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

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 [1,2] 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 .Such huge orbital angular momentum will be partially converted into spin polarization of the hadrons by spin-orbit coupling proposed by the pioneer works [3][4][5].The STAR collaboration at the Relativistic Heavy-Ion Collider (RHIC) has measured the global polarization of Λ and Λ hyperons [6].The results show that the vorticity of the quarkgluon plasma (QGP) generated in the collisions is as large as ω ∼ 10 22 s −1 , making it the fastest vortical system observed in nature to date.The global polarization has been well-understood by various phenomenological models [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].
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 [128].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 Λ and Λ hyperons.As mentioned previously, it is challenging to distinguish the local polarization of Λ hyperons induced by different sources through the experimental measurements.Remarkably, our previous study [127] has found that the local helicity polarization induced by thermal vorticity dominates over other contributions at √ s N N = 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 lowenergy 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 [23,129,130] 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 √ s N N = 7.7, 19.6, 39 GeV Au+Au collisions with Simulating Many Accelerated Strongly-interacting Hadrons (SMASH) [131][132][133][134][135][136][137] 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) [138][139][140][141] 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 [142,143].
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 µν = diag{+, −, −, −} and the projector ∆ µν = g µν − u µ u ν with u µ being fluid velocity.

II. THEORETICAL AND NUMERICAL FRAMEWORK
We follow Refs.[27,127,142,143] 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 Here, we parameterize the momentum of an on-shell particle as , where p T is the transverse momentum, Y is the momentum rapidity and φ p is the azimuthal angle, p =p/|p| is the unit vector along the direction of momentum, and S(p) is the spatial component of the single-particle mean spin polarization vector S µ (p).The spin polarization vector S µ (p) for fermionic systems can be evaluated by using the modified Cooper-Frye formula [9,10,56,111], under the assumption of local thermal equilibrium, where dΣ µ is the normal vector of the freeze-out hypersurface, m Λ denotes the mass of Λ hyperons, and J µ 5 (p, X) and N µ (p, X) stand for the axial-charge and number-density current in the phase space, respectively.
Inserting the J µ 5 (p, X) obtained from chiral kinetic theory up to O( ) [56] into the spin polarization vector S µ , we derive the helicity polarization [23,111,127], where 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 µ over temperature, respectively.Here, we introduce We also assume that the system reaches the local thermal equilibrium for simplicity, i.e. we choose For other decomposition of spin vector, we refer to Refs.[97,98,100,[108][109][110].For convenience, we further decompose helicity polarization induced by thermal vorticity S h thermal into two separate terms [127], denoting the polarization related to the gradient of temperature, and caused by the kinetic vorticity ω = ∇ × 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 [144][145][146][147][148][149][150][151][152] 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 [142,143].However, the event-byevent 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 More discussions on the properties of helicity polarization can be found in Refs.[127,142,143].
Similar to the local spin polarization in heavy ion collisions, e.g.see Refs.[19,20,23,110,127], we propose a possible physical observable followed Ref. [127], The prefactor 2 in numerator comes from experimental measurement of local spin polarization, which is proportional to 1 s S µ (p) with s = 1 2 for the Λ 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 S in Eq. ( 6) in the laboratory frame, while the one defined in the rest frame of Λ hyperons can easily be derived by the Lorentz transformation, S = S − (p•S)p EΛ(EΛ+mΛ) with E Λ being the energy of Λ hyperons.Similarly, the helicity polarization can also defined by S , i.e. p • S = mΛ EΛ p • S = mΛ EΛ 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. ( 5), where the superscripts indicate the respective sources.We utilize the (3+1)D CLVisc hydrodynamics [23,129,130] to simulate the evolution of the QGP at different collision energies.We employ the SMASH model [131][132][133][134][135][136][137] for initial conditions and adopt the NEOS-BQS equations of state [153,154].Later on, we will also check the results from the AMPT initial model [138][139][140][141].We have chosen the simulation parameters according to the studies [23,130], 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 with n being baryon number and w being enthalpy density [23,130].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 φ p .The integration bounds for Eq. ( 6) is chosen as p T ∈ [0.5, 3] GeV and Y ∈ [−1, 1].We take the mass of Λ hyperons, m Λ = 1.116GeV in Eq. ( 2), as well as the mass term in all (u • p) terms of Eq. ( 4).

III. NUMERICAL RESULTS FROM HYDRODYNAMICS APPROACHES
We present the numerical results for helicity polarization P H (φ p ) induced by various sources as a function of azimuthal angle in 20 − 50% centrality at √ s N N = 7.7, 19.6, 39 GeV Au+Au collisions.Similar to Refs.[24][25][26], we also compute the first Fourier sine coefficient of the helicity polarization of Λ hyperons, P H sin φ p , 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 [16,118,155,156].The experimental measure- 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 CB = 1.2.The shortened form "chemical", "shear", "accT", " and "ω × 0.1" stand for the P chemical  ment of global polarization agrees with this expectation; that is, the global polarization increases as the collision energy decreases [6].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 ω H 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 [157], 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 [20, 23, 109-112, 122, 123].Based on our previous studies [23], the magnitude of local spin polarization along the out-of-plane direction for Λ hyperons induced by shear viscous tensor, fluid acceleration, and ∇(µ/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 Λ 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 √ s N N = 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 ω H 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 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 7.7GeV 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 [23,130] in Fig. 2(c).We find that, regardless of the implemented initial conditions or the value of the baryon diffusion coefficient C B , P ω H 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 ω H dominates in P total H 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 Λ hyperons [24][25][26].We plot P H sin φ p as a function of p T in 20 − 50% centrality at √ s N N = 7.7 GeV as an example in Fig. 3.We find that the magnitude of P H sin φ p 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 ω H 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 [109] and isothermal equilibrium [110], on the helicity polarization.These two approximations are of great importance in delineating the local spin polarization of Λ hyperons at √ s N N = 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 Λ 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 P H sin φ p 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 ω H in the total helicity polarization still holds.

IV. SUMMARY
We have studied the helicity polarization of Λ hyperons and its first Fourier sine coefficient at RHIC-BES energies and observed that the helicity polarization induced by kinetic vorticity P ω H dominates at intermediate and low collision energies.The helicity polarization led by other sources is one order of magnitude smaller than P ω H and their net contributions approximately cancel out.Furthermore, the dominance of P ω H 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 helic-ity 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.

Figure 1 :
Figure 1: Helicity polarization as a function of azimuthal angle φp in 20 − 50% centrality at √ sNN = 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 ω only, i.e.P ω H .The green dashed lines represent the total helicity polarization excluding P ω H .

1 Figure 2 :
Figure 2: (a) Helicity polarization induced by various sources in 20 − 50% centrality at √ sNN = 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 CB = 1.2.The shortened form "chemical", "shear", "accT", " and "ω × 0.1" stand for the P chemical

Figure 3 :
Figure 3: The first Fourier sine coefficient of the helicity polarization as a function of pT in 20 − 50% centrality at √ sNN = 7.7 GeV Au+Au collisions with SMASH initial condition.The same color assignments as in Fig. 1.