[c,d]Rajeev Singh

# Study of spin polarization dependence on rapidity, transverse momentum, and
azimuthal angle

Wojciech Florkowski  Radoslaw Ryblewski

###### Abstract

We investigate the spacetime evolution of spin polarization within a
hydrodynamic framework based on the de Groot–van Leeuwen–van Weert expressions
for energy-momentum and spin tensors. The system’s deviation from boost
invariance results in the interplay of different spin polarization components,
impacting spin observables. We specifically examine the transverse momentum,
azimuthal angle, and rapidity dependence of the mean spin polarization vector
of $\Lambda$ hyperons. Our results qualitatively align with other models and
experimental data on global spin polarization rapidity dependence. While the
quadrupole structure is absent in the longitudinal component at midrapidity,
our analysis reveals non-trivial signal at forward rapidities that differs
from predictions based on the Bjorken expansion.

##  1 Introduction

In the past years, relativistic hydrodynamics has significantly matured into a
theory with diverse applications [1, 2]. This advancement has led to the
extension of standard hydrodynamic formalism and allowed determination of
certain quark-gluon plasma properties [3, 4]. Recent measurements of spin
polarization of particles produced in relativistic heavy-ion collisions have
offered new insights into these studies [5, 6, 7, 8]. The concept of local
thermodynamic equilibrium including spin degrees of freedom was proposed and
turned out to be very useful in explaining several spin polarization phenomena
like, for example, the collision energy dependence due to polarization-
vorticity coupling [9, 10, 11, 12, 13]. Despite early successes, spin-thermal
models have encountered difficulties to explain more detailed observables, in
particular, the measured transverse-momentum dependence of spin polarization
along the beam direction, which displayed an opposite trend to model
predictions [6, 14, 15, 16]. This discrepancy has stimulated further
theoretical exploration, raising questions about spin non-equilibrium effects
and their intrinsic dynamics. Among these developments, incorporating spin
degrees of freedom into hydrodynamic frameworks has received notable attention
as it opens avenues for probing quantum aspects of matter within a classical
hydrodynamic context [17].

The development of relativistic hydrodynamics with spin, based on quantum
kinetic theory, was initially proposed in [18] and has since been elaborated
in various studies [19, 20, 21, 22, 23, 24, 25, 26]. Other notable approaches
have utilized effective action methods [27, 28], entropy current analysis [29,
30, 31], and theories involving non-local collisions [32, 33, 34, 35, 36].
Generally, spin polarization dynamics is governed by a rank-two anti-symmetric
spin polarization tensor, $\omega^{\alpha\beta}$, which introduces six
additional Lagrange multipliers into hydrodynamics. Along with the standard
multipliers, they must be determined from the conservation laws. Notably, the
spin polarization tensor may be distinct from thermal vorticity, central
object in spin-thermal models [10, 37, 11, 38, 39, 15, 40, 41, 42, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. The hydrodynamics-with-spin formalism
[18, 19] is grounded in the energy-momentum and spin tensor forms introduced
by de Groot, van Leeuwen, and van Weert (GLW) [56] which have been linked to
canonical expressions derived from the Noether theorem through pseudo-gauge
transformations [20].

