We consider different implementations of momentum-dependent hadronic mean-fields in the relativistic quantum molecular dynamics (RQMD) framework. First, Lorentz scalar implementation of Skyrme type potential is examined. Then, full implementation of Skyrme type potential as a Lorentz vector in the RQMD approach is proposed. We find that scalar implementation of Skyrme force is too weak to generate repulsion explaining observed data of sideward flows at $sqrt{s_{NN}}<10$ GeV, while vector implementation gives collective flows compatible with the data for a wide range of beam energies $2.7 <sqrt{s_{NN}}<20$ GeV. We show that our approach reproduces the negative proton directed flow at $sqrt{s_{NN}}>10$ GeV discovered by the experiments. We discuss the dynamical generation mechanisms of the directed flow within a conventional hadronic mean-field. A positive slope of proton directed flow is generated predominantly during compression stages of heavy-ion collisions by the strong repulsive interaction due to high baryon densities. In contrast, at the expansion stages of the collision, the negative directed flow is generated more strongly over the positive one by the tilted expansion and shadowing by the spectator matter. At lower collision energies $sqrt{s_{NN}}<10$ GeV, the positive flow wins against the negative flow because of a long compression time. On the other hand, at higher energies $sqrt{s_{NN}}>10$ GeV, negative flow wins because of shorter compression time and longer expansion time. A transition beam energy from positive to negative flow is highly sensitive to the strength of the interaction.
Relativistic quantum molecular dynamics based on the relativistic mean field theory (RQMD.RMF) is extended by including momentum-dependent potential. The equation of state (EoS) dependence of the directed and the elliptic flow of protons in the beam energy range of $2.3 < sqrt{s_{NN}}< 20$ GeV is examined. It is found that the directed flow depends strongly on the optical potential at high energies,$sqrt{s_{NN}} > 3 $ GeV, where no information is available experimentally. The correlation between effective mass at saturation density and the optical potential is found: smaller values of effective mass require smaller strengths of the optical potential to describe the directed flow data.This correlation can also be seen in the beam energy dependence of the elliptic flow at $sqrt{s_{NN}}>3$ GeV, although its effect is rather weak. On the other hand, stiff EoS is required to describe the elliptic flow at lower energies.Experimental constraints on the optical potential from $pA$ collisions will provide important information on the EoS at high energies.The proton directed and the elliptic flow are well described in the RQMD.RMF model from $sqrt{s_{NN}}=2.3$ to 8.8 GeV. In contrast,to reproduce the collapse of the directed flow above 10 GeV, pressure has to be reduced, which indicates a softening of the EoS around $sqrt{s_{NN}} =10 $ GeV.
The McLerran-Venugopalan (MV) model is a Gaussian effective theory of color charge fluctuations at small-$x$ in the limit of large valence charge density, {it i}.{it e}., a large nucleus made of uncorrelated color charges. In this work, we explore the effects of the first non-trivial (even C-parity) non-Gaussian correction on the color charge density to the MV model (quartic term) in SU(2) and SU(3) color group in the non-perturbative regime. We compare our (numerical) non-perturbative results to (analytical) perturbative ones in the limit of small or large non-Gaussian fluctuations. The couplings in the non-Gaussian action, $barmu$ for the quadratic and $kappa_4$ for the quartic term, need to be renormalized in order to match the two-point function in the Gaussian theory. We investigate three different choices for the renormalization of these couplings: i) $kappa_{4}$ is proportional to a power of $barmu$; ii) $kappa_4$ is kept constant and iii) $barmu$ is kept constant. We find that the first two choices lead to a scenario where the small-$x$ action evolves towards a theory dominated by large non-Gaussian fluctuations, regardless of the system size, while the last one allows for controlling the deviations from the MV model.
