# Analytic solutions of relativistic dissipative spin hydrodynamics with
Bjorken expansion

Dong-Lin Wang<EMAIL_ADDRESS>Department of Modern Physics,
University of Science and Technology of China, Anhui 230026, China  Shuo Fang
<EMAIL_ADDRESS>Department of Modern Physics, University of Science
and Technology of China, Anhui 230026, China  Shi Pu<EMAIL_ADDRESS>Department of Modern Physics, University of Science and Technology of China,
Anhui 230026, China

###### Abstract

We have studied analytically the longitudinally boost-invariant motion of a
relativistic dissipative fluid with spin. We have derived the analytic
solutions of spin density and spin chemical potential as a function of proper
time $\tau$ in the presence of viscous tensor and the second order relaxation
time corrections for spin. Interestingly, analogous to the ordinary particle
number density and chemical potential, we find that the spin density and spin
chemical potential decay as $\sim\tau^{-1}$ and $\sim\tau^{-1/3}$,
respectively. It implies that the initial spin density may not survive at the
freezeout hyper-surface. These solutions can serve both to gain insight on the
dynamics of spin polarization in relativistic heavy-ion collisions and as
testbeds for further numerical codes.

##  I Introduction

In the non-central relativistic heavy ion collisions, huge orbital angular
momenta are generated and polarize the particles in the quark gluon plasma
(QGP) through the spin-orbital couplings (ZTL_XNW_2005PRL, ; ZTL_XNW_2005PLB,
). In 2017, the global polarization of $\Lambda$ and $\bar{\Lambda}$ hyperons
led by the initial huge orbital angular momentum has been measured by STAR
experiments and been well understood by various phenomenological models
(Karpenko_epjc2017_kb, ; XieYilong_prc2017_xwc, ; LiHui_prc2017_lpwx, ;
Sun:2017xhx, ; WeiDexian_prc2019_wdh, ; Shi_plb2019_sll, ; Fu:2020oxj, ;
Ryu:2021lnx, ). The experimental results also indicate that QGP is the most
vortical fluid (STAR:2017ckg, ) so far.

The experimental data in Au+Au collisions at $200$ GeV for the local spin
polarization, which is the azimuthal angle dependent spin polarization of the
$\Lambda$ and $\bar{\Lambda}$ hyperons along the beam and the out-plane
directions (Adam:2019srw, ), disagrees with many phenomenological models, e.g.
relativistic hydrodynamics model (Becattini_prl2018_bk, ; Fu:2020oxj, ) and
transport models (XieYilong_prc2017_xwc, ; Xia:2018tes, ;
WeiDexian_prc2019_wdh, ). Later on, people find that this disagreement cannot
be solved by considering the feed-down effects (Xia:2019fjf, ;
Becattini_epjc2019_bcs, ). Although the kinetic theory of massless fermions in
Ref. (Liu:2019krs, ) and results from a simple phenomenological model in Ref.
(Voloshin_2018epjWeb, ) can show the similar azimuthal angle dependence as
experimental data, it is still a puzzle in the relativistic heavy ion
community.

To solve this puzzle, it requires a deeper understanding of spin effects in
the QGP from both microscopic and macroscopic theories. One microscopic theory
for massive fermions is the quantum kinetic theory (QKT) for massive fermions,
which is a natural extension of the chiral kinetic theory for massless
fermions (Stephanov:2012ki, ; Son:2012zy, ; Chen:2012ca, ; Manuel:2013zaa, ;
Manuel:2014dza, ; Chen:2014cla, ; Chen:2015gta, ; Hidaka:2016yjf, ;
Mueller:2017lzw, ; Hidaka:2018mel, ; Huang:2018wdl, ; Gao:2018wmr, ;
Liu:2018xip, ; Lin:2019ytz, ; Lin:2019fqo, ). Recently, the QKT for massive
fermions with the collisional kernel has been developed (Gao:2019znl, ;
Weickgenannt:2019dks, ; Weickgenannt:2020aaf, ; Weickgenannt:2020sit, ;
Hattori:2019ahi, ; Yang:2020hri, ; Liu:2020flb, ; Weickgenannt:2021cuo, ;
Sheng:2021kfc, ). Also see the early works on the statistical model for the
relativistic particles with spin (Becattini:2007nd, ; Becattini:2007sr, ;
Becattini:2013fla, ) and Ref. (Zhang:2019xya, ) for a microscopic model for
spin polarization through particle collisions.

Very recently, many studies have shown that the polarization can also been
induced by shear viscous tensor, e.g. from early studies of massless fermions
(Hidaka:2017auj, ), the recent studies for massive fermions (Liu:2020dxg, ;
Liu:2021uhn, ) and statistic model (Becattini:2021iol, ). Those studies
(Fu:2021pok, ; Becattini:2021iol, ; Yi:2021ryh, ) show quantitative agreement
with experimental data by adding the shear induced polarization to the
original studies (Becattini:2013fla, ; Fang:2016uds, ).

