Probing the softest region of nuclear equation-of-state

Bao-An Li11Address after Aug. 10: Dept. of Chemistry and Physics, Arkansas State University, P.O. Box 419, State University, AR 72467-0419 and C. M. Ko Cyclotron Institute and Physics Department,
Texas A&M University, College Station, TX 77843
preprint: Phys. Rev. C (1998) in press.

An attractive, energy-dependent mean-field potential for baryons is introduced in order to generate a soft region in the nuclear equation-of-state, as suggested by recent lattice QCD calculations of baryon free matter at finite temperature. Based on a hadronic transport model, we find that although this equation-of-state has negligible effects on the inclusive hadronic spectra, it leads to a minimum in the energy dependence of the transverse collective flow and a delayed expansion of the compressed matter. In particular, the transverse flow changes its direction as the colliding system passes through the softest region in the equation-of-state.
PACS number(s): 25.75.+r

The primary goal of experiments on relativistic heavy-ion collisions is to create and study the Quark-Gluon-Plasma (QGP) (1). Lattice QCD calculations of baryon-free matter at finite temperature (2)have shown that although both the energy and entropy densities increase appreciably at about 150 MeV where the phase transition from the hadronic matter to the QGP occurs, the change in pressure is considerably weaker, leading thus to a small sound velocity which is given by the pressure gradient with respect to energy density, i.e., Vs2dP/desuperscriptsubscript𝑉𝑠2𝑑𝑃𝑑𝑒V_{s}^{2}\equiv dP/de . This soft region in the nuclear equation-of-state is expected to have significant influence on the collective dynamics of the hot and dense matter formed in relativistic heavy ion collisions. In particular, a small sound velocity delays the expansion of the compressed matter and leads also to a reduced transverse collective flow. Indeed, recent studies based on the hydrodynamical model have demonstrated that as one varies the incident energy of a heavy ion collision, the system may reach the softest region in the equation-of-state and forms a long-lived fireball, which then results in the appearance of a minimum in the energy dependence of the transverse collective flow (3); (4); (5); (6). Although the hydrodynamical model provides a convenient framework to study the effects of nuclear equation-of-state on the collective behavior of colliding nuclei (5); (7), it requires the introduction of an ad hoc freeze-out procedure to compare the calculated results with the experimental measurements.

In this Rapid Communication, we take a different approach to explore the effect of a softened equation-of-state on experimental observables by using the hadronic transport model ART (11), which treats consistently the final freeze out. The possibility of describing the main features of lattice QCD phase transition in a hadronic model has recently been demonstrated by allowing hadron masses to decrease in medium, which has been argued to be related to the restoration of chiral symmetry (8). Although neither QGP phase transition nor chiral symmetry restoration actually happens in our approach as in the hydrodynamical models, results from our studies are useful for understanding the signals for a real phase transition. Indeed, we find that the softened equation-of-state leads to both a minimum in the incident energy dependence of the transverse collective flow and a delayed expansion of the compressed matter as in the hydrodynamical model. Our results further show that such an equation-of-state has almost no effect on the inclusive hadron spectra.

To introduce a soft region in the equation-of-state, we consider for the baryon sector of the reaction system the following relation between the pressure P𝑃P and energy density e𝑒e in the mean-field approach (9); (10):

P𝑃\displaystyle P =\displaystyle= 23Ekρ+ρdWdρ-W,23subscript𝐸𝑘𝜌𝜌𝑑𝑊𝑑𝜌𝑊\displaystyle\frac{2}{3}E_{k}\rho+\rho\frac{dW}{d\rho}-W, (1)
e𝑒\displaystyle e =\displaystyle= (m+Ek)ρ+W,𝑚subscript𝐸𝑘𝜌𝑊\displaystyle(m+E_{k})\rho+W, (2)

where Eksubscript𝐸𝑘E_{k} , W𝑊W , ρ𝜌\rho , and m𝑚m are the average kinetic energy per baryon, potential energy density, baryon density, and nucleon mass, respectively. It was shown in ref. (10)that W𝑊W generally depends on both ρ𝜌\rho and Eksubscript𝐸𝑘E_{k} . We require that at low energy densities, it coincides with the normal soft equation-of-state obtained from the well-known Skyrme interaction, i.e.,

