Preamble

Light nuclei elliptic flow at mid-rapidity in GeV Au+Au collisions using coalescence model

Y. Xu, X. H. He, and Y. P. Zhang
1. School of Physics, Harbin Institute of Technology, Harbin 150001, China
2. Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, Gansu 730000, China
3. School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China

Light nuclei collective flow is an important probe for understanding their production mechanisms in heavy-ion collisions. The STAR collaboration has reported that the atomic mass number ($A$) scaling of light nuclei elliptic flow ($v_2$) is broken at $\sqrt{s_{NN}} = 3.0$ GeV. Observations reveal that, while protons maintain negative $v_2$ values at mid-rapidity at both 3.0 and 3.2 GeV, light nuclei exhibit a sign change from negative at 3.0 GeV to positive at 3.2 GeV. In this study, we investigate the $v_2$ of protons and deuterons in mid-central Au+Au collisions at $\sqrt{s_{NN}} = 3.0, 3.2, 3.5$, and $3.9$ GeV using the JAM2 microscopic transport model. Deuterons are formed via nucleon coalescence, with the spatial distance and momentum difference between constituent protons and neutrons serving as the coalescence criteria. Our calculations successfully reproduce the sign change in deuteron $v_2$ at 3.2 GeV. We observe a strong dependence of nucleon coalescence probability on the azimuthal angle relative to the reaction plane. This effect is primarily driven by the transverse momentum dependence of the mean spatial and momentum separations between nucleon pairs, which vary with the nucleon azimuthal angle. Moreover, our analysis demonstrates that the stiffness of the nuclear equation of state (EoS) plays a crucial role in determining the energy dependence of this sign change in deuteron $v_2$.

INTRODUCTION

The study of nuclear matter at extreme temperatures and densities offers valuable insights into the properties of strongly interacting many-body systems described by quantum chromodynamics (QCD). The macroscopic properties of nuclear matter under extreme conditions are most evident in measurable collective features, which represent the common dynamics of multiple particles produced in a single reaction. These collective features are manifested as collective flow, characterized by the motion of numerous outgoing particles exhibiting either aligned directional movement or uniform velocity magnitudes \cite{1}. The measurement of azimuthal anisotropies in particles produced from collisions has emerged as a crucial probe for elucidating the fundamental characteristics of the quark-gluon plasma and the architecture of hadrons \cite{2, 3}. The observation of hadron elliptic flow ($v_2$) shows approximate number-of-constituent-quark (NCQ) scaling at high beam energies at RHIC and LHC. This scaling is interpreted as a signature for the emergence of the quark-gluon plasma (QGP) formed during these collisions \cite{4-10}.

In heavy-ion collisions, the production mechanism of light nuclei (such as deuteron, triton, $^3$He, etc.) remains a topic of ongoing debate \cite{11-13}. A widely accepted theoretical model is that these nuclei are formed through the coalescence of nucleons, which can be either newly produced or transported from the colliding nuclei \cite{14-18}. According to this model, light nuclei emerge at the late stage of the collision evolution when the constituent nucleons come into close proximity in both coordinate and momentum space \cite{19-21}. A key feature of this model, analogous to the NCQ scaling in hadron flow, is that the collective flow of light nuclei is expected to exhibit scaling behavior with respect to their atomic mass number ($A$). The distinction between quark coalescence and nucleon coalescence lies in the fact that, in nucleon coalescence, both the momentum and spatial distribution of the constituent nucleons (protons and neutrons) can be directly measured in heavy-ion collision experiments alongside the resulting light nuclei.

