1 Introduction

The exploration of quantum chromodynamics (QCD) phase diagrams and nuclear matter properties has been one of the most important motivations for relativistic heavy-ion collisions [1,2,3,4,5,6]. The region of high temperature and vanishing baryon chemical potential of the QCD phase diagram, where the phase transition from hadronic matter to quark–gluon plasma (QGP) [7] is a smooth crossover, has been well studied in experiments at the RHIC [8,9,10] and LHC [11, 12]. Theoretical calculations suggest that there may exist a critical end point in the finite baryon chemical potential region [13]. It is well known that several observables, such as directed flow [14,15,16,17] and moments of distributions of conserved charges  [18], have been proposed to be sensitive to critical behavior. In the RHIC Beam Energy Scan phase I program (BES-I), the non-monotonic behavior of the net proton directed flow slope with respect to rapidity as a function of collision energy is observed, and the minimum of the distribution suggests the softest point of the equation of state (EoS) or prediction of the critical end point [19].

The collective flow [20, 21] has been extensively used to study the transverse properties of hot and dense matter created in heavy-ion collisions owing to their sensitivity to expansion dynamics. They are defined by the coefficients of the Fourier expansion of the azimuthal distribution of the emitted particles with respect to the reaction plane: [22]

$$\begin{aligned} E\dfrac{\mathrm {d}^{3}N}{\mathrm {d}^{3}p} = \dfrac{1}{2\pi } \dfrac{\mathrm {d}^{2}N}{p_{\text{T}}\mathrm {d}p_{\text{T}}\mathrm {d}y} \left[ 1+2\sum \limits _{n=1}^{\infty }v_{n}\cos [n(\phi -\Psi _\text {RP})] \right] . \end{aligned}$$
(1)

Here, \(\phi\) is the azimuthal angle of the emitted particles and \(\Psi _\text {RP}\) is the reaction plane angle. In this simulation, the reaction plane angle, \(\Psi _\text {RP}\), is zero. The coefficients, \(v_n\), can then be determined by \(v_{n} = \langle \cos [n(\phi )] \rangle\), where the average runs over all particles in all events. The first two coefficients, \(v_1\) and \(v_2\), are of particular interest as they are established early during the system evolution. Recently, the STAR experiment has reported new results for collective flow at \(\sqrt{s_{{\text{NN}}}}\) = 3 GeV [23, 24], which corresponds to a region with a high baryon density of 750 MeV [25]. The data suggest that hadronic interactions dominate the EoS in collisions at such energies. Exploring the phase structure in the high baryon density region is the main task of the second phase of the beam energy scan at the RHIC (BES-II) program [26], as well as the motivation for future facilities: such as the facility for proton and ion research (FAIR) [27], nuclotron-based ion collider facility (NICA) [28], and high-intensity heavy-ion accelerator facility (HIAF) [29, 30]. Regarding collectivity concerns, the responses of baryons and mesons are quite different. For example, we will focus on the collective measurements of protons, representative of baryons, and \(\pi ^{+}\)s, representative of produced mesons. In the case of pions, if the effect of isospin is neglected, there should be no differences in the flow of \(\pi ^{+}\) and \(\pi ^{-}\). Calculations of Au+Au collisions from a hadronic transport model UrQMD [31, 32] will be used in this study.

This paper is organized as follows: The basic features of the UrQMD model are briefly discussed in Sect. 2. In Sect. 3, we present a \(v_1\) and \(v_2\) comparison between the UrQMD calculations and recent STAR preliminary results at \(\sqrt{s_{{\text{NN}}}}\) = 3 GeV. The results of the energy dependence of \(v_1\) and \(v_2\) from UrQMD are also discussed. Finally, a summary is provided in Sect. 4.

