(3+1)-Dimensional Hydrodynamic Expansion with a Critical Point from Realistic Initial Conditions
J. Steinheimer, M. Bleicher, H. Petersen, S. Schramm, H. Stocker, D. Zschiesche
aa r X i v : . [ nu c l - t h ] O c t (3+1)-Dimensional Hydrodynamic Expansion with a Critical Point from RealisticInitial Conditions J. Steinheimer, M. Bleicher, H. Petersen,
1, 2
S. Schramm, H. St¨ocker,
1, 2, 3 and D. Zschiesche Institut f¨ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨at,Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany Frankfurt Institute for Advanced Studies (FIAS),Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany Gesellschaft f¨ur Schwerionenforschung (GSI), Planckstr. 1, D-64291 Darmstadt, Germany
We investigate a (3+1)-dimensional hydrodynamic expansion of the hot and dense system createdin head-on collisions of Pb+Pb/Au+Au at beam energies from 5 − A GeV. An equation of statethat incorporates a critical end point (CEP) in line with the lattice data is used. The necessaryinitial conditions for the hydrodynamic evolution are taken from a microscopic transport approach(UrQMD). We compare the properties of the initial state and the full hydrodynamical calculationwith an isentropic expansion employing an initial state from a simple overlap model. We findthat the specific entropy (
S/A ) from both initial conditions is very similar and only depends onthe underlying equation of state. Using the chiral (hadronic) equation of state we investigate theexpansion paths for both initial conditions. Defining a critical area around the critical point, weshow at what beam energies one can expect to have a sizable fraction of the system close to thecritical point. Finally, we emphasise the importance of the equation of state of strongly interactingmatter, in the (experimental) search for the CEP.
Heavy ion collisions at intermediate incident beam ener-gies (5 − A GeV) have recently attracted more andmore attention, since one expects to be able to scan awide - and by current expectations highly interesting[1, 2] - region of temperature T and baryo-chemical po-tential µ B in the QCD phase diagram. Following re-cent theoretical investigations (for recent lattice QCDresults see [3, 4, 5], for phenomenological studies see[6, 7, 8, 9, 10, 11, 12]), one hopes to find experimen-tal evidence for a phase transition from hadronic mat-ter (with broken chiral symmetry), to a Quark-GluonPlasma (QGP) phase (where chiral symmetry is re-stored). Especially the so called critical end point (CEP),a point in the phase diagram that terminates the phasetransition-line of the first order transition (which is ex-pected for high chemical potentials), is of great interest.Key observables like the directed and elliptic flow v and v [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,26, 27, 28, 29, 30] - but also particle multiplicities, ratiosand their fluctuations - are of imminent interest. Theyhave been predicted and sometimes already shown to besensitive to the properties of the QCD matter, i.e. to theEquation of State (EoS) and the active degrees of free-dom, in the early stage of the reaction. And indeed, theenergy dependences of various observables show anoma-lies at low SPS energies which might be related to theonset of deconfinement [7, 11].For a hydrodynamical modelling of heavy ion reactionsto study these observables one needs to specify initialconditions, i.e. an infinite set of space-time points withtheir corresponding energy- and baryon density. Sinceexperimental data provides mainly information that is in-tegrated over the systems time evolution, the initial statefor hydrodynamical simulations has to be inferred from model assumptions or by educated ’guessing’ in compar-ison to data. The latter approach - usually applied forrelativistic hydrodynamical simulations of nuclear colli-sions - is, however, by no means straight forward andhighly non-trivial: the connections between (observed)final state and the inferred initial conditions is blurredby the unknown equation of state, potential viscosity ef-fects, and freeze-out problems. Another issue concernsthe assumption of thermal equilibrium, which is proba-bly not true at least for the early stage of a heavy ioncollisions at intermediate energies.There have been attempts to solve these problemsby describing such collisions with viscous or multi-fluid-hydrodynamic models [22, 31, 32, 33, 34, 35, 36, 37, 38,39, 40, 41, 42, 43], but the practical application of thesemodels is difficult. To avoid (some of) these problems inthis paper, we describe the