The STAR collaboration has reported that the $A$-scaling of light nuclei holds in the low transverse momentum range ($p_T/A < 1.5$ GeV/c) at $\sqrt{s_{NN}} = 7.7 - 200$ GeV \cite{22}. Results from transport plus coalescence models for light nuclei are also consistent with these experimental measurements. However, recent measurements at $\sqrt{s_{NN}} = 3.0 - 3.9$ GeV Au+Au collisions by the STAR fixed-target experiments have revealed distinct scaling patterns for light nuclei anisotropic flow \cite{23-25}. The directed flow ($v_1$) of light nuclei clearly follows $A$-scaling, strongly supporting coalescence as the dominant production mechanism for these clusters. In contrast, their $v_2$ shows a breakdown of $A$-scaling behavior under the same collision conditions. Moreover, proton $v_2$ values are negative at mid-rapidity for collision energies of 3.0 GeV and 3.2 GeV, transitioning to positive values above 3.2 GeV \cite{25}. Similarly, for deuteron and $^3$He, the $v_2$ values are negative at mid-rapidity at 3.0 GeV but approach zero or become positive at 3.2 GeV.

The negative $v_2$ at low energies is attributed to the shadowing effect of the spectators in the collision. Given that the transition energy for the sign change of $v_2$ differs between protons and light nuclei, it is essential to investigate whether this shadowing effect exhibits mass-dependent behavior or if the final phase space distribution of nucleons plays a more dominant role. Such an investigation could provide critical insights into the formation time and mechanisms of light nuclei in heavy-ion collisions. In this paper, we employ the newly developed Jet AA Microscopic transport model (JAM2) \cite{26, 27}, combined with a nucleon coalescence model, to calculate the $v_2$ of protons and deuterons in mid-central Au+Au collisions at $\sqrt{s_{NN}} = 3.0, 3.2, 3.5$, and $3.9$ GeV. We investigate the nucleon coalescence probability as a function of azimuthal angle with respect to the reaction plane and explore the dependence of proton and deuteron $v_2$ on the stiffness of the equation-of-state (EoS).

METHOD

The calculation begins with event generation for Au+Au collisions using JAM2 at the specified energies. Within the JAM2 framework, the initial positions of incoming nucleons are sampled according to the nuclear density distribution. The nuclear collision is determined by summing the contributions of binary hadron-hadron collisions based on their closest approach distances. Particle production is modeled through resonance and string excitations, followed by their subsequent decays. For this analysis, we employ the mean-field model of JAM2 with incompressibility parameters of $\kappa = 210$ MeV and $380$ MeV. The spatial positions and momenta of protons and neutrons are recorded at a fixed time of 50 fm/c for subsequent coalescence into light nuclei.

During the afterburner coalescence stage, deuteron formation occurs when the phase-space distance between a proton-neutron pair falls below specified thresholds. For each pair, we calculate the relative spatial distance $\Delta R = |R_1 - R_2|$ and relative momentum distance $\Delta P = |P_1 - P_2|$ in the rest frame of the pair. A deuteron is formed when both $\Delta R < 4.5$ fm and $\Delta P < 0.3$ GeV/c are simultaneously satisfied. These parameters are based on the work in \cite{28}, where they effectively describe the yield of deuterons at $\sqrt{s_{NN}} = 3.0$ GeV. Our analysis focuses on the sign change between protons and deuterons; thus, we exclude light nuclei with $A > 2$ to simplify the estimation of $v_2$ for all proton-neutron pairs.

In JAM2, the default event-plane angle is set to zero. Consequently, the particle $v_2$ is calculated as $\langle \cos(2\phi) \rangle$, where $\phi$ represents the particle's azimuthal angle. Collision centrality is determined using the impact parameter $b$. To facilitate direct comparison with STAR experimental data, results are calculated for $b = 4.3 - 8.5$ fm, corresponding to the 10-40% centrality interval.

[FIGURE:1]

RESULTS AND DISCUSSION

Figure 1 shows the $v_2$ as a function of particle rapidity in Au+Au collisions at $\sqrt{s_{NN}} = 3.0, 3.2, 3.5$, and $3.9$ GeV. The results were calculated using the same transverse momentum range ($0.4 < p_T/A < 1.0$ GeV/c) as the STAR experimental analysis \cite{25}. At mid-rapidity, $v_2$ exhibits distinct energy-dependent behavior. Specifically, protons show negative $v_2$ values at $\sqrt{s_{NN}} = 3.0$ and $3.2$ GeV, transitioning to positive values at 3.5 and 3.9 GeV. In contrast, deuterons show negative $v_2$ values only at 3.0 GeV, becoming positive at 3.2 GeV and above. These results are quantitatively consistent with STAR measurements \cite{25}.

