What we have learned so far from 3-fluid hydrodynamics

Transcription

1 What we have learned so far from 3-fluid hydrodynamics and V.N. Russkikh Kurchatov Institute, Moscow, Russia Gesellschaft für Schwerionenforschung, Darmstadt, Germany Available data on heavy-ion collisions at AGS and SPS energies are analyzed using a 3-fluid dynamical model within a urely hadronic scenario. We investigate the roblems met in reroducing these data within this scheme. In articular, we try to indicate those data which could oint towards the occurrence of a hase transition into the quark-gluon hase. We also discuss the success of the model in reroducing the transverse-mass sectra of various hadrons. We argue that the simultaneous reroduction of the inverse-sloe arameters of all considered articles may imly that these articles belong to the same hydrodynamic flow at the instant of their freeze-out rather than that it signals the onset of a hase transition. PoS(CPOD7)8 Critical Point and Onset of Deconfinement 4th International Worksho July GSI Darmstadt,Germany Seaker. c Coyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. htt://os.sissa.it/

2 1. Introduction The interest in nucleus-nucleus collisions at incident energy range from AGS to SPS has been revived because the onset of the deconfinement hase transition is exected in this domain. Moreover, a critical end oint [1] of the transition line of the QCD hase diagram may be accessible in such reactions [2, 3]. The above exectations motivated new rojects at the roosed accelerator facilities FAIR at GSI [4, 5] and NICA at JINR [6]. The future SPS [7] and RHIC [8, 9, 1] rograms are also devoted to the same roblems. In Ref. [11] we introduced a 3-fluid dynamical (3FD) model which is suitable for simulating heavy-ion collisions at high incident energies. U to now the most extensive simulations [11, 12, 13, 14] were erformed with a urely hadronic equation of state (EoS) [15]. This EoS should serve as a reference for subsequent simulations with more sohisticated scenarios including a hase transition. In this talk we would like to investigate how good the available data from AGS and SPS can be understood without involving the concet of deconfinement. This analysis is based on success and mainly failures of our simulations. In articular, we would like to indicate those data which seem to require a hase transition into the quark-gluon hase for their reroduction. 2. The 3FD Model A direct way to address thermodynamic roerties of the matter roduced in nuclear collisions consists in alication of hydrodynamic simulations. However, finite nuclear stoing ower, observed at high incident energies, oints towards strong non-equilibrium effects which revent a straight alication of conventional hydrodynamics esecially at the initial stage of the reaction. The use of viscosity and thermal conductivity does not hel to overcome this difficulty, because by definition they are suitable for weak non-equilibrium cases only. A ossible way out is to emloy a multi-fluid aroximation to the collision dynamics. Unlike the conventional hydrodynamics, where a local instantaneous stoing of rojectile and target matter is assumed, a secific feature of the dynamic 3-fluid descrition is a finite stoing ower resulting in the counter-streaming regime of leading baryon-rich matter. The basic idea of a 3-fluid aroximation to heavy-ion collisions [11] is that at each sace-time oint x = (t, x) the generally nonequilibrium distribution function of baryon-rich matter, can be reresented as a sum of two distinct contributions, f bar. (x, ) = f (x, ) + f t (x, ), initially associated with constituent nucleons of the rojectile () and target (t) nuclei. In addition, newly roduced articles, oulating the mid-raidity region, are associated with a fireball (f) fluid. It is assumed that constituents within each distribution are locally equilibrated, both thermodynamically and chemically. This assumtion justifies the term fluids. Therefore, the 3-fluid aroximation is a minimal way to simulate the finite stoing ower at high incident energies. Our 3FD model [11] is a straightforward extension of the 2-fluid model with radiation of direct ions [16] and the (2+1)-fluid model [17]. We extend the above models in such a way that roduction of the baryon-free fireball fluid is delayed due to a certain formation time, during which the matter of the fluid roagates without interactions. PoS(CPOD7)8 2

