# Probing vortical structures in heavy-ion collisions at RHIC-BES energies
through helicity polarization

Cong Yi<EMAIL_ADDRESS>Department of Modern Physics, University of
Science and Technology of China, Hefei, Anhui 230026  Xiang-Yu Wu
<EMAIL_ADDRESS>Institute of Particle Physics and Key Laboratory
of Quark and Lepton Physics (MOE), Central China Normal University, Wuhan,
Hubei, 430079  Di-Lun Yang<EMAIL_ADDRESS>Institute of Physics,
Academia Sinica, Taipei, 11529  Jian-Hua Gao<EMAIL_ADDRESS>Shandong
Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial
Environment, Institute of Space Sciences, Shandong University, Weihai,
Shandong 264209  Shi Pu<EMAIL_ADDRESS>Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui 230026  Guang-You
Qin<EMAIL_ADDRESS>Institute of Particle Physics and Key Laboratory
of Quark and Lepton Physics (MOE), Central China Normal University, Wuhan,
Hubei, 430079

###### Abstract

We investigate the hydrodynamic helicity polarization of $\Lambda$ hyperons,
defined as the projection of the spin polarization vector along the directions
of particle momenta, at RHIC-BES energies by utilizing the relativistic (3+1)D
CLVisc hydrodynamics framework with SMASH initial conditions. As opposed to
local spin polarization at high energy collisions, our hydrodynamic
simulations demonstrate that the helicity polarization induced by the kinetic
vorticity dominates over other contributions at intermediate and low collision
energies. Our findings provide an opportunity to probe the fine structure of
local kinetic vorticity as a function of azimuthal angle at intermediate and
low collision energies by mapping our predictions to the future measurements
in experiments.

##  I Introduction

Spin, as a fundamental property of particles, plays a critical role in high-
energy physics, e.g. proton spin puzzles (see recent reviews (Leader:2013jra,
; Wakamatsu:2014zza, ) and references therein). Recently, a major breakthrough
related to the spin polarization in the relativistic heavy ion collisions has
drawn widespread attentions. In non-central heavy-ion collisions, two heavy
nuclei are accelerated to nearly the speed of light and collide with each
other. These collisions generates a large amount of orbital angular momentum,
estimated to be on the order of $10^{5}\hbar$. Such huge orbital angular
momentum will be partially converted into spin polarization of the hadrons by
spin-orbit coupling proposed by the pioneer works (Liang:2004ph, ;
Liang:2004xn, ; Gao:2007bc, ). The STAR collaboration at the Relativistic
Heavy-Ion Collider (RHIC) has measured the global polarization of $\Lambda$
and $\overline{\Lambda}$ hyperons (STAR:2017ckg, ). The results show that the
vorticity of the quark-gluon plasma (QGP) generated in the collisions is as
large as $\omega\sim 10^{22}\leavevmode\nobreak\ s^{-1}$, making it the
fastest vortical system observed in nature to date. The global polarization
has been well-understood by various phenomenological models(Becattini:2007nd,
; Becattini:2007sr, ; Becattini:2013fla, ; Fang:2016vpj, ; Karpenko:2016jyx, ;
Xie:2017upb, ; Li:2017slc, ; Sun:2017xhx, ; Shi:2017wpk, ; Wei:2018zfb, ;
Xia:2018tes, ; Shi:2019wzi, ; Fu:2020oxj, ; Ryu:2021lnx, ; Lei:2021mvp, ;
Vitiuk:2019rfv, ; Wu:2022mkr, ).

