Model Study of the Energy Dependence of Anisotropic Flow in Heavy-Ion Collisions at √ s N N = 2–4.5 GeV

: The anisotropic ﬂow is one of the important observables sensitive to the equation of state (EOS) and transport properties of the strongly interacting matter created in relativistic heavy-ion collisions. In this work, we report a detailed multi-differential study of the directed ( v 1 ), elliptic ( v 2 ), triangular ( v 3 ), and quadrangular ( v 4 ) ﬂow coefﬁcients of protons in relativistic heavy-ion collisions at √ s NN = 2.2–4.5 GeV using several hadronic transport models. Recent published results for Au + Au collisions at √ s NN = 2.4 GeV from HADES experiment and at √ s NN = 3.0 GeV from the STAR experiment have been used for comparison. The study motivates further experimental investigations of the anisotropic collective ﬂow of protons and neutrons in a high baryon density region.


Introduction
The exploration of the properties of strongly interacting QCD matter for a wide range of temperatures and densities is the main goal of the present and future Beam Energy Scan (BES) programs with relativistic heavy-ions [1]. The existing BES programs include: programs BES-I and BES-II of the STAR experiment at RHIC for Au + Au collisions at √ s NN = 3-200 GeV (collider + fixed target) [2], program of the NA61/SHINE fixed target experiment at SPS for different colliding systems (Be + Be, Ar + Sc Xe + La, Pb + Pb) at √ s NN = 5.1-17.3 GeV [3], program of the BM@N fixed target experiment at Nuclotron for different colliding systems (Ar + Sc, Xe + CsI, Ar + Pb, Bi + Bi) at √ s NN = 2.4-3.5 GeV [4] and program of the HADES fixed target experiment at SIS-18 for (Au + Au, Ag + Ag) at √ s NN = 2.4-2.55 GeV [5]. The main goal is to study the the high-density equation of state (EOS) and search for a possible phase transition in dense baryonic matter [6].
In the near future, the MPD experiment at NICA ( √ s NN = 4-11 GeV) [7] and the CBM experiment at FAIR ( √ s NN = 2.7-4.9 GeV) [8] will further explore the phase diagram at a high baryon density region with high statistics data. The anisotropic flow, as manifested by the anisotropic collective emission of particles in the plane transverse to the beam direction, is one of the important observables sensitive to the equation of state (EOS) and transport properties of the strongly interacting matter created in relativistic heavy-ion collisions [9]. Moreover, the neutron-proton elliptic flow ratio and the v 2 difference have been demonstrated to be a sensitive probe of the EoS of asymmetric matter [10,11]. The anisotropic flow can be quantified by Fourier coefficients v n [9] in the expansion of the particle azimuthal distribution as: where n is the order of the harmonic, ϕ is the azimuthal angle of particle of the given type, and Ψ R is the azimuthal angle of the reaction plane. The flow coefficients v n can be calculated as v n = cos[n(ϕ − Ψ R )] , where the brackets denote the average over the particles and events. The directed (v 1 ) and elliptic (v 2 ) flows are dominant flow signals in the energy range of 2 < √ s NN < 11 GeV [9,[12][13][14][15][16][17]. The most stringent currently available constraints on the high-density EOS of symmetric nuclear matter come from the measurements of v 1 and v 2 flow signals of protons in Au + Au collisions at √ s NN = 2.7-4.3 GeV by the E895 experiment at the AGS in Brookhaven [9,[12][13][14]18]. Compared to the relativistic transport model calculations, the v 1 results from E895 can be well reproduced by a soft EOS with the incompressibility K 0 = 210 MeV, while reproducing the v 2 measurements required larger values of K 0 = 300 MeV (and therefore a harder EOS) [18]. The same conclusion can be drawn from the recent work, where an attempt was made to describe the v 1 and v 2 results from the E895 experiment using a different hadronic transport code [19]. Such a large spread of K 0 values may come from a substantial disagreement between the E895 [12,13] and the recent STAR results for v 1 and v 2 of protons [15,16]. Therefore, high precision measurements of both the directed and elliptic flows from 2 < √ s NN < 5 GeV are required, in order to reduce the existing discrepancy between the present results and to further constrain the EOS for the symmetric matter. Recently, the HADES experiment has reported the first measurements of the higher order (v 3 and v 4 ) flow coefficients of protons in Au + Au collisions at √ s NN = 2.4 GeV [17]. The higher order flow coefficients are expected to be more sensitive to the EOS and may provide additional constraints. The ratio of the elliptic flow signals observed for neutrons and protons proposed to study the EoS of asymmetric matter [10]. However, an effective constraint for the EoS of asymmetric matter would require experimental values for the elliptic flow of both protons and neutrons of an accuracy of 1% or better [11]. At present, in spite of the rather rich experimental data on proton flow in the considering energy range of colliding nuclei, rather scarce data on neutron flow are available only at energies below 1 AGeV for Au + Au collisions [10]. The BM@N experiment opens an opportunity to measure the flow of protons by the existing magnetic spectrometer and neutrons by the neutron detector, which is now under development at the Nuclotron energy range, where no experimental data on neutron flow exist.
