The hadronic SU(3) Parity Doublet Model for Dense Matter, its extension to quarks and the strange equation of state
aa r X i v : . [ h e p - ph ] A ug The hadronic SU(3) Parity Doublet Model for Dense Matter, its extension to quarksand the strange equation of state
J. Steinheimer ∗ Institut f¨ur Theoretische Physik, Goethe-Universit¨at,Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany
S. Schramm † FIAS, Johann Wolfgang Goethe University, Frankfurt am Main, Germany
H. St¨ocker
Institut f¨ur Theoretische Physik, Goethe-Universit¨at,Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, GermanyFrankfurt Institute for Advanced Studies (FIAS),Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany andGSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, Planckstr. 1, D-64291 Darmstadt, Germany (Dated: August 14, 2018)A chiral model is introduced that is based on the parity doublet formulation of chiral symmetryincluding hyperonic degrees of freedom. The phase structure of the model is determined. Dependingon the masses of the chiral partners the transition to the chirally restored phase shows a first-orderline with critical endpoints as function of chemical potential and temperature in additional to thestandard liquid-gas phase transition of self-bound nuclear matter.We extend the parity doublet model to describe the deconfinement phase transition which is inquantitative agreement with lattice data at µ B = 0. The phase diagram of the model is presentedwhich shows a decoupling of chiral symmetry restoration and deconfinement. Loosening the con-straint of strangeness conservation we also investigate the phase diagram at net strangeness density.We calculate the strangeness per baryon fraction and the baryon strangeness correlation factor, twoquantities that are sensitive on deconfinement and that can be used to interpret lattice calculations. PACS numbers: 21.65.Mn,12.38.Aw,12.39.Fe,25.75.Nq
I. INTRODUCTION
The study of dense and hot hadronic matter is a centraltopic of nuclear physics. It is directly linked to the searchfor the phase transition to chirally restored and decon-fined matter in ultra-relativistic heavy-ion collisions aswell as to the study of extremely dense but rather coldmatter inside compact stars. In spite of several decades ofexperimental and theoretical research the phase structureof strongly interacting matter remains uncertain with theexception of the regime around cold saturated nuclearmatter and, to some extent the transition behavior atvanishing chemical potential, where lattice gauge calcu-lation indicate a cross-over transition to chirally restoredand deconfined matter, at a temperature currently deter-mined to be around 150 to 160 MeV [1, 2].At finite chemical potential the phase structure of QCDis even less clear. While early extensions of lattice studiesto finite µ B proposed the existence of a critical endpointat rather small chemical potential [3, 4], other latticeinvestigations cannot confirm evidence for its existence[5, 6]. ∗ Electronic address: [email protected] † Electronic address: schramm@fias.physik.uni-frankfurt.de
A central point of these investigations is the under-standing of the phase transition in the hadronic andquark-hadron matter. Recent lattice calculations andtheir analysis in terms of a hadron resonance gas hint tothe importance of hadronic degrees of matter in drivingthe phase transition to a quark-gluon plasma [2, 7, 8].Furthermore, the low temperature of the chiral transi-tion [2, 9], the good agreement with chiral perturbationtheory below T c [10] and the apparent sensitivity on thehadron properties [2] (caused by lattice discretization ef-fects) supports the idea that the chiral transition couldbe explained with hadronic interactions. Therefore, alsoa study of purely hadronic models and their properties,especially the restoration of chiral symmetry is impor-tant. One main benchmark for any useful comprehensivemodel of that kind is a reasonable description of satu-rated nuclear matter. In order to have a realistic de-scription of highly excited matter strange hadrons haveto be included in the model description. In a simple linearsigma-model it is not possible to have stable bound nu-clear matter. Therefore a number of extended approachesadding vector and dilaton fields were discussed [11–14],including extensions to flavor SU(3) [15–17]. II. THE PARITY DOUBLET MODEL
