdot
p)}{T}\partial_{\nu}T-p^{\sigma}[\partial_{<\sigma}u_{\nu>}+\frac{1}{3}\Delta_{\sigma\nu}(\partial\cdot
u)+u_{\nu}Du_{\sigma}].$ (4)

and

$f_{\lambda}^{(0)}=1/(e^{(u\cdot p-\mu_{\lambda})/T}+1),$ (5)

is the distribution function with $T$ local temperature, and $\mu_{\pm}$
chemical potentials for right or left-handed fermions, respectively. In
general, $f_{\lambda}$ should incorporate non-equilibrium corrections
depending on interaction to satisfy the kinetic equation near local
equilibrium, while the corrections are expected to be higher orders in the
gradient expansion and omitted here for simplicity Hidaka:2017auj ;
Hidaka:2018ekt . Here, electromagnetic fields are defined in the fluid-rest
frame,

$E_{\mu}\equiv u^{\nu}F_{\mu\nu},\quad
B^{\mu}\equiv\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}u_{\nu}F_{\alpha\beta}.$
(6)

We also decompose the derivative of $u_{\nu}$ as

$\displaystyle\partial_{\mu}u_{\nu}$ $\displaystyle=$
$\displaystyle\partial_{<\mu}u_{\nu>}+u_{\mu}Du_{\nu}+\frac{1}{3}\Delta_{\mu\nu}(\partial\cdot
u)+\omega_{\mu\nu},$ (7)

where

$\displaystyle\Delta^{\mu\nu}$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}-u^{\mu}u^{\nu},$ (8)

is the projector,

$D\equiv u\cdot\partial,$ (9)

$A^{<\mu\nu>}$ is the traceless part of an arbitrary tensor $A^{\mu\nu}$,

$A^{<\mu\nu>}\equiv\frac{1}{2}[\Delta^{\mu\alpha}\Delta^{\nu\beta}+\Delta^{\nu\beta}\Delta^{\mu\alpha}]A_{\alpha\beta}-\frac{1}{3}\Delta^{\mu\nu}\Delta^{\alpha\beta}A_{\alpha\beta},$
(10)

and $\omega^{\mu\nu}$ is the vorticity tensor

$\displaystyle\omega_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\epsilon_{\alpha\beta\mu\nu}u^{\mu}\omega^{\nu}+\frac{1}{2}(u_{\alpha}Du_{\beta}-u_{\beta}Du_{\alpha}),$
(11)

with vorticity defined as,

$\omega^{\mu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}u_{\nu}\partial_{\alpha}u_{\beta}.$
(12)

Using the following relation,

$\displaystyle u^{\mu}(p\cdot\omega)-\omega^{\mu}(u\cdot p)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}p_{\nu}\partial_{\alpha}u_{\beta}+\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}p_{\nu}u_{\alpha}Du_{\beta},$
(13)

we find

$\displaystyle u^{\mu}(p\cdot\omega)-\omega^{\mu}(u\cdot
p)-2S_{(u)}^{\mu\nu}\tilde{E}_{\nu}$ (14) $\displaystyle=$
$\displaystyle-\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}p_{\nu}T\partial_{\alpha}\frac{u_{\beta}}{T}+\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}p_{\nu}u_{\alpha}Du_{\beta}-\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}p_{\alpha}u_{\beta}\frac{1}{T}\partial_{\nu}T$
$\displaystyle-\frac{1}{(u\cdot
p)}\epsilon^{\mu\nu\alpha\beta}p_{\alpha}u_{\beta}E_{\nu}-\frac{T}{(u\cdot
p)}\epsilon^{\mu\nu\alpha\beta}p_{\alpha}u_{\beta}\partial_{\nu}\frac{\mu_{\lambda}}{T}+\frac{1}{(u\cdot
p)}\epsilon^{\mu\nu\alpha\beta}p_{\alpha}u_{\beta}p^{\sigma}\partial_{<\sigma}u_{\nu>}.$

In addition, given the equilibrium distribution functions in Eq. (5), we can
rewrite the last term in $\mathcal{J}_{\lambda}^{\mu}$ as

$\displaystyle 2\pi\textrm{sign}(u\cdot
p)\lambda\frac{\hbar}{4}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}\partial_{\nu}^{p}\delta(p^{2})f_{\lambda}^{(0)}\approx-2\pi\textrm{sign}(u\cdot
p)\lambda\frac{\hbar\delta(p^{2})}{4}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}u_{\nu}\partial_{p\cdot
u}f_{\lambda}^{(0)},$ (15)

up to $\mathcal{O}(\hbar)$. Note that in the above equation we have dropped
the total derivative term, which is irrelevant to our current study.