Interestingly, the polarization along the beam and out-of-plane directions,
namely the local spin polarization has been measured by STAR (Niida:2018hfw, ;
Adam:2019srw, ) and ALICE (ALICE:2021pzu, ), and are investigated by many
different models, e.g. the statistical methods (Becattini:2017gcx, ;
Becattini:2022zvf, ; Palermo:2023cup, ), quantum kinetic theory (Gao:2019znl,
; Weickgenannt:2019dks, ; Weickgenannt:2020aaf, ; Hattori:2019ahi, ;
Wang:2019moi, ; Yang:2020hri, ; Weickgenannt:2020sit, ; Li:2019qkf, ;
Liu:2020flb, ; Weickgenannt:2021cuo, ; Wang:2021qnt, ; Sheng:2021kfc, ;
Huang:2020wrr, ; Wang:2020dws, ; Stephanov:2012ki, ; Son:2012zy, ; Gao:2012ix,
; Chen:2012ca, ; Manuel:2013zaa, ; Manuel:2014dza, ; Chen:2014cla, ;
Chen:2015gta, ; Hidaka:2016yjf, ; Hidaka:2017auj, ; Mueller:2017lzw, ;
Hidaka:2018ekt, ; Hidaka:2018mel, ; Gao:2018wmr, ; Huang:2018wdl, ;
Liu:2018xip, ; Lin:2019ytz, ; Lin:2019fqo, ; Yamamoto:2020zrs, ; Gao:2020vbh,
; Gao:2020pfu, ; Hidaka:2022dmn, ; Fang:2022ttm, ), spin hydrodynamics
(Hattori:2019lfp, ; Fukushima:2020qta, ; Fukushima:2020ucl, ; Li:2020eon, ;
She:2021lhe, ; Montenegro:2017lvf, ; Montenegro:2017rbu, ; Florkowski:2017ruc,
; Florkowski:2018myy, ; Becattini:2018duy, ; Florkowski:2018fap, ;
Yang:2018lew, ; Bhadury:2020puc, ; Shi:2020qrx, ; Gallegos:2021bzp, ;
Hongo:2021ona, ; Florkowski:2017dyn, ; Florkowski:2018ahw, ;
Florkowski:2019qdp, ; Florkowski:2019voj, ; Bhadury:2020cop, ; Shi:2020htn, ;
Singh:2020rht, ; Wang:2021ngp, ; Liu:2020ymh, ; Peng:2021ago, ;
Florkowski:2021wvk, ; Copinger:2022jgg, ; Wang:2021wqq, ), other effective
theories (Zhang:2019xya, ; Liu:2020dxg, ; Liu:2021uhn, ) and phenomenological
simulations (Xie:2017upb, ; Xia:2018tes, ; Voloshin:2017kqp, ; Liu:2019krs, ;
Wei:2018zfb, ; Wu:2020yiz, ; Wu:2019eyi, ; Fu:2020oxj, ; Xia:2019fjf, ;
Becattini:2019ntv, ; Li:2021jvn, ; Florkowski:2021xvy, ; Sun:2021nsg, ). It is
found that the local spin polarization can be induced by various sources,
including the thermal vorticity, shear viscous tensor, fluid acceleration,
gradient of baryon chemical potential over temperature, and electromagnetic
fields (Hidaka:2017auj, ; Liu:2020dxg, ; Liu:2021uhn, ; Becattini:2021suc, ;
Fu:2021pok, ; Becattini:2021iol, ; Yi:2021ryh, ; Ryu:2021lnx, ;
Alzhrani:2022dpi, ; Buzzegoli:2022fxu, ; Palermo:2022lvh, ). These studies
have also been extended to low-energy collisions (Aziz:2021HADES, ;
STAR:2021beb, ; Ivanov:2019ern, ; Deng:2020ygd, ; Guo:2021udq, ;
Ayala:2021xrn, ; Deng:2021miw, ; Fu:2020oxj, ; Ryu:2021lnx, ; Wu:2022mkr, ;
Fu:2022myl, ; Fu:2022oup, ) and isobaric collisions (STAR:2023eck, ), as well
as to the discussion on the vortical smoke rings (Lisa:2021zkj, ;
Serenone:2021zef, ). Despite the global polarization has provided insight into
the kinetic or thermal vorticity as a function of collision energies, the fine
structure of the vorticity, such as its dependence on the azimuthal angle,
have not been fully explored. These information may not been accurately
captured by the local spin polarization due to the considerable influence of
other sources beyond the thermal vorticity. In this work, we extend our
previous studies (Yi:2021unq, ) and demonstrate that the helicity polarization
can help us to probe the fine structure of kinetic vorticity in low-energy
collisions.

Helicity polarization, defined as the projection of spin polarization vector
onto the direction of the particles’ momentum, is widely used for
characterizing the spin polarization in high energy physics (leader_2001, ).
In many high-energy scattering processes, there are no preferred quantization
directions for spin, unlike the case of global polarization where the
direction of the initial orbital angular momentum is naturally chosen as the
quantization direction. In such cases, helicity is often preferred over spin
to describe the spin polarization.

