Effects of LatticeQCD EoS and Continuous Emission on Some Observables Y. Hama�, R. Andrade�, F. Grassi�, O. Socolowski†, T. Kodama��, B. Tavares�� and S.S. Padula‡ �Instituto de Física, Universidade de São Paulo, C.P. 66318, 05315-970 São Paulo-SP, Brazil †Instituto Tecnológico da Aeronáutica, Praça Marechal Eduardo Gomes, 50 - Vila das Acácias, CEP 12228-900 São José dos Campos-SP, Brazil ��Instituto de Física, Universidade Federal do Rio de Janeiro, C.P. 68528, 21945-970 Rio de Janeiro-RJ, Brazil ‡Instituto de Física Teórica, Universidade Estadual Paulista, Rua Pamplona 145, Bela Vista - CEP 01405-000 São Paulo-SP, Brazil Abstract. Effects of lattice-QCD-inspired equations of state and continuous emission on some observables are discussed, by solving a 3D hydrodynamics. The particle multiplicity as wellv 2 are found to increase in the mid-rapidity. We also discuss the effects of the initial-condition fluctuations. Keywords: LatticeQCD equations of state, hydrodynamic model PACS: 24.10.Nz, 25.75.-q, 25.75.Ld 1. HYDRODYNAMIC MODELS Hydrodynamics is one of the main tools for studying the collective flow in high-energy nuclear collisions. Here, we shall examine some of the main ingredients of such a de- scription and see how likely more realistic treatment of these elements may affect some of the observable quantities. The main components of any hydrodynamic model are the initial conditions, the equations of motion, equations of state and some decoupling pre- scription. We shall discuss how these elements are chosen in our studies. Initial Conditions: In usual hydrodynamic approach, one assumes some highly sym- metric and smooth initial conditions (IC). However, since our systems are small, large event-by-event fluctuations are expected in real collisions, so this effect should be taken into account. We introduce such IC fluctuations by using an event simulator. As an ex- ample, we show here the energy density for central Au+Au collisions at 130A GeV, FIGURE 1. The initial energy density atη � 0 is plotted in units of GeV/fm3. One random event is shown vs. average over 30 random events (� smooth initial conditions in the usual hydro approach). 485 given by NeXuS1 [1]. Some consequences of such fluctuations have been discussed elsewhere[5, 6, 7]. We shall discuss some others in Sec. 2. Equations of Motion: In hydrodynamics, the flow is governed by the continuity equa- tions expressing the conservation of energy-momentum, baryon-number and other con- served charges. Here, for simplicity, we shall consider only the energy-momentum and the baryon number. Since our systems have no symmetry as discussed above, we de- veloped a special numerical code called SPheRIO (SmoothedParticle hydrodynamic evolution of Relativistic heavyIOn collisions) [8], based on the so called Smoothed- Paricle Hydrodynamics (SPH) algorithm [9]. The main characteristic of SPH is the parametrization of the flow in terms of discrete Lagrangian coordinates attached to small volumes (called “particles”) with some conserved quantities. Equations of State: In high-energy collisions, one often uses equations of state (EoS) with a first-order phase transition, connecting a high-temperature QGP phase with a low- temperature hadron phase. A detailed account of such EoS may be found, for instance, in [7]. We shall denote them 1OPT EoS. However, lattice QCD showed that the transition line has a critical end point and for small net baryon surplus the transition is of crossover type [10]. The following parametrization may reproduce this behavior, in practice: P � λ PH ��1�λ �PQ �2δ� � �PQ�PH� 2�4δ � (1) s � λ sH ��1�λ �sQ � (2) ε � λεH ��1�λ �εQ�2 � 1��µ�µc� 2� δ� � �PQ�PH� 2�4δ � (3) 10 -1 1 10 1 10 10 2 10 -3 10 -2 10 -1 1 10 0.1 0.15 0.2 0.1 0.15 0.2 µ=0 GeV ε (G eV /fm 3 ) µ=0.2 GeV µ=0.4 GeV s ( fm -3 ) P ( Ge V/f m3 ) T (GeV) 0.1 0.15 0.2 0.25 FIGURE 2. A comparison ofε�T �, s�T � andP�T � as given by our parametrization with a critical point (solid lines) and those with a first-order phase transition (dashed lines). 