initial stages of the collisionwith a non-equilibrium transport model (UrQMD). Wethen use the so obtained distributions for energy- andbaryon-density as initial conditions for a one-fluid butfully (3+1)-dimensional hydrodynamical calculation. Forthe hydrodynamical evolution an EoS with a first orderchiral phase transition and a CEP at finite µ B is applied.This paper is organised as follows: In the first part wedescribe the chiral equation of state that is used in moredetail and explain how the structure of the phase dia-gram with a critical end point is modelled. Afterwardstwo different scenarios for the initial conditions and thesubsequent evolution are compared. The full hydro evo-lution with microscopic initial conditions is contrastedwith lines of constant entropy per baryon number from asimple overlap model. Then, the results concerning theCEP are presented and the influence of the equation ofstate on the evolution is studied. The last part sum-marises the paper.Let us start with the discussion of the equation ofstate. The present chiral hadronic SU (3) Lagrangianincorporates the complete set of baryons from the low-est flavour- SU (3) octet, as well as the entire multipletsof scalar, pseudo-scalar, vector and axial-vector mesons[44]. In mean-field approximation, the expectation valuesof the scalar fields relevant for symmetric nuclear mattercorrespond to the non-strange and strange chiral quarkcondensates, namely the σ and its s ¯ s counterpart ζ , re-spectively, and further the ω and φ vector meson fields.Another scalar iso-scalar field, the dilaton χ , is intro-duced to model the QCD scale anomaly. However, if χ does not couple strongly to baryonic degrees of freedomit remains essentially “frozen” below the chiral transition[44]. Consequently, we focus here on the role of the quarkcondensates.Interactions between baryons and scalar (BM) or vec-tor (BV) mesons, respectively, are introduced as L BM = − X i ψ i ( g iσ σ + g iζ ζ ) ψ i , (1) L BV = − X i ψ i (cid:0) g iω γ ω + g iφ γ φ (cid:1) ψ i , (2)Here, i sums over the baryon octet ( N , Λ, Σ, Ξ). A term L vec with mass terms and quartic self-interaction of thevector mesons is also added: L vec = 12 a ω χ ω + 12 a φ χ φ + g ( ω + 2 φ ) . The scalar self-interactions are L = − k χ ( σ + ζ ) + k ( σ + ζ ) + k ( σ ζ )+ k χσ ζ − k χ − χ ln χ χ + δ χ ln σ ζσ ζ . (3)Interactions between the scalar mesons induce the spon-taneous breaking of chiral symmetry (first line) and thescale breaking via the dilaton field χ (last two terms).Non-zero current quark masses break chiral symmetryexplicitly in QCD. In the effective Lagrangian this corre-sponds to terms such as L SB = − χ χ (cid:20) m π f π σ + ( √ m K f K − √ m π f π ) ζ (cid:21) . (4)According to L BM (1), the effective masses of thebaryons, m ∗ i ( σ, ζ ) = g iσ σ + g iζ ζ , are generated throughtheir coupling to the chiral condensates, which attainnon-zero vacuum expectation values due to their self-interactions [44] in L (3). The effective masses of themesons are obtained as the second derivatives of themesonic potential V Meson ≡ −L − L vec − L SB about itsminimum.The baryon-vector couplings g iω and g iφ result frompure f -type coupling as discussed in [44], g iω = ( n iq − n i ¯ q ) g V , g iφ = − ( n is − n i ¯ s ) √ g V , where g V denotes thevector coupling of the baryon octet and n i the num-ber of constituent quarks of species i in a given hadron.The resulting relative couplings agree with additive quarkmodel constraints.All parameters of the model discussed so far are fixedby either symmetry relations, hadronic vacuum observ-ables or nuclear matter saturation properties (for detailssee [44]). In addition, the model also provides a sat-isfactory description of realistic (finite-size and isospinasymmetric) nuclei and of neutron stars [44, 45, 46].If the baryonic degrees of freedom are restricted to themembers of the lowest lying octet, the model exhibitsa smooth decrease of the chiral condensates (crossover)for both high T and high µ [44, 47]. However, addi-tional baryonic degrees of freedom may change this intoa first-order phase transition in certain regimes of the T - µ q plane, depending on the couplings [47, 48, 49]. Tomodel the influence of such heavy baryonic states, we adda single resonance with mass m R = m + g R σ and vec-tor coupling g Rω = r V g Nω . The mass parameters, m , g R and the relative vector coupling r V represent free pa-rameters, adjusted to reproduce the phase diagram dis-cussed above[86]. In principle, of course, one should cou-ple the entire spectrum of