# Hadronic effects on $\Lambda$ polarization in relativistic heavy ion
collisions

Haesom Sung<EMAIL_ADDRESS>Department of Physics and Institute of
Physics and Applied Physics, Yonsei University, Seoul 03722, Korea  Cyclotron
Institute, Texas $A\&M$ University, College Station, TX 77843, USA  Che Ming
Ko<EMAIL_ADDRESS>Cyclotron Institute, Texas $A\&M$ University, College
Station, TX 77843, USA  Department of Physics and Astronomy, Texas $A\&M$
University, College Station, TX 77843, USA  Su Houng Lee<EMAIL_ADDRESS>Department of Physics and Institute of Physics and Applied Physics, Yonsei
University, Seoul 03722, Korea

###### Abstract

The $\Lambda$ hyperon spin flip and non-flip cross sections are calculated in
a simple hadronic model by including both the $s$-channel process involving
the spin 3/2, positive parity $\Sigma^{*}(1358)$ resonance and the $t$-channel
process via the exchange of a scalar $\sigma$ meson. Although the $t$-channel
process gives a thermally averaged cross section that is about a factor of 1.3
larger than that from the $s$-channel process, the $\Lambda$ spin flip to non-
flip cross sections is negligibly small in the $t$-channel compared to the
constant value of 1/3.5 in the $s$-channel process. With these cross sections
included in a schematic kinetic model, the effects of hadronic scatterings on
the $\Lambda$ spin polarization in Au-Au collisions at $\sqrt{s}_{NN}=7.7$ GeV
are studied. It is found that the $\Lambda$ spin polarization only decreases
by about 7% during the hadronic stage of these collisions, which thus
justifies the assumption in theoretical studies that compare the $\Lambda$
polarization calculated at the chemical freeze out to the measured one at the
kinetic freeze out.

##  I Introduction

Lambda ($\Lambda$) hyperons produced in non-central Au-Au collisions at the
Relativistic Heavy ion Collider (RHIC) have been found by the STAR
Collaboration Adamczyk  _et al._ (2017a) to be partially polarized in the
direction perpendicular to the reaction plane, with a magnitude increasing
with decreasing collision energy. Both hydrodynamic Karpenko and Becattini
(2017) and coarse-grained transport Li  _et al._ (2017) models have been used
to describe the measured data by assuming that $\Lambda$ hyperons are in
thermal equilibrium with the vorticity field present at the chemical freeze-
out of the collisions. The experimental data can also be described by the non-
equilibrium chiral kinetic approach Sun and Ko (2017a), which takes into
account the effect of the vorticity field on the equations of motion and
scatterings of quarks and antiquarks, by assuming that $\Lambda$ hyperons are
formed from the coalescence of polarized $s$ quarks with polarized $u$ and $d$
quarks at hadronization Sun and Ko (2017b). Besides the above $\Lambda$ global
polarization, the STAR Collaboration has also measured the $\Lambda$ local
polarization along the beam direction and its azimuthal angle dependence in
the transverse plane of the collisions Niida (2019). Although both
hydrodynamic Karpenko and Becattini (2017) and coarse-grained transport Li
_et al._ (2017) models fail to describe the measured $\Lambda$ local
polarization by giving an opposite sign in the azimuthal angle dependence, the
correct sign is obtained in the chiral kinetic approach Sun and Ko (2019) due
to the redistribution of axial charges in the transverse plane. The latter
result has been confirmed by a study based on a covariant angular-momentum-
conserved chiral transport model Liu  _et al._ (2020). This dynamic effect in
the chiral kinetic approach can be included in the hydrodynamic approach by
adding the effect from the thermal shear field Liu and Yin (2021); Becattini
_et al._ (2021a); Yi  _et al._ (2021), as in the study based on the 3+1 MUSIC
hydrodynamic model Fu  _et al._ (2021) and the 3+1 viscous eHLLe and EVHO-QGP
hydrodynamic models Becattini  _et al._ (2021b).

Since both $\Lambda$ global and local spin polarizations decrease with
temperature as the hadronic matter cools Sun  _et al._ (2022), the fact that
the experimental data is consistent with the $\Lambda$ spin polarization at
the hadronization temperature indicates that it freezes out or decouples early
from the hadronic matter. It was pointed out in Ref. Ko (2023) that this would
be the case if the $\Lambda-\pi$ scattering is through the spin 3/2, parity
positive $\Sigma^{*}(1358)$ resonance since the ratio of $\Lambda$ spin non-
flip to flip probabilities in this scattering is 3.5 based on a simple
consideration of $\Lambda$ spin and pion orbital angular momentum couplings in
this scattering. A similar conclusion is reached in a recent study of the
temperature dependence of the $\Lambda$ spin relaxation time in a hot pion gas
based also on the $s$-channel $\Sigma^{*}(1358)$ resonance approximation
Hidaka  _et al._ (2023). In the present paper, the $\Lambda$ spin flip and
non-flip cross sections by a pion are explicitly calculated not only for the
$s$-channel process suggested in Ref. Ko (2023) but also for the $t$-channel
process involving the exchange of the scalar $\sigma$ meson. Including these
cross sections in a schematic kinetic equation makes it possible to quantify
the effect of hadronic scatterings on the $\Lambda$ spin polarization as the
hadronic matter expands from the chemical to kinetic freeze out. Results from
the present study shows that a spin polarized $\Lambda$ produced at the
chemical freeze-out of heavy ion collisions in Au-Au collisions at
$\sqrt{s_{NN}}=7.7$ GeV only looses its spin polarization by about 7% after it
decouples at the kinetic freeze-out. This result thus justifies the assumption
used in the literature of comparing the calculated $\Lambda$ spin polarization
at chemical freeze-out to the measured value in experiments.