Back to the heavy ion collisions, the use of helicity polarization allows us
to probe a distinct feature of the spin polarization for $\Lambda$ and
$\overline{\Lambda}$ hyperons. As mentioned previously, it is challenging to
distinguish the local polarization of $\Lambda$ hyperons induced by different
sources through the experimental measurements. Remarkably, our previous study
(Yi:2021unq, ) has found that the local helicity polarization induced by
thermal vorticity dominates over other contributions at $\sqrt{s_{NN}}=200$
GeV Au+Au collisions. We argue that helicity polarization induced by kinetic
vorticity, as part of thermal vorticity, will play the crucial role to the
total helicity polarization in the low-energy collisions. Consequently, we
further proposed that this finding can be utilized to probe the local strength
of kinetic vorticity as a function of azimuthal angle in low-energy collisions
by measuring helicity polarization.

To verify our conjecture, we utilize the relativistic (3+1)D CLVisc
hydrodynamics framework (Pang:2012he, ; Wu:2021fjf, ; Wu:2022mkr, ) to
investigate the azimuthal angle dependence of hydrodynamic helicity
polarization at RHIC-Beam Energy Scan (BES) energies in this work. We report
the numerical simulation of hydrodynamic helicity polarization at
$\sqrt{s_{NN}}=7.7,19.6,39\text{ GeV}$ Au+Au collisions with Simulating Many
Accelerated Strongly-interacting Hadrons (SMASH) (Weil:2016zrk, ;
Schafer:2019edr, ; Mohs:2019iee, ; Hammelmann:2019vwd, ; Mohs:2020awg, ;
Schafer:2021csj, ; Inghirami:2022afu, ) initial condition. As anticipated, the
helicity polarization induced by the kinetic vorticity is one order of
magnitude larger than other contributions in low-energy collisions. This
finding holds even when we choose A-Multi-Phase-Transport (AMPT) (Lin:2004en,
; Wu:2021hkv, ; Wu:2018cpc, ; Zhao:2017yhj, ) initial conditions or different
baryon diffusion coefficients. Our study presents a novel approach to
investigate the structure of kinetic vorticity in low-energy heavy-ion
collisions by connecting hydrodynamic simulations with the measurable helicity
polarization. The finding can also provide a baseline for the investigation on
local parity violation through the correlations of helicity polarization
proposed by (Becattini:2020xbh, ; Gao:2021rom, ).

This paper is organized as follows. In Sec. II, we briefly introduce the
theoretical framework and hydrodynamical setup for the helicity polarization.
We present our numerical results of the helicity polarization at various
collision energies, initial conditions, and baryon diffusion coefficients in
Sec. III and summarize our findings in Sec. IV. Throughout this work, we adopt
the Minkowski metric $g_{\mu\nu}=\textrm{diag}\\{+,-,-,-\\}$ and the projector
$\Delta^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}$ with $u^{\mu}$ being fluid
velocity.

Figure 1: Helicity polarization as a function of azimuthal angle $\phi_{p}$ in
$20-50\%$ centrality at $\sqrt{s_{NN}}=7.7,19.6,39$ GeV Au+Au collisions with
SMASH initial condition. The red dashed-dotted lines denote the helicity
polarization induced by the kinetic vorticity $\omega$ only, i.e.
$P_{H}^{\omega}$. The green dashed lines represent the total helicity
polarization excluding $P_{H}^{\omega}$.

##  II Theoretical and numerical framework

We follow Refs. (Becattini:2017gcx, ; Becattini:2020xbh, ; Gao:2021rom, ;
Yi:2021unq, ) to introduce the theoretical framework for the helicity
polarization in relativistic heavy ion collisions. The helicity polarization
for a relativistic particle with mass $m$ is defined as

$\displaystyle S^{h}$ $\displaystyle=$
$\displaystyle\text{$\widehat{\mathbf{p}}$$\cdot$}\mathcal{\bm{S}}(\mathbf{p}).$
(1)