resonances to the scalar andvector fields. However, to keep the number of additionalcouplings small, we effectively describe the couplings ofall higher baryonic resonances by a single state with ad-justable couplings, mass and degeneracy. This methodof replacing the influence of many states by an effectiveresonance has a long tradition in scattering theory andrelated fields (see e.g. [50]).In what follows, the meson fields are replaced by their(classical) expectation values, which corresponds to ne-glecting quantum and thermal fluctuations. Fermionshave to be integrated out to one-loop. The grand canon-ical potential can then be written asΩ /V = −L vec − L − L SB − V vac (5) − T X i ∈ B γ i (2 π ) Z d k h ln (cid:16) e − T [ E ∗ i ( k ) − µ ∗ i ] (cid:17)i + T X l ∈ M γ l (2 π ) Z d k h ln (cid:16) − e − T [ E ∗ l ( k ) − µ ∗ l ] (cid:17)i , where γ B , γ M denote the baryonic and mesonicspin-isospin degeneracy factors and E ∗ B,M ( k ) = q k + m ∗ B,M are the corresponding single particle en-ergies. In the second line we sum over the baryon octetstates plus the additional heavy resonance with degen-eracy γ R (assumed to be 16). The effective baryo-chemical potentials are µ ∗ i = µ i − g iω ω − g iφ φ , with µ i = ( n iq − n i ¯ q ) µ q + ( n is − n i ¯ s ) µ s . The chemical poten-tials of the mesons are given by the sum of the corre-sponding quark and anti-quark chemical potentials. Thevacuum energy V vac (the potential at ρ B = T = 0) hasbeen subtracted.By extremizing Ω /V one obtains self-consistent gapequations for the meson fields. Here, globally non-strangematter is considered and µ s for any given T and µ q is ad-justed to obtain a vanishing net strangeness. The domi-nant “condensates” are then the σ and the ω fields. Therehave also been first attempts to model the dynamical evo-lution of the condensates themselves instead of ’locking’them at their equilibrium values (see e.g. [51, 52]).Let us now turn to the explanation of the initial con-ditions that have been used. Two different ways to de-scribe the initial conditions and the expansion will bediscussed. In one setup the energy and density distri-butions obtained from the UrQMD transport simulationare mapped to the thermodynamic quantities, which thenserve as initial conditions for the (3+1) dimensional hy-drodynamic evolution. In the second setup initial energy-and baryon-densities as obtained from a simple overlapmodel are employed and then paths of constant entropyper baryon are followed.In the first scenario, the Ultra-relativistic QuantumMolecular Dynamics Model (UrQMD 1.3) is used to cal-culate the initial state for the hydrodynamical evolution.This has been done to account for the non-equilibriumin the very early stage of the collision. In this config-uration the effect of event-by-event fluctuations of theinitial state is naturally included. Due to the earlytransition time to hydrodynamics, only initial scatter-ings, i.e. baryon-baryon collisions and string excita-tions/fragmentations are relevant here. As many detailsof the UrQMD model are not relevant for the presentinitial state calculation we refer the interested reader forthe details [53, 54] of the UrQMD model and its thermo-dynamic properties [55, 56, 57]. The coupling betweenthe UrQMD initial state and the hydrodynamical evo-lution happens when the two Lorentz-contracted nucleihave passed through each other. This assures that (es-sentially) all initial baryon-baryon scattering have pro-ceeded and that the energy deposition has taken place.It should be noted that the present approach is differentfrom using a kinetic model for the freeze-out procedure[58, 59, 60, 61, 62, 63, 64, 65] which is not done in thepresent investigation, but it is in spirit similar to theNeXSPheRIO approach [66, 67, 68, 69, 70, 71, 72].To allow for a consistent and numerically stable map-ping of the ’point like’ particles from UrQMD to the 3-dimensional spatial-grid with a cell size of (0 . , eachhadron is represented by a Gaussian with a finite width.I.e. each particle is described by a three-dimensionalGaussian distribution of its total energy-, momentum-(in x-, y-, and z-direction) and baryon number-density.The width of these Gaussians is chosen to be σ = 1 fm. Asmaller Gaussian widths leads to numerical instabilities(e.g. entropy production) in the further hydrodynam-ical evolution, while a broader width would smear outthe initial fluctuations to a large extend. To account for FIG. 1: The initial temperature distribution in the z-y plane.Where z is the beam-axis and y the out of plane axis.