For the convenience, we decompose the $\mathcal{J}_{5}^{\mu}$ as,

$\mathcal{J}_{5}^{\mu}=\mathcal{J}_{\textrm{thermal}}^{\mu}+\mathcal{J}_{\textrm{shear}}^{\mu}+\mathcal{J}_{\textrm{accT}}^{\mu}+\mathcal{J}_{\textrm{chemical}}^{\mu}+\mathcal{J}_{\textrm{EB}}^{\mu},$
(16)

where

$\displaystyle\mathcal{J}_{\textrm{thermal}}^{\mu}$ $\displaystyle=$
$\displaystyle
a\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}p_{\nu}\partial_{\alpha}\frac{u_{\beta}}{T},$
$\displaystyle\mathcal{J}_{\textrm{shear}}^{\mu}$ $\displaystyle=$
$\displaystyle-a\frac{1}{(u\cdot
p)T}\epsilon^{\mu\nu\alpha\beta}p_{\alpha}u_{\beta}p^{\sigma}\partial_{<\sigma}u_{\nu>}$
$\displaystyle\mathcal{J}_{\textrm{accT}}^{\mu}$ $\displaystyle=$
$\displaystyle-a\frac{1}{2T}\epsilon^{\mu\nu\alpha\beta}p_{\nu}u_{\alpha}(Du_{\beta}-\frac{1}{T}\partial_{\beta}T).$
$\displaystyle\mathcal{J}_{\textrm{chemical}}^{\mu}$ $\displaystyle=$
$\displaystyle a\frac{1}{(u\cdot
p)}\epsilon^{\mu\nu\alpha\beta}p_{\alpha}u_{\beta}\partial_{\nu}\frac{\mu}{T},$
$\displaystyle\mathcal{J}_{\textrm{EB}}^{\mu}$ $\displaystyle=$ $\displaystyle
a\frac{1}{(u\cdot
p)T}\epsilon^{\mu\nu\alpha\beta}p_{\alpha}u_{\beta}E_{\nu}+a\frac{B^{\mu}}{T},$
(17)

with

$a=4\pi\hbar\textrm{sign}(u\cdot p)\delta(p^{2})f_{V}^{(0)}(1-f_{V}^{(0)}).$
(18)

Note that

$f_{V}^{(0)}=\frac{1}{2}(f_{+}^{(0)}+f_{-}^{(0)}),$ (19)

and we have set the same chemical potential for both left and right fermions
$\mu=\mu_{+}=\mu_{-}$ for simplicity. The subscripts, _thermal, shear, accT,
chemical_ and _EB_ , stand for the terms related to thermal vorticity, shear
viscous tensor, the fluid acceleration minus gradient of temperature
$Du_{\mu}-(\Delta_{\mu\nu}\partial^{\nu}T)/T$, the gradient of $\mu/T$, and
electromagnetic fields, respectively. Except for
$\mathcal{J}_{\textrm{thermal}}^{\mu}$ and part of
$\mathcal{J}_{\textrm{EB}}^{\mu}$ led by magnetic fields, all other terms in
$\mathcal{J}_{5}^{\mu}$ come from the corrections beyond global equilibrium.

Let us take a close look at the $\mathcal{J}_{\textrm{accT}}^{\mu}$, which is
usually neglected due to the following reason. One may utilize hydrodynamic
equations of motion up to the order of $\hbar$ or $\partial^{2}$,

$\displaystyle DT$ $\displaystyle=$
$\displaystyle-\frac{\epsilon+P}{p}\Delta^{\mu\nu}\partial_{\nu}u_{\mu}+\mathcal{O}(\hbar,\partial^{2}),$
$\displaystyle Du_{\mu}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{\mu\nu}\partial^{\nu}P}{\epsilon+P}+\mathcal{O}(\hbar,\partial^{2}),$
$\displaystyle D\bar{\mu}_{{\rm R/L}}$ $\displaystyle=$
$\displaystyle\mathcal{O}(\hbar,\partial^{2}),$ (20)

to replace the temporal derivatives $D$ upon thermodynamic parameters, where
$\epsilon$ and $P$ correspond to the energy density and pressure,
respectively. Here the $\hbar$ corrections in Eq. (20) are irrelevant since
they only contribute to higher-order terms at $\mathcal{O}(\hbar^{2})$ in
Wigner functions except for the off-equilibrium fluctuations led by
collisions.