The negative $v_2$ at low energies is attributed to the spectator shadowing effect, where the prolonged passage time of spectators along the impact parameter direction influences the anisotropic expansion. However, a discrepancy appears at 3.2 GeV: protons and deuterons have opposite signs at mid-rapidity. This arises from the dynamics of deuteron formation. In our calculation, deuterons form at 50 fm/c, a late stage when the influence of spectators has diminished. This delayed formation explains why deuterons maintain positive $v_2$ even when proton $v_2$ is negative.

To gain deeper insight, we examine the azimuthal angle ($\phi$) distributions. The collective flow coefficients are characterized through Fourier expansion of the particle azimuthal distribution:
$$ \frac{dN}{d\phi} \propto 1 + 2v_1\cos(\phi) + 2v_2\cos(2\phi) + \dots $$
[FIGURE:2]
[FIGURE:3]

In the energy range explored, the system exhibits strong $v_1$ and nearly vanishing $v_2$ at mid-rapidity. By comparing distributions before and after coalescence (Fig. 2 and Fig. 3), we can examine how coalescence probability varies with $\phi$.

[FIGURE:4]

Under the assumption that protons and neutrons have identical distributions, the coalescence model predicts:
$$ \frac{dN_d}{d\phi} \approx P_{coal}(\phi) \left( \frac{dN_p}{d\phi} \right)^2 $$
where $P_{coal}$ is the coalescence probability. Figure 4 shows that $P_{coal}$ has a clear azimuthal dependence, with a minimum near $\phi = \pi/2$ (perpendicular to the reaction plane). This indicates that nucleons moving parallel to the reaction plane are more likely to coalesce. Consequently, nucleons contributing to negative $v_2$ (near $\phi = \pi/2$) have a reduced probability of forming deuterons, explaining the sign change.

[FIGURE:5]
[FIGURE:6]

We analyze the average spatial ($\langle \Delta R \rangle$) and momentum ($\langle \Delta P \rangle$) separations as functions of proton $p_T$ (Fig. 5). Both show strong $p_T$ dependence. Higher momentum protons around $\phi = \pi/2$ exhibit larger separations, reducing their coalescence probability (Fig. 6). This azimuthal dependence of $P_{coal}$ explains the breaking of mass-number scaling and the sign change of deuteron $v_2$ at 3.2 GeV.

[FIGURE:7]

Figure 7 shows $v_2$ results using two nuclear incompressibility parameters ($\kappa = 210$ MeV and $380$ MeV). At 3.0 GeV, both proton and deuteron $v_2$ remain negative regardless of the EoS. However, at 3.2 GeV, deuteron $v_2$ shows strong sensitivity to EoS stiffness, remaining negative for the stiff EoS but turning positive for the soft EoS. This implies that the nuclear EoS plays a crucial role in determining the sign inversion of light nuclei $v_2$.

SUMMARY

Using the JAM2 transport model with a nucleon coalescence afterburner, we investigated the elliptic flow of protons and deuterons in Au+Au collisions at $\sqrt{s_{NN}} = 3.0 - 3.9$ GeV. Our model successfully reproduces the observed sign change in deuteron $v_2$ at 3.2 GeV. The analysis of $dN/d\phi$ distributions reveals that the coalescence probability is higher for nucleons moving along the reaction plane than for those moving perpendicularly. This azimuthal dependence is driven by the $p_T$ dependence of spatial and momentum separations between nucleon pairs. Furthermore, the transition energy for the deuteron $v_2$ sign change is sensitive to the stiffness of the nuclear equation of state. These findings provide deeper insights into the late-stage formation of light nuclei and offer a means to constrain the nuclear incompressibility parameter using experimental flow data.

Acknowledgments: We are grateful for discussions with Dr. Yasushi Nara. This work is supported in part by the National Key R&D Program of China (No. 2024YFA1610700) and the National Natural Science Foundation of China (No. 12205342).