The experimental data [5] clearly show a decrease in the magnitude of global
polarization with increasing center-of-mass energy, approaching zero at the
highest RHIC and LHC energies. This trend renders low- and mid-energy
collisions particularly intriguing for the study of polarization
phenomenology. In this case, the assumptions of the Bjorken model become
inappropriate [22]. Consequently, herein111In this paper, we adopt the ‘mostly
minus’ metric convention, $g_{\alpha\beta}=\hbox{diag}(+1,-1,-1,-1)$, and
define the scalar product of four-vectors $a^{\alpha}$ and $b^{\alpha}$ as
$a\cdot b=a^{\alpha}b_{\alpha}=a^{0}b^{0}-{\bm{a}}\cdot{\bm{b}}$, with three-
vectors in bold. The Levi-Civita tensor $\epsilon^{\alpha\beta\gamma\delta}$
has $\epsilon^{0123}=+1$. The Lorentz-invariant momentum space measure is
$dP=d^{3}p/(E_{p}(2\pi)^{3})$, with particle energy
$E_{p}=\sqrt{m^{2}+{\bm{p}}^{2}}$ and four-momentum
$p^{\mu}=(E_{p},{\bm{p}})$. Anti-symmetrization is denoted by square brackets,
e.g., $M_{[\mu\nu]}=\frac{1}{2}(M_{\mu\nu}-M_{\nu\mu})$ for a tensor $M$. The
Hodge dual of a tensor ${C}^{\alpha\beta}$ is marked with a tilde. Directional
derivatives are abbreviated as
$U^{\alpha}\partial_{\alpha}\equiv\ensurestackMath{\stackon[1pt]{(\phantom{x})}{\mkern
2.0mu\bullet}}$,
$X^{\alpha}\partial_{\alpha}\equiv\ensurestackMath{\stackon[1pt]{(\phantom{x})}{\mkern
2.0mu\blacksquare}}$,
$Y^{\alpha}\partial_{\alpha}\equiv\ensurestackMath{\stackon[1pt]{(\phantom{x})}{\mkern
2.0mu\square}}$,
$Z^{\alpha}\partial_{\alpha}\equiv\ensurestackMath{\stackon[1pt]{(\phantom{x})}{\mkern
2.0mu\circ}}$, and divergence of a four-vector $A$ as
$\partial_{\alpha}A^{\alpha}\equiv\theta_{A}$. Furthermore, we use natural
units where $c=\hbar=k_{B}=1$. we break the boost invariance in the beam
direction, however, we still maintain the assumption of transverse
homogeneity. For simplicity, our analysis focuses on an ideal relativistic gas
composed of classical massive particles with spin one-half [56, 3]. Moving
away from the assumption of boost invariance introduces complex effects due to
the longitudinal expansion of the system. This also results in the mixing of
different electric-like and magnetic-like sectors of spin coefficients. By
adopting a physics-motivated initial condition for the hydrodynamic background
and spin variables, we evolve the system until freeze-out, and then analyze
the impact of such dynamics on spin polarization observables. Specifically, we
examine the dependence of the mean spin polarization vector on transverse
momentum, azimuthal angle, and rapidity. Our findings reveal distinctive
patterns in rapidity that align with other studies, potentially offering new
insights for future spin polarization measurements [57].

##  2 Relativistic hydrodynamics with spin

This section outlines the hydrodynamic framework for spin-$\frac{1}{2}$
particles using GLW energy-momentum and spin tensor forms. Within this
framework, the spin effects are assumed to be small, meaning they do not
appear in the conservation laws for charge, energy, and linear momentum but
only in the angular momentum conservation [18, 20].

The conservation of baryon current is $\partial_{\alpha}N^{\alpha}(x)=0$,
where the net baryon current is $N^{\alpha}={\cal N}U^{\alpha}$ with $U^{\mu}$
being the fluid four-flow and ${\cal N}=4\sinh(\xi){\cal N}_{(0)}(T)$ is the
net baryon density with ${\cal N}_{(0)}(T)=kT^{3}z^{2}K_{2}\left(z\right)$
being the number density of neutral classical spinless and massive particles.
Here, $k=1/2\pi^{2}$, $z$ is the mass over temperature ratio, $z=m/T$, and
$K_{2}$ is the modified Bessel function of second kind. The term
$4\sinh(\xi)=2\left(e^{\xi}-e^{-\xi}\right)$ accounts for spin degeneracy and
the inclusion of both particles and antiparticles. Here, $\xi$ is the ratio of
baryon chemical potential ${\mu_{B}}$ and temperature $T$,
$\xi\equiv{\mu_{B}}/T$.

The energy-momentum conservation is expressed as
$\partial_{\alpha}T^{\alpha\beta}(x)=0$ where the energy momentum tensor,
$T^{\alpha\beta}=({\cal E}+{\cal P})U^{\alpha}U^{\beta}-{\cal
P}g^{\alpha\beta}$, has the perfect-fluid form with ${\cal E}=4\cosh(\xi){\cal
E}_{(0)}(T)$ and ${\cal P}=4\cosh(\xi){\cal P}_{(0)}(T)$ being the energy-
density and pressure, respectively. Similarly to ${\cal N}_{(0)}(T)$, we
define ${\cal
E}_{(0)}(T)=kT^{4}z^{2}\left[zK_{1}\left(z\right)+3K_{2}\left(z\right)\right]$
and ${\cal P}_{(0)}(T)=T{\cal N}_{(0)}(T)$ [18].