3 P/(n m N ) ex.soft 2P-EoS T= MeV T=1 MeV T=2 MeV T [GeV] K + K - E866, E917 (AGS) NA49 (SPS) n/n Figure 1: Baryon-density deendence of the ressure at various temeratures for different EoS s [15]: with incomressibility K = 21 MeV () and K = 1 MeV (ex.soft ), and 2P-EoS [18]. Lines corresonding to different temeratures are tagged by different symbols, as dislayed in the figure s 1/2 [GeV] Figure 2: Inverse-sloe arameters of the transverse-mass sectra of kaons, ions and rotons at midraidity roduced in central Au+Au and Pb+Pb collisions as functions of invariant incident energy. The 3FD results for are resented. Solid lines corresond to the λ = fit, while dashed lines resent results with λ = 1 for ions and with λ = 1 for rotons. Data are from Refs. [19, 2, 21, 22]. The equation of state is a key quantity for the hydrodynamic model. U to now the most extensive simulations were erformed with a urely hadronic EoS [15] with the incomressibility K = 21 MeV (below referred as ). It does well in reroducing major art of the available data. The density deendence of the ressure at different temeratures (T ) is dislayed in Fig. 1. In this figure we also resent an extra-soft hadronic EoS (ex.soft ) with K = 1 MeV and a two-hase EoS (2P-EoS) [18] which involves a 1st-order hase transition into the quark-gluon hase. In that 2P model, the saturation roerties of symmetric nuclear matter in the ground state are correctly reroduced. The quark-gluon hase in the 2P model is constructed as a system of massive quasi-articles interacting via a density-deendent otential which simulates the hard-thermal-loo interactions. The 2P-EoS is in quite reasonable agreement with lattice QCD data on temerature and baryon chemical otential deendence of relevant thermodynamic quantities. PoS(CPOD7)8 Note that the hase transition leads to a softening of the EoS at high baryon densities. The eculiarity of the ex.soft is that it is similar to the 2P-EoS in the quark-gluon hase. In this talk we are going to discuss the reroduction of data in the context of required softening/hardening of the EoS. In view of Fig. 1, required softening may indicate a hase transition into quark-gluon hase. All other arameters of the model were ket fixed as described in Ref. [11]: (i) the freezeout energy density is ε frz =.4 GeV/fm 3, (ii) the friction between the baryon-rich fluids is tuned to the value given in Ref. [11], (iii) the formation time of the fireball fluid is τ = 2 fm/c. 3

4 3. What we have learned from analysis of data With the simle we succeeded to reasonably reroduce a great body of exerimental data in the incident energy range (1 16)A GeV. The list includes raidity distributions [11], transverse-mass sectra [13, 14], and multilicities of various hadrons [11]. However, we also found out certain roblems. Precisely these roblems we are going to analyze. As an excetion, we are also going to discuss the success of the 3FD model in reroduction of transverse-mass sectra, since the excitation functions of inverse sloes of these sectra were interreted as an indication of a hase transition. 3.1 Transverse-Mass Sectra The 3FD results for inverse-sloe arameters of transverse-mass sectra of kaons, ions and rotons roduced in central Au+Au and Pb+Pb collisions are resented in Fig. 2. The inverse sloes T were deduced by fitting the calculated sectra by the formula d 2 N ( m T dm T dy (m T) λ ex m ) T, (3.1) T where m T and y are the transverse mass and raidity, resectively. Though the urely exonential fit with λ = does not always rovide the best fit of the sectra, it allows a systematic way of comaring sectra at different incident energies. In order to comly with exerimental fits at AGS energies (and hence with dislayed exerimental oints), we also resent results with λ = 1 for ions and with λ = 1 for rotons. In Ref. [13] it was shown that dynamical freeze-out descrition [14], alied to the 3FD model [11], naturally exlains the incident energy behavior of inverse-sloe arameters of transverse-mass sectra observed in exeriments. This freeze-out dynamics differs from conventionally used freezeout schemes. It effectively brings about a attern similar to that of a liquid gas hase transition. As seen from Fig. 2, the inverse-sloe arameters (with the urely exonential fit, λ = ) at mid raidity reveal a "ste-like" behavior. They increase with incident energy across the AGS energy domain and then saturate at SPS energies. In Refs. [23, 24] this saturation was associated with the deconfinement hase transition. This assumtion was indirectly confirmed by the fact that microscoic transort models (HSD and UrQMD [25], and GiBUU [26]), based on hadronic degrees of freedom, failed to reroduce the observed behavior of the inverse sloes of kaons. However, these transort models [25, 26] do well describe ion and roton transverse-mass sectra in a wide range of incident energies. Therefore, the failure with the kaon inverse sloes may be interreted as a signature that kaon interaction cross sections (in microscoic models) are not big enough in order to have the kaons be catured by matter in a common flow. Another ossibility is that multi-body collisions are imortant in the transort. This was checked within the GiBUU model [27], where three-body interactions were included in simulations. It was found that the three-body collisions indeed result in good reroduction of all transverse-mass sectra [27]. Therefore, to our mind, the simultaneous reroduction of inverse-sloes of all considered articles imlies that these articles belong to the same hydrodynamic flow at the instant of their freeze-out. PoS(CPOD7)8 4