In the ideal limit, one of the hydrodynamic equations of motion becomes
$Du_{\mu}=(\Delta_{\mu\nu}\partial^{\nu}T)/T$. Therefore, the
$\mathcal{J}_{\textrm{accT}}^{\mu}$ vanishes in the ideal hydrodynamics. In
the realistic hydrodynamic simulations, the dissipative corrections could
further modify the evolution of $Du_{\mu}$. In the viscous hydrodynamics, we
have,

$Du_{\alpha}=\frac{1}{\epsilon+P}\Delta_{\mu\alpha}\partial^{\mu}P-\frac{1}{\epsilon+P}\Delta_{\mu\alpha}\partial_{\nu}\pi^{\mu\nu}+\mathcal{O}(\hbar,\partial^{3})\approx\frac{\Delta_{\mu\alpha}}{T}\big{(}\partial^{\mu}T-s^{-1}\partial_{\nu}\pi^{\mu\nu}\big{)},$
(21)

where
$\pi^{\mu\nu}=2\Delta^{\mu\alpha}\Delta^{\nu\beta}\eta(\partial_{(\alpha}u_{\beta)}-g_{\alpha\beta}\partial_{\rho}u^{\rho}/3)+\mathcal{O}(\partial^{2})$
corresponds to the shear-stress tensor with $\eta$ being the shear viscosity
and $s$ denotes the entropy density. Here we take $\epsilon+P=Ts$ and omit the
correction from bulk viscosity for simplicity. We can further rewrite
$\mathcal{J}_{\textrm{accT}}^{\mu}$ as

$\displaystyle\mathcal{J}_{\textrm{accT}}^{\mu}$ $\displaystyle=$
$\displaystyle
a\frac{1}{2T^{2}s}\epsilon^{\mu\nu\alpha\beta}p_{\nu}u_{\alpha}\partial^{\rho}\pi_{\beta\rho}\approx
a\frac{\eta}{T^{2}s^{2}}\epsilon^{\mu\nu\alpha\beta}p_{\nu}u_{\alpha}\partial^{\rho}(s\hat{\pi}_{\beta\rho}),$
(22)

where $\hat{\pi}^{\mu\nu}=\pi^{\mu\nu}/(2\eta)$ and we have assumed $\eta/s$
is a constant to make the final approximation above. Consequently, we would
like to emphasize here that the contributions from
$\mathcal{J}_{\textrm{accT}}^{\mu}$ to the local spin polarization should
depend on the EoS and parameter $\eta/s$. Note that in the early work
(Karpenko:2018erl, ), the authors have discussed the contribution from fluid
acceleration $Du_{\beta}$ implicitly involved in
$\mathcal{J}_{\textrm{thermal}}^{\mu}$ to local polarization. Additionally,
the magnetic-field contribution in $\mathcal{J}_{\textrm{EB}}^{\mu}$ has been
also discussed in Ref. Fang_2016PRC_polar .

Although we discuss the polarization of massless fermions above, we may extend
our analysis to massive fermions. In the case of massive fermions, we may
generalize the on-shell condition $\delta(p^{2})$ in Eq. (18) to
$\delta(p^{2}-m_{i}^{2})$ with $m_{i}$ being the mass of fermions. Then, we
obtain the $\mathcal{S}^{\mu}$ in Eq. (2) from different sources,