The macroscopic theory for the spinness particles is the relativistic spin
hydrodynamics, which is a theory with the relativistic hydrodynamics equations
coupled to the conservation equation of total angular momentum. The expression
of spin hydrodynamics has been derived from the entropy principle
(Hattori:2019lfp, ; Fukushima:2020qta, ; Fukushima:2020ucl, ; Li:2020eon, ;
She:2021lhe, ), the Lagrangian effective theory (Montenegro:2017lvf, ;
Montenegro:2017rbu, ), kinetic approaches (Florkowski_prc2018_ffjs, ;
Florkowski:2018myy, ; Becattini_plb2019_bfs, ; Florkowski:2018fap, ;
Bhadury:2020puc, ; Shi:2020qrx, ), and general discussion from field theory
(Gallegos:2021bzp, ). The ideal spin hydrodynamics has been discussed in Ref.
(Florkowski_prc2018_ffjs, ; Florkowski:2018myy, ; Becattini_plb2019_bfs, ;
Florkowski:2018fap, ; Hattori:2019lfp, ; Bhadury:2020puc, ; Shi:2020qrx, ;
Montenegro:2017lvf, ; Montenegro:2017rbu, ; Li:2020eon, ). In the gradient
expansion, the dissipative spin hydrodynamics are derived in canonical
(Hattori:2019lfp, ) and Belinfante form (Fukushima:2020qta, ;
Fukushima:2020ucl, ). Also, see recent reviews (Wang:2017jpl, ;
Becattini:2020ngo, ; Becattini:2020sww, ; Gao:2020vbh, ; Liu:2020ymh, ) and
the reference therein.

Analogous to works on the relativistic magnetohydrodynamics (Pu:2016ayh, ;
Roy:2015kma, ; Pu:2016bxy, ; Pu:2016rdq, ; Siddique:2019gqh, ; Wang:2020qpx,
), it is necessary to derive the analytic solutions for the dissipative spin
dynamics. These analytic solutions provide the power law behavior of the
quantities related to the spin as a function of proper time $\tau$, such as
the spin density and spin chemical potential. On the other hand, the numerical
codes for the dissipative spin hydrodynamics has not been developed yet in our
community. It is worthwhile to search for analytic solutions for spin
hydrodynamics in some simple, but nevertheless realistic, test cases.

In this work, we study analytically the canonical relativistic spin
hydrodynamics in the time-honored longitudinal boost invariant Bjorken flow
(Bjorken:1982qr, ). We also consider the viscous effects coupled to the spin
hydrodynamics. Since relativistic hydrodynamics in the first order of the
gradient expansion are unstable and violate the causality (Hiscock:1983zz, ;
Hiscock:1985zz, ; Hiscock:1987zz, ; Denicol:2008ha, ; Koide:2009sy, ;
Pu:2009fj, ; Pu:2011vr, ), we solve the equations of spin hydrodynamic coupled
to the simplest relaxation time corrections for the spin dynamics. To see the
spin corrections to the ordinary terms, we take the pseudo gauge
transformation and check the results in the Belinfante form of spin dynamics.

We emphasize that in the current work we are searching for the self-consistent
analytic solutions of dissipative spin hydrodynamics, which are different with
the studies of spin hydrodynamics in Bjorken (Florkowski:2019qdp, ;
Singh:2020efc, ; Singh:2021man, ) and Gubser (Singh:2020rht, ) expanding
backgrounds.

The structure of this work is as following. In Sec. II, we will briefly review
the conservation equations of dissipative spin hydrodynamics in both canonical
and Belinfante form. In Sec. III.1, we study the canonical spin hydrodynamics
with non-vanishing viscous tensor in a longitudinally boost invariant fluid
and obtain the analytic solutions. Next, we consider the relaxation time
equations for spin dynamics and derive the analytic solutions in the gradient
expansion in Sec. III.2. We also take the pseudo gauge transformation and
compute the Belinfante form spin hydrodynamics in Sec. III.3 and discuss our
results in Sec. III.4. We summarize in Sec. IV.

Throughout this work, we adopt the metric
$g^{\mu\nu}=\textrm{diag}\\{+,-,-,-\\}$ and the fluid velocity $u^{\mu}$
satisfying $u^{2}=1$. We define the projector
$\Delta^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}$. For simplicity, for an arbitrary
tensor $A^{\mu\nu}$, we introduce the symmetric, anti-symmetric and traceless
parts as $A^{(\mu\nu)}=(A^{\mu\nu}+A^{\nu\mu})/2$,
$A^{[\mu\nu]}=(A^{\mu\nu}-A^{\nu\mu})/2$ and
$A^{\left\langle\mu\nu\right\rangle}=\frac{1}{2}[\Delta^{\mu\alpha}\Delta^{\nu\beta}+\Delta^{\nu\alpha}\Delta^{\mu\beta}]A_{\alpha\beta}-\frac{1}{3}\Delta^{\mu\nu}(A^{\rho\sigma}\Delta_{\rho\sigma})$.

##  II Main equations in relativistic dissipative spin hydrodynamics

In this section, we briefly review the main equations of dissipative spin
hydrodynamics in both canonical and Belinfante form. In canonical form of
relativistic spin hydrodynamics, the conserved total angular momentum (TAM)
can be decomposed as (Fukushima:2020ucl, )

$\displaystyle J^{\alpha\mu\nu}$ $\displaystyle=$ $\displaystyle
x^{\mu}T^{\alpha\nu}-x^{\nu}T^{\alpha\mu}+\Sigma^{\alpha\mu\nu},$ (1)

where $T^{\mu\nu}$ is the the canonical energy momentum tensor (EMT) and
$\Sigma^{\alpha\mu\nu}$ is a rank-3 spin tensor corresponding to the classical
spin part of TAM. Up to $\mathcal{O}(\partial^{1})$ in the gradient expansion,
the canonical EMT can be further decomposed as (Hattori:2019lfp, ;
Fukushima:2020qta, ; Li:2020eon, ),

$\displaystyle T^{\mu\nu}$ $\displaystyle=$ $\displaystyle
eu^{\mu}u^{\nu}-p\Delta^{\mu\nu}+2h^{(\mu}u^{\nu)}+2q^{[\mu}u^{\nu]}+\pi^{\mu\nu}+\phi^{\mu\nu},$
(2)

where $e,p,u^{\mu},h^{\mu}$ and $\pi^{\mu\nu}$ denote energy density,
pressure, fluid velocity, heat flow and viscous tensor, respectively. The
$q^{\mu}$ and $\phi^{\mu\nu}$ are the anti-symmetric EMT parts. Note that,
$h^{\mu},q^{\mu},\pi^{\mu\nu}$ and $\phi^{\mu\nu}$ are perpendicular to the
fluid velocity $u^{\mu}$.