The conservation equations for baryon current and energy-momentum constitute a
system of five partial differential equations for five variables: ${\mu_{B}}$,
$T$, and three independent components of $U^{\mu}$. Solving these perfect-
fluid equations provides the hydrodynamic background needed to determine spin
evolution. As our energy-momentum tensor is symmetric, the conservation of
total angular momentum implies the conservation of spin,
$\partial_{\alpha}S^{\alpha,\beta\gamma}(x)=0$, where [20]

$\displaystyle S^{\alpha,\beta\gamma}=\cosh(\xi)\left[U^{\alpha}\left({\cal
N}_{(0)}\omega^{\beta\gamma}+{\cal
A}_{(0)}U^{\delta}U^{[\beta}\omega^{\gamma]}_{\hphantom{\alpha}\delta}\right)+{\cal
B}_{(0)}\left(U^{[\beta}\Delta^{\alpha\delta}\omega^{\gamma]}_{\hphantom{\alpha}\delta}+2U^{(\alpha}\Delta^{{\delta)}[\beta}\omega^{\gamma]}_{\hphantom{\alpha}\delta}\right)\right]$
(1)

with thermodynamic coefficients ${\cal B}_{(0)}=-2({\cal E}_{(0)}+{\cal
P}_{(0)})/(Tz^{2})$ and ${\cal A}_{(0)}=2{\cal N}_{(0)}-3{\cal B}_{(0)}$.

###  2.1 Basis vectors and spin polarization tensor

While relaxing the boost-invariant description, it becomes necessary to
consider flow gradients that might develop in the longitudinal direction. To
accommodate this effect, we introduce the following parameterization:
$U^{\alpha}=\left(\cosh\Phi,0,\,0,\sinh\Phi\right)$ with the fluid rapidity
$\Phi=\vartheta(\tau,\eta)+\eta$.

Due to homogeneity in the $x$-$y$ plane, the transverse flow components are
zero. The other basis vectors are: $X^{\alpha}=\left(0,1,0,0\right),$
$Y^{\alpha}=\left(0,0,1,0\right),$ and
$Z^{\alpha}=\left(\sinh\Phi,0,0,\cosh\Phi\right)$. The directional derivatives
take the forms:
$U\cdot\partial=\cosh({\vartheta})\partial_{\tau}+\frac{\sinh({\vartheta})}{\tau}\partial_{\eta}$
and
$Z\cdot\partial=\sinh({\vartheta})\partial_{\tau}+\frac{\cosh({\vartheta})}{\tau}\partial_{\eta}$,
whereas the divergences are:
$\partial_{\alpha}U^{\alpha}=\frac{\cosh({\vartheta})}{\tau}+\ensurestackMath{\stackon[1pt]{\vartheta}{\mkern
2.0mu\circ}}$ and
$\partial_{\alpha}Z^{\alpha}=\frac{\sinh({\vartheta})}{\tau}+\ensurestackMath{\stackon[1pt]{\vartheta}{\mkern
2.0mu\bullet}}$. Given that all scalar functions depend on $\tau$
(longitudinal proper time) and $\eta$ (space-time rapidity), it follows that
the divergence of vectors $X$ and $Y$ vanishes, $\partial\cdot X=0$ and
$\partial\cdot Y=0$, and their directional derivatives are
$X\cdot\partial=\partial_{x}$ and $Y\cdot\partial=\partial_{y}$.

The spin polarization tensor $\omega^{\alpha\beta}$ in Eq. (1) can be
expressed in terms of spin polarization components ($C^{\prime}s$) [57]

$\displaystyle\omega_{\alpha\beta}=2\left(C_{\kappa
X}X_{[\alpha}U_{\beta]}+C_{\kappa Y}Y_{[\alpha}U_{\beta]}+C_{\kappa
Z}Z_{[\alpha}U_{\beta]}\right)+\epsilon_{\alpha\beta\gamma\delta}U^{\gamma}\left(C_{\omega
X}X^{\delta}+C_{\omega Y}Y^{\delta}+C_{\omega Z}Z^{\delta}\right).$ (2)

##  3 Background evolution

In scenarios involving non-boost-invariant expansion, we use the following
background equations: $\ensurestackMath{\stackon[1pt]{{\cal N}}{\mkern
2.0mu\bullet}}+{\cal N}~{}\theta_{U}=0$, $\ensurestackMath{\stackon[1pt]{{\cal
E}}{\mkern 2.0mu\bullet}}+({\cal E}+{\cal P})~{}\theta_{U}=0$, and $({\cal
E}+{\cal P})\ensurestackMath{\stackon[1pt]{U^{\alpha}}{\mkern
2.0mu\bullet}}-(\partial^{\alpha}-U^{\alpha}U^{\beta}\partial_{\beta}){\cal
P}=0$. Consequently, in the $\tau-\eta$ space, we solve three partial
differential equations. They determine temperature, baryon chemical potential,
and the correction to the longitudinal fluid rapidity.