1 Many other simulators, based on microscopic models,e.g. HIJING [2], VNI [3], URASiMA [4], � � �, show such event-by-event fluctuations. 486 6.5 7 7.5 8 8.5 9 0 0.2 0.4 0.6 0.8 1 0 2 µ=0 GeV s/ε (G eV -1 ) µ=0.2 GeV µ=0.4 GeV P ( Ge V/f m3 ) ε (GeV) 0 2 0 2 4 FIGURE 3. Plots ofs�ε andP as function ofε for the two EoS shown in Fig. 2. whereλ � �1� �PQ�PH�� � �PQ�PH� 2�4δ ��2 and suffixesQ andH denote those quantities given by the MIT bag model and the hadronic resonance gas, respectively, andδ � δ�µb� � δ0exp���µb�µc� 2�, with µc �const. As is seen, whenδ�µb� �� 0, the transition from hadron phase to QGP is smooth. We could chooseδ�µb� so to make it exactly 0 whenµb � µc , to guarantee the first-order phase transition there. However, in practice our choice above showed to be enough. We shall denote the EoS given above, with δ0 �� 0, CP EoS. Let us compare, in Fig. 2,ε�T�, s�T� and P�T �, given by the two sets of EoS. one can see that the crossover behavior is correctly reproduced by our parametrization for CP EoS, while finite jumps inε ands are exhibited by 1OPT EoS, at the transition temperature. It is also seen, as mentioned above, that atµb � 0�4GeV the two EoS are indistinguishable. Now, since in a real collision what is directly given is the energy distribution at a certain initial time (besidesnb, s, etc.), whereasT is defined with the use of the former, we plotted some quantities as function ofε in Fig. 3. One immediately sees there some remarkable differences between the two sets of EoS: naturallyp is not constant for CP EoS in the crossover region; moreover,s is larger. We will see in Sec.2 that these features affect the observables in non-negligible way. Decoupling Prescription: Usually, one assumes decoupling on a sharply defined hy- persurface. We call thisSudden Freeze Out (FO). However, since our systems are small, particles may escape from a layer with thickness comparable with the systems’ sizes. We proposed an alternative description calledContinuous Emission (CE) [11] which, as compared to FO, we believe is closer to what happens in the actual collisions. In CE, particles escape from any space-time pointxµ , according to a momentum-dependent escaping probability��x�k� � exp�� � ∞ τ ρ�x��σv dτ �� � To implement CE in SPheRIO code, we had to approximate it to make the computation practicable. We took� on the average,i.e., ��x�k�� ���x�k�� ���x� � exp � �κ s2��ds�dτ � � � (4) 487 exp. data CP EoS (EbE-CE) 1OPT EoS (EbE-CE) η d N /d η h++h- 0 100 200 300 400 500 600 700 800 -6 -4 -2 0 2 4 6 exp. data CP EoS (EbE-CE) 1OPT EoS (EbE-CE) pT (GeV) (2 πp T )-1 d 2 n /d p T d y (G eV -2 ) (h++h-)/2 10 -2 10 -1 1 10 10 2 10 3 0 0.5 1 1.5 2 2.5 3 FIGURE 4. η andpT distributions for the most central Au+Au at 200A GeV. Results of CP EoS and 1OPT EoS are compared. The data are from PHOBOS Collaboration[12]. The last equality has been obtained by making a linear approximation of the density ρ�x�� � α s�x�� and κ � 0�5α �σv� is estimated to be 0.3, corresponding to�σv� � 2 fm2. It will be shown in Sec. 2 that CE gives important changes in some observables. 2. RESULTS Let us now show results of computation of some observables, as described above, for Au+Au at 200A GeV. We start computingη andpT distributions for charged particles, to fix the parameters. Then,v2 and HBT radii are computed free of parameters. Pseudo-rapidity distribution: Figure 3 shows that the inclusion of a critical end point increases the entropy per energy. This means that, given the same total energy, CP EoS produces larger multiplicity, which is clearly shown in the left panel of Fig. 4, especially in the mid-rapidity region. Now, we shall mention that, once the equations of state are chosen, fluctuating IC produce smaller multiplicity, for the same decoupling prescription, as compared with the case of smooth averaged IC [7]. -4 -2 0 2 4 η -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 v2 ( ch ) CP EoS (EbE - CE) CP EoS (EbE - FO , T = 135 MeV) 1OPT EoS (EbE - CE) 1OPT EoS (EbE -FO , T = 135 MeV) Exp. data (15 - 25) % f f Average over 200 events -0.05 0 0.05 0.1 0.15 v 0 2 4 6 8 10 12 14 16 18 d N / d v 0.48 < η < 0.95 2 2 e v 200 events FIGURE 5. Left: η distribution ofv2 for charged particles in the centrality�15�25�% Au+Au at 200A GeV, computed with fluctuating IC. The vertical bars indicate dispersions. The data are from PHOBOS Collaboration [13]. Right:v2 distribution in the interval 0�48� η � 0�95, corresponding to CP EoS and CE. 488 0 0.2 0.4 0.6 0.8 1 1.2 1.4 kT (GeV) 1 2 3 4 5 6 7 8 R L ( fm ) π+π+ π−π− CP EoS (CE) CP EoS (FO) 1OPT EoS (CE) 1OPT EoS (FO) FIGURE 6. kT dependence of HBT radiusRL for π in the most central Au+Au at 200A GeV, computed with fluctuating IC. The data are from PHENIX Collaboration [14]. Transverse-Momentum Distribution: As discussed in Sec. 1, since the pressure does not remain constant in the crossover region, we expect that the transverse acceleration is larger for CP EoS, as compared with 1OPT EoS case. In effect, the right panel of Fig. 4 does show thatpT distribution is flatter for CP EoS, but the difference is small. The freezeout temperature suggested byη and pT distributions turned out to beTf 135�140MeV. Elliptic-Flow Parameter v2: We show, in Fig. 5, results for theη distribution ofv2 for Au+Au collisions at 200A GeV. As seen, CP EoS gives largerv2 , as a consequence of larger acceleration in this case as discussed in Sec.1. Notice that CE makes the curves narrower, as a consequence of earlier emission of particles, so with smaller acceleration, at large-�η � regions. Due to the IC fluctuations, the resulting fluctuations ofv2 are large, as seen in Figs. 5. It would be nice to measure such av2 distribution, which would discriminate among several microscopic models for the initial stage of nuclear collisions. HBT Radii: Here, we show our results for the HBT radii, in Gaussian approximation as often used, for the most central Au+Au collisions at 200A GeV. As seen in Figs. 6 and 0 0.2 0.4 0.6 0.8 1 1.2 1.4 kT (GeV) 2 3 4 5 6 R S ( fm ) π+π+ π−π− CP EoS (CE) CP EoS (FO) 1OPT EoS (CE) 1OPT EoS (FO) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 kT (GeV) 2 3 4 5 6 R S ( fm ) π+π+ π−π− CP EoS (CE) CP EoS (FO) 1OPT EoS (CE) 1OPT EoS (FO) FIGURE 7. kT dependence of HBT radiiRs andRo for pions in the most central Au+Au at 200A GeV, computed with event-by-event fluctuating IC. The data are from PHENIX Collaboration [14]. 489 7, the differences between CP EoS results and those for 1OPT EoS are small. ForRs , and especially forRo , one sees that CP EoS combined with continuous emission gives steeperkT dependence, closer to the data. However, there is still numerical discrepancy in this case. 3. CONCLUSIONS AND OUTLOOKS In this work, we introduced a parametrization of lattice-QCD EoS, with a first-order phase transition at largeµb and a crossover behavior at smallerµb . By solving the hy- drodynamic equations, we studied the effects of such EoS and the continuous emission. Some conclusions are:i) The multiplicity increases for these EoS in the mid-rapidity;ii) The pT distribution becomes flatter, although the difference is small;iii) v2 increases; CE makes theη distribution narrower;iv) HBT radii slightly closer to data. In our calculations, the effect of the continuous emission on the interacting component has not been taken into account. A more realistic treatment of this effect probably makes Ro smaller, since the duration for particle emission becomes smaller in this case. Another improvement we should make is the approximations we used for��x� p�. ACKNOWLEDGMENTS We acknowledge financial support by FAPESP (04/10619-9, 04/15560-2, 04/13309-0), CAPES/PROBRAL, CNPq, FAPERJ and PRONEX. REFERENCES 1. H.J. Drescher, F.M. Liu, S. Ostapchenko, T. Pierog and K. Werner,Phys. Rev. C 65, 054902 (2002). 2. M. Gyulassy, D.H. Rischke and B. Zhang,Nucl. Phys. 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