A Comparison Study of Collisions at Relativistic Energies Involving Light Nuclei

Abstract

We present extensive comparisons of ¹⁶O+¹⁶O collisions at a center-of-mass energy per nucleon pair $\sqrt{s_{NN}}=200$ GeV and ²⁰⁸Pb+¹⁶O collisions at $\sqrt{s_{NN}}=68.5$ GeV as well as ²⁰Ne+²⁰Ne collisions at $\sqrt{s_{NN}}=200$ GeV and ²⁰⁸Pb+²⁰Ne collisions at $\sqrt{s_{NN}}=68.5$ GeV based on a multiphase transport (AMPT) model. We recommend measuring the ratio of the elliptic flow to the triangular flow, which shows appreciable sensitivity to the structure of light nuclei, as also found in other studies. This is especially so if the observable is measured near the target rapidity in ²⁰⁸Pb+¹⁶O or ²⁰⁸Pb+²⁰Ne collisions, as originally found in the present study. Our study serves as a useful reference for understanding the effect of structure on observables in collisions involving light nuclei under analysis or on the schedule.

1. Introduction

Understanding the structure of nuclei is a fundamental goal of nuclear physics. Besides traditional methods, relativistic heavy-ion collisions provide an alternative way of extracting the density distribution of colliding nuclei [1]. In the past few years, various probes for nucleus deformation have been proposed and applied in experimental analysis. The basic idea is that the deformation of colliding nuclei may enlarge the anisotropy of the overlap region and enhance the fluctuation of the overlap area in the initial stage of the collision, which will lead to larger anisotropic flows and stronger transverse momentum fluctuations in the final stage of the collision [2,3]. Using this principle, researchers have successfully extracted the deformation parameters of heavy nuclei such as ⁹⁶Ru [4], ⁹⁶Zr [4], ¹⁹⁷Au [5], and ²³⁸U [6] as well as the shape of ¹²⁹Xe [7,8,9]. The focus of the community has now turned to collisions involving light nuclei [10,11,12,13,14,15,16], which have not only large deformations but also different $\alpha$-cluster configurations. For example, ¹⁶O may have a tetrahedral structure formed by four $\alpha$ clusters [17,18,19] with considerable octupole deformation, while ²⁰Ne may have a bowling-pin structure consisting of five $\alpha$ clusters [20,21] with considerable quadrupole and octupole deformation. The existence of $\alpha$-cluster structure may break the scaling relation between the deformation parameter and specific observables in relativistic heavy-ion collisions, such as the anisotropic flows and the transverse momentum fluctuation [22]. While these observables are sensitive to the deformation of colliding nuclei, it is of interest to directly probe the existence of $\alpha$-cluster structure in light nuclei through their collisions at relativistic energies [23]. On the experimental side, in addition to ¹⁶O+¹⁶O (light–light) collisions at the top energy of the Relativistic Heavy Ion Collider, which are currently under analysis [24], the System for Measuring Overlap with Gas in the Large Hadron Collider beauty experiment enables the study of fixed-target collisions at relativistic energies, and collisions involving light nuclei such as ²⁰⁸Pb+¹⁶O and ²⁰⁸Pb+²⁰Ne (heavy–light) collisions are scheduled for future experiments (see, e.g., Refs. [12,25,26,27]).

It is of interest to compare and understand the different effects of deformation and clustering in light–light collisions and heavy–light collisions on the collision dynamics and final observables. Compared to light–light collisions, heavy–light collisions are carried out at a lower center-of-mass collision energy but have a larger initial overlap area. The event-by-event fluctuation is expected to be stronger in light–light collisions than in heavy–light collisions. Therefore, different correspondence between initial anisotropies and final flows as well as between the initial overlap size and the final transverse momentum distribution may exist in light–light and heavy–light collisions. While it is difficult to probe directly the existence of $\alpha$-cluster structure in light–light collisions through mid-rapidity observables [23], it is of interest to investigate whether this is the case in heavy–light collisions. Furthermore, in asymmetric systems such as heavy–light collisions, it remains to be determined at what rapidity observables are most sensitive to the structure of light nuclei. The above will be studied in the present paper based on the AMPT model. We will compare observables in light–light and heavy–light collisions by assuming light nuclei have a spherical shape, a deformed Woods–Saxon (WS) shape, or an $\alpha$-cluster structure, in order to investigate the effects of deformation and $\alpha$-cluster structure in different collision systems.