Figure 1: Evolution of temperature (left) and fluid rapidity (right) shown as
a function of $\eta$ for various values of $\tau$. Solid lines indicate a non-
zero baryon chemical potential, while dashed lines represent its absence.
Black and red symbols mark freeze-out points at various times [57].

To properly include a non-trivial rapidity dependence of the initial energy
density profile, ${\cal E}_{0}(\eta)={\cal E}(\tau_{0},\eta)$, we introduce
the function: ${\cal E}_{0}(\eta)=\frac{{\cal
E}_{0}^{c}}{2}\left[\Theta(\eta)\left(\tanh(a-\eta
b)+1\right)+\Theta(-\eta)\left(\tanh(a+\eta b)+1\right)\right]$, where
$a=6.2$, $b=1.9$, and $\Theta$ is the Heaviside step function. In our model,
${\cal E}_{0}^{c}={\cal E}(T_{0}^{c},\mu_{B0}^{c})$ represents the initial
energy density at the center ($\eta=0$), with the initial central temperature
$T_{0}^{c}=T(\tau_{0},\eta=0)=0.26$ GeV and baryon chemical potential
$\mu_{B0}^{c}$. Regarding the baryon chemical potential profile, we consider a
constant value across rapidity, $\mu_{B0}(\eta)=\mu_{B0}^{c}=\rm{const.}$, and
explore two scenarios: either a vanishing baryon chemical potential
$\mu_{B0}^{c}=0$ (mimicking conditions typical in high-energy experiments), or
a non-zero value $\mu_{B0}^{c}=0.12$ GeV (to investigate effects associated
with baryon chemical potential at lower energies). Throughout our analysis,
the initial longitudinal flow profile is assumed to follow the Bjorken form,
$\Phi_{0}(\eta)=\eta$ [57].

Figure 1 illustrates the temporal evolution of temperature $T$ (left panel)
and longitudinal fluid rapidity correction $\vartheta$ (right panel) as
functions of space-time rapidity at various longitudinal proper times $\tau$,
starting from the initial time $\tau_{0}=0.25$ fm/c. Dashed lines represent
scenarios where $\mu_{B0}^{c}=0$, while solid lines indicate cases with
$\mu_{B0}^{c}=0.12$ GeV. Our analysis reveals that the temperature evolution
is symmetric with respect to $\eta$. Conversely, the evolution of the fluid
rapidity correction $\vartheta$ exhibits an asymmetry in $\eta$. At
midrapidity ($\eta=0$), we observe a decrease in temperature over time $\tau$,
similar to the Bjorken scenario [22]. Furthermore, at larger values of
$|\eta|$, gradients in energy density lead to the formation of gradients in
fluid velocity. It is also noteworthy that in the case of a non-zero baryon
chemical potential, there is a pronounced decrease of both $T$ and $\vartheta$
around $\eta\approx\pm 5$. Aside from this effect, the presence of baryon
chemical potential does not significantly impact the evolution of the
background parameters.

##  4 Spin components evolution

As the spin tensor given by Eq. (1) is antisymmetric in the last two indices,
we obtain six evolution equations for six independent spin components [57]