Both EMT and TAM are conserved, i.e.

$\displaystyle\partial_{\mu}T^{\mu\nu}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\partial_{\alpha}J^{\alpha\mu\nu}$ $\displaystyle=$
$\displaystyle 0,$ (3)

which lead to,

$\displaystyle\partial_{\alpha}\Sigma^{\alpha\mu\nu}$ $\displaystyle=$
$\displaystyle-2T^{[\mu\nu]}.$ (4)

It means the antisymmetric part of EMT acts as a source or absorption term for
spin current. Then we make a tensor decomposition for $\Sigma^{\alpha\mu\nu}$,

$\Sigma^{\alpha\mu\nu}=u^{\alpha}S^{\mu\nu}+\Sigma_{(1)}^{\alpha\mu\nu},$ (5)

where $S^{\mu\nu}$ is named as spin density and $\Sigma_{(1)}^{\alpha\mu\nu}$
refers to the higher order correction with
$u_{\alpha}\Sigma_{(1)}^{\alpha\mu\nu}=0$. Inserting Eqs. (2, 5) into Eq. (4)
yields,

$\displaystyle\partial_{\alpha}(u^{\alpha}S^{\mu\nu})$ $\displaystyle=$
$\displaystyle-2(q^{\mu}u^{\nu}-q^{\nu}u^{\mu}+\phi^{\mu\nu}).$ (6)

After introducing the spin degree of freedom , it is necessary to modify the
thermodynamic relations. We treat spin density $S^{\mu\nu}$ as particle number
density, then we also need to introduce the spin potential
$\omega_{\mu\nu}=-\omega_{\nu\mu}$. Now, the new thermodynamic relations
become,

$\displaystyle e+p$ $\displaystyle=$ $\displaystyle
Ts+\omega_{\mu\nu}S^{\mu\nu},$ $\displaystyle de$ $\displaystyle=$
$\displaystyle Tds+\omega_{\mu\nu}dS^{\mu\nu},$ $\displaystyle dp$
$\displaystyle=$ $\displaystyle sdT+S^{\mu\nu}d\omega_{\mu\nu},$ (7)

where $s$ is the entropy density and $T$ is the temperature. In general, we
can also have the charge currents in a system. For simplicity, we neglect
these currents throughout this work. For more general discussions, one can
also see Ref. (Hattori:2019lfp, ; Fukushima:2020qta, ; Li:2020eon, ).

The second law of thermodynamics requires that the entropy production rate
should always be positive definite. It gives the general expression for those
dissipative terms. The $q^{\mu}$ and $\phi^{\mu\nu}$ are given by,

$\displaystyle q^{\mu}$ $\displaystyle=$
$\displaystyle\lambda\left[\frac{1}{T}\Delta^{\mu\alpha}\partial_{\alpha}T+(u\cdot\partial)u^{\mu}-4\omega^{\mu\nu}u_{\nu}\right],$
$\displaystyle\phi^{\mu\nu}$ $\displaystyle=$
$\displaystyle-\gamma\left(\Omega^{\mu\nu}-2\frac{1}{T}\Delta^{\mu\alpha}\Delta^{\nu\beta}\omega_{\alpha\beta}\right),$
(8)

where
$\Omega^{\mu\nu}=-\Delta^{\mu\rho}\Delta^{\nu\sigma}\frac{1}{2}[\partial_{\rho}(\frac{u_{\sigma}}{T})-\partial_{\sigma}(\frac{u_{\rho}}{T})]$
is the thermal vorticity (Becattini:2013fla, ; Fang:2016uds, ) and
$\lambda,\gamma\geq 0$ are the transport coefficients. The ordinary viscous
tensor $\pi^{\mu\nu}$ is given by,

$\displaystyle\pi^{\mu\nu}$ $\displaystyle=$
$\displaystyle\eta_{s}\partial^{\left\langle\mu\right.}u^{\left.\nu\right\rangle}-\Pi\Delta^{\mu\nu},$
$\displaystyle\Pi$ $\displaystyle=$ $\displaystyle-\zeta(\partial\cdot u),$
(9)

where $\eta_{s}$ and $\zeta$ are the shear and bulk viscosities, respectively.

Note that, all of above results based on the assumption that the
$\omega^{\mu\nu}$ and/or $S^{\mu\nu}$ are at $\mathcal{O}(\partial^{1})$. If
both $\omega^{\mu\nu}$ and $S^{\mu\nu}$ are at the leading order of the
gradient expansion, then the decomposition of EMT can be very different, also
see Ref. (She:2021lhe, ).