The paper is organized as follows. In Sec. II, the $\Lambda-\pi$ spin-
dependent scattering cross sections are calculated. Their thermal averages are
given in Sec. III, which is followed by the presentation of the schematic
kinetic approach in Sec. IV. The time evolution of the temperature of the
hadronic matter and the pion fugacity in Au-Au collisions at
$\sqrt{s_{NN}}=7.7$ GeV are given in Sec. V. Results are then shown in Sec.
VI, and conclusions are drawn in Sec. VII.

##  II $\Lambda$ spin-dependent scattering cross sections with pion

Figure 1: Feynman diagrams for $\Lambda$ scattering by pion in (a) $s$-channel
and (b) $t$-channel processes with solid and dashed lines indicating $\Lambda$
and $\pi$, respectively. The double solid line in $s$-channel and the dotted
line in $t$-channel denote $\Sigma^{*}(1358)$ resonance and $\sigma$ meson,
respectively.

Since the hadronic matter produced in relativistic heavy ion collisions
consists mostly of pions, only the spin-dependent scattering cross sections of
$\Lambda$ with a pion are considered in this Section. The two Feynman diagrams
for the $s$\- and $t$-channel processes in this scattering are shown in Fig.
1.

###  II.1 The $s$-channel process

For the $s$-channel $\Lambda-\pi$ scattering through the $\Sigma^{*}(1358)$
resonance shown in Fig. 1(a), the spin averaged cross section can be
approximated by the Breit-Wigner formula,

$\displaystyle{\bar{\sigma}}(s)$ $\displaystyle=$
$\displaystyle\frac{2s_{\Sigma^{*}}+1}{(2s_{\Lambda}+1)(2s_{\pi}+1)}\frac{4\pi}{k^{2}}\frac{s\Gamma^{2}(s)}{(s-m_{\Sigma^{*}}^{2})^{2}+s\Gamma^{2}(s)}$
(1) $\displaystyle=$
$\displaystyle\frac{8\pi}{k^{2}}\frac{s\Gamma^{2}(s)}{(s-m_{\Sigma^{*}}^{2})^{2}+s\Gamma^{2}(s)},$

where $s_{\Lambda}=1/2$, $s_{\pi}=0$, and $s_{\Sigma^{*}}=3/2$ are used for
the spins of $\Lambda$, pion, and $\Sigma^{*}$, respectively. In Eq.(1), $s$
is the square of total center-of-mass energy, $k$ is the magnitude of the
momentum of $\Lambda$ and $\pi$ in their center-of-mass frame, i.e.,

$k=\sqrt{\frac{[s-(m_{\Lambda}+m_{\pi})^{2}][s-(m_{\Lambda}-m_{\pi})^{2}]}{4s}},$
(2)

$m_{\Sigma^{*}}=1382$ MeV is the mass of $\Sigma^{*}$ resonance, and
$\Gamma(s)$ is the width of $\Sigma^{*}$ if its mass is $\sqrt{s}$.

For a polarized $\Lambda$, its spin flip and non-flip scattering cross
sections by a pion in the $s$-channel through the $\Sigma^{*}$ resonance are
then proportional to the spin averaged scattering cross section in Eq.(1) with
their relative magnitude determined by the Clebsch-Gordan coefficients
$\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle$ for the coupling of the $\Lambda$
spin state $|j_{1},m_{1}\rangle$, the pion orbital angular momentum state
$|j_{1},m_{2}\rangle$ to the $\Sigma^{*}$ spin state $|J,M\rangle$. For a
$\Lambda$ having an initial spin pointing along the $z$-direction, the cross
section for its spin to remain pointing in this direction after the scattering
with a pion is .

$\displaystyle\sigma_{++}=\frac{A}{3}\sum_{m,M}\Big{|}\left\langle\frac{1}{2}\frac{1}{2}1m\Big{|}\frac{3}{2}M\right\rangle\Big{|}^{4}\bar{\sigma}$
$\displaystyle\hskip
17.34518pt=\frac{A}{3}\left[\Big{|}\left\langle\frac{1}{2}\frac{1}{2}11\Big{|}\frac{3}{2}\frac{3}{2}\right\rangle\Big{|}^{4}+\Big{|}\left\langle\frac{1}{2}\frac{1}{2}10\Big{|}\frac{3}{2}\frac{1}{2}\right\rangle\Big{|}^{4}\right.$
$\displaystyle\hskip
26.01724pt+\left.\Big{|}\left\langle\frac{1}{2}\frac{1}{2}1-1\Big{|}\frac{3}{2}-\frac{1}{2}\right\rangle\Big{|}^{4}\right]\bar{\sigma}$
$\displaystyle\hskip
17.34518pt=\frac{A}{3}\left[1+\left(\sqrt{\frac{2}{3}}\right)^{4}+\left(\sqrt{\frac{1}{3}}\right)^{4}\right]\bar{\sigma}=\frac{14A}{27}\bar{\sigma},$
(3)

where the factor of 1/3 comes from averaging over the three components of the
pion orbital angular momentum and $A$ is a normalization constant. Similarly,
the cross section for the $\Lambda$ to have its spin pointing in the negative
$z$-direction after scattering with the pion is