Here, we parameterize the momentum of an on-shell particle as
$p^{\mu}=(\sqrt{|\mathbf{p}|^{2}+m^{2}},\mathbf{p})=(\sqrt{p_{T}^{2}+m^{2}}\cosh
Y,p_{T}\cos\phi_{p},p_{T}\sin\phi_{p},\sqrt{p_{T}^{2}+m^{2}}\sinh Y)$, where
$p_{T}$ is the transverse momentum, $Y$ is the momentum rapidity and
$\phi_{p}$ is the azimuthal angle,
$\mathbf{\widehat{\mathbf{p}}=}\mathbf{p}/|\mathbf{p}|$ is the unit vector
along the direction of momentum, and $\mathcal{\bm{S}}(\mathbf{p})$ is the
spatial component of the single-particle mean spin polarization vector
$S^{\mu}(p)$. The spin polarization vector $S^{\mu}(p)$ for fermionic systems
can be evaluated by using the modified Cooper-Frye formula (Becattini:2013fla,
; Fang:2016vpj, ; Hidaka:2018mel, ; Yi:2021ryh, ), under the assumption of
local thermal equilibrium,

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

where $d\Sigma^{\mu}$ is the normal vector of the freeze-out hyper-surface,
$m_{\Lambda}$ denotes the mass of $\Lambda$ hyperons, and
$\mathcal{J}_{5}^{\mu}(p,X)$ and $\mathcal{N}^{\mu}(p,X)$ stand for the axial-
charge and number-density current in the phase space, respectively.

Inserting the $\mathcal{J}_{5}^{\mu}(p,X)$ obtained from chiral kinetic theory
up to $\mathcal{O}(\hbar)$ (Hidaka:2018mel, ) into the spin polarization
vector $\mathcal{S}^{\mu}$, we derive the helicity polarization (Yi:2021ryh, ;
Yi:2021unq, ; Wu:2022mkr, ),

$\displaystyle S_{\textrm{hydro}}^{h}(\mathbf{p})$ $\displaystyle=$
$\displaystyle
S_{\textrm{thermal}}^{h}(\mathbf{p})+S_{\textrm{shear}}^{h}(\mathbf{p})+S_{\textrm{accT}}^{h}(\mathbf{p})$
(3) $\displaystyle+S_{\textrm{chemical}}^{h}(\mathbf{p}),$

where

$\displaystyle S_{\textrm{thermal}}^{h}(\mathbf{p})$ $\displaystyle=$
$\displaystyle\int
d\Sigma^{\sigma}F_{\sigma}p_{0}\epsilon^{0ijk}\widehat{p}_{i}\partial_{j}\left(\frac{u_{k}}{T}\right),$
$\displaystyle S_{\textrm{shear}}^{h}(\mathbf{p})$ $\displaystyle=$
$\displaystyle-\int
d\Sigma^{\sigma}F_{\sigma}\frac{\epsilon^{0ijk}\widehat{p}^{i}p_{0}}{(u\cdot
p)T}(p^{\sigma}\pi_{\sigma j}u_{k}),$ $\displaystyle
S_{\textrm{accT}}^{h}(\mathbf{p})$ $\displaystyle=$ $\displaystyle\int
d\Sigma^{\sigma}F_{\sigma}\frac{\epsilon^{0ijk}\widehat{p}^{i}p_{0}u_{j}}{T}\left[(u\cdot\partial)u_{k}+\frac{\partial_{k}T}{T}\right],$
$\displaystyle S_{\textrm{chemical}}^{h}(\mathbf{p})$ $\displaystyle=$
$\displaystyle-2\int
d\Sigma^{\sigma}F_{\sigma}\frac{p_{0}\epsilon^{0ijk}\widehat{p}_{i}}{(u\cdot
p)}\partial_{j}\left(\frac{\mu}{T}\right)u_{k},$ (4)

stand for the contributions from thermal vorticity, the shear viscous tensor,
the fluid acceleration minus the gradient of temperature $T$, the gradient of
baryon chemical potential $\mu$ over temperature, respectively. Here, we
introduce $\pi_{\sigma
j}=\partial_{\sigma}u_{j}+\partial_{j}u_{\sigma}-u_{\sigma}(u\cdot\partial)u_{j}$
and $F^{\mu}=\hbar[8m_{\Lambda}\Phi(\mathbf{p})]^{-1}p^{\mu}f_{eq}(1-f_{eq}),$
$\Phi(\mathbf{p})=\int d\Sigma^{\mu}p_{\mu}f_{eq}$. We also assume that the
system reaches the local thermal equilibrium for simplicity, i.e. we choose
$f_{eq}=1/\left[\text{exp}[(p^{\mu}u_{\mu}-\mu)/T]+1\right]$. For other
decomposition of spin vector, we refer to Refs. (Liu:2019krs, ; Liu:2020dxg, ;
Liu:2021uhn, ; Fu:2021pok, ; Becattini:2021iol, ; Becattini:2021suc, ). For
convenience, we further decompose helicity polarization induced by thermal
vorticity $S_{\text{thermal }}^{h}$ into two separate terms Yi:2021unq ,