The canonical EMT is gauge dependent at the operator level and not symmetric.
One can choose Belinfante form EMT $\mathcal{T}^{\mu\nu}$, which is gauge
invariant and symmetric, through the following pseudo-gauge transformation,

$\displaystyle\mathcal{T}^{\mu\nu}$ $\displaystyle=$ $\displaystyle
T^{\mu\nu}+\partial_{\lambda}K^{\lambda\mu\nu},$ (10)

with

$K^{\lambda\mu\nu}=\frac{1}{2}\left(\Sigma^{\lambda\mu\nu}-\Sigma^{\mu\lambda\nu}+\Sigma^{\nu\mu\lambda}\right).$
(11)

According to the Ref. (Fukushima:2020ucl, ), up to
$\mathcal{O}(\partial^{1})$, we get,

$\displaystyle\mathcal{T}^{\mu\nu}$ $\displaystyle=$ $\displaystyle
eu^{\mu}u^{\nu}-p\Delta^{\mu\nu}+\frac{1}{2}\partial_{\lambda}(u^{\mu}S^{\nu\lambda}+u^{\nu}S^{\mu\lambda})$
(12) $\displaystyle=$ $\displaystyle(e+\delta
e)u^{\mu}u^{\nu}-p\Delta^{\mu\nu}+2(h^{(\mu}+\delta
h^{(\mu})u^{\nu)}+(\pi^{\mu\nu}+\delta\pi^{\mu\nu}),$

Here we have introduced the spin corrections to the energy density, $\delta
e$, the heat flow, $\delta h^{\mu}$ and the viscous tensor,
$\delta\pi^{\mu\nu}$,

$\displaystyle\delta e$ $\displaystyle=$ $\displaystyle
u_{\mu}\partial_{\lambda}S^{\mu\lambda},$ $\displaystyle\delta h^{\mu}$
$\displaystyle=$
$\displaystyle\frac{1}{2}(\Delta_{\beta}^{\mu}\partial_{\lambda}S^{\beta\lambda}+u_{\beta}S^{\beta\lambda}\partial_{\lambda}u^{\mu}),$
$\displaystyle\delta\pi^{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\lambda}(u^{\left\langle\mu\right.}S^{\left.\nu\right\rangle\lambda})+\frac{1}{3}\Delta_{\rho\sigma}\partial_{\lambda}(u^{\rho}S^{\sigma\lambda})\Delta^{\mu\nu}.$
(13)

The Belinfante TAM becomes

$\mathcal{J}^{\alpha\mu\nu}=J^{\alpha\mu\nu}+\partial_{\rho}(x^{\mu}K^{\rho\alpha\nu}-x^{\nu}K^{\rho\alpha\mu})=x^{\mu}\mathcal{T}^{\alpha\nu}-x^{\nu}\mathcal{T}^{\alpha\mu}.$
(14)

Both Belinfante TAM and EMT are conserved. Also see Ref. (Fukushima:2020ucl, ;
Fukushima:2020qta, ) for the discussion about the connection between canonical
and Belinfante form spin hydrodynamics.

##  III Analytic solutions in Bjorken expansion

In this section, we consider the time-honored Bjorken expansion of ordinary
relativistic hydrodynamics to the dissipative spin hydrodynamics. The basic
idea of Bjorken expansion is as follows. The fluid velocity is given by
(Bjorken:1982qr, ),

$u^{\mu}=\left(\frac{t}{\tau},0,0,\frac{z}{\tau}\right),$ (15)

where $\tau=\sqrt{t^{2}-z^{2}}$ is the proper time. The system is assumed to
be homogenous in transverse plane. As a consequence, all of the macroscopic
quantities only depend on the proper time $\tau$ and are independent on the
space rapidity $\eta=\frac{1}{2}\ln[(t+z)/(t-z)]$.

To close the system, the equations of state (EoS) are essential. We choose the
energy density $e$ and $\omega^{\mu\nu}$ as two kinds of thermodynamic
variables. In this work, we follow the simplest EoS for the relativistic fluid
in the high temperature limit,

$e=3p.$ (16)

In Analogy to number density and chemical potential in the high temperature
limit, we assume the EoS for $S^{\mu\nu}$ is,

$S^{\mu\nu}=a_{1}T^{2}\omega^{\mu\nu},$ (17)

where $a_{1}$ is constant.

Similar to our previous works on searching for the analytic solutions of
relativistic magnetohydrodynamics (Pu:2016ayh, ; Roy:2015kma, ; Pu:2016bxy, ;
Pu:2016rdq, ; Siddique:2019gqh, ; Wang:2020qpx, ), our strategy is as follows.
As initial conditions, we assume that the inital fluid velocity is given by
Eq. (15) and all the thermodynamic quantities are independent on $x,y$ and
$\eta$. Next, we will compute the dynamical evolution equations for fluid
velocity $u^{\mu}$, energy density $e$ and spin density $S^{\mu\nu}$. At last,
we search for the special configurations of $e$ and $\omega^{\mu\nu}$, which
can hold the initial Bjorken velocity (15). To keep the whole system boost
invariant, we therefore assume that only $\omega^{xy}$ is nonzero initially.
Then EoS (17) reduces to,

$S^{xy}=a_{1}T^{2}\omega^{xy}.$ (18)

Furthermore, to simplify the calculations and to highlight the spin effect, we
choose the Landau frame in which heat flow $h^{\mu}$ is always zero.

###  III.1 Canonical form spin hydrodynamics in Bjorken flow

Now, we discuss the canonical EMT and the corresponding conservation equations
for the Bjorken flow. The main equations are Eqs. (3, 6).