P=25Efρ-179ρ2/ρ0+164(ρρ0)7/6ρ  (MeV/fm3),𝑃25subscript𝐸𝑓𝜌179superscript𝜌2subscript𝜌0164superscript𝜌subscript𝜌076𝜌MeVsuperscriptfm3P=\frac{2}{5}E_{f}\rho-179\rho^{2}/\rho_{0}+164(\frac{\rho}{\rho_{0}})^{7/6}% \rho~{}~{}~{}~{}({\rm MeV/fm^{3}}), (3)

where Efsubscript𝐸𝑓E_{f} and ρ0subscript𝜌0\rho_{0} are, respectively, the Fermi energy and normal nuclear matter density. Above the soft region, the pressure is taken to be

P=200(ρρ0)1/3ρ-153  (MeV/fm3),𝑃200superscript𝜌subscript𝜌013𝜌153MeVsuperscriptfm3P=200(\frac{\rho}{\rho_{0}})^{1/3}\rho-153~{}~{}~{}~{}({\rm MeV/fm^{3}}), (4)

which is similar to that of a noninteracting quark gas confined in a MIT bag.

To satisfy the condition at high energy densities, the following potential energy density is introduced,

W=ρ[600(ρρ0)1/3-35Ef(ρρ0)2/3+153/ρ+C1],(MeV/fm3)𝑊𝜌delimited-[]600superscript𝜌subscript𝜌01335subscript𝐸𝑓superscript𝜌subscript𝜌023153𝜌subscript𝐶1MeVsuperscriptfm3W=\rho[600(\frac{\rho}{\rho_{0}})^{1/3}-\frac{3}{5}E_{f}(\frac{\rho}{\rho_{0}}% )^{2/3}+153/\rho+C_{1}],~{}~{}~{}~{}({\rm MeV/fm^{3}}) (5)

where C1subscript𝐶1C_{1} is a constant depending on the energy density at which Eq. ( 4) is reached, i.e., the upper boundary of the soft region.

To illustrate the effects of the soft region of nuclear equation-of-state, we consider the extreme case of a vanishing sound velocity. This can be achieved by keeping a constant pressure at Pcsubscript𝑃𝑐P_{c} while changing the energy density via the following potential energy density in Eq. ( 1):

W=ρ(C2-Pcρ-Ek),(MeV/fm3)𝑊𝜌subscript𝐶2subscript𝑃𝑐𝜌subscript𝐸𝑘MeVsuperscriptfm3W=\rho(C_{2}-\frac{P_{c}}{\rho}-E_{k}),~{}~{}~{}~{}({\rm MeV/fm^{3}}) (6)

where C2subscript𝐶2C_{2} is again a constant determined from the energy density of the lower boundary of transition region. In deriving the above expression we have assumed that Eksubscript𝐸𝑘E_{k} scales with ρ2/3superscript𝜌23\rho^{2/3} as in an adiabatic process.

The resulting equation-of-state is shown in Fig. 1as super-soft eos1 and eos2, with the softest region in the energy density range of 1<e<2GeV/fm31𝑒2GeVsuperscriptfm31<e<2~{}{\rm GeV/fm}^{3} and 1.5<e<2.5GeV/fm31.5𝑒2.5GeVsuperscriptfm31.5<e<2.5~{}{\rm GeV/fm}^{3} , respectively. For comparison, we have also included the kinetic pressure, corresponding to a pure cascade model, as a function of energy density. The mean-field potential to be used in the transport model can be obtained from the relation UW/ρ𝑈𝑊𝜌U\equiv\partial W/\partial\rho , which is repulsive outside but attractive inside the softest region. In particular, the mean-field potential in the softest region is energy-dependent in order to balance the increasing kinetic pressure as the energy density increases. We note that both parameters C1subscript𝐶1C_{1} and C2subscript𝐶2C_{2} have no effect on the dynamics as only the second derivative of the potential energy density appears in the equation of motion.

Effects of the softened equation-of-state can be studied using the relativistic transport model ART. We refer the reader to Ref. (11)for the detail of the model and its applications in studying various aspects of relativistic heavy-ion collisions at AGS energies. It has been found in these studies that the transverse collective flow, defined by

<Px/N>t=1N<Px/N>(y)dNdysgn(y)𝑑y,subscriptexpectationsubscript𝑃𝑥𝑁𝑡1𝑁expectationsubscript𝑃𝑥𝑁𝑦𝑑𝑁𝑑𝑦sgn𝑦differential-d𝑦<P_{x}/N>_{t}=\frac{1}{N}\int<P_{x}/N>(y)\cdot\frac{dN}{dy}\cdot{\rm sgn}(y)% \cdot dy, (7)

is sensitive to the nuclear equation-of-state. We expect that the super-soft equation-of-state we introduced here will also affect the transverse collective flow if the energy density achieved in the collisions reaches the softest region, as in the recent study based on the hydrodynamical model (5); (7). In the latter study, a minimum in the excitation function of <Px/N>tsubscriptexpectationsubscript𝑃𝑥𝑁𝑡<P_{x}/N>_{t} at a beam energy of about 6 GeV/nucleon has been found when a softened equation-of-state due to the QCD phase transition is used.