Pauli blocking is carefully investigated for the processes of $NN rightarrow N Delta$ and $Delta rightarrow N pi$ in heavy-ion collisions, aiming at a more precise prediction of the $pi^-/ pi^+$ ratio which is an important observable to constrain the high-density symmetry energy. We use the AMD+JAM approach, which combines the antisymmetrized molecular dynamics for the time evolution of nucleons and the JAM model to treat processes for $Delta$ resonances and pions. As is known in general transport-code simulations, it is difficult to treat Pauli blocking very precisely due to unphysical fluctuations and additional smearing of the phase-space distribution function, when Pauli blocking is treated in the standard method of JAM. We propose an improved method in AMD+JAM to use the Wigner function precisely calculated in AMD as the blocking probability. Different Pauli blocking methods are compared in heavy-ion collisions of neutron-rich nuclei, ${}^{132}mathrm{Sn}+{}^{124}mathrm{Sn}$, at 270 MeV/nucleon. With the more accurate method, we find that Pauli blocking is stronger, in particular for the neutron in the final state in $NN rightarrow N Delta$ and $ Delta to Npi$, compared to the case with a proton in the final state. Consequently, the $pi^-/pi^+$ ratio becomes higher when the Pauli blocking is improved, the effect of which is found to be comparable to the sensitivity to the high-density symmetry energy.
Relativistic quantum molecular dynamics with scalar and vector interactions based on the relativistic mean meson field theory, RQMD.RMF, is developed.It is implemented into the microscopic transport code JAM.The sensitivity of the directed and of the elliptic proton flow in high energy heavy-ion collisions on the stiffness of the RMF equation of state (EoS) is examined. These new calculations are compared to experimental data at mid-central Au + Au collisions in the beam energy range $2.5 < sqrt{s_{NN}} < 20$ GeV. This new RQMD model with the relativistic mean field scalar and vector meson interactions does describe consistently, with one RMF parameter set,the beam energy dependence of both the directed flow and the elliptic flow,from SIS18 to AGS and RHIC BES-II energies, $sqrt{s_{NN}}< 10$ GeV.There are different sensitivities of these different kinds of flow to the EoS: elliptic flow is most sensitive to the nuclear incompressibility constant,at the moderate beam energies $sqrt{s_{NN}}<3$ GeV,whereas the directed flow is most sensitive to the effective baryon mass at saturation density at $3< sqrt{s_{NN}}<5 $ GeV. Matters abruptly change in the next higher energy range,$sqrt{s_{NN}}gtrsim 10-20$ GeV:the directed flow data show a double change of sign of the slope of $v_1$, inverting twice in this energy range,in sudden contradiction to the RQMD.RMF calculation for a monotonous, stiff EoS. This surprising oscillating behavior,a double change of sign of the $v_1$ slope, points to the appearance of a hitherto unknown first-order phase transition in excited QCD matter at high baryon densities in mid-central Au + Au collisions.
The beam energy dependence of $v_4$ (the quadrupole moment of the transverse radial flow) is sensitive to the nuclear equation of state (EoS) in mid-central Au + Au collisions at the energy range of $3 < sqrt{s_{NN}} < 30$ GeV, which is investigated within the hadronic transport model JAM. Different equations of state, namely, a free hadron gas, a first-order phase transition and a crossover are compared. An enhancement of $v_4$ at $sqrt{s_{{NN}}}approx 6$ GeV is predicted for an EoS with a first-order phase transition. This enhanced $v_4$ flow is driven by both the enhancement of $v_2$ as well as the positive contribution to $v_4$ from the squeeze-out of spectator particles which turn into participants due to the admixture of the strong collective flow in the shocked, compressed nuclear matter.