Firstly, let us consider the the acceleration equation
$\Delta_{\nu\alpha}\partial_{\mu}T^{\mu\nu}=0$, which provides the dynamical
evolution of fluid velocity,

$\displaystyle(u\cdot\partial)u_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{(e+p)}\left[\Delta_{\alpha}^{\mu}\partial_{\mu}p-(q\cdot\partial)u_{\alpha}+q_{\alpha}(\partial\cdot
u)+\Delta_{\nu\alpha}(u\cdot\partial)q^{\nu}\right.$ (19)
$\displaystyle\left.-\Delta_{\nu\alpha}\partial_{\mu}\phi^{\mu\nu}-\Delta_{\nu\alpha}\partial_{\mu}\pi^{\mu\nu}\right].$

If $\omega^{xy}$ depends on the proper time $\tau$ only, with the Bjorken
velocity (15), the $q^{\mu}$ and $\phi^{\mu\nu}$ becomes,

$\displaystyle q^{\mu}$ $\displaystyle=$
$\displaystyle-4\lambda\omega^{\mu\nu}u_{\nu},$ $\displaystyle\phi^{\mu\nu}$
$\displaystyle=$
$\displaystyle\frac{2\gamma}{T}[\omega^{\mu\nu}+2u^{[\mu}\omega^{\nu]\beta}u_{\beta}].$
(20)

Inserting Eq. (20) into Eq. (19), we find that the Bjorken velocity (15) will
not be modified under our assumption.

Next, let us consider the energy conservation equation
$u_{\nu}\partial_{\mu}T^{\mu\nu}=0$, i.e,

$(u\cdot\partial)e+(e+p)\partial\cdot u+\partial\cdot
q+q^{\nu}(u\cdot\partial)u_{\nu}+u_{\nu}\partial_{\mu}\phi^{\mu\nu}-\pi^{\mu\nu}\partial_{\mu}u_{\nu}=0.$
(21)

Using Eqs. (20), the above equation becomes,

$\frac{d}{d\tau}e+\frac{4}{3}e\frac{1}{\tau}-s\left(\frac{2}{3}\frac{\eta_{s}}{s}+\frac{\zeta}{s}\right)\frac{1}{\tau^{2}}=0,$
(22)

where the $s$ is the entropy. Note that, we assume that the dimensionless
quantities $\eta_{s}/s$ and $\zeta/s$ are constant.

Thirdly, from Eq. (20), the spin density in Eq. (6) reads,

$\displaystyle\frac{dS^{xy}}{d\tau}+S^{xy}\frac{1}{\tau}$ $\displaystyle=$
$\displaystyle-\frac{4\gamma}{T}\omega^{xy}.$ (23)

Note that, in this case, the $q^{\mu}$ does not contribute to the $S^{xy}$.

To solve Eqs. (22, 23), we can express the $s$ and $e$ as functions of $T$ and
$\omega^{xy}$ by using EoS (16, 18) and the thermodynamic relations (7),

$\displaystyle e(T,\omega^{xy})$ $\displaystyle=$ $\displaystyle
c_{1}T^{4}+3a_{1}T^{2}\omega_{xy}^{2},$ $\displaystyle s(T,\omega^{xy})$
$\displaystyle=$ $\displaystyle\frac{4}{3}c_{1}T^{3}+2a_{1}T\omega_{xy}^{2},$
(24)

where the constant $c_{1}$ can be determined by the the initial conditions,
i.e.

$c_{1}=\left[\frac{e_{0}}{T_{0}^{4}}-3a_{1}\left(\frac{\omega_{0}^{xy}}{T_{0}}\right)^{2}\right],$
(25)

with $e_{0}=e(\tau_{0})$, $T_{0}=T(\tau_{0})$ and $\omega_{0}^{xy}$ being the
energy density, temperature and spin chemical potential at the initial proper
time $\tau_{0}$, respectively.

After inserting Eqs.(24) into Eq. (22), we find that the solutions cannot be
written into a compact form. On the other hand, as mentioned in Sec. II, we
have always assume that the $\omega^{xy}\sim\mathcal{O}(\partial^{1})$ in the
gradient expansion in current formalism and it implies that

$\omega^{xy}/T\ll 1.$ (26)

Therefore, we can consider a power expansion of $\omega^{xy}/T$, which is
equivalent to the gradient expansion. Eqs. (22, 23) with Eqs. (24) becomes,

$\displaystyle\frac{d}{d\tau}T+\frac{1}{3}\frac{T}{\tau}-\frac{1}{3}\left(\frac{2}{3}\frac{\eta_{s}}{s}+\frac{\zeta}{s}\right)\frac{1}{\tau^{2}}+\mathcal{O}\left((\omega_{xy}/T)^{2}\right)$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
T\frac{d}{d\tau}\omega^{xy}+2\omega^{xy}\frac{d}{d\tau}T+T\omega^{xy}\frac{1}{\tau}+\frac{4\gamma}{a_{1}T^{2}}\omega^{xy}+\mathcal{O}\left((\omega_{xy}/T)^{2}\right)$
$\displaystyle=$ $\displaystyle 0.$ (27)

The solutions are,

$\displaystyle T(\tau)$ $\displaystyle=$ $\displaystyle
T_{0}\left(\frac{\tau_{0}}{\tau}\right)^{1/3}-\frac{1}{2\tau}\left(\frac{2}{3}\frac{\eta_{s}}{s}+\frac{\zeta}{s}\right)\left[1-\left(\frac{\tau}{\tau_{0}}\right)^{2/3}\right]+\mathcal{O}\left((\omega_{0}^{xy}/T_{0})^{2}\right),$
$\displaystyle\omega^{xy}(\tau)$ $\displaystyle=$
$\displaystyle\omega_{0}^{xy}\left(\frac{\tau_{0}}{\tau}\right)^{1/3}\exp\left[-\frac{2\gamma\tau_{0}}{a_{1}T_{0}^{3}}\left(\frac{\tau^{2}}{\tau_{0}^{2}}-1\right)\right]\left\\{1+\left(\frac{2}{3}\frac{\eta_{s}}{s}+\frac{\zeta}{s}\right)\frac{1}{T_{0}^{4}}\right.$
(28)
$\displaystyle\left.\times\left[\frac{T_{0}^{3}}{\tau_{0}}\left(\left(\frac{\tau_{0}}{\tau}\right)^{2/3}-1\right)+\frac{\gamma}{a_{1}}\left(3\left(\frac{\tau}{\tau_{0}}\right)^{2}-\frac{9}{2}\left(\frac{\tau}{\tau_{0}}\right)^{4/3}+\frac{3}{2}\right)\right]\right\\}$
$\displaystyle+\mathcal{O}\left((\omega_{0}^{xy}/T_{0})^{2},(\eta_{s}/s)^{2},(\zeta/s)^{2},(\eta_{s}\zeta/s^{2})\right),$

where for $\omega^{xy}(\tau)$, we only keep the linear terms of $\eta_{s}/s$
and $\zeta/s$.