5 <P x > [GeV/c] Au+Au =2A GeV =6A GeV E895 (measured) E895 (reflected) 3FD, b=6 fm ex.soft =4A GeV =8A GeV (y- yc.m.)/yc.m. v v 1(st) v1(3) v1(st) v1(2) v1(3) Pb+Pb (4A GeV) Pb+Pb (158A GeV) 3FD, b=5.6 fm ex.soft y-y c.m. Figure 3: Directed flow of rotons as a function of raidity for mid-central collisions at AGS (left anels) and SPS energies (right anels). The AGS data are from E895 collaboration [31]. At SPS energies, the NA49 data [32] obtained by two different methods are dislayed: by the standard method (v(st)) and by the method of n-article correlations (v(n)). Full symbols corresond to measured data, while oen symbols are those reflected with resect to the midraidity. 3.2 Directed Flow Let us roceed to roblems which we met. The directed flow (P x or v 1 ) is a quantity that is sensitive to the resence of a hase transition. As demonstrated in Refs. [28, 29, 3], the 1st-order hase transition leads to a significant reduction of the directed flow [28] and even develos an antiflow behavior in the midraidity region [29, 3]. Analyzing flow data within the 3FD model [12] we found that the directed flow data favor a steady softening of the EoS with increasing beam energy. As seen from Fig. 3, this EoS softening essentially occurs across the AGS energy range, while at higher (SPS) energies the same extrasoft EoS remains referable. In view of above the mentioned redictions [28, 29, 3], this EoS softening can be viewed as a signal of the deconfinement transition. PoS(CPOD7)8 3.3 Raidity Distributions The roton raidity distributions basically reflect the stoing ower achieved in the nuclear collision. Therefore, their reroduction would indicate that a model roerly describes at least the global features of the collision rocess. Till the beginning of this year we reorted success of the 3FD model with in reroduction of the raidity distributions, see Fig. 4. However, when new data by the NA49 collaboration [37] were then ublished, we saw that the situation is not that evident, see Fig. 5, left anel. In site of fitting raidity sectra at reference energies of 1A and 158A GeV, the failed to reroduce the net-baryon distributions at energies in-between. We studied the effect of the stiffness of the EoS on these distributions, see Fig. 5. It turned out that the net-baryon raidity sectrum at 4A GeV is almost erfectly reroduced with the soft H- EoS with the incomressibility K = 15 MeV, while ex.soft with K = 1 MeV is required at energies 2A and 3A GeV. For the energies 8A and 158A GeV the best result is still achieved 5

6 dn/dy =2A GeV =4A GeV 3FD, b=2 fm E895, 5% E895, 5% (y- y c.m. )/y c.m =6A GeV E917 =8A GeV E917 4 fm 6 fm 8 fm 11 fm b=2 fm 4 fm 6 fm 8 fm 11 fm b=2 fm (y- y c.m. )/y c.m. =1.6A GeV b=2 fm E =158A GeV 4 fm 6 fm 8 fm 11 fm NA49, 5% 3FD, b=2.5 fm (y- y c.m. )/y c.m. Figure 4: Proton raidity sectra (solid lines) at AGS and SPS energies for various imact arameters (b) calculated with the. Exerimental oints are taken from [33] at 2A and 4A GeV, and [34] at 6A, 8A and 1.5A GeV. The NA49 data for = 158A GeV are from Refs. [35, 36]. The ercentage indicates the fraction of the total reaction cross section, corresonding to exerimentally selected events. _ (B-B), dn/dy 1A GeV(â8) 1 3 2A GeV(â6) 1 2 4A GeV(â12) 3A GeV(â4) 4A GeV(â2) 8A GeV(â1.5) 158A GeV 3FD, _ (B-B), dn/dy 1A GeV(â8) 1 3 2A GeV(â6) 1 2 4A GeV(â12) 3A GeV(â4) 4A GeV(â2) 8A GeV(â1.5) 158A GeV 3FD, soft _ (B-B), dn/dy 1 2 4A GeV(â12) 1A GeV(â8) 1 3 2A GeV(â6) 3A GeV(â4) 4A GeV(â2) 8A GeV(â1.5) 158A GeV 3FD, ex.soft PoS(CPOD7) y-y c.m y-y c.m y-y c.m. Figure 5: Net-baryon raidity sectra from central collisions of Au+Au at AGS energies and Pb+Pb at SPS energies as functions of raidity for with K = 21 MeV (left anel), soft with K = 15 MeV (middle anel) and ex.soft with K = 1 MeV (right anel). Data at SPS energies are from Ref. [35, 36, 37]. Net-baryon data at AGS energies are obtained from the corresonding roton data (Ref. [33] at 4A GeV and Ref. [34] at 1.5A GeV) by multilying them by the factor A/Z. For clarity of reresentation, data sets from bottom to to are scales by additional factors dislayed in the figure. with our standard. At the same time, raidity sectra at AGS energies are quite insensitive to the stiffness. Thus, we conclude that similarly to the directed flow the net-baryon raidity sectra at energies 2A 4A also require a softer EoS. This may be a signal of deconfinement transition. However, the fact that the standard remains favorable for raidity sectra at 8A and 158A GeV aarently contradicts the trend found in the directed-flow data. This contradiction can be an artifact of a biased fit to the data: we have changed only the stiffness while keeing other quantities (the freezeout energy density, the friction, and the formation time) fixed. It would be of interest to try an 6