The recently published v n results for Au + Au collisions at √ s NN = 2.4 GeV from the HADES experiment [17] and at √ s NN = 3.0 GeV from the STAR experiment [15] have been used for comparison. We seek to establish whether the selected transport models can reproduce the distinctive features of v n of protons. We predict the energy and system size (Au + Au, Xe + Cs, Ag + Ag) dependence of v n of protons in relativistic heavy-ion collisions at √ s NN = 2.2-4.5 GeV. The results would be useful as predictions for the upcoming beam energy scan experiments: BM@N at Nuclotron and MPD at NICA Collider.

Short Description of the Transport Models
In this section, we shortly summarise the main features of the selected hadronic transport models.
The Ultrarelativistic Quantum Molecular Dynamics (UrQMD) model [20,21] is a microscopic transport approach based on the binary elastic and inelastic scattering of hadrons, resonance excitations and decays as well as string dynamics and strangeness exchange reactions. We have used the version 3.4 of the UrQMD with the default set of parameters in the cascade mode (no potentials, EoS = 0) and mean-field with a hard Skyrme equation of the state (EoS = 1) with K 0 = 380 MeV.
The Parton-Hadron-Quantum-Molecular Dynamics (PHQMD) [22] is a n-body dynamical transport approach designed to provide a microscopic dynamical description for the formation of light and heavy clusters and hypernuclei as well as for hadrons in relativistic heavy-ion collisions. The propagation of baryons is based on the n-body QMD dynamics while the description of mesons was taken from the PHSD model. The PHQMD includes mutual 2-body density-dependent Skyrme type potentials for interactions among baryons. Two sets of parameters for the nuclear equation of the state have been used; see Table I from Ref. [22]. The parameters for the "soft EOS" lead to a nuclear incompressibility of K 0 = 200 MeV and for the "hard EOS" to a K 0 = 380 MeV [22].
DCM-QGSM-SMM is a hybrid heavy-ion event generator based on the Dubna Cascade Model (DCM), the Quark-Gluon String Model (QGSM) and the Statistical Multifragmentation Model (SMM) [23]. JET AA Microscopic Transportation Model (JAM) has been developed to simulate high energy nuclear collisions [24][25][26]. We have used the version JAM 1.9 [26] with five different EOS implementations: soft momentum independent NS2 (K 0 = 210 MeV), hard momentum independent NS1 (K 0 = 380 MeV), soft momentum dependent MD4 (K 0 = 210 MeV) and hard momentum dependent MD2 and MD3 (K 0 = 380 MeV) with different values of the optical potential; see Table I from Ref. [26] for the detailed description of the EOS implementations. The JAM approach has been found to reproduce the directed and integral elliptic flow data at 2.3 < √ s NN < 8 GeV simultaneously with the parameter set MD2 [26].

