1 Introduction

In relativistic heavy-ion collisions, properties of the collision system are often investigated using the identified particle spectra, collective motion in transverse momentum plane (collective flow) [1-4], jet-quenching [5, 6], Hanbury–Brown–Twiss (HBT) correlation [7, 8], etc. Many experimental results suggest that hot-dense matter at the partonic level with large collective motion is created in the early stages of relativistic heavy-ion collisions; this is a new type of strong-coupling quark–gluon plasma (QGP) matter with almost the nearly lowest shear viscosity.

On the other hand, global rotation of the collision system is important to understand the global and local polarization of the quark matter. By using a multi-phase transport (AMPT) model and hard-sphere model, evolution of angular momentum and vorticity fields have been calculated for Au + Au collisions [9]. Some theoretical works have predicted globally polarized QGP in relativistic heavy-ion collisions [10, 11], and it was suggested that system vorticity and particle production mechanism should be studied by measuring global spin alignment of vector mesons in relativistic heavy-ion collisions [12, 13]. Recently, RHIC-STAR collaboration [14] reported the first measurements for global Λ hyperon polarization in Au + Au collisions, indicating that the vorticity of the QGP might reach ω ~ (9±1) × 10²¹/s, which far surpasses the vorticity of all other known fluids.

The initial geometry distribution could influence initial dynamical fluctuation and intrinsic structure in the collided system, which affects the collective flow, HBT correlation, etc. Hydrodynamical models [15-18] as well as transport models [19-23] were employed to investigate these effects. Some observables or physical quantities that are sensitive to the initial geometry distribution were proposed, such as collective flows [24-26], fluctuation of conserved quantities [27, 28], density fluctuations [29], and charge separation [30]. In References [19, 20] the carbon was considered with a 3-α structure and collided against a heavy nucleus at very high energies; the results implied that the final collective flow was sensitive to the intrinsic geometry distribution of carbon. The ratio of triangular flow to elliptic flow can be used to detect the intrinsic structure of α-clustering nuclei within an α-clustered ¹²C colliding against heavy-ion by using AMPT model [31]. The α-cluster model, which was originally proposed by Gamow [32], considered some light nuclei made of N-α, such as ¹²C with 3-α and ¹⁶O with 4-α. It was suggested that α clustering configurations can be identified by giant dipole resonance [33, 34] or photo-nuclear reaction in the quasi-deuteron region [35, 36] by an extended quantum molecular dynamics (EQMD) model.

In this work, the angular momentum, inertia, and average angular velocity for an asymmetric collision system, namely ¹²C + ¹⁹⁷Au, was investigated for comparing the three different clustering configurations of ¹²C. In the second section, an introduction to AMPT model and the calculation methods for angular momentum and other physical quantities in the many-body system is presented. In the third section, we first discuss the main calculation results of the angular momentum, inertia, and velocity for the three clustering configurations. Then, we explore the density distribution of the system in detail. The paper is concluded in the fourth section.

2 model and calculation methods

A step-by-step simulation of heavy-ion collisions in phase-space was obtained by AMPT model [37]. AMPT is a successful model used to study colliding energy ranges of relativistic heavy-ion collisions at RHIC [37] and LHC [38]. The model can be used to study pion-HBT correlations [39], collective flow [40, 41], di-hadron azimuthal correlations [42], strangeness production [43, 28], and chiral magnetic effects [44, 45], etc.

AMPT model was developed to simulate the collision system. It is applied to different transport theories to describe the many-body interaction from the initial to final states. The model is initialized by employing the HIJING model [46, 47] that generated hard mini-jet partons and soft strings to form initial states. It was followed by a parton cascade model (ZPC) [48] that simulates the interaction of the melted partons. Then, a quark coalescence model was adopted to describe the formation of hadrons; these hadrons participate in hadronic rescattering using a relativistic transport (ART) model [49].