In Fig. 2, we show the excitation function for the average transverse momentum predicted by the ART model using the four equation-of-states shown in Fig. 1. For the case of either the cascade or the normal soft equation-of-state, details can be found in our previous publications (11). It is interesting to see that the super-soft equation-of-state with the softest region leads to a minimum in the excitation function of the transverse flow, as in hydrodynamical calculations. Furthermore, the incident energy at which the minimum occurs depends on the location of the softest region in the equation-of-state. For both super-soft eos1 and eos2, the average transverse momenta near the minima are even smaller than those in the cascade calculations as attractive mean-field potentials have been introduced in these equation-of-states to counterbalance the positive gradient of the thermal pressure. This is more clearly seen in Fig. 3, where we compare the average transverse momentum distribution using the cascade model and the super-soft eos1 for the Au+Au reaction at Pbeam/A=12subscript𝑃beam𝐴12P_{\rm beam}/A=12 GeV/c. It is seen that the flow parameter, defined as the slope of the transverse momentum distribution at mid-rapidity, changes from positive to negative sign as a result of the attractive mean-field potential in the softest region. Our results also show that the flow parameter changes gradually from positive to negative then to positive as the beam energy increases. A change in the sign of flow parameter as the beam energy increases is another clear indication of the softening of the nuclear equation-of-state. To observe this effect experimentally, one needs to measure the absolute direction of the transverse flow besides its strength. This can be done by studying the shadowing effect on pion flow (12); (13); (14); (15)and the circular polarization of the γ𝛾\gamma ray emitted from the reaction (16), as in heavy ion collisions at low energies.

A softened equation-of-state is expected not only to slow down the expansion of the system but also to affect the amount of compression in the collision. Since the pressure decreases from the normal soft, super-soft to the cascade at the same energy density, one expects to reach the highest compression in the cascade model. Shown in Fig. 4are the radial density distributions of nucleons, pions and kaons at 20 fm/c from the Au+Au reaction at an impact parameter of 2 fm and a beam momentum of 12 GeV/c. At this already rather late time of the reaction, the normal soft equation-of-state gives the most extend particle distributions, indicating thus the fastest compression and expansion. The particle distributions from calculations using the super-soft eos1 are more compact, implying thus a slower compression and expansion.

It is worth mentioning that we have also studied other inclusive hadronic observables using the four equations of state and have found that it is almost impossible to distinguish the predictions using the softened equation-of-state from others.

In summary, we have studied effects of the softened nuclear equation-of-state on heavy ion collisions at relativistic energies within a hadronic transport model. To keep a zero pressure gradient in the softest region, an attractive mean field for baryons has been introduced. The softened equation-of-state is found to lead to a minimum in the excitation function of transverse flow and a delayed expansion but has almost no effect on the inclusive hadronic observables. Although it is questionable to use the hadronic degrees of freedom at very high energy densities, our results do indicate that a super-soft equation-of-state can produce the effects expected from the phase transition. Experiments on the measurement of the transverse collective flow at different incident energies will be extremely useful in our search for the signatures of chiral and deconfinement transitions.

This work was supported in part by NSF Grant No. PHY-9509266, the Robert A Welch foundation under Grant A-1358, and the Texas Advanced Research Program.

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Figure 1: Nuclear equation-of-state at zero temperature.

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Figure 2: Excitation function of total in-plane transverse momentum for the reaction of Au+Au at an impact parameter of 2 fm.

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Figure 3: In-plane transverse momentum distribution for the reaction of Au+Au at Pbeam/A=12subscript𝑃beam𝐴12P_{\rm beam}/A=12 GeV/c and an impact parameter of 2 fm.

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Figure 4: Radial density distribution of nucleons, pions, and kaons in the reaction Au+Au at Pbeam/A=12subscript𝑃beam𝐴12P_{\rm beam}/A=12 GeV/c and an impact parameter of 2 fm. The solid, dot, and dash lines are calculated using the soft eos, the super-soft eos1, and the cascade model, respectively.