$\displaystyle\mathcal{S}_{\textrm{thermal}}^{\mu}(\mathbf{p})$
$\displaystyle=$ $\displaystyle\frac{\hbar}{8m_{\Lambda}N}\int
d\Sigma^{\sigma}p_{\sigma}f_{V}^{(0)}(1-f_{V}^{(0)})\epsilon^{\mu\nu\alpha\beta}p_{\nu}\partial_{\alpha}\frac{u_{\beta}}{T},$
$\displaystyle\mathcal{S}_{\textrm{shear}}^{\mu}(\mathbf{p})$ $\displaystyle=$
$\displaystyle-\frac{\hbar}{4m_{\Lambda}N}\int d\Sigma\cdot
pf_{V}^{(0)}(1-f_{V}^{(0)})\frac{\epsilon^{\mu\nu\alpha\beta}p_{\alpha}u_{\beta}}{(u\cdot
p)T}\frac{1}{2}\left\\{p^{\sigma}(\partial_{\sigma}u_{\nu}+\partial_{\nu}u_{\sigma})-Du_{\nu}\right\\}$
$\displaystyle\mathcal{S}_{\textrm{accT}}^{\mu}(\mathbf{p})$ $\displaystyle=$
$\displaystyle-\frac{\hbar}{8m_{\Lambda}N}\int d\Sigma\cdot
pf_{V}^{(0)}(1-f_{V}^{(0)})\frac{1}{T}\epsilon^{\mu\nu\alpha\beta}p_{\nu}u_{\alpha}(Du_{\beta}-\frac{1}{T}\partial_{\beta}T),$
$\displaystyle\mathcal{S}_{\textrm{chemical}}^{\mu}(\mathbf{p})$
$\displaystyle=$ $\displaystyle\frac{\hbar}{4m_{\Lambda}N}\int d\Sigma\cdot
pf_{V}^{(0)}(1-f_{V}^{(0)})\frac{1}{(u\cdot
p)}\epsilon^{\mu\nu\alpha\beta}p_{\alpha}u_{\beta}\partial_{\nu}\frac{\mu}{T},$
$\displaystyle\mathcal{S}_{\textrm{EB}}^{\mu}(\mathbf{p})$ $\displaystyle=$
$\displaystyle\frac{\hbar}{4m_{\Lambda}N}\int d\Sigma\cdot
pf_{V}^{(0)}(1-f_{V}^{(0)})\left(\frac{1}{(u\cdot
p)T}\epsilon^{\mu\nu\alpha\beta}p_{\alpha}u_{\beta}E_{\nu}+\frac{B^{\mu}}{T}\right),$
(23)

where $N=\int d\Sigma^{\mu}p_{\mu}f_{V}^{(0)}$ and now
$p^{0}=\sqrt{|\mathbf{p}|^{2}+m_{i}^{2}}$. In this work, we will only evaluate
the spin polarization of $\Lambda$ and neglect $\bar{\Lambda}$.

##  III Results and discussion

###  III.1 Setup for Simulations

In this section, we implement the (3+1) dimensional viscous hydrodynamic
CLVisc (Pang:2012he, ; Pang:2018zzo, ) with AMPT initial conditions
(Lin:2004en, ; Wu:2019eyi, ; Wu:2020yiz, ) to generate the freezeout hyper-
surface and the profile of fluid velocity and temperature at that hyper-
surface. Unless noted otherwise, we choose the EoS “ _s95p-pce_ ”
(Huovinen:2009yb, ) instead of the EoS (Denicol:2018wdp, ) used in Refs.
(Fu:2020oxj, ; Fu:2021pok, ).

Figure 1: The polarization $\mathcal{P}^{z}$ and $\mathcal{P}^{y}$ as a
function of $\phi_{p}$ for $\Lambda$ and $s$ equilibrium scenarios shown in
the left and right panels. We have chosen $\eta/s=1/(4\pi)$ and the freezeout
temperature is $165\,\textrm{MeV}$. Red, green ,and blue curves stand for the
contributions from the thermal vorticity, shear induced polarization, and the
acceleration terms, respectively. The black curve denotes the total
polarization.

Figure 2: The polarization $\mathcal{P}^{z}$ and $\mathcal{P}^{y}$ as a
function of $\phi_{p}$ for $\Lambda$ hypersons and $s$ quarks in the lad frame
and their own rest frames. We have chosen $\eta/s=1/(4\pi)$ and the freezeout
temperature is $165\,\textrm{MeV}$. The red and black curves stand for the
total polarization in the lab and the $\Lambda$ (or $s$ quark) rest frame,
respectively. The green and blue curves denote the shear induced polarization
in the lab and the $\Lambda$ (or the $s$ quark) rest frame, respectively.

We introduce the polarization along the beam direction as $\mathcal{P}^{z}(p)$
and along the out-plane direction as $\mathcal{P}^{y}(p)$, which are obtained
by integrating over the medium rapidity range $[-1,+1]$,

$\displaystyle\mathcal{P}^{z}(p)$ $\displaystyle=$
$\displaystyle\int_{-1}^{+1}dY\mathcal{S}^{z}(p),$
$\displaystyle\mathcal{P}^{y}(p)$ $\displaystyle=$
$\displaystyle\int_{-1}^{+1}dY\mathcal{S}^{y}(p),$ (24)

where $\mathcal{S}^{\mu}$ is given by Eqs. (23). We again use the subscripts,
_thermal, shear, accT, chemical_ , and _EB_ for $\mathcal{P}^{\mu}(p)$ with
$\mu=y,z$ to specify the contributions to the polarization from thermal
vorticity, shear viscous tensor, $Du_{\beta}-\frac{1}{T}\partial_{\beta}T$,
the gradient of $\mu/T$, and electromagnetic fields, respectively.