The initial nucleon distribution in ¹²C would be a Woods–Saxon distribution, which originated from the HIJING model [46, 47], or α-cluster configurations [33, 34, 50]. The latter configurations include two cases: the three α-clustering chain structure and the three α-clustering triangle structure. The parameters for the α-clustered ¹²C were configured from the EQMD model [33, 34, 50] and are discussed in our previous work [31] in detail. Therefore, we can obtain the phase-space in ¹²C + ¹⁹⁷Au collisions at $\sqrt{s_{\text{NN}}}$ = 200 GeV for the calculations using AMPT model.

In this work, ¹²C is configured as a chain structure, triangle structure with 3-α and the Woods–Saxon distribution of nucleons from the HIJING model [46, 47] packaged in AMPT model. For ¹⁹⁷Au, its initial configuration was just the simple Woods–Saxon distribution. The distribution of the radial center of the α clusters in ¹²C is assumed to be a Gaussian function, $e^{-0.5{\left(\frac{r-r_{\text{c}}}{{\sigma }_{r_{\text{c}}}}\right)}^2}$, where r_c is average radial center of an α cluster and σ_rc is the width of the distribution. In addition, the nucleon inside each α cluster will be given by the Woods–Saxon distribution. The parameters of r_c and σ_rc can be obtained using the EQMD calculations [33-36]. For the triangle structure, r_c = 1.8 fm and σ_rc = 0.1 fm. For the chain structure, r_c = 2.5 fm and σ_rc = 0.1 fm for the two outer α clusters, and the other cluster will be at the center in ¹²C. Once the radial center of the α cluster is determined, the centers of the three clusters will be placed in an equilateral triangle for the triangle structure or in a line for the chain structure. Further details can be found in our previous work [31, 51].

In heavy-ion collisions, impact parameter b is defined as the perpendicular distance between the path center of projectile nucleus and the target nucleus. The plane formed by b and the beam direction is called the reaction plane. Because the direction of b is random for every collision, the reaction plane direction will also be random. Hence, the observables that are sensitive to the reaction plane direction should be corrected to that. The participant plane is considered as the reasonable proximation to the reaction plane, and the participant plane angle Ψ_nPP is defined by the following equation [52-54]:

where Ψ_nPP is the nth-order participant plane angle, r and ϕ_part are the coordinate position and azimuthal angle of participants in the collision zone in the initial state, respectively, and the average 〈…〉 denotes the density weighting. For the calculation, the system will be rotated to the participant plane direction in the transverse plane.

In AMPT model, we consider a many-body system with discrete particles, and the total angular momentum $\overrightarrow{J}$ can be calculated by summing each particle’s contribution as done for low-energy heavy-ion collisions [55].

where ${\overrightarrow{r}}_i$ and ${\overrightarrow{p}}_i$ are the coordinates and momentum of each particle, respectively. The relation between vorticity ${\overrightarrow{\omega }}_i$ and velocity ${\overrightarrow{v}}_i$ is ${\overrightarrow{v}}_i$ = ${\overrightarrow{\omega }}_i\times {\overrightarrow{r}}_i$ for the particle and ${\overrightarrow{v}}_i$ = ${\overrightarrow{p}}_i/E_i$, where Ei is the particle’s energy. In this model, the system is symmetric around the system rotational axis $\widehat{\omega }$ with a constant vorticity $\overrightarrow{\omega }$. If the system is considered as an approximate rigid body in a fixed stage, each particle’s vorticity ${\overrightarrow{\omega }}_i$ has a component projected onto the axis $\widehat{\omega }$, namely the $\overrightarrow{\omega }$ and the vertical component will cancel each other out within the system. From the above discussion, the angular momentum can be written as

where $I_0={\displaystyle \sum {}_i}(|{\overrightarrow{r}}_i{|}^2-{(\widehat{\omega }\cdot {\overrightarrow{r}}_i)}^2)E_i$ is the moment-of-inertia. Because the phase-space of the system is corrected to event plane angle, the event plane direction will be along the y-axis, and $\overrightarrow{\omega }$ will also be along the y-axis. The above discussion and deducing are in the assumption of approximate rigid body system and for the expanding system with discrete particles, the averaged vorticity component along the y-axis can be defined using a similar method from [9] as

In the earlier stage, our observation focused on the u($\overline{u}$), d($\overline{d}$), and s($\overline{s}$) quarks, which are primarily composed of partons; therefore, implicated most physics in the collision system. After hardronization, the system experiences a phase transfer to later stages; then, we focused on (anti-)protons and other mesons, including π^± and K^±. The phases we considered for the transport simulation of the collision are the initial partons, freeze-out partons, hadrons without hadronic rescattering, and final-state hadrons.