Results
We need to define the centrality of the collision to make the proper comparison with the published experimental v n results. In both STAR [27] and HADES [28] experiments, the measured multiplicity distribution of the produced particles dN/dN ch is divided into percentile centrality classes, with the most central class defined by X% of events with the highest value of N ch , which corresponds to small values of the impact parameter b. The correlation between measured N ch and b of the collision is then inferred from the comparison of dN/dN ch with the Monte-Carlo Glauber (MC-Glauber) simulations [27][28][29]. Following the analysis procedure of the STAR experiment [27], the centrality has been determined by the reference multiplicity N ch of the produced charged particles, which is the number of charged particles within the pseudorapidity range |η| < 0.5. As an example, Figure 1 shows the reference multiplicity distribution for Au + Au collisions at √ s NN = 2.4-4.5 GeV for the mean-field mode of JAM (left panel) and UrQMD (right panel) models. The selection of centrality classes is based on MC-Glauber approach [29]. As an example, Figure 2 shows the reference multiplicity distributions of charged particles for Au + Au collisions at √ s NN  [29]. A total of 10% centrality classes are indicated with black vertical lines in Figure 2. Let us start the comparison with published results from the cascade version of the selected hadronic transport models. From the previous studies, we know that cascade models can roughly capture the overall magnitude and trends of the measured directed (v 1 ) and elliptic (v 2 ) flow signals of protons in Au + Au collisions at √ s NN = 4.5-7.7 GeV [30][31][32].  The directed flow (v 1 ) can probe the very early stages of the collision, as it is generated during the passage time of the two colliding nuclei t pass = 2R/(γ s β s ), where R is the radius of the nucleus at rest, β s is the spectator velocity in c.m. and γ s is the corresponding Lorentz factor, respectively. v 1 is expected to increase with increasing the passage time (decreasing collision energy). At lower beam energies √ s NN < 4 GeV, shadowing effects by the spectator matter play an important role in the generation of elliptic flow. For a broad range of energies ( √ s NN = 2-4 GeV), the v 2 results can be understood in terms of a delicate balance between (i) the ability of pressure developed early in the compressed overlap zone, to effect a rapid transverse expansion of nuclear matter, and (ii) the passage time t pass at which the accelerated nuclei interpenetrate each other [12,14,18]. In the energy range √ s NN = 4-2 GeV, the passage time t pass increases from 7 fm/c to 30 fm/c. If the passage time t pass is long compared to the expansion time, spectator nucleons serve to block the path of participant hadrons emitted toward the reaction plane, and nuclear matter is squeezedout perpendicular to this plane, giving rise to negative elliptic flow (v 2 < 0). The nuclear mean-field effects will significantly contribute to the observed azimuthal anisotropies in this energy range [12,14,15,18].
Recently, the HADES experiment at SIS-18 has reported the first detailed differential measurements of anisotropic flow coefficients v 1 , v 2 , v 3 and v 4 of protons in Au + Au collisions at √ s NN = 2.4 GeV [17]. Figure 4 shows the p T dependence of anisotropic flow       Recent v 1 and v 2 results from the STAR experiment were used to test the model calculations for Au + Au collisions at √ s NN = 3.0 GeV [15]. Figure 10 shows the rapidity (y cm ) dependence of v 1 (top panels) and v 2 (bottom panels) of proton (left panels), charged pions (middle panels) and charged kaons (right panels) in 10-40% central Au + Au collisions at √ s NN = 3.0 GeV. The symbols denote the published data from the STAR experiment [15].
The blue (MD2), purple (MD4), red (NS1) and yellow (NS2) bands represent the results from the mean-field mode of the JAM model with different EOS, as indicated. Again, with the mean-field, the JAM model qualitatively reproduces the rapidity dependence of v 1 and v 2 for protons and pions. However, the JAM model fails to reproduce the rapidity dependence of v 1 and v 2 for kaons. Figure 11 shows the p T dependence of elliptic flow    Figures 12 and 13 show the comparison between the PHQMD model calculations and the STAR results. Again, the symbols denote the published data from the STAR experiment [15]. The red and green bands represent the results from PHQMD model with hard (K 0 = 380 MeV) and soft (K 0 = 200 MeV) EOS, respectively. The purple bands represent the results from the Hadron-String-Dynamics (HSD) model [33]. PHQMD is close to the v 1 and v 2 data for kaons, but deviates for protons and pions. Considering only scattering and neglecting the kaon-nuclear potential does not allow the models to describe the kaon flow data [34] The detailed differential measurements from the STAR experiment may help to constrain the kaon-nuclear potential.  were obtained using the JAM model with hard momentum dependent mean-field MD2 (K 0 = 380 MeV) EOS. Figure 14 shows the p T dependence of anisotropic flow coefficients v 1 , v 2 , v 3 and v 4 of protons (from top to bottom panels) in the 10-40% central Au + Au collisions at √ s NN = 2.2-4.5 GeV. All v n coefficients have a strong p T dependence. All proton v n (p T ) values decrease with increasing the collision energy. The passage time decreases with the increasing collision energy. This will reduce the shadowing effects by the spectator matter [12,14,18]. Figure 14 shows that v n (p T ) of protons at midrapidity |y cm | < 0.2 transits from out-of-plane (v 2 < 0) to in-plane (v 2 > 0) in the collision energy around √ s NN ∼ 3.3 GeV.