$\displaystyle\ensurestackMath{\stackon[1pt]{\alpha_{x1}}{\mkern
2.0mu\bullet}}+\ensurestackMath{\stackon[1pt]{\beta_{y1}}{\mkern
2.0mu\circ}}=-\alpha_{x1}\,\theta_{U}-\frac{\alpha_{x2}U\ensurestackMath{\stackon[1pt]{Z}{\mkern
2.0mu\circ}}}{2}-\beta_{y1}\theta_{Z}+\beta_{y2}U\ensurestackMath{\stackon[1pt]{Z}{\mkern
2.0mu\bullet}},$ (3)
$\displaystyle\ensurestackMath{\stackon[1pt]{\alpha_{y1}}{\mkern
2.0mu\bullet}}-\ensurestackMath{\stackon[1pt]{\beta_{x1}}{\mkern
2.0mu\circ}}=-\alpha_{y1}\theta_{U}-\frac{\alpha_{y2}U\ensurestackMath{\stackon[1pt]{Z}{\mkern
2.0mu\circ}}}{2}+\beta_{x1}\theta_{Z}-\beta_{x2}U\ensurestackMath{\stackon[1pt]{Z}{\mkern
2.0mu\bullet}},$ (4)
$\displaystyle\ensurestackMath{\stackon[1pt]{\alpha_{z1}}{\mkern
2.0mu\bullet}}=-\alpha_{z1}\theta_{U},$ (5)
$\displaystyle\frac{\ensurestackMath{\stackon[1pt]{\alpha_{y2}}{\mkern
2.0mu\circ}}}{2}+\ensurestackMath{\stackon[1pt]{\beta_{x2}}{\mkern
2.0mu\bullet}}=-\frac{\alpha_{y2}\theta_{Z}}{2}+\alpha_{y1}Z\ensurestackMath{\stackon[1pt]{U}{\mkern
2.0mu\bullet}}-\beta_{x2}\theta_{U}-\beta_{x1}Z\ensurestackMath{\stackon[1pt]{U}{\mkern
2.0mu\circ}},$ (6)
$\displaystyle\frac{\ensurestackMath{\stackon[1pt]{\alpha_{x2}}{\mkern
2.0mu\circ}}}{2}-\ensurestackMath{\stackon[1pt]{\beta_{y2}}{\mkern
2.0mu\bullet}}=-\frac{\alpha_{x2}\,\theta_{Z}}{2}+\alpha_{x1}\,Z\ensurestackMath{\stackon[1pt]{U}{\mkern
2.0mu\bullet}}+\beta_{y2}\theta_{U}+\beta_{y1}\,Z\ensurestackMath{\stackon[1pt]{U}{\mkern
2.0mu\circ}},$ (7)
$\displaystyle\ensurestackMath{\stackon[1pt]{\beta_{z2}}{\mkern
2.0mu\bullet}}=-\beta_{z2}\theta_{U},$ (8)

where $U\ensurestackMath{\stackon[1pt]{Z}{\mkern
2.0mu\circ}}=\cosh\vartheta\left(1+\frac{\partial\vartheta}{\partial\eta}\right)/\tau+\sinh(\vartheta)\frac{\partial\vartheta}{\partial\tau}$
and $U\ensurestackMath{\stackon[1pt]{Z}{\mkern
2.0mu\bullet}}=\sinh\vartheta\left(1+\frac{\partial\vartheta}{\partial\eta}\right)/\tau+\cosh(\vartheta)\frac{\partial\vartheta}{\partial\tau}$.
Above, we have also introduced the notation

$\displaystyle\alpha_{i1}=\alpha_{i2}=-\cosh({\xi}){\cal B}_{(0)}C_{\kappa
i},\quad\beta_{i1}=\frac{\cosh({\xi}){\cal B}_{(0)}}{2}C_{\omega
i},\quad\beta_{i2}=\cosh({\xi})\left({\cal N}_{(0)}-{\cal
B}_{(0)}\right)C_{\omega i}.$ (9)

It is important to highlight that, unlike the Bjorken expansion scenario [22],
the current model exhibits coupling between certain spin components [57].
Specifically, from Eqs. (3) and (7), we observe a coupling between the
components $C_{\kappa X}$ and $C_{\omega Y}$. In a similar manner, $C_{\kappa
Y}$ and $C_{\omega X}$ are also coupled, as evident from Eqs. (4) and (6).
However, this coupling effect does not extend to the longitudinal spin
components $C_{\kappa Z}$ and $C_{\omega Z}$, which continue to evolve
independently of the other components.

In our numerical simulations, we adopt an initialization scheme for the spin
components $C$ suggested by the physical considerations elaborated in Refs.
[22, 57, 58]. Given that the initial longitudinal fluid rapidity correction
$\vartheta_{0}(\eta)$ is zero (reflecting a Bjorken flow profile), the non-
zero $y$-component of the spin angular momentum at the initial time is
associated with the component $C_{\omega Y}$. This necessitates that
$C_{\omega Y}$ exhibits symmetry in $\eta$ [58]. Therefore, at the initial
time, it is sufficient to choose only $C^{0}_{\omega Y}(\eta)$ and keep
remaining spin components vanishing:

$\displaystyle C^{0}_{\omega Y}(\eta)=C_{\omega
Y}(\tau_{0},\eta)=0.1/{\cosh(\eta)}.$ (10)

Figure 2: Spin polarization components, $C_{\kappa X}$ (left) and $C_{\omega
Y}$ (right), evolution as a function of $\eta$ [57].