$\displaystyle\sigma_{+-}=\frac{A}{3}\sum_{m,M}\Big{|}\left\langle\frac{1}{2}-\frac{1}{2}1m\Big{|}\frac{3}{2}M\right\rangle\left\langle\frac{3}{2}M\Big{|}\frac{1}{2}\frac{1}{2}1m\right\rangle\Big{|}^{2}\bar{\sigma}$
$\displaystyle\hskip
17.34518pt=\frac{A}{3}\left[\Big{|}\left\langle\frac{1}{2}-\frac{1}{2}11\Big{|}\frac{3}{2}\frac{1}{2}\right\rangle\left\langle\frac{3}{2}\frac{1}{2}\Big{|}\frac{1}{2}\frac{1}{2}10\right\rangle\Big{|}^{2}\right.$
$\displaystyle\hskip
26.01724pt+\left.\Big{|}\left\langle\frac{1}{2}-\frac{1}{2}10\Big{|}\frac{3}{2}-\frac{1}{2}\right\rangle\left\langle\frac{3}{2}-\frac{1}{2}\Big{|}\frac{1}{2}\frac{1}{2}1-1\right\rangle\Big{|}^{2}\right]\bar{\sigma}$
$\displaystyle\hskip
17.34518pt=\frac{A}{3}\left[\left(\sqrt{\frac{1}{3}}\sqrt{\frac{2}{3}}\right)^{2}+\left(\sqrt{\frac{2}{3}}\sqrt{\frac{1}{3}}\right)^{2}\right]\bar{\sigma}=\frac{4A}{27}\bar{\sigma}.$
(4)

The ratio of the spin non-flip to spin flip cross sections of a polarized
$\Lambda$ is thus $R=\sigma_{++}/\sigma_{+-}=3.5$ as given in Ref. Ko (2023).
Requiring $\sigma_{++}+\sigma_{+-}=\frac{2A}{3}\bar{\sigma}=\bar{\sigma}$
gives $A=3/2$, which leade to $\sigma_{++}=\frac{7}{9}\bar{\sigma}$ and
$\sigma_{+-}=\frac{2}{9}\bar{\sigma}$. Shown in Fig. 2 by the solid line is
the $s$-channel spin-averaged $\Lambda-\pi$ scattering cross section as a
function of $\sqrt{s}-\sqrt{s_{0}}$ with $\sqrt{s_{0}}=m_{\Lambda}+m_{\pi}$,
which is seen to peak at $\sqrt{s}=m_{\Sigma^{*}}$ as expected.

Figure 2: Spin averaged $\Lambda+\pi\rightarrow\Lambda+\pi$ cross section for
the $s$-channel (solid line) and the $t$-channel (dashed line) process as
functions of $\sqrt{s}-\sqrt{s_{0}}$. Also shown in the inset is the ratio of
spin flip to non-flip cross sections for the $t$-channel process.

###  II.2 The $t$-channel process

The $t$-channel contribution to $\Lambda-\pi$ scattering shown by the Feynman
diagram in Fig. 1(b) can be evaluated by using the following interaction
Lagrangian densities,

$\displaystyle{\cal
L}_{\sigma\Lambda\Lambda}=g_{\sigma\Lambda\Lambda}\bar{\Lambda}\Lambda\sigma,$
$\displaystyle{\cal L}_{\sigma\pi\pi}=g_{\sigma\pi\pi}\sigma\pi\pi.$ (5)

For the coupling constant $g_{\sigma\Lambda\Lambda}$, it has the value
$g_{\sigma\Lambda\Lambda}=\frac{2}{3}g_{NN\sigma}=7.07$ if one uses the
$g_{\sigma NN}$ in Ref. Machleidt  _et al._ (1987) and the light quark
counting rule for the relation between the couplings of light baryons and
strange baryons to light mesons. The $g_{\sigma\pi\pi}$ can be determined from
the $\sigma$ meson decay width according to

$\Gamma_{\sigma}=\frac{g^{2}_{\sigma\pi\pi}}{8\pi
m^{2}_{\sigma}}\frac{\sqrt{m^{2}_{\sigma}-4m_{\pi}^{2}}}{2}.$ (6)

Using $\Gamma_{\sigma}(s=m_{\sigma}^{2})\approx 200$ MeV Patrignani  _et al._
(2016), the coupling constant $g_{\sigma\pi\pi}$ is found to have the value
$g_{\sigma\pi\pi}=2.37$ GeV.