The rest of the paper is organized as follows. Section 2 gives the density distributions of ¹⁶O and ²⁰Ne and briefly describes the framework of the AMPT model. Section 3 presents extensive comparisons of the initial anisotropies as well as final anisotropic flows and transverse momentum fluctuations for different collision systems and different initial configurations. We give a brief conclusion in Section 4.

2. Theoretical Framework

The density distributions of ¹⁶O and ²⁰Ne are obtained based on a Bloch–Brink wave function approach, where empirical nucleon–nucleon interactions are used and the nucleon wave function inside each $\alpha$ cluster is approximated as a Gaussian form. It is assumed that four $\alpha$ clusters form a tetrahedron structure in ¹⁶O and five $\alpha$ clusters form a bowling-pin structure in ²⁰Ne. The distance parameters in the corresponding $\alpha$-cluster configurations are determined by minimizing the total energy after the angular-momentum projection of the whole wave function. For details of the framework in obtaining the wave functions and the density distributions in ¹⁶O and ²⁰Ne, we refer the reader to Ref. [22].

Figure 1a and Figure 1b display respectively the density contours of ¹⁶O and ²⁰Ne with $\alpha$-cluster structure obtained from the above framework. In order to distinguish the $\alpha$-cluster effect from the deformation effect on final observables, we have also constructed density distributions of the deformed WS form for both ¹⁶O and ²⁰Ne as follows:

$$\rho (r,\theta )={\displaystyle \frac{{\rho }_0}{1+exp\left\{\frac{r-R_0[1+{\beta }_2{Y}_{2,0}\left(\theta \right)+{\beta }_3{Y}_{3,0}\left(\theta \right)]}{d}\right\}}}.$$

(1)

In the above equation, ${\rho }_0$ is the normalization constant, $R_0$ is the radius parameter, d is the diffuseness parameter, ${\beta }_2$ and ${\beta }_3$ are the deformation parameters, and $Y_{2,0}$ and $Y_{3,0}$ are the spherical harmonics. The deformation parameters ${\beta }_n$ are determined in such a way that the WS-form density distributions have the same intrinsic multipole moments

$$Q_n=\int \rho \left(\overrightarrow{r}\right)r^n{Y}_{n,0}\left(\theta \right)d^{3}r$$

(2)

as the density distributions $\rho \left(\overrightarrow{r}\right)$ with $\alpha$-cluster structure. $R_0$ and d are determined in such a way that the deformed WS distributions have the same RMS radii (characterized by $\langle r^2\rangle$) and the same surface diffuseness (characterized by $\langle r^4\rangle$) as the density distributions $\rho \left(\overrightarrow{r}\right)$ with $\alpha$-cluster structure, where the lth-order moment of r is defined as $\langle r^l\rangle =\int \rho \left(\overrightarrow{r}\right)r^l{d}^{3}r/\int \rho \left(\overrightarrow{r}\right)d^{3}r$. In this way, the density distributions of ¹⁶O and ²⁰Ne with the above mentioned deformed WS form have the same global shape as the more realistic ones with $\alpha$-cluster structure. The values of $R_0$, d, ${\beta }_2$, and ${\beta }_3$ in the WS-form density distributions for both ¹⁶O and ²⁰Ne are listed in

Table 1. Parameter values for the deformed WS density distributions of ¹⁶O and ²⁰Ne.

	${\mathit{R}}_0$ (fm)	d (fm)	${\mathit{\beta }}_2$	${\mathit{\beta }}_3$
16O	1.973	0.507	0	0.223
20Ne	2.160	0.580	0.666	0.250

. It can be seen that the density distribution of ¹⁶O has a considerable ${\beta }_3$, and that of ²⁰Ne has considerable ${\beta }_2$ and ${\beta }_3$, as intuitively expected. The resulting density contours for the deformed WS distributions are displayed in Figure 1c,d. In order to investigate separately the deformation effects of colliding nuclei on final observables, we have further considered the distributions of the spherical WS form by simply setting ${\beta }_n=0$ in Equation (1). This way, the nucleus size is the same for the three cases, while the central density in the spherical WS form is higher than that from the spherical Skyrme–Hartree–Fock calculation [28].