Inserting solutions (28) into Eqs. (24) and EoS (17) yields,

$\displaystyle e(\tau)$ $\displaystyle=$ $\displaystyle
e_{0}\left(\frac{\tau_{0}}{\tau}\right)^{4/3}-2\frac{e_{0}\tau_{0}}{T_{0}\tau^{2}}\left(\frac{2}{3}\frac{\eta_{s}}{s}+\frac{\zeta}{s}\right)\left[1-\left(\frac{\tau}{\tau_{0}}\right)^{2/3}\right]$
(29)
$\displaystyle+\mathcal{O}\left((\omega_{0}^{xy}/T_{0})^{2},(\eta_{s}/s)^{2},(\zeta/s)^{2},(\eta_{s}\zeta/s^{2})\right),$

and

$\displaystyle S^{xy}(\tau)$ $\displaystyle=$ $\displaystyle
a_{1}\omega_{0}^{xy}T_{0}^{2}\left(\frac{\tau_{0}}{\tau}\right)\exp\left[-\frac{2\gamma\tau_{0}}{a_{1}T_{0}^{3}}(\frac{\tau^{2}}{\tau_{0}^{2}}-1)\right]$
(30)
$\displaystyle\times\left\\{1+\left(\frac{2}{3}\frac{\eta_{s}}{s}+\frac{\zeta}{s}\right)\frac{3\gamma}{2a_{1}T_{0}^{4}}\left[2\left(\frac{\tau}{\tau_{0}}\right)^{2}-3\left(\frac{\tau}{\tau_{0}}\right)^{\frac{4}{3}}+1\right]\right\\}$
$\displaystyle+\mathcal{O}\left((\omega_{0}^{xy}/T_{0})^{2},(\eta_{s}/s)^{2},(\zeta/s)^{2},(\eta_{s}\zeta/s^{2})\right).$

The dissipative effects from $\phi^{\mu\nu}$ contribute an extra factor
$\exp[-2\gamma(\tau^{2}-\tau_{0}^{2})/(a_{1}T_{0}^{3}\tau_{0})]$ to Eqs. (28,
30). Both $\omega^{xy}$ and $S^{xy}$ decay much rapidity in a finite
$\phi^{\mu\nu}$ system than in a vanishing $\phi^{\mu\nu}$ system. Note that
the factor related to $\gamma$ looks similar to the factor
$\sim\exp[-\sigma(\tau-\tau_{0})]$ caused by the electric conducting flow in
magnetohydrodynamics with $\sigma$ the electric conductivity
(Siddique:2019gqh, ). From Eq. (30), we find that the viscous corrections to
$S^{xy}(\tau)$ are always positive when $\tau/\tau_{0}\geq 1$ and increase
when $\tau_{0}$ grows. It means that the viscous effects accelerate the decay
of spin density.

###  III.2 Corrections from relaxation time for spin transport

In this section, we compute the higher order effects for the spin transport.
For simplicity, we neglect $\pi^{\mu\nu}$ and concentrate on the spin effects.
It is well-known that the relativistic dissipative hydrodynamics in
$\mathcal{O}(\partial^{1})$ violate causality and are unstable, e.g. see Ref.
(Hiscock:1983zz, ; Hiscock:1985zz, ; Hiscock:1987zz, ; Denicol:2008ha, ;
Koide:2009sy, ; Pu:2009fj, ; Pu:2011vr, ) and the references therein.
Therefore, one needs to introduce the second order corrections. In this work,
we only consider a standard second order correction to the dissipative terms
in Eqs. (8),

$\tau_{\phi}\frac{d}{d\tau}\phi^{\mu\nu}+\phi^{\mu\nu}=-\gamma\left(\Omega^{\mu\nu}-2\frac{1}{T}\Delta^{\mu\alpha}\Delta^{\nu\beta}\omega_{\alpha\beta}\right),$
(31)

where $\tau_{\phi}$ is the relaxation time for $\phi^{\mu\nu}$. The relaxation
time $\tau_{\phi}$ describes how fast the system could reach to the
equilibrium again after taking some perturbations $\phi^{\mu\nu}$ to an
equilibrium system. It is similar to the relaxation time equations for viscous
terms. Also see Ref. (Hattori:2020htm, ; Gallegos:2021bzp, ; Li:2020eon, ;
She:2021lhe, ) for other possible second order corrections of spin
hydrodynamics.