For the $t$-channel process in
$\Lambda(p_{1})+\pi(p_{2})\rightarrow\Lambda(p_{3})+\pi(p_{4})$ scattering,
its invariant amplitude is

$i\mathcal{M}_{t}=g_{\Lambda\Lambda\sigma}g_{\sigma\pi\pi}\bar{u}(p_{3})\frac{i}{t-m_{\sigma}^{2}+im_{\sigma}\Gamma_{\sigma}}u(p_{1}),$
(7)

where $t=(p_{1}-p_{3})^{2}=-2p^{2}(1-\cos\theta)$ with $p$ and $\theta$ being
the magnitude of and angle between $\Lambda$ initial and final momenta in
their center-of-mass frame, respectively. The spin-averaged squared invariant
amplitude for the $t$-channel process is then

$\displaystyle|\overline{\mathcal{M}_{t}}|^{2}$ $\displaystyle=$
$\displaystyle\frac{g_{\Lambda\Lambda\sigma}^{2}g_{\sigma\pi\pi}^{2}}{2}\frac{{\rm{Tr}}\left[(\not{p}_{3}+m_{\Lambda})(\not{p}_{1}+m_{\Lambda})\right]}{(t-m_{\sigma}^{2})^{2}+m_{\sigma}^{2}\Gamma^{2}_{\sigma}}$
(8) $\displaystyle=$ $\displaystyle
2g_{\Lambda\Lambda\sigma}^{2}g_{\sigma\pi\pi}^{2}\frac{p_{1}\cdot
p_{3}+m_{\Lambda}^{2}}{(t-m_{\sigma}^{2})^{2}+m_{\sigma}^{2}\Gamma^{2}_{\sigma}}$
$\displaystyle=$ $\displaystyle
g_{\Lambda\Lambda\sigma}^{2}g_{\sigma\pi\pi}^{2}\frac{4m_{\Lambda}^{2}-t}{(t-m_{\sigma}^{2})^{2}+m_{\sigma}^{2}\Gamma^{2}_{\sigma}}.$

To take into account the finite size of hadrons, form factors are introduced
at the $\sigma\Lambda\Lambda$ and $\sigma\pi\pi$ vertices. Using the monopole
form factor of the form

$F(t)=\frac{\Lambda^{2}+t}{\Lambda^{2}+m_{\sigma}^{2}},$ (9)

with the value $\Lambda=1.8$ GeV as in Ref. Sung  _et al._ (2021) for meson-
meson scatterings, the spin-averaged differential $t$-channel cross section
can then be calculated from

$\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^{2}s}|\overline{\mathcal{M}_{t}}|^{2}F^{4}(t).$
(10)

The total spin-averaged $t$-channel cross section after integrating the
scattering angles is shown in Fig. 2 as a function of $\sqrt{s}-\sqrt{s_{0}}$
by the dashed line. Its value is larger than that of the $s$-channel cross
section except at energy near the $\Sigma^{*}$ resonance.

Although it is possible to calculate the $\Lambda$ spin flip and non-flip
cross sections in the $t$-channel process by using the $\Lambda$ spin
projection operator, the explicit spin-dependent spinors of the $\Lambda$ are
used in the present study. In this approach, the spin-dependent amplitude is
proportional to the scalar product of $\Lambda$ spinors with initial spin
direction $s_{i}$ and final spin direction $s_{f}$, i.e.,

$\displaystyle\bar{u}_{s_{f}}(p_{3})u_{s_{i}}(p_{1})=(E_{\Lambda}+m_{\Lambda})\begin{pmatrix}\chi^{\dagger}_{s_{f}},\chi^{\dagger}_{s_{f}}\frac{{\bf
p}_{3}\cdot{\boldsymbol{\sigma}}}{E_{\Lambda}+m}\end{pmatrix}$
$\displaystyle\hskip 72.26999pt\times\begin{pmatrix}\chi_{s_{i}}\\\
-\frac{{\bf
p}_{1}\cdot{\boldsymbol{\sigma}}}{E_{\Lambda}+m_{\Lambda}}\chi_{s_{i}}\end{pmatrix}$
$\displaystyle\hskip
36.135pt=(E_{\Lambda}+m_{\Lambda})\left\\{\left[1-\frac{{\bf p}_{3}\cdot{\bf
p}_{1}}{(E_{\Lambda}+m_{\Lambda})^{2}}\right]\chi^{\dagger}_{s_{f}}\chi_{s_{i}}\right.$
$\displaystyle\hskip 43.36243pt\left.-i\frac{{\bf p}_{3}\times{\bf
p}_{1}}{(E_{\Lambda}+m_{\Lambda})^{2}}\cdot\chi^{\dagger}_{s_{f}}{\boldsymbol{\sigma}}\chi_{s_{i}}\right\\}.$
(11)

The $\Lambda$ spin flip and non-flip amplitudes can be explicitly evaluated
from Eq.(II.2) by taking the $x-z$ plane to be the scattering plane with the
$\Lambda$ initial momentum ${\bf p}_{1}=p\hat{z}$ and final momentum ${\bf
p}_{3}=p(\sin\theta\hat{x}+\cos\theta\hat{z})$, which leads to ${\bf
p}_{3}\times{\bf p}_{1}=-p^{2}\sin\theta\hat{y}$ and thus ${\bf
p}_{3}\times{\bf
p}_{1}\cdot\chi^{\dagger}_{s_{f}}{\boldsymbol{\sigma}}\chi_{s_{i}}=-p^{2}\sin\theta\chi^{\dagger}_{s_{f}}\sigma_{y}\chi_{s_{i}}$.
Along a direction characterized by the polar angles $(\theta_{s},\phi_{s})$,
the $\Lambda$ spin up and down states are