To describe non-equilibrium dynamics in collisions involving light nuclei, transport models are favored over hydrodynamics models. We will compare ¹⁶O+¹⁶O collisions at $\sqrt{s_{NN}}=200$ GeV under experimental analysis and ²⁰⁸Pb+¹⁶O collisions at $\sqrt{s_{NN}}=68.5$ GeV on the schedule, as well as the hypothetical ²⁰Ne+²⁰Ne collisions at $\sqrt{s_{NN}}=200$ GeV and ²⁰⁸Pb+²⁰Ne collisions at $\sqrt{s_{NN}}=68.5$ GeV on the schedule, based on the AMPT model. The density distribution of ²⁰⁸Pb is set as the empirical spherical WS form in AMPT. The coordinates of participant nucleons in ¹⁶O or ²⁰Ne are sampled according to the density distributions shown in Figure 1 with recentering and random orientations. In the string melting version of the AMPT model used in the present study, hadrons generated by the Heavy-Ion Jet Interacting Generator model [29] from collisions of participant nucleons are converted to partons according to the flavor and spin structures of their valence quarks. The momentum spectrum of these initial partons is described by the Lund string fragmentation function

$$f\left(z\right)\propto z^{-1}{(1-z)}^{a}exp(-b{m}_{\perp }^2/z),$$

(3)

with z being the light-cone momentum fraction of the produced hadron of transverse mass $m_{\perp }$ with respect to that of the fragmenting string, while a and b are two parameters. We set $a=0.5$ and $b=0.9$ GeV⁻² in the present study [30,31]. Sub-nucleon effects [32,33,34,35,36,37,38,39,40,41,42], which may affect the effective area of the nuclear overlap and the initial parton production, are not considered in the present study. The evolution of the partonic phase is described by Zhang’s parton cascade model [43], including two-body elastic collisions with the differential cross section

$${\displaystyle \frac{d\sigma }{dt}}\approx {\displaystyle \frac{9\pi {\alpha }_s^2}{2{(t-{\mu }^2)}^2}},$$

(4)

where t is the standard Mandelstam variable for four-momentum transfer. In the present study, we set the strong coupling constant ${\alpha }_s$ to 0.33 and the screening mass $\mu$ to 3.2 fm⁻¹, corresponding to a parton scattering cross section of 1.5 mb [30,31]. After the kinetic freeze-out of these partons, a spatial coalescence model is used to combine quarks or antiquarks into hadrons according to their constituents. A relativistic transport model [44] including various elastic, inelastic, and decay channels is then used to describe the evolution of the hadronic phase until the kinetic freeze-out of all hadrons. For further details of the AMPT model, we refer the reader to Ref. [45]. Once the experimental data of, for example, the charged-particle multiplicity and the anisotropic flows in the corresponding collision systems are available, we can further calibrate the above mentioned parameters used in the AMPT model, which is beyond the scope of the present study.

3. Results and Discussion

In this section, we extensively compare observables in ¹⁶O+¹⁶O collisions at $\sqrt{s_{NN}}=200$ GeV and ²⁰⁸Pb+¹⁶O collisions at $\sqrt{s_{NN}}=68.5$ GeV with different initializations of ¹⁶O, as well as ²⁰Ne+²⁰Ne collisions at $\sqrt{s_{NN}}=200$ GeV and ²⁰⁸Pb+²⁰Ne collisions at $\sqrt{s_{NN}}=68.5$ GeV with different initializations of ²⁰Ne. We only compare results in central (0–10%) collisions with the maximum multiplicity and the maximum effect of structure, with the centrality determined by ordering the charged-particle multiplicities in all events.