Overlap i.c UrQMD i.c. S / A E lab [AGeV] FIG. 2: Excitation functions of
S/A for the geometrical over-lap model and UrQMD initial conditions. the Lorentz-contraction of the nuclei in the longitudinaldirection, a gamma-factor (in longitudinal direction) isincluded. The resulting distribution function, e.g. forthe energy density, then reads: ǫ cf ( x, y, z ) = N exp ( x − x p ) + ( y − y p ) + ( γ z ( z − z p )) σ , (6) where N = ( π ) γ z σ E lab provides the proper normal-isation, ǫ cf and E cf are the energy density and to-tal energy of the particle in the computational frame,while ( x p , y p , z p ) is the position vector of the particle.Summing over all single particle distribution functionsleads to distributions of energy-, momentum- and baryonnumber-densities in each cell.This is done for Pb+Pb/Au+Au collisions at E lab =5 − A GeV with impact parameter b = 0 fm. To relatethe distributions of energy and baryon number-densityto thermodynamic quantities like pressure, temperature,chemical potential or entropy-density, the equation ofstate described above is used. As an example, Fig. 1shows the initial temperature distribution obtained for E lab = 10 A GeV.We contrast the microscopically calculated initial con-ditions described above with a simplified overlap geome-try initial condition. Therefore, we assume that the en-tire initial beam energy and baryon number equilibratesin a Lorentz-contracted volume determined by the over-lap of projectile and target in the center-of-mass frame.This allows to obtain a straightforward estimate for theinitial baryon number and energy density: ρ initial B = 2 γ CMS ρ , (7) ǫ initial = √ s ρ γ CMS . (8)It was shown in [35] that in the energy range of interesthere, this rather simple approach reproduces the specificentropy production from a three-fluid model quite well.For the subsequent hydrodynamical evolution of thesystem we apply a fully (3+1)-dimensional one-fluidmodel. The hydrodynamical equations are solved bymeans of the SHASTA (SHarp And Smooth TransportAlgorithm) as described in [73]. The EoS with a CEP isprovided in tabulated form with a fixed step size in en-ergy and baryon density: ∆ ǫ = 0 . n = 0 .
05 where ǫ and n are given in units of nuclear ground state densi-ties ( ǫ ≈ . n ≈ . − ). This hydro-dynamical model has been tested vigorously and appliedsuccessfully for various initial conditions and physics in-vestigations [73, 74, 75, 76, 77, 78, 79, 80].Fig. 2 depicts the excitation function of the total en-tropy S per baryon number A for the initial stage ofthe hydrodynamical evolution for both initial conditions(UrQMD, solid circles; overlap model, dashed line). Wehave checked that both quantities are separately con-served throughout the whole hydrodynamical evolution,so S/A is a time independent constant[87]. Interestingly,the simple geometric overlap model and the UrQMD ini-tial conditions yield basically the same value of
S/A fora given incident energy.The (isentropic) expansion paths for different beam en-ergies in the ǫ - n plane are shown in Fig. 3. Here n is the baryon-number density. Again lines of constant S/A for overlap initial conditions (blue open circles andlines), and the (3+1)-dimensional hydrodynamical evolu-tion with UrQMD initial conditions are compared. Themean values are obtained by weighting the value of a spe-cific quantity in a given cell with the energy density ofthat cell. E.g., the mean number density is calculated
Overlap i.c. with lines of const S/A UrQMD i.c. with hydro evolution Multiple UrQMD events for E lab = 10AGeV / n/n E lab [AGeV]= 16010040105 FIG. 3: Isentropic expansion paths in units of ground statedensities ( ǫ = 138 . n = 0 . − ). Red circlescorrespond to multiple UrQMD events for the same beamenergy and impact parameter. The black region indicates theregion below T ≤ from: < n > = P i,j,k n i,j,k · ǫ i,j,k P i,j,k ǫ i,j,k , (9)where i, j, k represent the cell indices. The mean val-ues for ǫ , T and µ q (quark chemical potential) are cal-culated accordingly. This has been done for equal timeintervals of ∆ t = 2 . S/A . The variation in energy andbaryon density due to the variation of the initial statein UrQMD even for a fixed impact parameter and fixedbeam energy are studied for E lab = 10 A GeV and indi-cated by the red full circles.As a next step the hydrodynamic evolution of the sys-tem is shown in the T − µ q plane in Fig. 4. Also in-dicated is the first order phase transition line and theCEP of the employed chiral EoS. Included are againlines of constant S/A and hydrodynamical evolutionpaths for the same beam energies (from left to right:160 , , , , A GeV) as in Fig. 3. As one can see, themean temperatures and chemical potentials of the hydro-dynamical evolution are not identical to the respectivelines of constant entropy. This is due to the averagingprocedure while a single cell does follow the isentropicpath. The time evolution of a central cell at the originin T and µ q is depicted in comparison (green dashed linewith green open circles).It is worthwhile to connect the present discussion of
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Lines of const. S/A UrQMD i.c. with hydro evolution Hydro evolution of central cell T [ M e V ] q [MeV] Endpoint byFodor & Katz (2004)