Since the electromagnetic fields decay rapidity (Bloczynski:2012en, ;
Deng:2012pc, ; Roy:2015coa, ), we can neglect the contributions from
electromagnetic fields $\mathcal{S}_{\textrm{EB}}^{\mu}$ to the polarization
vector $\mathcal{P}_{\textrm{EB}}^{\mu}$. Due to the limitation of the EoS in
“ _s95p-pce_ ”, we cannot get the sufficient information of chemical potential
and its gradient. Therefore, in the current study, we will only consider the
polarization induced by the thermal vorticity
$\mathcal{S}_{\textrm{thermal}}^{i}$, shear viscous tensor
$\mathcal{S}_{\textrm{shear}}^{i}$, and $\mathcal{S}_{\textrm{accT}}^{i}$ and
evaluate

$\mathcal{P}_{\textrm{total}}^{\mu}=\mathcal{P}_{\textrm{thermal}}^{\mu}+\mathcal{P}_{\textrm{shear}}^{\mu}+\mathcal{P}_{\textrm{accT}}^{\mu}.$
(25)

The possible contribution from $\mathcal{S}_{\textrm{chemical}}^{\mu}$ will
also be briefly discussed later.

Figure 3: The same setup and color assignments as Fig. 1 except for changing
the freezeout temperature to $157\,\textrm{MeV}$.

Figure 4: The same setup and color assignments as Fig. 1 except for changing
the shear viscosity to entropy density ratio to $\eta/s=2/(4\pi)$.

Figure 5: The polarization $10^{4}\times\mathcal{P}_{\textrm{chemcial}}^{z}$
and $10^{5}\times\mathcal{P}_{\textrm{chemcial}}^{y}$ as a function of
$\phi_{p}$ for $\Lambda$ and $s$ equilibrium scenarios. We have chosen
$\eta/s=1/(4\pi)$ and the freezeout temperature as $165$ MeV or $157$ MeV. See
color assignments for the results in the plots.

We consider two scenarios in this work. In the first scenario, since the
$\Lambda$ hyperons are produced at the chemical freezeout, we assume that one
can still utilize the macroscopic variables from hydrodynamics to describe the
thermodynamical states of $\Lambda$ hyperons, e.g. the temperature and its
gradient, i.e. we assume that $\Lambda$ hyperons are almost at local
equilibrium. We name it as $\Lambda$ _equilibrium scenario_ for short.

In the second scenario, as proposed in Ref. (Fu:2021pok, ), since the spin of
$s$ quark dominates over the total spin of $\Lambda$ hyperon in the parton
model, one can compute the polarization of $s$ quarks and assume the spin
polarization of the s quark is smoothly passed to the polarization of
$\Lambda$. We name it as the $s$ _equilibrium scenario_. Since the $s$ quarks
are much lighter than $\Lambda$ hyperons, polarization induced by shear
viscous tensor will be greatly enhanced in the $s$ equilibrium scenario as
opposed to the $\Lambda$ equilibrium scenario, where
$\mathcal{S}_{\textrm{shear}}^{\mu}$ is suppressed by $(u\cdot p)\sim
m_{\Lambda}$ in the denominator shown in Eq. (23).

Note the polarization vectors in Eq. (24) are shown in the laboratory frame.
In order to compare the results with the experimental data, we eventually
transform them to the rest frame of the $\Lambda$ hyperon for the $\Lambda$
equilibrium scenario or of the $s$ quark in the $\Lambda$ hyperon for the $s$
equilibrium scenario. In principle, we also need to consider the evolution of
$\Lambda$ hyperons before kinetic freezeout, whereas we neglect the evolution
of $\Lambda$ hyperons after the chemical freezeout for simplicity.

We would like to emphasize that we always choose the factor $m$ in the
denominator of the right-hand side of Eq. (2) as $m_{\Lambda}$ in both
$\Lambda$ and $s$ equilibrium scenarios. The mass factor in Ref. (Fu:2021pok,
) is instead chosen as $m_{s}$ in the $s$ equilibrium scenario, which will
make an enhancement to the overall magnitude of polarization in the $s$
equilibrium scenario. For numerical simulations, we choose
$m_{\Lambda}=1.116\,\textrm{GeV}$ for the mass of $\Lambda$ hyperons and
$m_{s}=0.3\,\textrm{GeV}$ for the mass of constituent $s$ quarks.