$\displaystyle\chi_{+}=\left(\begin{matrix}\cos\frac{\theta_{s}}{2}\cr
e^{i\phi_{s}}\sin\frac{\theta_{s}}{2}\end{matrix}\right),~{}\chi_{-}=\left(\begin{matrix}\sin\frac{\theta_{s}}{2}\cr-e^{i\phi_{s}}\cos\frac{\theta_{s}}{2}\end{matrix}\right).$
(12)

Using these expressions and $\sigma_{y}=\left(\begin{matrix}0&-i\cr
i&0\end{matrix}\right)$, one has

$\displaystyle\chi^{\dagger}_{+}\sigma_{y}\chi_{+}$ $\displaystyle=$
$\displaystyle\sin\phi_{s}\sin\theta_{s},$
$\displaystyle\chi^{\dagger}_{-}\sigma_{y}\chi_{+}$ $\displaystyle=$
$\displaystyle-\cos\theta_{s}\sin\phi_{s}-i\cos\phi_{s},$ (13)

which lead to

$\displaystyle\bar{u}_{+}(p_{3})u_{+}(p_{1})=(E_{\Lambda}+m_{\Lambda})\Bigg{[}1-\frac{p^{2}\cos\theta}{(E_{\Lambda}+m_{\Lambda})^{2}}$
$\displaystyle\hskip
72.26999pt+i\frac{p^{2}\sin\theta}{\left(E+m\right)^{2}}\sin\phi_{s}\sin\theta_{s}\Bigg{]},$
(14)
$\displaystyle\bar{u}_{-}(p_{3})u_{+}(p_{1})=\frac{p^{2}\sin\theta}{E_{\Lambda}+m_{\Lambda}}(\cos\phi_{s}-i\cos\theta_{s}\sin\phi_{s}).$

With above results, one obtains the following expression for the squared
$\Lambda$ spin non-flip and flip amplitudes after including the coupling
constants from the vertices and the propagator of exchanged $\sigma$ meson,

$\displaystyle|\mathcal{M}_{t++}|^{2}=\frac{g_{\Lambda\Lambda\sigma}^{2}g_{\sigma\pi\pi}^{2}}{(t-m_{\sigma}^{2})^{2}+m_{\sigma}^{2}\Gamma_{\sigma}^{2}}$
$\displaystyle\hskip
50.58878pt\times(E_{\Lambda}+m_{\Lambda})^{2}\Bigg{[}\left(1-\frac{p^{2}\cos\theta}{(E_{\Lambda}+m_{\Lambda})^{2}}\right)^{2}$
(16) $\displaystyle\hskip
50.58878pt+\frac{p^{4}\sin^{2}\theta}{\left(E+m\right)^{4}}\sin^{2}{\theta_{s}}\sin^{2}{\phi_{s}}\Bigg{]},$
$\displaystyle|\mathcal{M}_{t+-}|^{2}=\frac{g_{\Lambda\Lambda\sigma}^{2}g_{\sigma\pi\pi}^{2}}{(t-m_{\sigma}^{2})^{2}+m_{\sigma}^{2}\Gamma_{\sigma}^{2}}\frac{p^{4}\sin^{2}\theta}{(E_{\Lambda}+m_{\Lambda})^{2}}$
$\displaystyle\hskip
50.58878pt\times(\cos^{2}\theta_{s}\sin\phi^{2}_{s}+\cos^{2}\phi_{s}).$

It can be easily shown from Eqs.(8), (16), and (16) that
$|\mathcal{M}_{t++}|^{2}+|\mathcal{M}_{t+-}|^{2}=|\overline{\mathcal{M}_{t}}|^{2}$
as expected.

Eq.(16) shows that, with the $\Lambda$ having initial momentum in the
$z$-direction and final momentum in the $x-z$ plane, it has the largest spin-
flip cross section if it is initially longitudinally polarized along the
$z$-direction ($\theta_{s}=0$) or transversely polarized along the
$x$-direction ($\theta_{s}=\pi/2$ with $\phi_{s}=0$). In these cases, Eqs.(16)
and (16) become

$\displaystyle|\mathcal{M}_{t++}|^{2}=\frac{g_{\Lambda\Lambda\sigma}^{2}g_{\sigma\pi\pi}^{2}}{(t-m_{\sigma}^{2})^{2}+m_{\sigma}^{2}\Gamma_{\sigma}^{2}}$
$\displaystyle\hskip
50.58878pt\times(E_{\Lambda}+m_{\Lambda})^{2}\left[1-\frac{p^{2}\cos\theta}{(E_{\Lambda}+m_{\Lambda})^{2}}\right]^{2},$
(17)
$\displaystyle|\mathcal{M}_{t+-}|^{2}=\frac{g_{\Lambda\Lambda\sigma}^{2}g_{\sigma\pi\pi}^{2}}{(t-m_{\sigma}^{2})^{2}+m_{\sigma}^{2}\Gamma_{\sigma}^{2}}\frac{p^{4}\sin^{2}\theta}{(E_{\Lambda}+m_{\Lambda})^{2}}.$
(18)

The ratio of spin flip to non-flip cross sections calculated from Eq.(10)
using Eqs.(17) and (18) in Eq.(10) and after integrating the scattering angle
is shown in the inset of Fig. 2, which is seen to be negligibly small even for
the cases that the polarized $\Lambda$ has the largest spin-flip cross
section. This result is not surprising because the flip of $\Lambda$ spin due
to the exchange of a scalar meson is a relativistic effect, which is small
because of the small $\Lambda$ velocity in the hadronic matter as a result of
its large mass.