2 The UrQMD model

The ultra-relativistic quantum molecular dynamics model (UrQMD) [31, 32] is a microscopic transport model for simulating the process of relativistic heavy-ion collisions. It has been widely and successfully used for studying pp, pA, and AA collisions in heavy-ion collision physics over a broad energy range from a few GeV to CERN LHC energies. The particle production in UrQMD is similar to other transport models, which include resonance excitation and decay, as well as string dynamics and strangeness exchange reactions. The propagation of these hadrons and their exacted states is based on binary elastic and inelastic scattering, in which the scattering cross sections are obtained from experimental data and models. The cascade version of the UrQMD model successfully describes the particle production, as well as directed flow measurements in heavy-ion collisions at collision energies \(\sqrt{s_{{\text{NN}}}}\) > 7 GeV [33, 34]. However, at lower collision energies, it is necessary to include the mean-field potential to describe the collective flow in the high baryon density region [35,36,37,38,39]. A baryonic Skyrme potential  [35, 36], based on the relative baryon density, is popularly used for heavy-ion collisions, which defines the nuclear EoS:

$$\begin{aligned} U = \alpha \left( \frac{\rho }{\rho _{0}} \right) + \beta \left( \frac{\rho }{\rho _{0}} \right) ^{\gamma }. \end{aligned}$$
(2)

Here, \(\rho\) and \(\rho _{0}\) are the baryon density in the ground state and baryon density, respectively. By changing the parameters \(\alpha\), \(\beta\), and \(\gamma\), the stiffness of the nuclear EoS can be controlled. In this study, the parameters \(\alpha\), \(\beta\), and \(\gamma\) are -124MeV, 71 MeV, and 2, respectively, corresponding to a nuclear incompressibility of \(\kappa\) = 380 MeV. This value of \(\kappa\) was also used in this study [38], which is consistent with the HADES collective flow measurements at \(\sqrt{s_{{\text{NN}}}}\) = 2.4 GeV [40]. Note that the value of \(\kappa\) depends on the collision energy and can be model dependent. As discussed in Ref. [39, 41,42,43,44], the extracted values of \(\kappa\) are below 300 MeV for Au+Au collisions at lower energies. The momentum-dependent potential is not included in the mean field in this study.

In this work, we use version 3.4 of the UrQMD model to generate Monte Carlo event samples of Au+Au collisions at center of mass energies of \(\sqrt{s_{{\text{NN}}}}\) = 2.11, 2.22, 2.32, 2.4, 2.51, 2.7, 2.86, 3.0, 3.5, 4.5, and 4.9 GeV, corresponding to beam kinetic energies of 0.5, 0.7, 1.0, 1.23, 1.5, 2.0, 2.5, 2.91, 4.65, 8.9, and 11 GeV per nucleon, respectively. Both cascade and mean-field modes of UrQMD are used to systematically study the collective flow in the high baryon density region.

3 Results and discussion

Fig. 1
figure 1

(Color online) Reference multiplicity distribution in UrQMD with cascade mode in Au+Au collisions at \(\sqrt{s_{{\text{NN}}}}\) = 2.11–4.9 GeV. Similar to the experimental setup, the charged particles were measured within \(|\eta |<\)0.5

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Following the STAR experimental analysis procedures, the collision centrality in this study is determined by the reference multiplicity, which is the number of charged pions, kaons, and protons within the pseudorapidity range \(|\eta |\) < 0.5 [45]. In Fig. 1, the reference multiplicity distribution in Au+Au collisions with the UrQMD cascade version from the center-of-mass energy \(\sqrt{s_{{\text{NN}}}}\) = 2.11–4.9 GeV is shown. These hadrons are produced more copiously at higher collision energies. The collision centrality bins for each energy can be determined by fitting the Glauber model [46]. In the following discussion, we report results from the 10–40% centrality bin where the collectivity is expected to have the maximum strength, especially the second order of the Fourier coefficient \(v_2\).

According to the recent STAR collective flow measurements at \(\sqrt{s_{{\text{NN}}}}\) = 3 GeV [23, 24], the UrQMD model results from both mean-field and cascade modes are compared with experimental data for the same centrality interval and kinematic selection criteria. Figure 2 presents the rapidity dependence of \(v_1\) and \(v_2\) for proton and \(\pi ^{+}\) in 10–40% centrality Au+Au collisions at \(\sqrt{s_{{\text{NN}}}}\) = 3 GeV. The symbols denote the STAR data. The red and blue bands represent the results from the UrQMD mean-field and cascade mode, respectively. Due to the experimental acceptance, only the negative rapidity range results are shown [23].