3 Results and discussion

For non-central collisions, the total angular momentum of the collision system would be mainly along the y-axis [9], which in the case of ¹²C + ¹⁹⁷Au collision would be of the order of magnitude of 10⁵. Most of the total angular momentum was carried by the spectators on the outer edge of the initial nucleus that passed on without being intervened, while the colliding parts, known as the participants, formed the QGP matter under high temperature and pressure [31, 37, 51].Therefore, because we only considered the participants using Eq. (3), the remnant angular momentum is reduced to the order of magnitude of 10³, as indicated in Fig. 3(a).

Fig. 3.Full-screen View

(Color online) Same as Fig. 1, except for the average angular velocity along the y-axis.

From the hydrodynamic perspective, the colliding system was nearly isolated [56, 57]. Therefore, the angular momentum would be conserved during the system evolvement after the collision. From Fig. 1(a) to (d), it can clearly be seen that Jy did not change significantly during the four stages: initialed participants, partons at freeze-out stage, hadrons without hadron rescattering, and hadrons after hadron rescattering. However, the QGP matter soon cooled down and then froze out, scattering out the particles that had a high momentum [51, 56]. Therefore, the system expanded, and the total angular inertia increased significantly from the order of magnitude of 10⁵ to 10⁷, which is shown in Fig. 2 (a) to (d). On the other hand, the decreasing average angular velocity ⟨ω_y⟩ calculated using Eq. (4) could also manifest the system expansion in Fig. 3, corresponding to the increasing I₀.

Fig. 1.Full-screen View

(Color online) Angular momentum Jy at different impact parameters in the collision process for ¹²C + ¹⁹⁷Au collision at $\sqrt{s_{\text{NN}}}=200$ GeV. From left to right panel:(a) the initial partons, (b) the freeze-out partons, (c) the hadrons without hadronic rescattering, and (d) the final-state hadrons. The three colors represent the three different α-cluster structures of the ¹²C.

Fig. 2.Full-screen View

(Color online) Same as Fig. 1, except for the angular inertia.

In Fig. 1 (a), the angular momentum of the participants first increased then decreased when the impact parameter b was increased gradually, which is in agreement with the results of other works [9]. This tendency can be understood from Eq. (2); in near-central collision, the angular momentum was dominated by the distance between each discrete particle and the center of mass, the increase of which contributed to the rising total angular momentum. However, the ¹⁹⁷Au nucleus employed the Woods–Saxon distribution [21-23], which possessed the Gaussian density drop against the radius; therefore, fewer initial nucleons (participants) could be contributed to the colliding process as the impact parameter increased. Because ¹²C occupied a smaller region than ¹⁹⁷Au [19, 20], the Woods–Saxon distribution of Au also induced the decreasing tendency of the total angular inertia at a larger impact parameter, as shown in Fig. 2. While for peripheral collision with larger b, the number of participants decreased rapidly, and at approximately 6 fm, which is approximately equal to the radius of the nucleus of ¹⁹⁷Au in the Woods–Saxon model [37], leading to a more rapid decay for a larger impact parameter.

Studies on heavy-ion collision [9, 58] have revealed a global rotation system with large angular momentum. In Fig.4, we showed the distribution of the average angular velocity along the y-axis ωy in the rapidity-transverse momentum plane, which implied that most participants carried a small angular velocity of the order of magnitude of approximately 10^-2. Correspondingly, in the second panel of Fig. 3, for the collective behavior, the total system at the parton freeze-out state was estimated to have an average angular velocity approximately equal to 0.021/fm, where we discussed the case of a middle range of impact parameter at b=4.5 fm.