In this case, the conservation equations become,

$\displaystyle\frac{d}{d\tau}e+\frac{4}{3}e\frac{1}{\tau}$ $\displaystyle=$
$\displaystyle 0,$ (32)

and,

$(u\cdot\partial)u_{\alpha}=\frac{1}{(e+p)}[\Delta_{\alpha}^{\mu}\partial_{\mu}p-\Delta_{\nu\alpha}\partial_{\mu}\phi^{\mu\nu}].$
(33)

The evolution equations for the spin density become,

$\displaystyle\frac{dS^{xy}}{d\tau}+S^{xy}\frac{1}{\tau}$ $\displaystyle=$
$\displaystyle-2\phi^{xy},$
$\displaystyle\tau_{\phi}\frac{d}{d\tau}\phi^{xy}+\phi^{xy}$ $\displaystyle=$
$\displaystyle\frac{2\gamma}{T}\omega^{xy}.$ (34)

We find that the fluid velocity will not be modified when $\phi^{xy}$ only
depends on the $\tau$. It is straightforward to get the solution for Eq. (32),

$\displaystyle e(\tau)$ $\displaystyle=$ $\displaystyle
e_{0}\left(\frac{\tau_{0}}{\tau}\right)^{4/3},$ $\displaystyle T(\tau)$
$\displaystyle=$ $\displaystyle
T_{0}\left(\frac{\tau_{0}}{\tau}\right)^{1/3}+\mathcal{O}\left((\omega_{0}^{xy}/T_{0})^{2}\right).$
(35)

Then, we can solve the $\phi^{xy},S^{xy}$ and $\omega^{xy}$,

$\displaystyle\phi^{xy}(\tau)$ $\displaystyle=$ $\displaystyle
e^{-\frac{\tau}{2\tau_{\phi}}}f(\tau)+\mathcal{O}\left((\omega_{0}^{xy}/T_{0})^{2}\right),$
$\displaystyle\omega^{xy}(\tau)$ $\displaystyle=$
$\displaystyle\left(\frac{\tau_{0}}{\tau}\right)^{\frac{1}{3}}\frac{T_{0}}{2\gamma}e^{-\frac{\tau}{2\tau_{\phi}}}\left[\frac{1}{2}f(\tau)-\left(\frac{4\gamma\tau_{\phi}^{2}}{a_{1}\tau_{0}T_{0}^{3}}\right)^{\frac{1}{3}}g(\tau)\right]+\mathcal{O}\left((\omega_{0}^{xy}/T_{0})^{2}\right),$
$\displaystyle S^{xy}(\tau)$ $\displaystyle=$ $\displaystyle
a_{1}\left(\frac{\tau_{0}}{\tau}\right)\frac{T_{0}^{3}}{2\gamma}e^{-\frac{\tau}{2\tau_{\phi}}}\left[\frac{1}{2}f(\tau)-\left(\frac{4\gamma\tau_{\phi}^{2}}{a_{1}\tau_{0}T_{0}^{3}}\right)^{\frac{1}{3}}g(\tau)\right]+\mathcal{O}\left((\omega_{0}^{xy}/T_{0})^{2}\right),$
(36)

where $f(\tau)$ and $g(\tau)$ are shown in Appendix A.

In the $\tau_{\phi}\rightarrow 0$ limit, we prove that the solutions (36)
reduce to Eqs. (28, 30). In the $\tau_{\phi}\rightarrow\infty$ limit, the
solutions (36) become,

$\lim_{\tau_{\phi}\rightarrow\infty}\omega^{xy}(\tau)=\omega_{0}^{xy}(\tau_{0}/\tau)^{1/3},\qquad\lim_{\tau_{\phi}\rightarrow\infty}S^{xy}(\tau)=a_{1}T_{0}^{2}\omega_{0}^{xy}(\tau_{0}/\tau),.$
(37)

Therefore, the new term proportional to $\tau_{\phi}$ in Eq. (31) slows down
the decay caused by the $\phi^{\mu\nu}$ as expected.

###  III.3 Results for Belinfante form EMT

By using the pseudo gauge transformation (11), we can obtain the Belinfante
form EMT $\mathcal{T}^{\mu\nu}$. With the solutions (30) in Sec. III.1 and
solutions (36) in Sec. III.2, we find that

$\displaystyle\frac{1}{2}\partial_{\lambda}(u^{\mu}S^{\nu\lambda}+u^{\nu}S^{\mu\lambda})$
$\displaystyle=$ $\displaystyle\mathcal{O}(\partial^{2}),$ (38)

up to the order of $\mathcal{O}(\partial^{2})$. Therefore, Belinfante EMT
reduces to the ordinary EMT without spin effects. We have checked these spin
corrections in Eqs. (13) and found that all of them vanish under our
assumption.

We emphasize that it does not mean that there are no the spin effects in
Belinfante form of dissipative spin hydrodynamics. In order to observe these
spin corrections in Eqs. (13), we have to consider the inhomogeneity in the
transverse plane.

###  III.4 Discussion

Let us take a close look to the leading order of $\omega^{xy}$ and $S^{xy}$ in
Eqs. (28, 30). We find that $\omega^{xy}$ and $S^{xy}$ decay as
$\sim\tau^{-1/3}$ and $\sim 1/\tau$, respectively. From Ref.
(Becattini:2013fla, ; Fang:2016uds, ), one can compute the spin polarization
of $\Lambda$ hyperons caused by the thermal vorticity by the modified Cooper-
Frye formula,

$\mathcal{S}^{\mu}({\bf p})=\frac{\int d\Sigma\cdot
pf(1-f)\epsilon^{\mu\nu\alpha\beta}p_{\nu}\left[\partial_{\alpha}\left(\frac{u_{\beta}}{T}\right)+...\right]}{8m_{\Lambda}\int
d\Sigma\cdot pf},$ (39)

where the $m_{\Lambda}$ is the mass of $\Lambda$ hyperon, $\Sigma^{\mu}$ is
the freezeout hypersurface, $f=f(x,p)$ is the distribution functions of
particles and $...$ denotes the other possible corrections from shear viscous
tensor and others (Hidaka:2018mel, ; Liu:2020dxg, ; Liu:2021uhn, ; Fu:2021pok,
; Becattini:2021iol, ; Yi:2021ryh, ). In general, the spin density
$S^{\mu\nu}$ and $\omega^{\mu\nu}$ contribute to the distribution function
$f(x,p)$. Our results (28, 30) imply that the initial spin density decays
rapidly and may not play an important role to the polarization vector
$\mathcal{S}^{\mu}({\bf p})$ at the freezeout hypersurface. Although the huge
initial orbital angular momentum could transfer to the spin, it may still be
difficult to accumulate the net spin density in such an expanding system.
Therefore, one may still neglect the contributions from $S^{\mu\nu}$ or
$\omega^{\mu\nu}$ to the calculations of polarization vector
$\mathcal{S}^{\mu}({\bf p})$.