Fig. 2
figure 2

(Color online) Rapidity dependence of \(v_1\) (top panel) and \(v_2\) (bottom panel) of the proton and \(\pi ^{+}\) in the 10–40% centrality Au+Au collision at \(\sqrt{s_{{\text{NN}}}}\) = 3 GeV. The symbols denote STAR data. The red and blue bands represent the results from the UrQMD mean-field and cascade mode, respectively

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Fig. 3
figure 3

(Color online) Transverse momentum dependence of \(v_1\) (top panel) and \(v_2\) (bottom panel) of the proton and \(\pi ^{+}\) in the 10–40% centrality Au+Au collision at \(\sqrt{s_{{\text{NN}}}}\) = 3 GeV. The symbols denote STAR data. The red and blue bands represent the results from the UrQMD mean-field and cascade mode, respectively

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Negative \(v_2\) values were observed at mid-rapidity for both protons and \(\pi ^{+}\) at \(\sqrt{s_{{\text{NN}}}}\) = 3 GeV, indicating the shadowing effect of the passing spectators. Meanwhile, the proton \(v_2\) is positive in the forward rapidity region. For the \(v_1\) results, a strong \(v_1\) signal was observed compared to the high-energy results [47]. The proton \(v_1\) increases from the forward to mid-rapidity region, whereas \(\pi ^{+}\) \(v_1\) exhibited the opposite trend. The standard UrQMD cascade mode failed to describe the experimental data. However, by including the mean-field potential, the UrQMD results reproduce the rapidity dependence and negative \(v_2\) values at mid-rapidity for both protons and \(\pi ^{+}\). These agreements between the experimental data and UrQMD calculations with mean-field potential imply that hadronic interactions play an important role in the collective flow development at such collision energies. Similar to Fig. 2, we present the \(p_{\text{T}}\) dependence of \(v_1\) and \(v_2\) for protons and \(\pi ^{+}\) in 10–40% centrality Au+Au collisions at \(\sqrt{s_{{\text{NN}}}}\) = 3 GeV in Fig. 3. The mid-rapidity range of \(-0.5< y < 0\) for protons and \(\pi ^{+}\) are used in this analysis, which is consistent with the STAR experimental selection. Both \(v_1\) and \(v_2\) are negative and decrease with increasing \(p_{\text{T}}\). Again, the red bands shown in the figure indicate the UrQMD results with mean-field potential, which can reproduce the \(p_{\text{T}}\) dependence and qualitatively describe the experimental data. However, quantitative differences are visible in some \(p_{\text{T}}\) bins for both protons and \(\pi ^{+}\).

Fig. 4
figure 4

(Color online) Particle production yield as a function of freeze-out time for proton (left panel) and \(\pi ^{+}\) (right panel) from UrQMD 3 GeV 0–10% Au+Au collisions. The red and blue lines represent the results from the mean-field and cascade modes, respectively

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We systematically studied the mean-field potential effect on the particle production, directed flow, and elliptic flow development in the high baryon density region. The particle production yield as a function of kinetic freeze-out time for protons and \(\pi ^{+}\) in 3 GeV Au+Au central collisions (0–10%) at mid-rapidity (\(|y|<\) 0.5) is presented in Fig. 4. The red and blue lines represent the results from the UrQMD mean-field and cascade modes, respectively. Note that these two distributions were normalized by the number of events. Overall, hadrons are produced mostly during kinetic freeze-out in the time interval 10–25 fm/c. The mean value of the distribution is shown in each panel using the corresponding colors. It can be seen that the mean value from the mean-field version is smaller than that of the cascade version. This can be explained by the larger pressure generated by the density-dependent fields, which drives the strong expansion. Thus, these hadrons are pushed outside the system and freeze out earlier than those from the cascade mode.