We begin by comparing the spatial distributions of partons in the initial stage of different collision systems based on AMPT model calculations in Figure 2. Here, the nth-order anisotropic coefficient is calculated from

$${ϵ}_n={\displaystyle \frac{\sqrt{{[{\sum }_i{r}_{\perp ,i}^{n}cos\left(n{\varphi }_i\right)]}^2+{[{\sum }_i{r}_{\perp ,i}^{n}sin\left(n{\varphi }_i\right)]}^2}}{{\sum }_i{r}_{\perp ,i}^n}},$$

(5)

where $r_{\perp ,i}=\sqrt{x_i^2+y_i^2}$ and ${\varphi }_i=arctan(y_i/x_i)$, respectively, are the polar coordinate and the polar angle of the ith parton in the transverse plane. The fluctuation of the overlap’s inverse area is defined as

$$\delta d_{\perp }^2={(d_{\perp }-\langle d_{\perp }\rangle )}^2,$$

(6)

where $d_{\perp }=1/\sqrt{\overline{x^2}\overline{y^2}}$ is the overlap’s inverse area [46], with $\overline{(...)}$ representing the average over all particles in one event, while $\langle ...\rangle$ represents the average over all events. A smaller number of initial partons and a smaller overlap area are observed in light–light than heavy–light collisions, and this leads to stronger fluctuations and thus systematically larger $\langle {ϵ}_n\rangle$ and $\langle \delta d_{\perp }^2\rangle$ in light–light collisions. In some cases, the anisotropic coefficients from light–light and heavy–light collision systems show the same sensitivity to the structure of light nuclei. $\langle {ϵ}_3\rangle$ in ¹⁶O+¹⁶O and ²⁰⁸Pb+¹⁶O collisions is rather insensitive to the octupole deformation of ¹⁶O, owing to the similar contribution from fluctuations, but is enhanced by $\alpha$-cluster structure in ¹⁶O with a regular tetrahedron configuration. On the other hand, $\langle {ϵ}_2\rangle$ is more sensitive to the quadrupole deformation of colliding nuclei and is enhanced with deformed or $\alpha$-clustered ²⁰Ne in ²⁰Ne+²⁰Ne and ²⁰⁸Pb+²⁰Ne collisions. However, $\langle {ϵ}_n\rangle$ values do not necessarily show the same sensitivity to the density distribution of light nuclei in light–light and heavy–light collision systems. It can be seen that $\langle {ϵ}_2\rangle$ generated from fluctuations in ¹⁶O+¹⁶O collisions is insensitive to the density distribution of ¹⁶O, while the $\alpha$-cluster structure in ¹⁶O with a regular tetrahedron configuration slightly reduces fluctuations and thus $\langle {ϵ}_2\rangle$ in ²⁰⁸Pb+¹⁶O collisions. $\langle {ϵ}_3\rangle$ is insensitive to the density distribution of ²⁰Ne in ²⁰⁸Pb+²⁰Ne collisions with more initial partons and larger overlap area, but it is enhanced with deformation or $\alpha$-cluster structure in ²⁰Ne in ²⁰Ne+²⁰Ne collisions due to fewer initial partons and smaller initial overlap area. Consequently, the ratio $\langle {ϵ}_2\rangle /\langle {ϵ}_3\rangle$ is reduced with $\alpha$-cluster structure in ¹⁶O+¹⁶O and ²⁰⁸Pb+¹⁶O collisions, and it is enhanced with deformed or $\alpha$-clustered ²⁰Ne in ²⁰Ne+²⁰Ne and ²⁰⁸Pb+²⁰Ne collisions. $\langle \delta d_{\perp }^2\rangle$ is insensitive to the density distribution of light nuclei in heavy–light collision systems due to the large overlap area, but it shows some sensitivity to the density distribution of light nuclei in light–light collision systems.

We now move to a comparison of the pseudorapidity distributions of the final charged particles in different collision systems shown in Figure 3. For the simplicity of the discussion, the pseudorapidity in the following results represents that in the laboratory frame, i.e., $\eta \sim {\eta }_{lab}$. It can be seen that the major multiplicity of charged particles is in the pseudorapidity range of $-1.5<\eta <1.5$ in light–light collisions, while it is in the pseudorapidity range of $4<\eta <7$ in heavy–light collisions, where a significantly higher peak is observed. Although there is no doubt that mid-rapidity observables are generally used as deformation probes of colliding nuclei in symmetric collision systems, it is challenging to choose the appropriate rapidity range for observables in asymmetric collision systems. For heavy–light collision systems, one expects that observables near target rapidities, e.g., $2<\eta <5$, will be more sensitive to the structure of target light nuclei. The multiplicity of charged particles is more sensitive to the density distribution of ²⁰Ne than to that of ¹⁶O, likely due to the quadrupole deformation of ²⁰Ne, but this is not adequate to probe the deformation of ²⁰Ne in experimental analysis.