$\displaystyle S_{\nabla T}^{h}(\mathbf{p})$ $\displaystyle=$
$\displaystyle\int
d\Sigma^{\sigma}F_{\sigma}\frac{p_{0}}{T^{2}}\widehat{\mathbf{p}}\cdot(\mathbf{u}\times\nabla
T),$ $\displaystyle S_{\omega}^{h}(\mathbf{p})$ $\displaystyle=$
$\displaystyle\int
d\Sigma^{\sigma}F_{\sigma}\frac{p_{0}}{T}\widehat{\mathbf{p}}\cdot\bm{\omega},$
(5)

denoting the polarization related to the gradient of temperature, and caused
by the kinetic vorticity $\bm{\omega}=\nabla\times\mathbf{u}$, respectively.
Later on, we will see that the above decomposition can improve our
understanding of helicity polarization.

Since the electromagnetic fields generating by the collisions decay rapidly
(Deng:2012pc, ; Roy:2015coa, ; Roy:2015kma, ; Pu:2016ayh, ; Pu:2016bxy, ;
Inghirami:2016iru, ; Shi:2017cpu, ; Siddique:2019gqh, ; Peng:2022cya, ) and
are negligible at the late stage, we omit the helicity polarization induced by
electromagnetic fields for simplicity. In general, an axial chemical
potential, characterising local parity violation, near the freeze-out
hypersurface also contributes to the helicity polarization (Becattini:2020xbh,
; Gao:2021rom, ). However, the event-by-event averaged axial density or axial
chemical potential is almost vanishing. We therefore exclude these
contributions in the analysis of helicity polarization. We also notice that in
the absence of the axial charge the system has the space reversal symmetry,
which leads to
$S_{\textrm{hydro}}^{h}(Y,\phi_{p})=-S_{\textrm{hydro}}^{h}(-Y,\phi_{p}+\pi).$
More discussions on the properties of helicity polarization can be found in
Refs. (Becattini:2020xbh, ; Gao:2021rom, ; Yi:2021unq, ).

Similar to the local spin polarization in heavy ion collisions, e.g. see Refs.
(Fu:2020oxj, ; Becattini:2021iol, ; Ryu:2021lnx, ; Yi:2021unq, ; Wu:2022mkr,
), we propose a possible physical observable followed Ref. (Yi:2021unq, ),

$\displaystyle P_{H}(\phi_{p})$ $\displaystyle=$
$\displaystyle\frac{2\int_{Y_{\text{min}}}^{Y_{\text{max}}}dY\int_{p_{T\text{min}}}^{p_{T\text{max}}}p_{T}dp_{T}[\Phi(\mathbf{p})S_{\textrm{hydro}}^{h}]}{\int_{Y_{\text{min}}}^{Y_{\text{max}}}dY\int_{p_{T\text{min}}}^{p_{T\text{max}}}p_{T}dp_{T}\Phi(\mathbf{p})}.$
(6)

The prefactor $2$ in numerator comes from experimental measurement of local
spin polarization, which is proportional to
$\frac{1}{s}\mathcal{S}^{\mu}(\mathbf{p})$ with $s=\frac{1}{2}$ for the
$\Lambda$ hyperons. Note that the $S^{h}$ defined in Eq. (1) is not a Lorentz
scalar. In the current study, we define the polarization three-vector
$\bm{\mathcal{S}}$ in Eq. (6) in the laboratory frame, while the one defined
in the rest frame of $\Lambda$ hyperons can easily be derived by the Lorentz
transformation,
$\bm{\mathcal{S}}^{\prime}=\bm{\mathcal{S}}-\frac{(\mathbf{p}\cdot\bm{\mathcal{S}})\mathbf{p}}{E_{\Lambda}(E_{\Lambda}+m_{\Lambda})}$
with $E_{\Lambda}$ being the energy of $\Lambda$ hyperons. Similarly, the
helicity polarization can also defined by $\bm{\mathcal{S}}^{\prime}$, i.e.
$\mathbf{\widehat{\mathbf{p}}}\cdot\bm{\mathcal{S}}^{\prime}=\frac{m_{\Lambda}}{E_{\Lambda}}\mathbf{\widehat{\mathbf{p}}}\cdot\bm{\mathcal{S}}=\frac{m_{\Lambda}}{E_{\Lambda}}S^{h}$.