Fig. 5
figure 5

(Color online) \(v_{1}\) slopes (\(\mathrm {d}v_{1}/\mathrm {d}y|_{y=0}\)) and \(v_{2}\) at mid-rapidity as a function of freeze-out time for proton (left panel) and \(\pi ^{+}\) (right panel) from UrQMD 3 GeV 10–40% Au+Au collisions. The red and blue bands represent the results from the mean-field and cascade modes, respectively

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Fig. 6
figure 6

(Color online) Mean transverse momentum \(\langle p_{\text{T}} \rangle\) as a function of collision energy for proton (squares) and \(\pi ^{+}\) (circles) from UrQMD 10–40% Au+Au collisions. The red and blue symbols represent the results from the mean-field and cascade modes, respectively

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Fig. 7
figure 7

(Color online) \(v_{1}\) slopes (\(\mathrm{d}v_{1}/\mathrm{d}y|_{y=0}\)) at mid-rapidity as a function of collision energy for proton (left panel) and \(\pi ^{+}\) (right panel) from UrQMD 10–40% Au+Au collisions. The red and blue symbols represent the results from the mean-field and cascade modes, respectively. World experimental data are represented by black symbols

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To quantify the strength of directed flow at mid-rapidity, a linear fit is employed to extract the slope of the \(v_{1}(y)\) distribution [19, 47], as shown in Fig. 2. The directed flow is exactly zero at \(y = 0\) owing to the momentum conservation effect. Figure 5 presents the \(v_1\) slopes \(\mathrm {d}v_{1}/\mathrm {d}y|_{y=0}\) and elliptic flow \(v_2\) at mid-rapidity as a function of kinetic freeze-out time (t) for proton and \(\pi ^{+}\) in 3 GeV Au+Au mid-central collisions (10–40%); the red and blue bands represent the results from the UrQMD mean-field and cascade mode, respectively. Both \(v_1\) and \(v_2\) are established in the very early stage of the system evolution and show a rapid increase for t \(\lesssim\) 15 fm/c. They then increased slightly with increasing freeze-out time. As can be seen in the figure, the \(v_1\) values for the mid-rapidity proton and pion start from below zero before turning into positive values at a later time, as experimentally observed. Owing to the baryonic mean field, at \(t\sim 15\) fm/c, the \(v_1\) slope for the proton is much larger than that of \(\pi ^{+}\), especially in the results for the mean-field mode. Conversely, while the cascade predicts an in-plane expansion, for example, positive \(v_2\) for both protons and pions, the mean field tends to keep the incoming nucleons together. As a result, shadowing seems to be sustained throughout the collision period, leading to negative values of \(v_2\) and out-of-plane expansion at this energy. In short, compared to the cascade mode, the mean-field potential provides a stronger expansion in the \(x-z\) plane (\(v_1\)) and an effective blocking that leads to the expansion in the \(y-z\) plane (\(v_2\)) in Au+Au collisions at 3 GeV.

The mean transverse momentum \(\langle p_{\text{T}} \rangle\) as a function of collision energy \(\sqrt{s_{{\text{NN}}}}\) at a mid-rapidity of \(|y|<\) 0.5 for protons and \(\pi ^{+}\) in UrQMD 10–40% Au+Au collisions is shown in Fig. 6. The red and blue symbols represent the results from the cascade and mean-field modes, respectively. It can be observed that \(\langle p_{\text{T}} \rangle\) increases with increasing collision energy for both particle species owing to the higher energies distributed in the transverse direction. The mean-field potential slightly increases \(\langle p_{\text{T}} \rangle\) for all particles as a result of the increased radial flow. The increase in \(\langle p_{\text{T}} \rangle\) is greater for heavier particle protons. The enhancement in the mean-field mode is due to the more repulsive interactions in the high baryon density region.