We then extensively compare the pseudorapidity distributions of final-state observables such as the elliptic flow $\langle v_2^2\rangle$, the triangular flow $\langle v_3^2\rangle$, and the transverse momentum fluctuation $\langle \delta p_T^2\rangle$ in light–light and heavy–light collisions for different initial density distributions of light nuclei, and they are calculated according to

$$\begin{array}{ccc}\hfill \langle v_n^2\rangle & =& {\langle cos\left[n({\phi }_i-{\phi }_j)\right]\rangle }_{i,j},\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill \langle \delta p_T^2\rangle & =& {\langle (p_{T,i}-\langle \overline{p_T}\rangle )(p_{T,j}-\langle \overline{p_T}\rangle )\rangle }_{i,j}.\hfill \end{array}$$

(8)

Here, ${\langle ...\rangle }_{i,j}$ represents the average over all possible combinations of $i,j$ for all events, while $p_{T,i}=\sqrt{p_{x,i}^2+p_{y,i}^2}$ and ${\phi }_i=arctan(p_{y,i}/p_{x,i})$, respectively, are the momentum and its polar angle for the ith particle in the transverse plane. To calculate $\langle v_n^2\rangle$ and $\langle \delta p_T^2\rangle$ at a certain pseudorapidity $\eta$, particles i and j are chosen from the range of $(\eta -1.5,\eta +1.5)$ with a gap of $\left|\Delta \eta \right|>1$. Figure 4 compares the results in ¹⁶O+¹⁶O and ²⁰⁸Pb+¹⁶O collisions, and Figure 5 compares the results in ²⁰Ne+²⁰Ne and ²⁰⁸Pb+²⁰Ne collisions. While in most cases $\langle v_2^2\rangle$, $\langle v_3^2\rangle$, and $\langle \delta p_T^2\rangle$, respectively, are strongly correlated with $\langle {ϵ}_2\rangle$, $\langle {ϵ}_3\rangle$, and $\langle \delta d_{\perp }^2\rangle$ as shown in Figure 2, the detailed behavior depends on the pseudorapidity range and other effects such as particle multiplicities, non-flow, etc. In Figure 4, the similar $\langle v_2^2\rangle$ for different cases in ¹⁶O+¹⁶O collisions and the smaller $\langle v_2^2\rangle$ with $\alpha$-cluster structure in ²⁰⁸Pb+¹⁶O collisions are consistent with the behaviors of the corresponding $\langle {ϵ}_2\rangle$ shown in Figure 2a. The relative difference in $\langle v_3^2\rangle$ from different initializations of ¹⁶O depends on the pseudorapidity and is generally difficult to distinguish from the statistical error. $\langle \delta p_T^2\rangle$ shows no sensitivity to the initialization of ¹⁶O in either ¹⁶O+¹⁶O or ²⁰⁸Pb+¹⁶O collisions. In Figure 5, the larger $\langle v_2^2\rangle$ values from deformed or $\alpha$-clustered ²⁰Ne in both ²⁰Ne+²⁰Ne and ²⁰⁸Pb+²⁰Ne collisions are consistent with the behaviors of the corresponding $\langle {ϵ}_2\rangle$ shown in Figure 2e. Compared to the spherical case, $\langle v_3^2\rangle$ slightly decreases after incorporating the deformation or $\alpha$-cluster structure in ²⁰Ne in both ²⁰⁸Pb+²⁰Ne and ²⁰Ne+²⁰Ne collisions, in contrast to the relative difference in $\langle {ϵ}_3\rangle$ for different scenarios shown in Figure 2f. This is due to the effect of the quadrupole deformation of ²⁰Ne, which may reduce the triangular flow based on AMPT simulations as shown in Figure 3 (d) of Ref. [22]. Except that $\langle \delta p_T^2\rangle$ is slightly smaller with spherical ²⁰Ne in ²⁰Ne+²⁰Ne collisions, $\langle \delta p_T^2\rangle$ shows almost no sensitivity to the initialization of ²⁰Ne in either ²⁰Ne+²⁰Ne or ²⁰⁸Pb+²⁰Ne collisions.