To investigate the helicity polarization originating from various sources, we
utilize Eq. (3) to express $P_{H}$ as the sum of four terms and further
decompose the thermal-vorticity contribution in light of Eq. (II),

$\displaystyle P_{H}^{\textrm{total}}$ $\displaystyle=$ $\displaystyle
P_{H}^{\text{thermal}}+P_{H}^{\text{shear}}+P_{H}^{\text{accT}}+P_{H}^{\text{chemical}},$
(7) $\displaystyle P_{H}^{\text{thermal}}$ $\displaystyle=$ $\displaystyle
P_{H}^{\omega}+P_{H}^{\nabla T},$ (8)

where the superscripts indicate the respective sources.

We utilize the (3+1)D CLVisc hydrodynamics (Pang:2012he, ; Wu:2021fjf, ;
Wu:2022mkr, ) to simulate the evolution of the QGP at different collision
energies. We employ the SMASH model (Weil:2016zrk, ; Schafer:2019edr, ;
Mohs:2019iee, ; Hammelmann:2019vwd, ; Mohs:2020awg, ; Schafer:2021csj, ;
Inghirami:2022afu, ) for initial conditions and adopt the NEOS-BQS equations
of state (Monnai:2019hkn, ; Monnai:2021kgu, ). Later on, we will also check
the results from the AMPT initial model (Lin:2004en, ; Wu:2021hkv, ;
Wu:2018cpc, ; Zhao:2017yhj, ). We have chosen the simulation parameters
according to the studies Wu:2021fjf ; Wu:2022mkr , where the hydrodynamic
model has been shown to successfully reproduce the pseudo-rapidity
distribution of charged hadrons, as well as the transverse momentum spectra of
protons, pions, and kaons measured in the experiments. We would like to
emphasize that the final results for the spectra of mesons in low energy
collisions are insensitive to a parameter $C_{B}$, which connects to a baryon
diffusion coefficient
$\kappa_{B}=\frac{C_{B}}{T}n\left[\frac{1}{3}\cot\left(\frac{\mu}{T}\right)-\frac{nT}{w}\right]$
with $n$ being baryon number and $w$ being enthalpy density Wu:2021fjf ;
Wu:2022mkr . We set $C_{B}=0$ in most of our simulations unless explicitly
stated otherwise. From the simulations, we obtain the profile of temperature,
chemical potential and fluid velocity at the chemical freeze-out hypersurface.
By inputting these quantities into Eqs. (2, 3, 6), we derive the helicity
polarization as a function of the azimuthal angle $\phi_{p}$. The integration
bounds for Eq. (6) is chosen as $p_{T}\in[0.5,3]$ GeV and $Y\in[-1,1]$. We
take the mass of $\Lambda$ hyperons, $m_{\Lambda}=1.116$ GeV in Eq. (2), as
well as the mass term in all $(u\cdot p)$ terms of Eq. (4).

Figure 2: (a) Helicity polarization induced by various sources in $20-50\%$
centrality at $\sqrt{s_{NN}}=7.7$ GeV Au+Au collisions with SMASH initial
condition. (b) The results were obtained using the same parameters as in Fig.
2(a) except for the initial condition given by AMPT model. (c) The results are
obtained using the same parameters as in Fig. 2(a) except for $C_{B}=1.2$. The
shortened form "chemical", "shear", "accT", "$\nabla T$" and "$\omega\times
0.1$" stand for the
$P_{H}^{\text{chemical}},P_{H}^{\text{shear}},P_{H}^{\text{accT}},P_{H}^{\nabla
T}$ and $0.1\times P_{H}^{\omega}$, respectively.  Figure 3: The first Fourier
sine coefficient of the helicity polarization as a function of $p_{T}$ in
$20-50\%$ centrality at $\sqrt{s_{NN}}=7.7$ GeV Au+Au collisions with SMASH
initial condition. The same color assignments as in Fig. 1.