Figure 2 shows the outcomes of our numerical simulations for the spin
components $C_{\kappa X}$ and $C_{\omega Y}$. As previously mentioned, even
though $C_{\kappa X}$ is initially set to zero, it undergoes significant
evolution due to its coupling with $C_{\omega Y}$, as dictated by Eqs. (3) and
(7). It is important to note that all other spin components remain equal zero.
The symmetry in $\eta$ for both $C_{\kappa X}$ and $C_{\omega Y}$ is preserved
during the evolution, which is steered by the evolution equations (3) and (7),
and the initial condition [57].

The impact of the background evolution on spin, which manifests in the spin
equations of motion through the thermodynamic coefficients (9), is evident.
Similarly to the temperature evolution (as shown in Fig. 1), the coefficient
$C_{\omega Y}$ diminishes over time at the center. However, interestingly, at
large rapidities (around $\eta\approx\pm 5$), where the system approaches the
large mass limit, the dynamics of the spin reverses, leading to an increase in
the magnitude of $C_{\omega Y}$ with the proper time $\tau$. Furthermore, Fig.
2 reveals that the presence of a homogeneous non-zero (albeit small) baryon
chemical potential has negligible influence on the spin dynamics. This is
understandable, given that the baryon chemical potential primarily affects the
thermodynamic coefficients (9) through the term $\cosh(\xi)$ [57].

##  5 Spin polarization at freeze-out

Following our investigation of the spin polarization components, we proceed
now to calculate the average spin polarization of particles emitted at freeze-
out. This step is crucial for understanding the potential impact of spin
dynamics on experimentally measurable observables. Notably, the freeze-out
times $(\tau_{\rm FO})$ vary depending on the space-time rapidity of the fluid
cells, as illustrated by the black and red symbols in Fig. 1. In this context,
it becomes necessary to first define the freeze-out hypersurface
$(\Sigma_{\mu})$, which plays a pivotal role in the phase-space density of the
PL four-vector

$E_{p}\frac{d\Pi_{\mu}^{\mbox{*}}(p)}{d^{3}p}=-\frac{1}{(2\pi)^{3}m}\int\cosh(\xi)\Delta\Sigma_{\lambda}p^{\lambda}\,e^{-\beta\cdot
p}\,(\widetilde{\omega}_{\mu\beta}p^{\beta})^{\mbox{*}},$ (11)

where $\Delta\Sigma\cdot p=m_{T}\left[\tau_{\rm
FO}(\eta)\cosh\left(y_{p}-\eta\right)-\tau^{\prime}_{\rm
FO}(\eta)\sinh\left(y_{p}-\eta\right)\right]dxdyd\eta$, $\beta\cdot
p=(m_{T}/T)\cosh\left(y_{p}-\Phi\right)$, and
$(\widetilde{\omega}_{\mu\beta}p^{\beta})^{\mbox{*}}$ is the Lorentz
transformed tensor $\widetilde{\omega}_{\mu\beta}p^{\beta}$ to the particle
rest frame (PRF) [57]

$(\widetilde{\omega}_{\mu\beta}p^{\beta})^{\mbox{*}}=m\alpha_{p}p_{y}\left[\begin{array}[]{c}\phantom{-}0\\\
\\\ p_{x}\left[C_{\kappa X}\sinh(\Phi)+C_{\omega Y}\cosh(\Phi)\right]\\\ \\\
p_{y}\left[C_{\kappa X}\sinh\Phi\\!+\\!C_{\omega
Y}\cosh\Phi\right]-\frac{m_{T}}{m\alpha_{p}p_{y}}\left[C_{\kappa
X}\sinh\left(\Phi\\!-\\!y_{p}\right)\\!+\\!C_{\omega Y}\cosh\left(\Phi-
y_{p}\right)\right]\\\ \\\ -\left[m_{T}\left(C_{\kappa X}\cosh\left(\Phi-
y_{p}\right)+C_{\omega
Y}\sinh\left(\Phi\\!-\\!y_{p}\right)\right)+m\left(C_{\kappa
X}\cosh\Phi\\!+\\!C_{\omega Y}\sinh\Phi\right)\right]\\\ \end{array}\right],$
(12)

with $\alpha_{p}\equiv 1/(m^{2}+mE_{p})$ [19]. It is important to point out
that we have included here only the components $C_{\kappa X}$ and $C_{\omega
Y}$. This decision is based on the fact that, within the scope of our current
numerical analysis, these are the only non-vanishing components. The mean spin
polarization per particle, $\langle\pi_{\mu}\rangle_{p}$, is the ratio of the
momentum density of the total PL four-vector (11) to the particle momentum
density [20]