###  III.2 Numerical Results and Discussions

Here we briefly summarize the results presented in each figure. In Fig. 1, we
show the local spin polarization coming from different sources in the lab
frame. In Fig. 2, we compare the results in the particles’ rest frames to
those in Fig. 1. In Fig. 3, we present the results with reduced freezeout
temperature. In Fig. 4, the results with larger $\eta/s$ but the same
freezeout temperature as Fig. 1 are presented. Finally, we show the spin
polarization led by a small and constant chemical potential from
$\mathcal{P}^{\mu}_{\rm chemical}$ for a heuristic discussion. The detailed
results and discussions are presented below.

In Fig. 1, we plot the local spin polarization in both $\Lambda$ and $s$
equilibrium scenarios with $\eta/s=1/(4\pi)$ and the freezeout temperature
$T=165$ MeV. We observe that the
$\mathcal{\mathcal{S}_{\textrm{shear}}^{\mu}}$ always leads to the “same” sign
contribution, qualitatively consistent with experimental data, to the local
spin polarization in both $z$ and $y$ directions and in the two scenarios.
This observation is in accordance with Ref. (Fu:2021pok, ). Similar to thermal
vorticity, $\mathcal{S}_{\textrm{accT}}^{\mu}$ induced by non-vanishing
$Du_{\mu}-(\Delta_{\mu\nu}\partial^{\nu}T)/T$ leads to the “opposite” sign
contribution to the local polarization. Along the beam direction, the
magnitude of polarization $\mathcal{P}_{\textrm{accT}}^{z}$ is much smaller
than the magnitudes of $\mathcal{P}_{\textrm{thermal }}^{z}$ and
$\mathcal{P}_{\textrm{shear }}^{z}$ in the two scenarios. Along the out-plan
direction, the $\mathcal{P}_{\textrm{accT}}^{y}$ is almost canceled with
$\mathcal{P}_{\textrm{shear}}^{y}$ in $\Lambda$ equilibrium scenario, while
$|\mathcal{P}_{\textrm{accT}}^{y}|<|\mathcal{P}_{\textrm{shear}}^{y}|$ in the
$s$ equilibrium scenario. As a consequence of the competition between
$\mathcal{P}_{\textrm{shear}}^{\mu}$ and
$\mathcal{P}_{\textrm{thermal}}^{\mu}+\mathcal{P}_{\textrm{accT}}^{\mu}$, the
$\mathcal{P}_{\textrm{total}}^{z,y}$ in both $\Lambda$ and $s$ equilibrium
scenarios disagree with the experimental data.

In contrast, the hydrodynamic simulations in Ref. (Fu:2021pok, ) find that the
$\mathcal{P}_{\textrm{total}}^{\mu}$ in $s$ equilibrium scenarios agree with
the experimental data qualitatively. As mentioned previously, we have chosen a
different EoS against the one in Ref. (Fu:2021pok, ).

One possible reason leading to the different results may come from the
following fact. In the EoS of _sp95-pce_ , the speed of sound connecting QGP
phase and PCE hadronic phase is not smooth and may lead to un-physical
vorticity structure near to the freeze-out surface. We notice that the authors
in Ref. Fu:2020oxj  have studied the EoS dependence in the global
polarization.

We have also checked the results by using another EoS _lattice-wb2014_
Borsanyi:2013bia  and the total polarization has a similar “sgin”” as the
experimental data qualitatively. But, $\mathcal{P}^{z,y}_{\textrm{total}}$ as
a function of $\phi_{p}$ are almost flat $|\mathcal{P}^{z}_{\textrm{total}}|$
is quantitatively close to zero due to the highly suppression led by the
factor $m_{\Lambda}$ in the denominator of the right-hand side of Eq. (2).
Therefore, our results show that the spin polarization with local-equilibrium
corrections from hydrodynamic simulations are sensitive to the choice of EoS.

In Fig. 2, we compare the polarization in the laboratory frame and in
particles’ rest frames. It is found the difference of the polarization in the
two frames is mild for $\mathcal{P}^{z}_{\textrm{total}}$. Although the
overall magnitudes and peaks of $\mathcal{P}_{\textrm{total}}^{y}$ are reduced
in the particles’ rest frames, in general, the choices of frames do not change
the polarization qualitatively.