This is in a good agreement with the results of the E895 experiment at AGS [12,14]. The model predicts that the higher order (v 3 and v 4 ) flow coefficients will be close to zero at √ s NN > 3.3 GeV. One can observe the similar trends in the rapidity dependence (y cm ) dependence of anisotropic flow coefficients v 1 , v 2 , v 3 and v 4 of protons; see Figure 15.  The rapidity dependence of anisotropic flow coefficients v 1 , v 2 , v 3 and v 4 of protons for different energies becomes less complicated if one uses the scaled rapidity y = y cm /y beam , since for the colliding beams one then always has y beam = ±1 in the center-of-mass frame. The passage time t pass can be estimated as t pass = 2R/sinh(y beam ), where R is the radius of the nucleus. The scaled rapidity (y = y cm /y beam ) dependence of anisotropic flow coefficients v n may reflect the partial scaling of v n with t pass in this energy range, see Figure 16.  Figure 17 shows the scaled rapidity (y = y cm /y beam ) dependence of v 2 of protons in the central 0-10% (left panels), midcentral 10-40% (central panels) and peripheral 40-80% (right panels) Au + Au collisions at √ s NN = 2.4, 3.3, 4.0 GeV. One can clearly observe that the transition of v 2 from out-of-plane (v 2 < 0) to in-plane (v 2 > 0) has a strong centrality dependence. This is in a qualitative agreement with the results of the E895 experiment at AGS [14].  Figure 18 shows the collision energy dependence of v 2 of protons in the transverse momentum interval 0.4 < p T < 2.0 GeV/c for different intervals in collision centrality of Au + Au collisions, as indicated. Again, we observe the a strong centrality dependence for the transition of v 2 from out-of-plane (v 2 < 0) to in-plane (v 2 > 0).         The left panel of Figure 27 shows the differential v 2 (p T ) of protons in Au + Au collisions at √ s NN = 2.4 GeV for different bins in the collision centrality. The results have been obtained using the JAM model with hard momentum dependent mean-field MD2.
The v 2 (p T ) results exhibit the familiar increase, as collisions become more peripheral and the p T increase [12,14,15,17]. A specifically produced particle moving with the transverse velocity v t will be shadowed by the spectator matter during the passage time t pass . The simple geometrical estimate then leads to the condition [35]: v t > (2R − b)/t pass , where R is the radius of the nucleus and b is the impact parameter. it is easier to fulfill this condition for the particle with high p T and for peripheral collisions [35]. We have tested these data for the "integral flow scaling", observed for v 2 in heavy-ion collisions at RHIC [36][37][38], by dividing the differential values v 2 (centrality, p T ) shown in the left panel Figure 27 by the v 2 of protons integrated over the p T range 0.4-2.0 GeV/c for each of the indicated centrality selections (|v 2 (int,centrality)|). The right panel of Figure 27 shows that the "integral flow scaling" can be a general feature of anisotropic flow and can be observed at different colliding energies. Figure 28 shows the resulting scaled v 2 values for Xe + Cs and Au + Au collisions at √ s NN = 2.4 GeV (left panel) and √ s NN = 2.7 GeV (right panel). These scaled v 2 values are clearly independent of the colliding system size and show essentially perfect scaling for the full range of centralities. The recent v n measurements for Au + Au and Ag + Ag collisions from HADES experiment at SIS-18 and v n measurements from the upcoming Xe + Cs(I) run of the BM@N experiment at Nuclotron will be used to test these scaling relations for v n .

Conclusions
In summary, we have presented a detailed multi-differential study of collective flow coefficients v n of protons in relativistic heavy-ion collisions at √ s NN = 2.2-4.5 GeV using several hadronic transport models. The directed (v 1 ), elliptic (v 2 ) and higher order (v 3 and v 4 ) flow coefficients of protons have been studied as a function of transverse momentum p T , rapidity (y cm ) for several intervals in collision centrality. We have found that transport models JAM and UrQMD in the mean-field mode can qualitatively reproduce the recently published v n measurements of protons from Au + Au collisions at √ s NN = 2.4 GeV (HADES experiment) and √ s NN = 3.0 GeV (STAR experiment). In case of the cascade mode calculations, the absence of a repulsive potential significantly reduces the v 1 and v 2 signals and results in essentially zero signals for the higher order (v 3 and v 4 ) flow coefficients. The v n results for different colliding systems (Au + Au, Xe + Cs, Ag + Ag) and collision energies has been compared using different scaling variables. A similar approach and models will be used for neutron flow simulation nucleus-nucleus collisions at the energy range of the BM@N experiment. The results would be useful as predictions for the upcoming beam energy scan experiments: BM@N at Nuclotron and MPD at NICA Collider.