##  IV Conclusion

In this work, we have derived the analytic solutions of a longitudinally
boost-invariant dissipative spinness fluid with finite viscous tensor and
second order relaxation time corrections for the spin. Our main results are
shown in Eqs. (28, 30) and (36).

We find that the spin density and spin chemical potential decay as
$\sim\tau^{-1}$ and $\tau^{-1/3}$, respectively. It implies that the initial
spin density may not survive at the late time. Although the huge initial
orbital angular momentum could transfer to the spin, it may be difficult to
accumulate the net spin density in such an expanding system.

We have also computed the Belinfante EMT and have not found any corrections
from spin in this work. To observe the possible spin corrections to the
ordinary terms in Eqs. (13), one needs to consider the inhomogeneity in the
transverse plane, e.g. similar to the Ref. (Pu:2016bxy, ) for the
magnetohydrodynamics.

###### Acknowledgements.

D.L. Wang and S. Fang are grateful to the School of Science in Huzhou
University for its hospitality during the completion of this work. This work
is supported by National Nature Science Foundation of China (NSFC) under
Grants No. 12075235.

##  Appendix A Expression for Eqs. (36)

The $f(\tau)$ and $g(\tau)$ in Eqs. (36) are

$\displaystyle f(\tau)$ $\displaystyle=$ $\displaystyle
C_{1}\mathrm{Ai}\left[\mathcal{H}(\tau)\right]+C_{2}\mathrm{Bi}\left[\mathcal{H}(\tau)\right],$
$\displaystyle g(\tau)$ $\displaystyle=$ $\displaystyle
C_{1}\mathrm{Ai}^{\prime}\left[\mathcal{H}(\tau)\right]+C_{2}\mathrm{Bi}^{\prime}\left[\mathcal{H}(\tau)\right],$
(40)

where

$\mathcal{H}(\tau)=2^{-\frac{4}{3}}\left(\frac{1}{4\tau_{\phi}^{2}}-\frac{4\gamma\tau}{a_{1}\tau_{0}\tau_{\phi}T_{0}^{3}}\right)\left(-\frac{\gamma}{a_{1}\tau_{0}T_{0}^{3}\tau_{\phi}}\right)^{-\frac{2}{3}},$
(41)

The $\mathrm{Ai}(z)$ and $\mathrm{Bi}(z)$ are named as Airy functions, which
are the solutions of the differential equation $y^{\prime\prime}(z)\pm
zy(z)=0$, respectively. Note that, here both $\mathrm{Ai}(z)$ and
$\mathrm{Bi}(z)$ are entire Airy functions of $z$ with no branch cut
discontinuities. We also use the notation
$\mathrm{Ai}^{\prime}(x)=\frac{d}{dx}\mathrm{Ai}(x)$ and
$\mathrm{Bi}^{\prime}(x)=\frac{d}{dx}\mathrm{Bi}(x)$.

The coefficients $C_{1,2}$ are determined by the initial condition
$\phi^{xy}(\tau_{0})=\phi_{0}^{xy}$ and
$\omega^{xy}(\tau_{0})=\omega_{0}^{xy}$,

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\mathcal{A}^{-1}e^{\frac{\tau_{0}}{2\tau_{\phi}}}\left[\mathrm{Bi}(\mathcal{U})\mathcal{C}-\phi_{0}^{xy}\mathrm{Bi^{\prime}}(\mathcal{U})\right],$
$\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\mathcal{A}^{-1}e^{\frac{\tau_{0}}{2\tau_{\phi}}}\left[\mathrm{Ai}^{\prime}(\mathcal{U})\phi_{0}^{xy}-\mathrm{Ai}(\mathcal{U})\mathcal{C}\right],$
(42)

where

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\mathrm{Ai^{\prime}}(\mathcal{U})\mathrm{Bi}(\mathcal{U})-\mathrm{Ai}(\mathcal{U})\mathrm{Bi^{\prime}}(\mathcal{U}),$
$\displaystyle\mathcal{C}$ $\displaystyle=$
$\displaystyle\left(\frac{a_{1}\tau_{0}T_{0}^{3}}{4\gamma\tau_{\phi}^{2}}\right)^{\frac{1}{3}}\left(\frac{1}{2}\phi_{0}^{xy}-\frac{2\gamma}{T_{0}}\omega_{0}^{xy}\right),$
$\displaystyle\mathcal{U}$ $\displaystyle=$
$\displaystyle\left(\frac{a_{1}\tau_{0}T_{0}^{3}}{4\gamma}\right)^{\frac{2}{3}}\left(\frac{1}{4}\tau_{\phi}^{-\frac{4}{3}}-\frac{4\gamma}{a_{1}T_{0}^{3}}\tau_{\phi}^{-\frac{1}{3}}\right).$
(43)

Note that, here we only take the real solutions and neglect imaginary
solutions.

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