Let us now focus on observables in typical pseudorapidity ranges; the results are shown in Figure 6. For light–light collisions, we focus on the pseudorapidity range with the largest multiplicity of final charged particles, i.e., $-1.5<\eta <1.5$. For heavy–light collisions, we focus on the pseudorapidity range with the largest multiplicity of final charged particles, i.e., $4<\eta <7$, and the range near target rapidities, i.e., $2<\eta <5$, at which we expect that the results could be more sensitive to the structure of target light nuclei. Systematically, light–light collisions have stronger anisotropic flows and transverse momentum fluctuations than heavy–light collisions. The behaviors of $\langle v_2^2\rangle$ are roughly similar to those of $\langle {ϵ}_2^2\rangle$ as shown in Figure 2a,e, while the relative $\langle v_3^2\rangle$ from different initializations of light nuclei are different compared to those in Figure 2b,f. This shows that the elliptic flow manifests the initial geometry more clearly, while the triangular flow can be affected by other effects in dynamics of collisions involving light nuclei. Taking the ratios of the elliptic flow to the triangular flow in different collision systems reveals interesting behaviors: while $\langle v_2\rangle /\langle v_3\rangle$ shows almost no sensitivity to the structure of ¹⁶O in the pseudorapidity range with the largest particle multiplicity in both ¹⁶O+¹⁶O and ²⁰⁸Pb+¹⁶O collisions, it shows appreciable sensitivity to the structure of ¹⁶O near target rapidities in ²⁰⁸Pb+¹⁶O collisions, with the spherical case giving the largest $\langle v_2\rangle /\langle v_3\rangle$ and the deformation or $\alpha$-clustered ¹⁶O leading to smaller $\langle v_2\rangle /\langle v_3\rangle$. On the other hand, $\langle v_2\rangle /\langle v_3\rangle$ is enhanced by deformation or $\alpha$-cluster structure in ²⁰Ne in both ²⁰Ne+²⁰Ne and ²⁰⁸Pb+²⁰Ne collisions, and the effect is stronger near target rapidities in ²⁰⁸Pb+²⁰Ne collisions. We note that the centrality dependence of $\langle v_2\rangle /\langle v_3\rangle$ has also been shown to be sensitive to the $\alpha$-cluster structure of ¹⁶O in ¹⁶O+¹⁶O collisions in Refs. [13,47]. The transverse momentum fluctuation shows no sensitivity to the initialization of light nuclei. It can be seen that these observables are mainly sensitive to the global shape of light nuclei, while it is difficult to distinguish directly the density distribution of light nuclei with and without $\alpha$-cluster structure in both light–light and heavy–light collisions, since the detailed $\alpha$-cluster structure in light nuclei is mostly obscured by complicated dynamics in the partonic and hadronic phase.

4. Summary

We present extensive comparisons of ¹⁶O+¹⁶O collisions at $\sqrt{s_{NN}}=200$ GeV and ²⁰⁸Pb+¹⁶O collisions at $\sqrt{s_{NN}}=68.5$ GeV as well as ²⁰Ne+²⁰Ne collisions at $\sqrt{s_{NN}}=200$ GeV and ²⁰⁸Pb+²⁰Ne collisions at $\sqrt{s_{NN}}=68.5$ GeV based on the AMPT model; the main focus is on the sensitivity of observables to the structure of light nuclei. Light–light collisions systematically produce stronger anisotropic flows and transverse momentum fluctuations than heavy–light collisions. We found that the ratio of the elliptic flow to the triangular flow is a good probe of the structure of light nuclei as also mentioned in other studies, while the transverse momentum fluctuation shows weak sensitivity. In heavy–light collisions, observables near the target rapidity show stronger sensitivity to the structure of target light nuclei, as found for the first time in the present study. Observables are generally only sensitive to the global shape of colliding nuclei in both light–light and heavy–light collisions, while the detailed $\alpha$-cluster structure in light nuclei cannot be distinguished from the constructed deformed Woods–Saxon shape after the complicated dynamics. Our study serves as a useful reference for understanding the structure effect on observables in collisions involving light nuclei under analysis or on the schedule.