FIG. 4: Isentropic expansion paths in the T − µ q plane forvery central Pb+Pb/Au+Au reactions. UrQMD initial con-ditions with (3+1)-dimensional chiral hydrodynamical evolu-tion (averaged, full red line with circles; central cell, dashedgreen line with circles), isentropic expansion from the overlapmodel initial conditions are shown as full line in blue. Beamenergies are from left to right: 160 , , , , A GeV. Thephase boundary of the model is shown as full black line withthe critical end point, the Fodor and Katz critical end pointis shown separately with error bars [85]. the evolution path with UrQMD initial conditions to theresults obtained with various models (however withoutphase transition) that are investigated in [12]. The cur-rent findings with the chiral equation state support themain statement given there that it might be possible toreach the phase boundary to the QGP/chiral restorationalready at moderate beam energies ( E lab ∼ − A GeV).A further discussion about the effect of different equa-tions of state will be presented below. It should also benoted that we do not observe a focussing of the hydro-dynamical trajectories towards the critical end point incontrast to the findings by [81, 82]. However, see also thediscussion in [83, 84].Having at hand an EoS with a critical end point it ispossible to explore, which fraction of the evolving systemstays for how long close to the critical end point. The en-ergy dependence of this exposure tine is also investigated.Therefore, we define a ’critical volume’ by adding all cellsthat have a temperature of T CEP ±
10 MeV and a chem-ical potential µ CEP ±
10 MeV for each time step. Fig. 5shows the time evolution of this critical volume at vari-ous incident energies for the critical end point obtainedby the chiral EoS. Despite the surprising fact that themaximal volume is reached for the highest beam energies( E lab = 160 − A GeV) a quite large critical volume ofaround ∼
200 fm is already predicted at lower energiesof E lab = 60 A GeV. E lab [AGeV]=
60 100 160 180 200 c r i t i ca l v o l u m e [f m ] t [fm/c] FIG. 5: Time evolution of the critical volume for differentbeam energies for the CEP obtained with the chiral EoS. E lab [AGeV]=
60 100 160 180 200 c r i t i ca l v o l u m e [f m ] t [fm/c] FIG. 6: Time evolution of the critical volume for differentbeam energies the CEP obtained by Fodor and Katz [85].
Fig. 6 shows the critical volume for the T CEP and µ CEP values obtained by lattice QCD calculations [85], howeverusing the chiral EoS for the dynamics. In contrast to thechiral values in Fig. 5 the time for the maximum doesnot change with the energy in this case. The highest val-ues for the critical volume are still reached at the highestbeam energies, but one has to keep in mind that the crit-ical end point for the volume calculation and the criticalpoint in the evolution are different. The influence of dif-ferent EoS on the critical volume will be discussed below.To pin down the beam energy to maximise the critical
60 80 100 120 140 160 180 200010002000300040005000600070008000 V o l u m e i n t e g r a l [f m / c ] E lab [AGeV] Chiral eos endpoint Fodor & Katz endpoint
FIG. 7: Excitation function of the critical space-time volumefor the two different CEPs. volume for the longest period of time, the total 4-volumeis obtained by integration of the critical volume over thetime. The space-time volume is shown in Fig. 7 as afunction of E lab .Of course the actual size of the critical volume de-pends on the chosen ∆ T and ∆ µ intervals around T c and µ c . Also the choice of the Gaussian width of theparticles in the initial condition does have a small influ-ence on the critical volume, since at larger σ the distri-butions for T and µ q are wider and more ’smeared out’.However, we have checked that for the CEP obtainedwith the chiral EoS the excitation function is indepen-dent of σ . The integrated critical volume (for the CEPobtained from the chiral model) does steadily increaseup to E lab ≈ A GeV. Having the maximum criticalspace-time volume in the excitation at such an unexpect-edly high beam energy is due to the way the lines ofconstant entropy behave at the phase boarder (see Fig.4). With the present chiral EoS, the trajectories com-ing from higher temperatures turn right (towards lowertemperatures) at the phase transition line and thereforemuch higher energies are needed as if they would turn tosmaller µ q (left) and go up the phase line.As a last step, we modify our chiral EoS by adding allknown resonances (except the baryonic decuplett, sinceit is included by means of a so called ’test’-resonance)with vacuum-masses up to 2 GeV as a free gas. This al-lows to complete the baryon resonance spectrum withoutchanging the phase structure of the EoS. The positionand type of the phase transition in the phase diagramstays unchanged, because the added resonances have noinfluence on the chiral condensate σ .In Fig. 8 isentropic paths for the two chiral EoS (with
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Chiral EoS "No Hadgas" Pure Hadron-Gas EoS Chiral EoS "With Hadgas" T [ M e V ] q [MeV]Isentropes for UrQMD i.c. Endpoint Fodor & Katz (2006) E lab = 160 AGeV , 40 AGeV FIG. 8: Isentropic expansion paths for different beam ener-gies. The dashed lines resemble a pure hadron gas EoS. Theshort-dashed and straight lines refer to the chiral EoS withoutand with the completed resonance spectrum. and without completed baryon resonance spectrum) arecompared to those calculated with a free hadron-gas EoS(all particles have vacuum masses - which is of course notconsistent (and only shown for discussional purposes),since the free hadron-gas EoS does not have any phasetransition or CEP). As one can see, a beam energy of40 A GeV is more than sufficient for an hadron gas EoSto reach the CEP, while the energy needed with the chiralEoS is 160 A GeV (without heavy resonances) and E lab =40 − A GeV when heavy resonances are included.