We will now discuss the energy dependence of directed and elliptic flows from UrQMD and compare this with world experimental data. Figure 7 shows the energy dependence of the mid-rapidity slope of directed flow (\(\mathrm {d}v_{1}/\mathrm {d}y|_{y=0}\)) in 10–40% centrality Au+Au collisions. The transverse momentum range in the UrQMD calculations for proton and \(\pi ^{+}\) is 0.4 \(< p_{\text{T}}<\) 2.0 GeV/c and 0.2 \(< p_{\text{T}}<\) 1.6 GeV/c, respectively, which is consistent with the STAR 3 GeV results [23]. The black symbols denote the world experimental data [23, 48, 49]. The red and blue symbols represent the calculations from the UrQMD mean-field and cascade modes, respectively. For the \(\pi ^{+}\) \(v_1\) and \(v_2\) measurements, only the STAR results from \(\sqrt{s_{{\text{NN}}}}\) = 3 and 4.5 GeV are available. The proton \(\mathrm {d}v_{1}/\mathrm {d}y|_{y=0}\) values increase with decreasing collision energy for both the UrQMD cascade and mean-field calculations, while the mean-field potential significantly enhances the \(\mathrm {d}v_{1}/\mathrm {d}y|_{y=0}\) values, allowing comparison with the experimental data for both protons and \(\pi ^{+}\). However, there are still some differences relating to the choice of the \(p_{\text{T}}\) range. As we can see in Fig. 2, \(v_1\) has a strong \(p_{\text{T}}\) dependence at such a low energy.

The energy dependence of the elliptic flow at mid-rapidity in 10–40% centrality Au+Au collisions is shown in Fig. 8. The black symbols denote the world experimental data [23, 49, 50]. Both the sign and absolute value of \(v_2\) reflect the medium properties. \(v_2\) increases as a function of the collision energy. The results from the cascade mode are always above the results from the mean-field mode, but both converge at a high energy of approximately 5 GeV. At approximately 2.4 GeV, the proton \(v_2\) has already changed from a negative to positive value in the cascade mode, and the change occurs at a much higher energy in the case of the mean-field calculation. According to world data, the proton elliptic flow transits from in-plane expansion to out-of-plane expansion at \(\sqrt{s_{{\text{NN}}}}\) \(\sim\) 3.6 GeV as a result of the shadowing effect, where particles are blocked in the reaction plane by the spectators and are emitted mainly in the out-of-plane direction. In Fig. 8, the UrQMD calculations with the mean-field potential qualitatively describe the experimental data for both protons and \(\pi ^{+}\) and are quantitatively consistent with the data for \(\sqrt{s_{{\text{NN}}}}<\) 3.6 GeV.

Fig. 8
figure 8

(Color online) Similar to Fig. 7: \(v_{2}\) at mid-rapidity as a function of the collision energy for protons (left panel) and \(\pi ^{+}\) (right panel) from UrQMD to 10–40% Au+Au collisions

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4 Summary

A hadronic transport model, UrQMD, was employed to study the directed and elliptic flows in the high baryon density region. The model provides a good description of the recent STAR results of 10–40% Au+Au collisions at \(\sqrt{s_{{\text{NN}}}}\) = 3 GeV by including the mean-field potential. This indicates that hadronic interactions play an important role in the collective flow development in the high baryon density region. Other measurements from the 3 GeV Au+Au collisions, including strangeness production [51] and high moments of protons [52], also suggest that hadronic interactions dominate the dynamics in the high baryon density region.

The results of the time evolution of directed and elliptic flows show that the collective flow is fully developed at \(\sim 10-15\) (fm/c).

The mean-field potential slightly enhances \(\langle p_{\text{T}} \rangle\) because of the stronger radial expansion and shifts the freeze-out to an earlier time. In addition, compared to the cascade mode, the mean-field option delays the proton \(v_2\) crossing zero as a function of collision energy, from \(\sqrt{s_{{\text{NN}}}}\) = 2.5 to 3.5 GeV. A negative or positive \(v_2\) corresponds to an out-of-plane or in-plane expansion, respectively.

These results will be useful for understanding experimental data from upcoming experimental facilities focused on the high baryon density region.