An elegant and alternative description of a transitionto chirally restored matter is the parity doublet model. Inthis approach an explicit mass term for baryons is pos-sible, where the signature for chiral symmetry restora-tion is the degeneracy of the baryons and their respec-tive parity partners. There are several SU(2) studies ofnuclear matter adopting this approach showing that itis possible to generate saturated matter in the paritydoublet approach [18–23]. A SU(3) parity-doublet de-scription of hadronic matter was still missing. In [24]hyperonic decays in vacuum have been studied in suchan approach. In the following we outline the basic SU(3)parity model. With this ansatz we study nuclear mattersaturation in order to fulfill one of the benchmarks for auseful model as mentioned above. Subsequently we cal-culate the phase diagram of isospin-symmetric matter byvarying the baryonic chemical potential and temperatureof the system.In the parity doublet model positive and negative par-ity states of the baryons are grouped in doublets. Thetwo components of the fields defining the parity part-ners, ϕ + and ϕ − transform in opposite way regardingchiral transformations: ϕ ′ + R = Rϕ + R ϕ ′ + L = Lϕ + L ϕ ′− R = Lϕ − R ϕ − L = Rϕ − L , (1)where L and R are rotations in the left- and right handedsubspaces. This allows for a chirally invariant mass termin the Lagrangian of the general form: m ( ¯ ϕ − γ ϕ + − ¯ ϕ + γ ϕ − ) = m ( ¯ ϕ − L ϕ + R − ¯ ϕ − R ϕ + L − ¯ ϕ + L ϕ − R + ¯ ϕ + R ϕ − L ) , (2)where m represents a mass parameter. The generalSU(3) extension of the approach using the non-linear rep-resentation of the fields is quite straightforward as shownin [24]. As outlined in [15] one constructs SU(3)-invariantterms in the Lagrangian including the meson-baryon andmeson-meson self-interaction terms assuming a nonlin-ear realization of chiral symmetry. The part of the La-grangian coupling the baryon and the mesonic fields rel-evant in a mean-field approximation reads L B = Tr(¯Ξ i∂/ Ξ) + m Tr( (cid:0) ¯Ξ γ τ Ξ (cid:1) + D (1) s Tr(¯Ξ { Σ , Ξ } )+ F (1) s Tr(¯Ξ [Σ , Ξ]) + S (1) s Tr(Σ)Tr(¯ΞΞ)+ D (2) s Tr(¯Ξ τ { Σ , Ξ } ) + F (2) s Tr(¯Ξ τ [Σ , Ξ])+ S (2) s Tr(Σ)Tr(¯Ξ τ Ξ) + D v Tr(¯Ξ γ µ { V µ , Ξ } )+ F v Tr(¯Ξ γ µ [ V µ , Ξ]) + S v Tr( V µ )Tr(¯Ξ γ µ Ξ) . (3)Here Ξ is the baryon octet whereby each field is a dou-blet consisting of the baryon and its negative parity part-ner. Σ and V µ are the multiplets of the scalar and vectormesons. The Pauli matrices τ i act on the doublets. In TABLE I: Model parameters for different values of the massof the nucleonic parity partner k k k ǫ (370 . . M N ∗ [MeV] 1400 1440 1490 1535 g σ -7.27 -7.467 -7.714 -7.933 g σ -0.765 -0.792 -0.823 -0.850 α σ g Nω general the various sets D ( i ) , F ( i ) , S ( i ) correspond to theD-type and F-type SU(3) invariant baryon-meson cou-plings. Note that the parity doublet models allow for twodifferent scalar coupling terms i = 1 ,
2. In order not tobe overwhelmed by coupling constants we will restrict theset of non-zero couplings in the actual calculations. Asthe term proportional to m mixes the upper and lowercomponents of the parity doublets, one diagonalizes thematrix by introducing new fields B with a diagonal massmatrix. Taking along only the diagonal meson contribu-tions, the scalar and vector condensates in the mean fieldapproximation, the resulting Lagrangian L B then reads L B = X i ( ¯ B i i∂/B i ) + X i (cid:0) ¯ B i m ∗ i B i (cid:1) + X i (cid:0) ¯ B i γ µ ( g ωi ω µ + g ρi ρ µ + g φi φ µ ) B i (cid:1) . (4)The effective masses of the baryons (assuming isospinsymmetric matter) read m ∗ i ± = rh ( g (1) σi σ + g (1) ζi ζ ) + ( m + n s m s ) i ± g (2) σi σ ± g (1) ζi ζ. (5) E / A [ M e V ] B [1/fm ] FIG. 1: Negative binding energy per particle in MeV as func-tion of density of the system for the case M N ∗ = 1535 MeV.A reasonable nuclear matter ground state can be achieved. T [ M e V ] B [MeV] FIG. 2: Phase transition lines of the chiral and liquid gastransitions for a mass m N ∗ = 1535MeV. Solid lines markfirst-order transitions whereas dashed lines indicate a cross-over. The circles mark the critical end points. The liquid-gas and chiral symmetry