SUMMARY
We showed, that calculating the initial conditions ofan heavy ion collision with the UrQMD model, yieldsvery similar
S/A values for a given beam energy as thesimple overlap model. We then compared an isentropicexpansion scenario within a full (3+1)dimensional idealhydrodynamic evolution with a chiral EoS including aCEP with constant
S/A lines. For the hydro evolution,the systems mean values of energy- and baryon-densityfollow isentropic paths in the ǫ − n phase-diagram, whilein the T − µ plane, a single cell follows the isentropic path,while the averaged quantities deviate from the isentropicexpectation. Most importantly it was shown, that con-cerning the search for the critical end point, it might notbe sufficient to apply a free hadron gas EoS to estimatethe energy needed to generate a system that, during itsexpansion, goes through the critical region. Applyingdifferent EoS (as we have done) can very much changepredictions at what beam energy that CEP is reached. Acknowledgements
This work was supported by BMBF and GSI. The com-putational resources were provided by the Frankfurt Cen-ter for Scientific Computing (CSC). We would like tothank Dr. Adrian Dumitru for helpful discussions. H.Petersen thanks the Deutsche Telekom Stiftung for thescholarship and the Helmholtz Research School on QuarkMatter Studies for additional suppport. [1] U. W. Heinz and M. Jacob, arXiv:nucl-th/0002042.[2] M. Gyulassy and L. McLerran, Nucl. Phys. A , 30(2005) [arXiv:nucl-th/0405013].[3] Z. Fodor and S. D. Katz, JHEP , 014 (2002)[arXiv:hep-lat/0106002].[4] Z. Fodor, S. D. Katz and C. Schmidt, JHEP , 121(2007) [arXiv:hep-lat/0701022].[5] F. Karsch, J. Phys. G , S633 (2005)[arXiv:hep-lat/0412038].[6] M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys.Rev. Lett. , 4816 (1998) [arXiv:hep-ph/9806219].[7] M. Gazdzicki and M. I. Gorenstein, Acta Phys. Polon. B , 2705 (1999) [arXiv:hep-ph/9803462].[8] M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys.Rev. D , 114028 (1999) [arXiv:hep-ph/9903292].[9] L. V. Bravina et al. , Phys. Rev. C , 024904 (1999)[arXiv:hep-ph/9906548].[10] L. V. Bravina et al. , Phys. Rev. C , 064902 (2001)[arXiv:hep-ph/0010172].[11] M. Gazdzicki et al. [NA49 Collaboration], J. Phys. G ,S701 (2004) [arXiv:nucl-ex/0403023].[12] I. C. Arsene et al. , Phys. Rev. C , 034902 (2007)[arXiv:nucl-th/0609042].[13] J. Hofmann, H. Stoecker, U. W. Heinz, W. Scheid andW. Greiner, Phys. Rev. Lett. , 88 (1976).[14] H. Stoecker and W. Greiner, Phys. Rept. , 277 (1986).[15] H. Sorge, Phys. Rev. Lett. , 2048 (1999)[arXiv:nucl-th/9812057].[16] J. Y. Ollitrault, Phys. Rev. D , 229 (1992).[17] C. M. Hung and E. V. Shuryak, Phys. Rev. Lett. ,4003 (1995) [arXiv:hep-ph/9412360].[18] D. H. Rischke, Nucl. Phys. A , 88C (1996)[arXiv:nucl-th/9608024].[19] H. Sorge, Phys. Rev. Lett. , 2309 (1997)[arXiv:nucl-th/9610026].[20] H. Heiselberg and A. M. Levy, Phys. Rev. C , 2716(1999) [arXiv:nucl-th/9812034].[21] S. Soff, S. A. Bass, M. Bleicher, H. Stoecker andW. Greiner, arXiv:nucl-th/9903061.[22] J. Brachmann et al. , Phys. Rev. C , 024909 (2000)[arXiv:nucl-th/9908010].