$\displaystyle{\langle\pi_{\mu}\rangle}_{p}=\frac{E_{p}\frac{d\Pi_{\mu}^{\mbox{*}}(p)}{d^{3}p}}{E_{p}\frac{d{\cal{N}}(p)}{d^{3}p}},\quad\text{where}\quad
E_{p}\frac{d{\cal{N}}(p)}{d^{3}p}=\frac{4}{(2\pi)^{3}}\int\Delta\Sigma_{\lambda}p^{\lambda}\cosh(\xi)\,e^{-\beta\cdot
p}.$ (13)

Using Eq. (13) we compute the average spin polarization per particle as a
function of momentum. The results of this computation, particularly for the
scenario with a vanishing baryon chemical potential at midrapidity and forward
rapidity, are shown in Figs. 3–4. As previously demonstrated in Fig. 2, the
presence of a non-vanishing baryon chemical potential exerts minimal influence
on the dynamics of spin. This observation extends to the momentum-dependent
polarization as well. Therefore, in our discussion we have chosen not to
present these results.

Figure 3: ${\langle\pi_{x}\rangle}_{p}$ (left) and
${\langle\pi_{y}\rangle}_{p}$ (right) components at mid-rapidity as a function
of $p_{x}$ and $p_{y}$ [57].

Figure 4: ${\langle\pi_{x}\rangle}_{p}$ (left), ${\langle\pi_{y}\rangle}_{p}$
(middle), and ${\langle\pi_{z}\rangle}_{p}$ (right) components at forward
rapidity as a function of $p_{x}$ and $p_{y}$ [57].

Figure 3 (left panel) shows the ${\langle\pi_{x}\rangle}_{p}$ component of the
mean polarization vector, exhibiting a quadrupole structure with alternating
signs across the quadrants [57]. The quadrupole structure primarily results
from the $p_{x}p_{y}$ term in the $x$ component of
$(\widetilde{\omega}_{\mu\beta}p^{\beta})^{\mbox{*}}$, Eq. (12), and its
magnitude diminishes with increasing rapidity, see Fig. 4 (left panel). The
${\langle\pi_{y}\rangle}_{p}$ component, Fig. 3 (right panel), is negative,
reflecting the initial spin angular momentum direction set in the hydrodynamic
equations. Its magnitude also decreases as rapidity increases, eventually
showing no dependence on $\phi_{p}$, see Fig. 4 (middle panel).

Of particular interest from the experimental perspective is the longitudinal
spin polarization, $\langle\pi_{z}\rangle_{p}$, along the beam ($z$) direction
[6, 14]. Its behavior, which is still seeking a definitive explanation, can be
understood using symmetry considerations in Eqs. (11) and (12). The symmetric
integration range in $\eta$ means that only $\eta$-even integrands contribute.
Given that $C_{\kappa X}$ and $C_{\omega Y}$ are odd and even functions of
$\eta$ respectively, $\langle\pi_{z}\rangle_{p}$ is zero at midrapidity.
However, at forward rapidities, we observe a distinct pattern in longitudinal
polarization, as illustrated in Fig. 4 (right panel). It is noteworthy that
the results presented here do not replicate the quadrupole structure of
longitudinal spin polarization primarily due to the assumption of homogeneity
in the transverse plane.

Figure 5: Momentum integrated spin polarization, $\langle\pi_{y}\rangle$, as a
function of rapidity (left), and in terms of $p_{T}$ (middle) and $\phi_{p}$
(right), with dashed lines for zero and solid lines for non-zero baryon
chemical potential, respectively [57].