##  III Numerical results from hydrodynamics approaches

We present the numerical results for helicity polarization $P_{H}(\phi_{p})$
induced by various sources as a function of azimuthal angle in $20-50\%$
centrality at $\sqrt{s_{NN}}=7.7,19.6,39\text{ GeV}$ Au+Au collisions. Similar
to Refs. (Niida:2018hfw, ; Adam:2019srw, ; ALICE:2021pzu, ), we also compute
the first Fourier sine coefficient of the helicity polarization of $\Lambda$
hyperons, $\langle P_{H}\sin\phi_{p}\rangle$, as a function of transverse
momentum.

Let us start by discussing the impact of collision energy on helicity
polarization. As the collision energy decreases, nuclear stopping effects
become more prominent and the larger portion of orbital angular momenta of
colliding nuclei is transferred to the remnants. This leads to an expected
growth of the kinetic vorticity with decreasing collision energies, as has
been suggested by previous studies (Deng:2016gyh, ; Wei:2018zfb, ;
Deng:2020ygd, ; Huang:2020xyr, ). The experimental measurement of global
polarization agrees with this expectation; that is, the global polarization
increases as the collision energy decreases (STAR:2017ckg, ). Naturally, one
may expect that the helicity polarization induced by kinetic vorticity also
follows the same trend, which is clearly observed in Fig. 1.

Furthermore, another important observation from Fig. 1 is that the
$P_{H}^{\omega}$ dominates at intermediate and low collision energies. Unlike
the global spin polarization successfully mainly described by the contribution
from thermal vorticity in global equilibrium satisfying the Killing condition
(Becattini:2012tc, ), other corrections such as the shear viscous tensor,
gradient of baryon chemical potential over temperature should be incorporated
in local thermal equilibrium, which play an important role to the local spin
polarization (Fu:2021pok, ; Becattini:2021iol, ; Ryu:2021lnx, ; Yi:2021ryh, ;
Fu:2022myl, ; Fu:2022oup, ; Alzhrani:2022dpi, ; Wu:2022mkr, ). Based on our
previous studies (Wu:2022mkr, ), the magnitude of local spin polarization
along the out-of-plane direction for $\Lambda$ hyperons induced by shear
viscous tensor, fluid acceleration, and $\nabla(\mu/T)$ at intermediate and
low collision energies is much smaller than that caused by thermal vorticity.
This conclusion holds if we choose the mass of particles in Eqs. (2) and (4)
as the mass for $\Lambda$ hyperons. Naturally, we anticipate that the helicity
polarization induced by the thermal or kinetic vorticity dominates over other
contributions.

As an example, we examine the behavior of $P_{H}$ in $\sqrt{s_{NN}}=7.7$ GeV
Au+Au collisions to demonstrate the contributions from various sources, as
shown in Fig. 2(a). We observe that the magnitude of $P_{H}^{\omega}$ is
approximately $10$ times greater than that induced by other sources, which is
consistent with the above analysis for the corrections out of global
equilibrium. Furthermore, we observe that the dependence of
$\\{P_{H}^{\text{chemical}},P_{H}^{\text{shear}}\\}$ or
$\\{P_{H}^{\text{accT}},P_{H}^{\nabla T}\\}$ on $\phi_{p}$ resembles that of
the sine or negative sine function, respectively. Eventually,
$P_{H}^{\text{chemical}},P_{H}^{\text{shear}},P_{H}^{\text{accT}},P_{H}^{\nabla
T}$ nearly cancel each other out, highlighting the dominant role of kinetic
vorticity in helicity polarization.

A natural question that arises is whether the dominant role of kinetic
vorticity in helicity polarization persists for different initial conditions
or parameters. To verify it, we have studied the helicity polarization at
$7.7$GeV Au+Au collisions as an example using the AMPT initial conditions in
Fig. 2(b), and with a different baryon diffusion coefficient $C_{B}=1.2$
Wu:2021fjf ; Wu:2022mkr  in Fig. 2(c). We find that, regardless of the
implemented initial conditions or the value of the baryon diffusion
coefficient $C_{B}$, $P_{H}^{\omega}$ is always significantly larger than the
helicity polarization induced by other sources even though the magnitudes of
helicity polarization from all sources including kinetic vorticity are
together enhanced by a nonzero $C_{B}$. Therefore, our conclusion that
$P_{H}^{\omega}$ dominates in $P_{H}^{\textrm{total}}$ is independent of
initial conditions or $C_{B}$. The helicity polarization induced by other
sources, excluding the kinetic vorticity, also approximately cancel out with
each other.