[23] L. P. Csernai and D. Rohrich, Phys. Lett. B , 454(1999) [arXiv:nucl-th/9908034].[24] B. Zhang, M. Gyulassy and C. M. Ko, Phys. Lett. B ,45 (1999) [arXiv:nucl-th/9902016].[25] P. F. Kolb, J. Sollfrank and U. W. Heinz, Phys. Rev. C , 054909 (2000) [arXiv:hep-ph/0006129].[26] M. Bleicher and H. Stoecker, Phys. Lett. B , 309(2002) [arXiv:hep-ph/0006147]. [27] P. F. Kolb and U. W. Heinz, arXiv:nucl-th/0305084.[28] H. Stoecker, Nucl. Phys. A , 121 (2005)[arXiv:nucl-th/0406018].[29] X. l. Zhu, M. Bleicher and H. Stoecker, Phys. Rev. C ,064911 (2005) [arXiv:nucl-th/0509081].[30] H. Petersen, Q. Li, X. Zhu and M. Bleicher, Phys. Rev.C , 064908 (2006) [arXiv:hep-ph/0608189].[31] I. N. Mishustin, V. N. Russkikh and L. M. Satarov, Sov.J. Nucl. Phys. , 454 (1988) [Yad. Fiz. , 711 (1988)].[32] I. N. Mishustin, V. N. Russkikh and L. M. Satarov, Nucl.Phys. A , 595 (1989).[33] U. Katscher, D. H. Rischke, J. A. Maruhn, W. Greiner,I. N. Mishustin and L. M. Satarov, Z. Phys. A , 209(1993).[34] J. Brachmann, A. Dumitru, J. A. Maruhn, H. Stoecker,W. Greiner and D. H. Rischke, Nucl. Phys. A , 391(1997) [arXiv:nucl-th/9703032].[35] M. Reiter, A. Dumitru, J. Brachmann, J. A. Maruhn,H. Stoecker and W. Greiner, Nucl. Phys. A , 99(1998) [arXiv:nucl-th/9806010].[36] M. Bleicher et al. , Phys. Rev. C , 1844 (1999)[arXiv:hep-ph/9811459].[37] J. Brachmann, A. Dumitru, H. Stoecker and W. Greiner,Eur. Phys. J. A , 549 (2000) [arXiv:nucl-th/9912014].[38] A. Dumitru et al. , Heavy Ion Phys. , 121 (2001)[arXiv:nucl-th/0010107].[39] V. N. Russkikh, Yu. B. Ivanov, E. G. Nikonov, W. Noren-berg and V. D. Toneev, Phys. Atom. Nucl. , 199 (2004)[Yad. Fiz. , 195 (2004)] [arXiv:nucl-th/0302029].[40] Yu. B. Ivanov, V. N. Russkikh and V. D. Toneev, Phys.Rev. C , 044904 (2006) [arXiv:nucl-th/0503088].[41] V. D. Toneev, Yu. B. Ivanov, E. G. Nikonov, W. Noren-berg and V. N. Russkikh, Phys. Part. Nucl. Lett. , 288(2005) [Pisma Fiz. Elem. Chast. Atom. Yadra , 43(2005)].[42] R. Baier and P. Romatschke, Eur. Phys. J. C , 677(2007) [arXiv:nucl-th/0610108].[43] H. Song and U. W. Heinz, arXiv:0709.0742 [nucl-th].[44] P. Papazoglou, D. Zschiesche, S. Schramm, J. Schaffner-Bielich, H. Stoecker and W. Greiner, Phys. Rev. C ,411 (1999) [arXiv:nucl-th/9806087].[45] S. Schramm, Phys. Lett. B , 164 (2003)[arXiv:nucl-th/0210053].[46] S. Schramm, Phys. Rev. C , 064310 (2002)[arXiv:nucl-th/0207060].[47] D. Zschiesche, S. Schramm, H. Stoecker and W. Greiner,Phys. Rev. C , 064902 (2002) [arXiv:nucl-th/0107037].[48] J. Theis, G. Graebner, G. Buchwald, J. A. Maruhn,W. Greiner, H. Stoecker and J. Polonyi, Phys. Rev. D , 2286 (1983).[49] D. Zschiesche, G. Zeeb, S. Schramm and H. Stoecker, J.Phys. G , 935 (2005) [arXiv:nucl-th/0407117].[50] N.F. Mott and H.S.W. Massey, The Theory of AtomicCollisions, Oxford Press (1971), pp437.[51] K. Paech, H. Stoecker and A. Dumitru, Phys. Rev. C ,044907 (2003) [arXiv:nucl-th/0302013].[52] K. Paech and A. Dumitru, Phys. Lett. B , 200 (2005)[arXiv:nucl-th/0504003].[53] S. A. Bass et al. , Prog. Part. Nucl. Phys. ,255 (1998) [Prog. Part. Nucl. Phys. , 225 (1998)][arXiv:nucl-th/9803035].