##  III Thermally averaged cross sections

Figure 3: Thermally averaged $\Lambda-\pi$ scatting cross section in
$s$-channel (solid line) and $t$-channel (dashed line) as functions of
temperature. Also shown in the inset is the ratio of thermally averaged
$\Lambda$ spin flip to non-flip cross sections as a function of temperature.

For $\Lambda$ scattering in a thermalized hadronic matter, the time evolution
of its spin polarization in the kinetic approach depends on the thermally
averaged $\Lambda-\pi$ scattering cross sections,

$\displaystyle\langle\sigma_{ab\rightarrow cd}v_{ab}\rangle$ $\displaystyle=$
$\displaystyle\frac{\int d^{3}{\bf p}_{a}d^{3}{\bf p}_{b}f_{a}({\bf
p}_{a})f_{b}({\bf p}_{b})\sigma_{ab\rightarrow cd}v_{ab}}{\int d^{3}{\bf
p}_{a}d^{3}{\bf p}_{b}f_{a}({\bf p}_{a})f_{b}({\bf p}_{b})}.$

In the above, $f_{i}({\bf p}_{i})$ is the Boltzmann momentum distribution of
particle species $i=a,b$, i.e., $f_{i}({\bf p}_{i})=g_{i}e^{-\sqrt{{\bf
p}_{i}^{2}+m_{i}^{2}}/T}$ with $m_{i}$ being the particle mass, and $v_{ab}$
is the relative velocity between particles $a$ and $b$. Figure 3 shows the
thermally averaged $s$\- and $t$-channel spin-averaged scattering cross
sections of $\Lambda$ by a pion. It is seen that the one from the $t$-channel
process is about a factor of 1.3 larger than that from the $s$-channel process
at all temperature of the hadronic matter, indicating that the average
invariant mass of a $\Lambda$ and $\pi$ pair in the hadronic matter is below
the $\Sigma^{*}$ mass. Also shown in the inset the ratio of thermally averaged
$\Lambda$ spin flip to non-flip cross sections, which is again negligibly
small as in the case for their spin-averaged cross sections.

##  IV Kinetic equations for $\Lambda$ spin evolution

In the kinetic approach, the numbers of spin-up and spin-down $\Lambda$
hyperons, denoted by $N_{\Lambda_{+}}$ and $N_{\Lambda_{-}}$, respectively,
change with time according to the kinetic equations,

$\displaystyle\frac{dN_{\Lambda_{+}}}{d\tau}=-\langle\sigma_{\Lambda_{+}\pi\rightarrow\Lambda_{-}\pi}v\rangle
z_{\pi}n^{(0)}_{\pi}N_{\Lambda_{+}}$ $\displaystyle\hskip
39.74872pt+\langle\sigma_{\Lambda_{-}\pi\rightarrow\Lambda_{+}\pi}v\rangle
z_{\pi}n^{(0)}_{\pi}N_{\Lambda_{-}},$ (20)
$\displaystyle\frac{dN_{\Lambda_{-}}}{d\tau}=\langle\sigma_{\Lambda_{+}\pi\rightarrow\Lambda_{-}\pi}v\rangle
z_{\pi}n^{(0)}_{\pi}N_{\Lambda_{+}}$ $\displaystyle\hskip
39.74872pt-\langle\sigma_{\Lambda_{-}\pi\rightarrow\Lambda_{+}\pi}v\rangle
z_{\pi}n^{(0)}_{\pi}N_{\Lambda_{-}}.$ (21)

In the above, $n^{(0)}_{\pi}$ is the density of thermally equilibrated pions,
given by $n_{\pi}^{(0)}=g_{\pi}\int\frac{d^{3}{\bf p}}{(2\pi)^{3}}f_{\pi}(p)$
with $g_{\pi}=3$ due to its isospin degeneracy, and $z_{\pi}$ is its fugacity
to take into account the fact that the effective pion number, which includes
both free pions and those from resonance decays, remains unchanged during the
expansion and cooling of the hadronic matter according to the statistical
hadronization model Andronic  _et al._ (2006).

Using the relation
$\langle\sigma_{\Lambda_{+}\pi\rightarrow\Lambda_{-}\pi}v\rangle=\langle\sigma_{\Lambda_{-}\pi\rightarrow\Lambda_{+}\pi}v\rangle\equiv\langle\sigma
v\rangle$, the number difference between spin-up and spin-down $\Lambda$
hyperons, $\Delta N(\tau)=N_{\Lambda_{+}}(\tau)-N_{\Lambda_{-}}(\tau)$, then
satisfies the equation,

$\frac{d\Delta N}{d\tau}=-2\langle\sigma v\rangle z_{\pi}n^{(0)}_{\pi}\Delta
N,$ (22)

which can be solved to give

$\Delta N(\tau)=\Delta N_{0}e^{-2\int\left<\sigma
v\right>z_{\pi}n^{(0)}_{\pi}d\tau},$ (23)

where $\Delta N_{0}$ denotes the initial number difference between spin up and
spin down $\Lambda$ hyperons.