Another possible relevant observable in experiments is the first Fourier sine
coefficient of the helicity polarization of $\Lambda$ hyperons (Niida:2018hfw,
; Adam:2019srw, ; ALICE:2021pzu, ). We plot $\langle P_{H}\sin\phi_{p}\rangle$
as a function of $p_{T}$ in $20-50\%$ centrality at $\sqrt{s_{NN}}=7.7\text{
GeV}$ as an example in Fig. 3. We find that the magnitude of $\langle
P_{H}\sin\phi_{p}\rangle$ increases with growing $p_{T}$. Moreover, we observe
that helicity polarization induced by kinetic vorticity still dominates over
other contributions, which is consistent with the results in Fig. 1.

As a remark, it is noteworthy that $P_{H}^{\omega}$ dominates the helicity
polarization, especially in low energy collisions. Therefore, by mapping the
hydrodynamic simulations to the helicity polarization measured in future
experiments, one can extract the structure of kinetic vorticity. This provides
us with a new opportunity to probe the structure of kinetic vorticity through
helicity polarization.

Before ending this section, we would like to discuss the potential impact of
two crucial approximations for the local spin polarization, namely strange
memory scenario (Fu:2021pok, ) and isothermal equilibrium (Becattini:2021iol,
), on the helicity polarization. These two approximations are of great
importance in delineating the local spin polarization of $\Lambda$ hyperons at
$\sqrt{s_{NN}}=200$ GeV Au+Au collisions. However, in low-energy collisions,
it is unclear whether the quark degrees of freedom are released from the
hadrons in the fireball. Therefore, it is plausible to consider the helicity
polarization of $\Lambda$ hyperons rather than $s$ quarks. We have also
checked numerically that in strange memory scenarios, the helicity
polarization induced by other effects, excluding kinetic vorticity, remains
negligible and contributes only to a small percentage of the total helicity
polarization. The sign of $\langle P_{H}\sin\phi_{p}\rangle$ remains unchanged
in the strange memory scenarios. In the isothermal equilibrium, the
temperature gradient near the chemical freeze-out hypersurface is assumed to
be vanishing. We have numerically checked that even if we drop contributions
from temperature gradient, the dominance of $P_{H}^{\omega}$ in the total
helicity polarization still holds.

##  IV Summary

We have studied the helicity polarization of $\Lambda$ hyperons and its first
Fourier sine coefficient at RHIC-BES energies and observed that the helicity
polarization induced by kinetic vorticity $P_{H}^{\omega}$ dominates at
intermediate and low collision energies. The helicity polarization led by
other sources is one order of magnitude smaller than $P_{H}^{\omega}$ and
their net contributions approximately cancel out. Furthermore, the dominance
of $P_{H}^{\omega}$ remains unchanged by variations in initial conditions and
baryon diffusion coefficient. Such a hierarchy for helicity polarization is
also unchanged even when adopting the approximations of the strange memory
scenarios and isothermal equilibrium in low-energy collisions.

Based on our results, we propose a novel approach to probe the fine structure
of kinetic vorticity by linking hydrodynamic simulations to the measurements
of helicity polarization in future low-energy nuclear collision experiments.
On the other hand, at low-energy collisions, the helicity polarization
provides a robust baseline for the equilibrium contribution to spin
polarization. The sizable mismatch for comparing our predictions with future
experimental measurements could reveal the potential role of non-equilibrium
contributions from collisional effects or even more exotic sources to local
spin polarization.

## Acknowledgments

This work is supported in part by the National Key Research and Development
Program of China under Contract No. 2022YFA1605500 and is also supported by
National Natural Science Foundation of China (NSFC) under Grants No. 12075235,
12135011, 12225503, 12175123, 11890710, 11890711, 11890713, 11935007 and
11475104. D.-L. Y. was supported by National Science and Technology Council
(Taiwan) under Grant No. MOST 110-2112-M-001-070-MY3.

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