[54] M. Bleicher et al. , J. Phys. G , 1859 (1999)[arXiv:hep-ph/9909407].[55] S. A. Bass et al. , Phys. Rev. Lett. , 4092 (1998) [arXiv:nucl-th/9711032].[56] M. Belkacem et al. , Phys. Rev. C , 1727 (1998)[arXiv:nucl-th/9804058].[57] L. V. Bravina et al. , J. Phys. G , 351 (1999)[arXiv:nucl-th/9810036].[58] C. Anderlik et al. , Phys. Rev. C , 3309 (1999)[arXiv:nucl-th/9806004].[59] A. Dumitru, S. A. Bass, M. Bleicher, H. Stoeckerand W. Greiner, Phys. Lett. B , 411 (1999)[arXiv:nucl-th/9901046].[60] S. A. Bass, A. Dumitru, M. Bleicher, L. Bravina,E. Zabrodin, H. Stoecker and W. Greiner, Phys. Rev.C , 021902 (1999) [arXiv:nucl-th/9902062].[61] V. K. Magas et al. , Heavy Ion Phys. , 193 (1999)[arXiv:nucl-th/9903045].[62] S. A. Bass and A. Dumitru, Phys. Rev. C , 064909(2000) [arXiv:nucl-th/0001033].[63] D. Teaney, J. Lauret and E. V. Shuryak, Nucl. Phys. A , 479 (2002) [arXiv:nucl-th/0104041].[64] D. Teaney, J. Lauret and E. V. Shuryak,arXiv:nucl-th/0110037.[65] C. Nonaka and S. A. Bass, Phys. Rev. C , 014902(2007) [arXiv:nucl-th/0607018].[66] S. Paiva, Y. Hama and T. Kodama, Phys. Rev. C ,1455 (1997).[67] C. E. Aguiar, Y. Hama, T. Kodama and T. Osada, Nucl.Phys. A , 639 (2002) [arXiv:hep-ph/0106266].[68] O. . J. Socolowski, F. Grassi, Y. Hama andT. Kodama, Phys. Rev. Lett. , 182301 (2004)[arXiv:hep-ph/0405181].[69] F. Grassi, Y. Hama, O. Socolowski and T. Kodama, J.Phys. G , S1041 (2005).[70] R. Andrade, F. Grassi, Y. Hama, T. Kodama, O. . J. So-colowski and B. Tavares, Eur. Phys. J. A , 23 (2006)[arXiv:nucl-th/0511021].[71] R. Andrade, F. Grassi, Y. Hama, T. Kodama andO. . J. Socolowski, Phys. Rev. Lett. , 202302 (2006)[arXiv:nucl-th/0608067].[72] C. E. Aguiar, T. Kodama, T. Koide and Y. Hama, Braz. J. Phys. , 95 (2007).[73] D. H. Rischke, S. Bernard and J. A. Maruhn, Nucl. Phys.A , 346 (1995) [arXiv:nucl-th/9504018].[74] R. Waldhauser, D. H. Rischke, U. Katscher,J. A. Maruhn, H. Stoecker and W. Greiner, Z. Phys. C , 459 (1992).[75] V. Schneider, U. Katscher, D. H. Rischke, B. Waldhauser,J. A. Maruhn and C. D. Munz, J. Comput. Phys. ,92 (1993).[76] D. H. Rischke, Y. Pursun and J. A. Maruhn, Nucl. Phys.A , 383 (1995) [Erratum-ibid. A , 717 (1996)][arXiv:nucl-th/9504021].[77] D. H. Rischke and M. Gyulassy, Nucl. Phys. A , 701(1996) [arXiv:nucl-th/9509040].[78] M. Gyulassy, D. H. Rischke and B. Zhang, Nucl. Phys.A , 397 (1997) [arXiv:nucl-th/9609030].[79] H. Stoecker, B. Betz and P. Rau, PoS C POD2006 , 029(2006) [arXiv:nucl-th/0703054].[80] B. Betz, P. Rau and H. Stocker, arXiv:0707.3942 [hep-th].[81] C. Nonaka and M. Asakawa, Phys. Rev. C , 044904(2005) [arXiv:nucl-th/0410078].[82] M. Asakawa and C. Nonaka, Nucl. Phys. A , 753(2006) [arXiv:nucl-th/0509091].[83] O. Scavenius, A. Mocsy, I. N. Mishustin andD. H. Rischke, Phys. Rev. C , 045202 (2001)[arXiv:nucl-th/0007030].[84] M. A. Stephanov, Prog. Theor. Phys. Suppl. ,139 (2004) [Int. J. Mod. Phys. A , 4387 (2005)][arXiv:hep-ph/0402115].[85] Z. Fodor and S. D. Katz, JHEP , 050 (2004)[arXiv:hep-lat/0402006].[86] Note that instead of an explicit mass term we could havecoupled the resonance to the dilaton χχ