##  V Time evolution of temperature and pion fugacity in hadronic matter

Since the largest $\Lambda$ polarization reported by the STAR Collaboration at
beam energy scan experiments at RHIC is from Au-Au central 20-50% collisions
at $\sqrt{s}_{NN}=7.7$ GeV Adamczyk  _et al._ (2017a), the hadronic scattering
effects on $\Lambda$ polarization in these collisions are considered in the
present study. According to the statistical hadronization model based on the
grand canonical ensemble fit to the experimentally measured yields of produced
particles in 30-40% collision centrality of these collisions Ref. Adamczyk
_et al._ (2017b), the extracted chemical freeze-out temperature, baryon
chemical potential, and strangeness chemical potential are about $T_{\rm
ch}=146$ MeV, $\mu_{B}=376$ MeV, and $\mu_{S}=88$ MeV, respectively. Also,
using the blast-wave model to fit the transverse momentum spectra of measured
particles, a kinetic freeze-out temperature of $T_{\rm k}=129$ MeV is found in
Ref. Adamczyk  _et al._ (2017b). The small charged chemical potential given in
Ref. Adamczyk  _et al._ (2017b) is, however, neglected in the present study.

$\sqrt{s}_{NN}$ | $T_{\rm ch}$ | $T_{\rm k}$ | $\tau_{\rm ch}$ | $\tau_{\rm k}$
---|---|---|---|---
(GeV) | (MeV) | (MeV) | (fm/c) | (fm/c)
7.7 | 146 | 129 | 5.13 | 6.46
Table 1: Parameter values in Eq.(24) for the time evolution of the temperature
of the hadronic matter produced in Au-Au collisions at $\sqrt{s_{NN}}=7.7$
GeV.

For the time evolution of the temperature of the hadronic matter from $T_{\rm
ch}=146$ MeV to $T_{\rm k}=129$ MeV, it can be obtained from the result in
Ref. Xu and Ko (2017), which is based on the AMPT model simulation of Au-Au
collisions at $\sqrt{s}_{NN}$=7.7 GeV, by parameterizing it as follows,

$T(\tau)=T_{\rm ch}-(T_{\rm ch}-T_{\rm k})\left(\frac{\tau-\tau_{\rm
ch}}{\tau_{\rm k}-\tau_{\rm ch}}\right)^{0.9}.$ (24)

In the above, $\tau_{\rm ch}$ and $\tau_{\rm k}$ are the chemical freeze-out
time and the kinetic freeze-out time, respectively. Values of these parameters
are given in TABLE 1. The time evolution of the hadronic matter temperature in
Au-Au collisions at $\sqrt{s}_{NN}=7.7$ GeV is shown by the solid line in the
upper panel of Fig. 4.

Figure 4: Time evolution of temperature (upper panel) and pion (solid line),
kaon (dashed line) and nucleon (dash-dotted line fugacities (lower panel) of
the hadronic matter in Au+Au collisions at $\sqrt{s}_{NN}=7.7$ GeV.

For the pion fugacity $z_{\pi}$ in Eqs.(20) and (21), it can be determined
together with the kaon fugacity $z_{k}$, and nucleon fugacity $z_{N}$ by
requiring the constancy of the effective pion, kaon, and nucleon numbers
during the expansion of the hadronic matter under the condition of constant
entropy per hadron Xu and Ko (2017) as described in detail in Ref. Sung  _et
al._ (2023). Specifically, one first expresses the effective pion, kaon, and
nucleon density at temperature $T$ in terms of the thermally equilibrated
hadron densities $n_{i}^{(0)}=g_{i}\int\frac{d^{3}{\bf
p}}{(2\pi)^{3}}f_{i}({\bf p})$, where $g_{i}$ is the spin and isospin
degeneracy of hadron species $i$. and the pion, kaon, and nucleon fugacities
as follows,

$\displaystyle n^{\rm
eff}_{\pi}(T)=z_{\pi}n^{(0)}_{\pi}+z_{\pi}^{2}n^{(0)}_{\rho}+z_{\pi}z_{K}n^{(0)}_{K^{*}}+z_{\pi}z_{N}n^{(0)}_{\Delta}+\cdots,$
$\displaystyle n^{\rm
eff}_{K}(T)=z_{K}n^{(0)}_{K}+z_{\pi}z_{K}n^{(0)}_{K^{*}}+z_{\pi}^{2}z_{K}n^{(0)}_{K_{1}}+z_{K}^{2}n^{(0)}_{\phi}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\cdots,$ $\displaystyle
n^{\rm eff}_{N}(T)=z_{N}n^{(0)}_{N}+z_{\pi}z_{N}n^{(0)}_{\Delta}+\cdots,$ (25)