restoration cross-over lines merge athigher temperature. where the various coupling constants g ( j ) i are given ascombinations of the original parameters D ( j ) , F ( j ) , S ( j ) in equation 3 and further adding a SU(3) breaking massterm that generates an explicit mass corresponding tothe strangeness n s of the baryon.The scalar meson interaction driving the spontaneousbreaking of chiral symmetry can be written in terms ofSU(3) invariants I = T r (Σ) , I = T r (Σ ) , I = det (Σ) , I = T r (Σ ): V = V + 12 k I − k I − k I + k ln( I ) (6)where V is fixed by demanding a vanishing potential inthe vacuum. The explicit symmetry breaking term thatgenerates the correct pion and kaon masses with theircorresponding decay constants can be written as L SB = m π f π σ + (cid:16) √ m k f k − √ m π f π (cid:17) ζ, (7)The set of scalar coupling constants are fitted in or-der to reproduce the vacuum masses of the nucleon, andthe Lambda, Sigma, and Xi hyperons, whereas the vec-tor couplings are chosen to reproduce reasonable valuesfor nuclear ground state properties. The resulting bind-ing energy per particle as function of density is shown inFig. 1. For all listed parameters the ground state en-ergy per baryon is between -15 MeV and -16 MeV, theground state density has a value of ρ = 0 . f m − andthe compressibility lies between 300 MeV and 310 MeV.The latter value is somewhat large. Here, a more detailedand extensive parameter study might likely lead to more
900 1200 1500 1800800100012001400160018002000 900 1200 1500 1800800100012001400160018002000 NN ** e ff e c t i v e m a ss [ M e V ] B [MeV] * FIG. 3: Masses of baryons as function of chemical potential.The degeneracy of the various parity doublets can be observedat high µ B . satisfactory values. Note that the value of the mass pa-rameter is set to m = 810 MeV for all parametrizations.Such a choice corresponds to a rather large bare mass ofthe baryons. In principle such a large m could also begenerated dynamically through a coupling to the dilatonfield (see e.g. [25]). Such a coupling can be introducedin our model in a straight forward way. In the presentinvestigation however we intended to have as few free pa-rameters as possible, allthough such an extension will besubject of future investigations.One candidate for the parity partner of the nucleon isthe N(1535) resonance. However, this assignment is un-clear, the state might also be a broad structure, so essen-tially the mass of the particle (assuming its existence) isnot determined. Resulting parameters for several valuesare shown in Table 1. A SU(3) description, in addition toenhance the number of degrees of freedom, also necessar-ily increases the number of parameters. In order not to beoverwhelmed by too many new parameters, for simplicitywe assume that the splitting of the various baryon speciesand their respective parity partners is of the same valuefor all baryons, which is achieved by setting g (2) σi ≡ g (2) and g (2) ζi = 0. This should be sufficient for a first inves-tigation of the model approach. This assumption agreesquite well with the even less certain assignments of theparity partners of the hyperons. Obvious candidates arethe Λ(1670) and Σ(1750), whose masses roughly followthe equal splitting approximation, assuming the nucle-onic parity partner to be the N(1535). In the case of theΞ ∗ the data are unclear.In another simplification the hyperonic vector inter-actions were tuned to generated reasonable optical po-tentials of the hyperons in ground state nuclear matter,with U Λ ( ρ o ) −
28 MeV and U Ξ ( ρ o ) −
18 MeV . The valuefor the strange quark mass was fixed at m s = 150 MeV.The numbers used are summarized in Table 1. A more
50 100 150 200 250 3000.00.20.40.60.81.050 100 150 200 250 3000.00.20.40.60.81.0 PT pions T =270 MeV T =220 MeV Lattice data: asqtad N =12 HISQ N =8 stout N = 12, f K stout N = 12 / T [MeV]