Next, we consider the dependence of freezeout temperature. We set the
freezeout temperature $T=157$ MeV and keep $\eta/s=1/(4\pi)$, and show the
results in Fig. 3. The magnitudes of $\mathcal{P}_{\textrm{accT}}^{\mu}$
increase significantly when freezeout temperature decreases 333In fact, the
magnitude of $\mathcal{P}_{\textrm{thermal}}^{\mu}$ also depends on the
freezeout temperature since $\mathcal{J}_{\textrm{thermal}}^{\mu}$ implicitly
incorporate the term related to $Du^{\mu}$. But this term has been previously
included in the hydrodynamic simulations for
$\mathcal{P}_{\textrm{thermal}}^{\mu}$ in global equilibrium. It is worthwhile
to note the full Wigner functions in local equilibrium, derived in Ref.
Hidaka:2017auj , actually does not contain a term associated with $Du^{\mu}$..
Since now
$|\mathcal{J}_{\textrm{accT}}^{\mu}|\sim\mathcal{O}\big{(}T^{-2}p\partial^{2}\big{)}$
as a crude estimation for the magnitude of $\mathcal{J}_{\textrm{accT}}^{\mu}$
coming from the dissipative correction, it may be expected that
$|\mathcal{P}_{\textrm{thermal}}^{\mu}|$ should increase with the reduced
freezeout temperature.

On the other hand, we observe that the shear induced polarization
$\mathcal{P}_{\textrm{shear}}^{\mu}$ is not sensitive to the freezeout
temperature, while the magnitude of thermal vorticity induced polarization
$\mathcal{P}_{\textrm{thermal}}^{\mu}$ is slightly enhanced. The difference
between $\mathcal{P}_{\textrm{total}}^{\mu}$ in the two scenarios and
experimental data increases due to the growth of the magnitude of
$\mathcal{P}_{\textrm{accT}}^{\mu}$. It turns out that the
$\mathcal{P}_{\textrm{total}}^{\mu}$ is also sensitive to the freezeout
temperature.

Thirdly, we discuss the dependence of $\eta/s$. In Fig. 4, we set the
$\eta/s=2/(4\pi)$ and freezeout temperature $T=165$ MeV. In comparison with
the case of $\eta/s=1/(4\pi)$ in Fig. 1, we find that all contributions to
$\mathcal{P}^{z}$ in the two scenarios have not changed much. On the contrary,
the magnitudes of both $\mathcal{P}_{\textrm{thermal}}^{y}$ and
$\mathcal{P}_{\textrm{accT}}^{y}$ in the two scenarios increase when $\eta/s$
grows. The magnitudes of $\mathcal{P}_{\textrm{shear}}^{y}$ in the two
scenarios become smaller than those in Fig. 1. Eventually, the
$\mathcal{P}_{\textrm{total}}^{y}$ in the two scenarios are still different
with the observation in experiments. It is found that the local polarization
at least for $\mathcal{P}_{\textrm{total}}^{y}$ also depends on $\eta/s$.

At last, we compute the possible contribution from the gradient of chemical
potential over temperature led by $\mathcal{S_{\textrm{chemical}}^{\mu}}$ in
Eq. (23). For simplicity, we assume that the chemical potential is constant
near the freezeout hypersurface and $\nabla\mu/T\simeq\mu\nabla(1/T)$. In Fig.
5, we choose the quark chemical potential $\mu=10$ MeV. Similar to the shear
viscous tensor, the $\mathcal{S_{\textrm{chemical}}^{\mu}}$ leads to the
“same” sign contribution, qualitatively consistent with experimental data, to
the local spin polarization in both $z$ and $y$ directions and in the two
scenarios. However, $\mathcal{P}_{\textrm{chemical}}^{\mu}\propto\mu$ is
highly suppressed by other contributions in
$\mathcal{P}_{\textrm{total}}^{\mu}$. So far, they are almost negligible in
the current study. However, $\mathcal{P_{\textrm{chemical}}^{\mu}}$ might be
important in the low-energy collisions. We will leave it for the future
studies.

As a concluding remark, we have shown that
$\mathcal{P}_{\textrm{total}}^{\mu}$ is generally sensitive to EoS, freezeout
temperature $T$, and the ratio $\eta/s$. We find that both
$\mathcal{S}_{\textrm{shear}}^{\mu}$ and $\mathcal{S}_{\textrm{accT}}^{\mu}$
play important roles to local spin polarization. The behavior of
$\mathcal{P}_{\textrm{total}}^{\mu}$ strongly depends on the predominant term
in the competition between $\mathcal{P}_{\textrm{shear}}^{\mu}$ and
$\mathcal{P}_{\textrm{thermal}}^{\mu}+\mathcal{P}_{\textrm{accT}}^{\mu}$.
Although $\mathcal{S}_{\textrm{shear}}^{\mu}$ may lead to the “same” sign to
the polarization, we still cannot get similar azimuthal angle dependence of
local spin polarization as the experimental data with the EoS we have chosen.