where $\cdots$ denotes the contribution from strong decays of other
resonances, which includes all particles of masses up to 1.7 GeV for mesons
and 2 GeV for baryons in the particle data book. In obtaining the above
equations, one has used the relations $z_{\rho}=z_{\pi}^{2}$,
$z_{K^{*}}=z_{\pi}z_{K}$, $z_{K_{1}}=z_{\pi}^{2}z_{K}$,
$z_{\Delta}=z_{\pi}z_{N}$, etc. because of expected chemical equilibrium
between $\pi$ and $\rho$, among $K^{*}$, $K$ and $\pi$, among $\Delta$, $N$
and $\pi$, etc. as a result of the large cross sections for the reactions
$\pi\pi\leftrightarrow\rho$, $K^{*}\leftrightarrow K\pi$,
$\Delta\leftrightarrow N\pi$ etc. One then calculates the entropy density and
total particle density according to $s(T)=-\sum_{i}g_{i}\int\frac{d^{3}{\bf
p}}{(2\pi)^{3}}(z_{i}f_{i})\ln(z_{i}f_{i})$ and
$n(T)=\sum_{i}z_{i}n_{i}^{(0)}$. Requiring $n^{\rm
eff}_{\pi,K,N}(T)V(T)=n^{\rm eff}_{\pi,K,N}(T_{\rm ch})V(T_{\rm ch})$ and
$s(T)/n(T)=s(T_{\rm ch})/n(T_{\rm ch})$ and with unity pion, kaon and nucleon
fugacities at $T_{\rm ch}$, these four equations can be solved to give the
pion, kaon and nucleon fugacities as well as the volume ratio $V(T)/V(T_{\rm
ch})$ at a given $T$. Shown in the lower panel of Fig. 4 are the pion fugacity
$z_{\pi}$ (solid line), the kaon fugacity $z_{k}$ (dashed line) and the
nucleon fugacity $z_{N}$ (dash-dotted line) as functions of time. One notes
that the constant entropy per hadron in this study is 6.31.

##  VI Results

Figure 5: Time evolution of the normalized number difference between spin up
and spin down $\Lambda$ hyperons in Au-Au collisions at $\sqrt{s}_{NN}=7.7$
GeV.

The solid line in Fig. 5 shows the time evolution of the normalized number
difference between spin up and spin down $\Lambda$ hyperons, which is defined
by taking $\Delta N_{0}=1$, in Au-Au collisions at $\sqrt{s}_{NN}=7.7$ GeV. It
is seen that its value decreases to 0.93 or is reduced by 7% at the end of
hadronic evolution after 1.33 fm/$c$, which justifies the neglect of hadronic
scatterings on the $\Lambda$ spin polarization assumed in the literature. The
hadronic effect on $\Lambda$ polarization remains small even one doubles the
lifetime of the hadronic phase in these collisions to 2.66 fm/$c$ as the value
of $\Delta N$ is now 0.88 or is reduced by 12% in this case.

##  VII Conclusions

The effects of hadronic scatterings on the $\Lambda$ spin polarization in Au-
Au collisions at $\sqrt{s}_{NN}=7.7$ GeV have been studied in a schematic
kinetic approach by using thermally averaged $\Lambda$ spin flip and non-flip
cross sections. These cross sections are calculated by including contributions
from both the $s$-channel process through the $\Sigma^{*}$ resonance and the
$t$-channel process via the exchange a scalar $\sigma$ meson. Although the
thermally averaged cross sections due to the $t$-channel process is about a
factor of 1.3 larger than those of the $s$-channel process, the ratio of the
spin flip to non-flip cross sections is negligibly small compared to that from
the $s$-channel process, which has a constant value of 1/3.5. Because of the
latter small value, the $\Lambda$ spin polarization is found to decrease by
only about 7% during the hadronic stage of these collisions. This result thus
justifies the assumption in theoretical studies of $\Lambda$ polarization that
compare its value calculated at the chemical freeze out to the measured one at
the kinetic freeze out.

The present study has neglected the interference between the $s$-channel and
$t$-channel processes. Because of the negligible spin flip to non-flip cross
sections in the $t$-channel process, including the interference terms is not
expected to affect the results and conclusion from the present study. Also,
the effect of $\Lambda$ scattering by nucleons is not considered in the
present study. The $\Lambda-N$ spin flip to non-flip scattering due to the
$\sigma$ meson exchange is also likely to be small as in $\Lambda-\pi$
scattering because of the small $\Lambda$ velocity, which makes the
relativistic spin flip effect negligible. Including $\Lambda-N$ scattering
through the $\sigma$ meson exchange is also not expected to affect the
conclusion of present study. However, $\Lambda-\pi$ and $\Lambda-N$
scatterings can also go through the exchange of an $\omega$ meson. For
$\Lambda-\pi$ scattering via the $\omega$-exchange, its contribution to
$\Lambda$ spin flip cross section is expected to be suppressed as it involves
creating a massive $\rho$ meson in the final state. Although the contribution
to $\Lambda$ spin flip cross section due to $\Lambda-N$ scattering via the
$\omega$ exchange may not be small, its effect on $\Lambda$ spin polarization
could be small compared to the $s-$channel $\Lambda-\pi$ scattering through
the $\Sigma^{*}$ resonance because the smaller nucleon than pion numbers in
Au-Au collisions at $\sqrt{s}_{NN}=7.7$ GeV Adamczyk  _et al._ (2017a).

## ACKNOWLEDGEMENTS

This work was supported by the Korea National Research Foundation under Grant
No. 2023R1A2C300302311, 2023K2A9A1A0609492411 (S.H.L.) and the U.S. Department
of Energy under Award No. DE-SC0015266 (C.M.K.). S. H. Lee also acknowledges
the support from the Samsung Science and Technology Foundation under Project
No. SSTF-BA1901-04. H. Sung thanks the Cyclotron Institute of Texas A&M
University for its hospitality during her stay as a visiting scholar supported
by the Korean Women Graduate Research in the US Program of the National
Research Foundation of Korea through funds from the Korean Ministry of Science
and ICT (2022K1A3A1A12097807).

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