Fig. 4.Full-screen View

(Color online) Angular velocity (ωy) distribution along the y-axis in the transverse momentum (p_T) and rapidity (η) plane in ¹²C + ¹⁹⁷Au collision at $\sqrt{s_{\text{NN}}}=200$ GeV. The impact parameter is b=4.5 fm for the mid-central collision. The left to right panels correspond to the freeze-out partons, the hadrons without hadronic rescattering, and the final-state hadrons, respectively. While from top to bottom are the three different α-clustering structures listed as the Woods–Saxon structure, the chain structure, and the triangle structure.

In most panel of Fig.4, the range of ωy distribution in rapidity at relatively high p_T was large compared to the low transverse momentum, although most of them remained around the mid-rapidity range. However, for the particles with relatively lower transverse momentum, rapidity range was narrower and at a larger magnitude, implying a strong collective structure, which could probably be seen as collective flow [31, 51]. In addition, from the density distribution in Fig.5, it can be seen that the center of collision region that composed of participants stayed with collective flow before being scattered out in the early stage.

Fig. 5.Full-screen View

(Color online) Same as Fig. 4 but for number density.

In the scenario [31, 51], the ¹⁹⁷Au nucleus would move towards the negative direction along the rapidity axis, while ¹²C took the opposite direction, which after collision transferred to a QGP matter and formed the global rotation [9]. In Fig. 5, for the impact parameter at 4.5 fm, most of the particle density was distributed in the range of low p_T and mid-rapidity in Fig.5. In the left column of Fig. 5, the two most dense regions at the parton freeze-out state represent the original ¹²C and ¹⁹⁷Au nucleus [19-23]. With time evolution, the two nuclei in opposite directions, which is shown in the middle panels, and their transverse momentum expanded towards a larger range. After the particles scattered out to the final-state hadrons, the system had a lower density that was chiefly distributed in the range of low p_T and mid-rapidity.

Comparing the left and the middle column of Fig. 4, the angular velocity ωy moved towards the region of larger p_T, which manifested the expansion of the system. At the same time, the magnitude of ωy decrease to 10^-3, which corresponds to the collective ⟨ω_y⟩ in Fig. 3 (c) and (d). In the right column, some scattered hadrons at a higher p_T and rapidity obtained higher angular momentum, which consequently led to a larger ωy. Few hadrons at large rapidity edge even carried large ωy that represent the high velocity scattered particles at the edge of rotation.

In a previous work on the collective flow [51], we reported the configurations of Woods–Saxon distribution and two α-clustering distributions for ¹²C to have very distinctive behaviors in the collective flow such as v₂ and v₃. However, in this study, for angular velocity, we found that the differences among the three structures were not as evident as the previous cases. For instance, global behaviors of the system such as I₀ and 〈ωy〉 in Fig. 2-3 perceived a rather small discrepancy among the three different cases of ¹²C configuration. On the other hand, Jy seemed to show a more significant difference, where Woods–Saxon distribution possessed the largest angular momentum Jy, followed by the triangle clusters structure, and then the chain clusters. The discrepancy between Woods–Saxon distribution and chain could be as large as approximately 20-30% at the peak, with an impact parameter equal to 6 fm. The internal configurations of α-cluster inside ¹²C [33, 34] determined the initial geometry of the collision system. At and after the system phase transition, it is highly probable that such geometry remained in the system [51] and could be manifested in later evolvement. Even in the final state, shown in (d) of Fig. 1, the discrepancy of the angular momentum could be large as 20% at peak. However, because of the expansion for flow, the average speed of system was expected to gradually decrease, making the distinction difficult. It is much likely that more interesting mechanism could be suggested in physical behavior of collective behavior.

4 Summary

In summary, we simulated the collisions of ¹²C against ¹⁹⁷Au at $\sqrt{s_{\text{NN}}}$ = 200 GeV using AMPT model, where three different structure patterns of C were considered separately and compared. We found the effects on the angular momentum of the chain and triangle α-clustered cases as well as the Woods–Saxon distribution case to have a discrepancy. Other collective behaviors, including angular inertia and average angular velocity, were also examined and were found to have much less distinction. This is consistent with the density distribution and angular velocity distribution for the three patterns, suggesting possible analysis for experimental study.