##  IV Conclusion

In this work, we first review the spin polarization (pesudo) vector
$\mathcal{S}^{\mu}({\bf p})$ derived from Wigner functions in Ref.
(Hidaka:2017auj, ). We decompose $\mathcal{S}^{\mu}({\bf p})$ into
$\mathcal{S}_{\textrm{thermal}}^{\mu},\mathcal{S}_{\textrm{shear}}^{\mu},\mathcal{S}_{\textrm{accT}}^{\mu},\mathcal{S}_{\textrm{chemical}}^{\mu}$
and $\mathcal{S}_{\textrm{EB}}^{\mu}$, which are led by thermal vorticity,
shear viscous tensor, fluid acceleration minus the gradient of temperature,
the gradient of chemical potential over temperature, and electromagnetic
fields, respectively. We then implement of (3+1) dimensional viscous
hydrodynamic CLVisc with AMPT initial conditions to obtain the numerical
results for local spin polarization with zero electromagnetic fields and a
vanishing chemical potential and focus on
$\mathcal{P}_{\textrm{thermal}}^{\mu},\mathcal{P}_{\textrm{shear}}^{\mu}$, and
$\mathcal{P}_{\textrm{accT}}^{\mu}$ contributed by
$\mathcal{S}_{\textrm{thermal}}^{\mu},\mathcal{S}_{\textrm{shear}}^{\mu}$, and
$\mathcal{S}_{\textrm{accT}}^{\mu}$. Inspired by Ref. (Fu:2021pok, ), we also
consider $\Lambda$ and $s$ equilibrium scenarios. As opposed to Ref.
(Fu:2021pok, ), we have chosen the EoS “ _s95p-pce_ ” as a different EoS and
investigate the influence of $\eta/s$ upon spin polarization.

Our numerical results show that both $\mathcal{P}_{\textrm{shear}}^{\mu}$ and
$\mathcal{P}_{\textrm{accT}}^{\mu}$ affect
$\mathcal{P}_{\textrm{total}}^{\mu}$ in addition to
$\mathcal{P}_{\textrm{thermal}}^{\mu}$ that has been well studied in the
literature. Although $\mathcal{P}_{\textrm{shear}}^{\mu}$ can lead to the
“same” sign contribution, qualitatively consistent with experimental data, to
the local spin polarization, $\mathcal{P}_{\textrm{accT}}^{\mu}$ also plays a
crucial role and could change the behavior of
$\mathcal{P}_{\textrm{total}}^{\mu}$ especially with lower freezeout
temperature. It turns out that $\mathcal{P}_{\textrm{total}}^{\mu}$ is rather
sensitive to EoS, $\eta/s$ and freezeout temperature $T$. With the adopted EoS
in our study, we do not observe the similar azimuthal angle dependence of
local spin polarization as the experimental data.

We conclude that although the shear induced polarization alone may result in
sizable effects on local spin polarization qualitatively consistent with
experimental observations, it may not be always the dominant effect over other
contributions in local equilibrium.

Although we have also checked the total local spin polarization by using
another EoS and get the similar behavior as in Ref. Fu:2021pok , the total
local spin polarization in s equilibrium scenario could still be highly
suppressed if one chooses the $m_{\Lambda}$ instead of $m_{s}$ in the
denominator of the right-hand side of Eq. (2).

Consequently, the so-called “sign” problem remains an open question. Moreover,
from the theoretical perspective, the Wigner function in local equilibrium
without the inclusion of off-equilibrium corrections pertinent to interaction
is not a self-consistent solution for a kinetic equation. As indirectly
supported by the influence from $\mathcal{P}_{\textrm{accT}}^{\mu}$, even in
the near-equilibrium condition, the second-order gradient terms, such as the
non-equilibrium corrections qualitatively studied in e.g. Refs. Hidaka:2017auj
; Hidaka:2018ekt  from CKT, may potentially give rise to sizable contributions
in numerical simulations. To comprehensively investigate the spin polarization
of fermions near local equilibrium and its direct connection to the “sign”
problem , it will be still imperative to conduct the theoretical and numerical
studies of spin hydrodynamics and the QKT with collisions.

###### Acknowledgements.

We are grateful to thank Yi Yin, Longgang Pang, Shuai Y. F. Liu, Baochi Fu and
Huichao Song for helpful discussions. S.P. is supported by National Nature
Science Foundation of China (NSFC) under Grants No. 12075235.

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