7 unbiased fit of the SPS data in order to find out if these data are comatible with a soft EoS..1 Pb+Pb (158A GeV) Pb+Pb (4A GeV) v 2.5 v v 2 (st) v 2 (2) y c.m. 3FD, b=5.6 fm ex.soft hard y c.m. Figure 6: Ellitic flow of rotons (uer anels) and ions (lower anels) as a function of raidity in mid-central (b = 5.6 fm) Pb+Pb collisions. 3FD calculations with, ex.soft, and hard (K = 38 MeV) are resented. The NA49 data [32] obtained by two different methods are dislayed: by the standard method (v(st)) and by the method of n-article correlations (v(n)). Full symbols corresond to measured data, while oen symbols are those reflected with resect to the midraidity. 3.4 Ellitic Flow In Ref. [12] we found that it is imossible to simultaneously reroduce the directed and ellitic flow with the same EoS. The directed flow requires a softer EoS, while the ellitic flow demands for a harder one. In that aer we assigned this deficiency to a lack of the roer descrition of the nonequilibrium transverse-momentum anisotroy at the initial stage of nuclear collision. Later we became aware that the ellitic flow is strongly affected by a ost-hydro cascade (so called afterburner ) [38, 39]. This afterburner essentially reduces (aroximately twice at SPS energies) the v 2 values achieved during the hydrodynamic stage. In articular, this reduction may hel to bring the 3FD results in corresondence with the exerimental data at 158A GeV. In Fig. 6 we resent urely hydrodynamic redictions for the ellitic flow based on the s with different stiffnesses, keeing in mind that they will be aroximately reduced to half the value by the ost-hydro cascade. However, the model still strongly overestimates the roton v 2 at 4A GeV (even on account of the afterburner effect). Considering that v 2 at 4A and 158A GeV only weakly deends on the stiffness, see Fig. 6, we cannot associate this collase of the roton v 2 at 4A GeV with an onset of the hase transition. In Ref. [4] this collase was assumed to be a signal of the critical oint in the quark-gluon hase diagram. PoS(CPOD7)8 3.5 Hadron Ratios The hadron ratios of hadron abundances arouse interest because the observed maximum in the K + /π + ratio can be interreted as a signal of the onset of the hase transition into quarkgluon hase [41, 42, 43, 44]. In addition, these ratios most distinctly demonstrate discreancies in descrition of hadron multilicities. Fig. 7 resents hadron ratios in the midraidity region (i.e. 7