From the experimental point of view, it is not only beneficial to calculate
differential spin observables but also to consider the integrated
$\langle\pi_{\mu}\rangle$ components, where the momentum index $p$ is
intentionally omitted. To obtain the spin polarization
$\langle\pi_{\mu}\rangle$, we integrate Eq. (13) over the momentum
coordinates. Our findings indicate that, given the adopted initialization
parameters and the symmetry properties of the spin polarization components,
only $\langle\pi_{y}\rangle$ component is non-vanishing and it is negative.
Figure 5 (left panel) illustrates the variation of global polarization as a
function of rapidity. Notably, at midrapidity the magnitude of
$\langle\pi_{y}\rangle$ reaches its peak and subsequently diminishes at
forward rapidities, suggesting that the primary source of hyperon polarization
is the midrapidity region. This pattern aligns qualitatively with other models
[40] and is a focus of forthcoming experimental investigations [59]. The
magnitude of $\langle\pi_{y}\rangle$ at $y_{p}=0$ is comparable, in
qualitative terms, to global polarization measurements [5].

The behavior of spin polarization in relation to transverse momentum $(p_{T})$
and azimuthal angle $(\phi_{p})$ is also of significant interest.
Understanding this relationship can provide crucial insights into the dynamics
of spin polarization within the transverse-momentum plane [58]. By integrating
Eq. (13) over transverse momentum or azimuthal angle, we can obtain the global
polarization as a function of $\phi_{p}$ or $p_{T}$, respectively [58],

$\displaystyle\langle\pi_{\mu}(\phi_{p})\rangle=\frac{\int
p_{T}dp_{T}E_{p}\,\frac{d\Pi_{\mu}^{\mbox{*}}(p)}{d^{3}p}}{\int
d\phi_{p}p_{T}dp_{T}E_{p}\,\frac{d{\cal{N}}(p)}{d^{3}p}},\quad\langle\pi_{\mu}(p_{T})\rangle=\frac{\frac{1}{2\,\pi}\int
d\phi_{p}\sin(2\phi_{p})E_{p}\,\frac{d\Pi_{\mu}^{\mbox{*}}(p)}{d^{3}p}}{\int
d\phi_{p}E_{p}\,\frac{d{\cal{N}}(p)}{d^{3}p}}\,.$ (14)

Figure 5 illustrates the variation of $\langle\pi_{y}\rangle$ in $p_{T}$
(middle panel) and $\phi_{p}$ (right panel). We observe a strong dependence of
polarization on $p_{T}$, which is more pronounced than in other models [40]
and experimental findings [60]. This may be attributed to our assumption of a
non-zero initial spin polarization that evolves over time. In the cases where
$p_{T}$ dependence is weak, the polarization may originate from spin-orbit
coupling, a factor not currently incorporated in our framework.

The $\phi_{p}$ dependence of polarization is particularly significant at
midrapidity. Within the range of $0<\phi_{p}<\pi/2$, this behavior
qualitatively aligns with the polarization trends observed in experiments [60,
40]. Overall, our analysis indicates that the impact of a non-zero baryon
chemical potential on $\langle\pi_{y}\rangle$ is relatively minor. However, it
is interesting to note that this effect manifests differently at forward
rapidity compared to midrapidity, with opposing influences in these regions
[57].

##  6 Summary

In our study, utilizing the spin hydrodynamics framework developed in Ref.
[20], we have examined the space-time evolution of spin polarization in one-
dimensional, non-boost-invariant and transversely homogeneous systems. This
work extends our previous study [22] focused on spin polarization evolution in
the Bjorken model. Our current analysis reveals interdependent behavior of
different spin coefficients. Our calculations include both momentum-dependent
and momentum-averaged components of the mean spin polarization vector of
$\Lambda$ particles at mid ($y_{p}=0$) and forward ($y_{p}=2$) rapidities. As
anticipated, the momentum-averaged spin polarization $y$-component is the only
non-zero contribution to the total polarization vector. Additionally, we
observe interesting characteristics in the $p_{T}$ and $\phi_{p}$ dependencies
of spin polarization, in particular, a distinctive decay of
$\langle\pi_{y}\rangle$ at forward rapidities [57]. Our findings suggest that
a more realistic description of measured quantities requires breaking of the
translational symmetry in the transverse plane and adopting a full
(3+1)-dimensional model. Further investigations in this direction are planned
for future studies.

Acknowledgements. This work was supported in part by the Polish National
Science Centre Grants No. 2022/47/B/ST2/01372 (W.F.) and No.
2018/30/E/ST2/00432 (R.R.). R.S. also acknowledges the support of Polish NAWA
Bekker program No.: BPN/BEK/2021/1/00342.

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