The beam energy dependence of the elliptic flow,$v_2$, is studied in mid-central Au+Au collisions in the energy range of $3leq sqrt{s_{NN}} leq 30$ GeV within the microscopic transport model JAM. The results of three different modes of JAM are compared; cascade-,hadronic mean field-, and a new mode with modified equations of state, with a first order phase transition (1.O.P.T.) and with a crossover transition. The standard hadronic mean field suppresses $v_2$, while the inclusion of the effects of a 1.O.P.T. (and also of a crossover transition) does enhance $v_2$ at $sqrt{s_{NN}}<30$ GeV. The enhancement or suppression of the scaled energy flow, dubbed elliptic flowis understood as being due to out of plane- flow, i.e. $v_2<0$, dubbed out of plane - squeeze-out, which occurs predominantly in the early, compression stage. Subsequently, the in-plane flow dominates, in the expansion stage, $v_2 > 0$. The directed flow, dubbed bounce- off, is an independent measure of the pressure, which quickly builds up the transverse momentum transfer in the reaction plane. When the spectator matter leaves the participant fireball region, where the highest compression occurs, a hard expansion leads to larger $v_2$. A combined analysis of the three transverse flow coefficients, radial $v_0$-, directed $v_1$- and elliptic $v_2$- flow, in the beam energy range of $3leqsqrt{s_{NN}}leq10$ GeV, distinguishes the different compression and expansion scenarios: a characteristic dependence on the early stage equation of state is observed. The enhancement of both the elliptic and the transverse radial flow and the simultaneous collapse of the directed flow of nucleons offers a clear signature if 1.O.P.T. is realized at the highest baryon densities created in high energy heavy-ion collisions.
We study the sensitivities of the directed flow in Au+Au collisions on the equation of state (EoS), employing the transport theoretical model JAM. The EoS is modified by introducing a new collision term in order to control the pressure of a system by appropriately selecting an azimuthal angle in two-body collisions according to a given EoS. It is shown that this approach is an efficient method to modify the EoS in a transport model. The beam energy dependence of the directed flow of protons is examined with two different EoS, a first-order phase transition and crossover. It is found that our approach yields quite similar results as hydrodynamical predictions on the beam energy dependence of the directed flow; Transport theory predicts a minimum in the excitation function of the slope of proton directed flow and does indeed yield negative directed flow, if the EoS with a first-order phase transition is employed. Our result strongly suggests that the highest sensitivity for the critical point can be seen in the beam energy range of $4.7leqsrtNNleq11.5$ GeV.
We analyze the directed flow of protons and pions in high-energy heavy-ion collisions in the incident energy range from $sqrt{s_{{scriptscriptstyle NN}}}=7.7$ to 27 GeV within a microscopic transport model. Standard hadronic transport approaches do not describe the collapse of directed flow below $sqrt{s_{{scriptscriptstyle NN}}}simeq 20$ GeV. By contrast, a model which simulates effects of a softening of the equation of state, well describes the behavior of directed flow data recently obtained by the STAR Collaboration~cite{STARv1}. We give a detailed analysis of how directed flow is generated. Particularly, we found that softening of effective equation of state at the overlapping region of two nuclei, i.e. the reaction stages where the system reaches high baryon density state, is needed to explain the observed collapse of proton directed flow within a hadronic transport approach.
We investigate the effects of nuclear mean-field as well as the formation and decay of nuclear clusters on the directed flow $v_1$ in high energy nucleus-nucleus collisions from $sqrt{s_{NN}}=7.7$ GeV to 27 GeV incident energies within a transport model. Specifically, we use the JAM transport model in which potentials are implemented based on the framework of the relativistic quantum molecular dynamics. Our approach reproduces the rapidity dependence of directed flow data up to $sqrt{s_{NN}}approx 8$ GeV showing the significant importance of mean-field. However, the slopes of $dv_1/dy$ at mid-rapidity are calculated to be positive at $sqrt{s_{NN}}=11.7$ and 19.6 GeV, and becomes negative above 27 GeV. Thus the result from the JAM hadronic transport model with nuclear mean-field approach is incompatible with the data. Therefore within our approach, we conclude that the excitation function of the directed flow cannot be explained by the hadronic degree of freedom alone.