8 Ratio dn/dy (y c.m. =) / / / (L+S )/- E895, E917 NA49 E877 NA49 NA57 1 s 1/2 [GeV] / - K - / - 3FD K + / + g S s 1/2 [GeV] E866, E917 NA49 NA44 1 s 1/2 [GeV] Ratio dn/dy (y c.m. =) K K FD 1 s 1/2 [GeV] E866, E917 NA49 Figure 7: Excitation functions of ratios of midraidity yeilds (left block of anels) and total yields (right block of anels) of hadrons roduced in central collisions of Au+Au at AGS energies and Pb+Pb at SPS energies. 3FD results are resented for. Bold lines corresond to calculations with additional strangeness suression γ S (see insertion in the left block of anels), while thin lines to those without suression. ratios of midraidity densities of articles) and those of total multilicities calculated within the 3FD model with, as well as their comarison with exerimental data. It is immediately seen that the strangeness roduction at low incident energies is evidently overestimated within the 3FD model. This is not surrising, since the hadronic-gas EoS used at the freeze-out [11] is based on the grand canonical ensemble. This shortcoming can be easily curied by introduction of a henomenological factor γ S (see, e.g., Refs. [45]), which accounts for an additional strangeness suression due to constraints of canonical ensemble. The resulting γ S factor is resented in the inserted anel of Fig. 7. As seen, at > 1A GeV there is no need for additional strangeness suression. The 3FD model with reroduces hadron ratios aroximately to the same extent as transort models HSD and UrQMD [43] and GiBUU [27] based on hadronic degrees of freedom do. In articular, the 3FD model also fails to reroduce the maximum in the K + /π + ratio. We found that calculated hadron ratios are quite insensitive to stiffness of the EoS. Therefore, the observed maximum in the K + /π + ratio can hardly be a signal of the onset of a hase transition. This may be a signal of the critical oint in the hase diagram. Indeed, relaxation rocesses are exected to be essentially slowed down near the critical oint. If it concerns also chemical equilibration, the resulting hadron ratios would reflect the chemical content of the system at earlier stages of its evolution. Then the K + /π + ratio would turn out to be enhanced near the critical oint. E877 NA49 PoS(CPOD7)8 4. Summury In these work we analyzed how good the available data from AGS and SPS can be understood within a urely hadronic scenario. The analysis is based on failures of our simulations of heavy-ion collisions within the 3FD model with a urely hadronic EoS. We tried to cure the roblems in the reroduction of various observables by means of changing the stiffness of the hadronic EoS. Note 8

9 that in terms of hydrodynamics, a softening of the EoS can be a manifestation of a hase transition. Our results are as follows: Directed flow data (at to AGS and SPS energies) and net-baryon raidity sectra (at low SPS energies) favor a softer EoS. This required softening may be interreted as an indication of the deconfinement transition. The Collase of the roton ellitic flow at 4A GeV and observed maximum in the K + /π + ratio hardly signal an onset of a hase transition, since these quantities are quite insensitive to the stiffness of the EoS. This observations may signal a roximity of a critical oint in the hase diagram. In this work we discussed roblems of the hadronic scenario but with one excetion. This excetion is the success of the 3FD model in reroduction of transverse-mass sectra of various hadrons (in articular, kaons), since the excitation functions of inverse sloes of kaon sectra were interreted as an indication of a hase transition [23, 24]. Our conclusion is as follows: The simultaneous reroduction of the inverse-sloes of all considered articles (, π and K) within the 3FD model suggests that these articles belong to the same hydrodynamic flow at the instant of their freeze-out rather than signals an onset of a hase transition. 3FD simulations with 2P-EoS [18], which involves the 1st-order hase transition into quarkgluon hase, are now in rogress. We are grateful to J. Knoll for stimulating discussions and for critical reading the text of this aer. This work was suorted in art by the Deutsche Forschungsgemeinschaft (DFG roject 436 RUS 113/558/-3), the Russian Foundation for Basic Research (RFBR grant NNIO_a), Russian Federal Agency for Science and Innovations (grant NSh ). PoS(CPOD7)8 References [1] M. Asakawa and K. Yazaki, Nucl. Phys. A54 (1989) 668. [2] Z. Fodor and S.D. Katz, JHEP 23 (22) 14; JHEP 44 (24) 5. [3] M. Stehanov, K. Rajagoal and E. Shuryak, Phys. Rev. D 6 (1999) J. Phys. G3 (24) S71. [4] P. Siller and G. Franchetti, Nucl. Instrum. Meth. A561 (26) 35. [5] V. Friese, The CBM Exeriment at FAIR, Talk at this worksho. [6] V.D. Toneev, The NICA/MPD Project at JINR (Dubna), Talk at this worksho. [7] A. Laszlo, NA61/SHINE at the CERN SPS - New exeriment using ugraded NA49 detector, Talk at this worksho. [8] A. Cho, Science 312 (26) 19; T. Satogata, RHIC Low-Energy Oerations: First Exerience and Ugrade Plans, Talk at this worksho. [9] D. Cebra, The Caabilities of the STAR Detector During the Sub-injection Energy Runs at RHIC, Talk at this worksho. 9

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