The UrQMD model is a microscopic multi-body transport method that can be used to study proton-proton (pp), proton-nucleus (pA) and nucleon-nucleus (AA) interactions in the energy range from SIS to LHC. The transport model is based on covariant propagation of color strings, constituent quarks, and double quarks (as string ends) with meson and baryon degrees of freedom [28]. It can combine different reaction mechanisms and give theoretical simulation results of various experimental observations. In this model, by introducing the formation time of hadrons produced by string fragments, the degree of freedom of subhadrons is entered [29–31]. They predominate in the early stages of heavy ion collisions (HICs) with high SPS and RHIC energies.

The UrQMD model and quantum molecular dynamics (QMD) model are based on the parallel principle: hadrons are represented by Gaussian wave packets in phase space, and the phase space of hadrons propagates according to Hamiltonian equations of motion [32], r i ⃗ ̇ = ∂ H ∂ p i ⃗ , p i ⃗ ̇ = − ∂ H ∂ r i ⃗ . (1) Here, r i ⃗ and p i ⃗ are the coordinate and momentum of the hadron i, respectively. The Hamiltonian H consists of the kinetic energy T and the effective interaction potential energy U, H = T + U . (2)

This microscopic transport approach simulates multiple interactions of in-going and newly produced particles, the excitation and fragmentation of color strings and the formation and decay of hadronic resonances. For higher energies, the treatment of subhadronic degrees of freedom is very important. In the current model, these degrees of freedom enter by introducing the formation time of hadrons produced by string fragments. The phase transition to the quark-gluon state is not explicitly incorporated into the model dynamics. However, a detailed analysis of the model in equilibrium state gives an effective Hagedorn type equation of state [33].

In this paper, we mainly study the effect of the correlations between coordinate and momentum spaces on the yields and the transverse momentum distribution of deuteron and anti-deuteron with the cascade mode in the RHIC energy region. In the nextwork, we will focus on the influence of potential on production of light particles in this energy region.

The coalescence model describes the formation of hadronic clusters in the kinetic freeze-out stage of a heavy-ion collision. A pair of final (anti-) nucleons with similar momentum can merge to form a deuteron or anti-deuteron with total momentum P [34]. In the calculation, we use a conventional phase space clustering model [35] to construct clusters, in which hadrons with relative distances less than R ₀ and relative momentum less than P ₀ are considered to belong to a cluster. As a rule of thumb, the parameter set (R ₀, P ₀) can be selected in the range of (3-4 fm, 0.25–0.35 GeV/c) to describe the experimental data of HICs [26]. In this article, we will investigate the effects of different set of R ₀ and P ₀ on the yield of (anti-)deuteron over the evolutional time.

The time dependence of the production of (anti-)deuterons in the mid-rapidity (|y| < 0.3) for 0, −,10% centrality should be considered since they might be produced by different mechanisms. The time evolution of the yield of (anti-)deuterons at s N N = 14.5, 62.4 and 200 GeV are shown in Figure 1, Figure 2 and Figure 3, respectively. The blue area represents the range of experimental data. In the view ranges of R ₀ and P ₀, it is clear that too many deuterons are produced before 40 fm/c at s N N = 14.5 and at 62.4 GeV, which are unstable and will subsequently fission. In the meantime, too many anti-deuterons are produced before 50 fm/c. If we select parameter sets of (R ₀, P ₀) (3.8 fm, 0.3 GeV/c), the data can be well described. For s N N = 200 GeV, the parameter sets of (R ₀, P ₀) (3.575 fm, 0.285 GeV/c) can be selected. These parameter sets of (R ₀, P ₀) are commonly used by researchers and are appropriate for this study [7, 26].

Figure 4 show the yields of (anti-)deuterons as the time evolution in the 0, −,10% centrality Au + Au collisions at s N N = 14.5, 62.4 and 200 GeV at mid-rapidity (|y| < 0.3). The red lines are the results calculated from the cascade mode of UrQMD model. The shaded bands are the experimental data. It can be found that the stopping times should be 50 fm/c for deuterons and 60 fm/c for anti-deuterons for s N N = 14.5 and 62.4 GeV, and the corresponds stopping times for s N N = 200 GeV should be 30 fm/c for deuterons and 45 fm/c for anti-deuterons. Therefore, these stopping times are adopted in the following calculations. From Figure 4, one can also find that the deuterons produced at the lower energies need a longer time to be spatially separated [36].

The scanning of R ₀ and P ₀ located in the colored regions of Figure 1, Figure 2 and Figure 3 is useful because they are reliable in the (anti-) deuteron data description of the mid-rapidity region. It is clear that the (anti-)deuteron production rate of RHIC can be well described by the cooperative method of UrQMD + coalescence if the UrQMD stop times are properly combined and the parameter set of (R ₀, P ₀) in the coalescence is chosen.

Figures 5–7 show the transverse momentum spectra for deuterons at mid-rapidity (|y| < 0.3) in Au + Au collisions at s N N = 14.5, 62.4 and 200 GeV with 0, −,10%, 10, −,20%, 20, −,40%, 40, −,60% and 60, −,80% centralities. The signs + lines are the results calculated from the cascade mode of the UrQMD model, and the circles are the experimental data. It is found that the calculated results of UrQMD model agree well with the experimental data except for s N N = 14.5 GeV at the 40, −,60% and 60, −,80% centralities. At s N N = 14.5 GeV for the 40, −,60% and 60, −,80% centralities, most of our calculations are lower than the experimental data. We know that the deuterons produced at large impact parameter have plenty of time to react with other particles [36], and some of observed deuterons come from the nuclear fragments [27]. The impact of this effect will be further studied.

Figures 8–10 show the transverse momentum spectra for anti-deuterons at mid-rapidity (|y| < 0.3) in Au + Au collisions at s N N = 14.5, 62.4 and 200 GeV for 0, −,10%, 10, −,20%, 20, −,40%, 40, −,60% and 60, −,80% centralities. The signs + lines are our calculated results using the UrQMD model with cascade mode shown in the every panel. The circles are the experimental data. It is found that the calculations of the UrQMD model are in keeping with the experimental data well at s N N = 200 GeV. At s N N = 14.5 GeV, due to anti-deuterons are mainly produced in fireball shells, and the antideuterons produced have a low probability of interacting with other particles, the transverse momentum spectra of anti-deuterons is more. Since the relative suppression of anti-nucleons recedes with increasing energy, anti-deuterons can form much closer to the fireball center. Deuteron and antideuteron formation have the same geometry at energies around s N N = 200 GeV [34]. Therefore, the theoretical calculation results can describe the experimental data well at s N N = 200 GeV.