FIG. 4: (Color online) Normalized value of the chiral conden-sate as a function of temperature, at µ B = 0. The solid lineis the model result for T = 220 MeV and the dashed line for T = 270 MeV. The grey dashed line depicts results for thepion contribution to the chiral condensate from chiral pertur-bation theory [26]. The symbols denote lattice data, wherethe open symbols represent older data with the asqtad and p4action and the colored symbols are more recent results (seetext). exhaustive study of various parameter setups will be per-formed in future work.The equations of motion following from Eqs. (4, 6,7) are then solved self-consistently in mean field approx-imation by minimizing the grand canonical potential asfunction of baryonic chemical potential and temperature.The resulting phase diagram for the transition fromchirally broken to chirally restored phase is shown in Fig.2. One can observe two first-order transition lines at highdensities. The first one is the liquid-gas phase transition,which indicates that the model exhibits a bound nuclearground state. The second one is the chiral transition,also signaling the onset of the population of the baryonicparity partners. The lines stop at second-order criticalend points, at (T c , µ c ) = (21 MeV, 1560 MeV) and (17.5MeV, 905 MeV). The critical point of the chiral transitionis very low in temperature. At values of M N ∗ below 1460MeV the first-order transition becomes a cross-over forall values of T and µ . Both cross-over lines join at atemperature T ≈
120 MeV.Fig. 3 shows the effective masses of the baryons withtheir parity partners as function of baryochemical poten-tial. One can observe the effect of the two phase tran-sitions, leading to essentially degenerate opposite paritystates.Here the calculations were done for isospin symmetricmatter with strangeness zero, which would be the logi-cal first-order assumption for matter created in heavy-ion collisions. The study of star matter in beta equilib-rium including leptons for ensuring charge neutrality isin progress. Results will be presented in a forthcomingpublication.
100 200 300 4000.00.20.40.60.81.0100 200 300 4000.00.20.40.60.81.0
Lattice data: HISQ N =6 HISQ N =8 asqtad N =12 stout cont.
T [MeV] T =270 MeV T =220 MeV FIG. 5: (Color online) Value of the Polyakov loop as a func-tion of temperature at µ B = 0. The solid line is the modelresult for T = 220 MeV and the dashed line for T = 270MeV. The symbols denote lattice data, where the open sym-bols represent older data with the asqtad and p4 action andthe colored symbols are more recent results. III. INCLUDING MESONS AND QUARKS
Recent results from different lattice collaborations in-dicate that the chiral phase transition at µ q = 0 occurs ata rather low temperature. Although there are still con-siderable systematic lattice effects there seems to be aconsensus that the crossover temperature is between 150and 160 MeV. This can be seen in figure (4) where thesymbols denote the different lattice actions. In particu-lar intriguing is the fact that the value of the chiral con-densate already drops to about 80 to 90% of its vaccumvalue at temperatures less than 150 MeV. In the paritydoublet model such an early decrease of the chiral con-densate can be hardly accommodated by only couplingthe baryons to the chiral fields as their mass is ratherlarge and they are not thermodynamically activated atsuch low temperatures. On the other hand calculationsin chiral perturbation theory have shown that the pioniccontribution to the chiral condensate is considerable atsuch low temperatures because of the small pion mass.This is also shown in figure (4), as the grey dashed linedepicts the chiral perturbation theory results for the pionself interaction taken from [26]. One can see, that at lowtemperature the behavior of the chiral condensate seemsdominated by the pseudoscalar contributions while onlyat larger temperature the baryon interactions become im-portant. Consequently we would like to include effects ofthe pseudoscalars in our parity doublet model to accom-modate the low temperature behavior of the chiral con-densate at small net baryon densities. To this end we takeinto account the coupling of the scalar field, which origi-nates from the explicit symmetry breaking term, Eq. (7)(see Ref. [27]). Including the pseudoscalar mesons thisterm generates a mass for the pions as m π = m π, σ/σ .
100 200 300 40002468100 200 300 40002468 T =270 MeV T =220 MeV HRG ( e - ) / T T [MeV]
FIG. 6: (Color online) The interaction measure (defined as( e − p ) /T ) as a function of temperature at µ B = 0. The solidline is the model result for T = 220 MeV and the dashed linefor T = 270 MeV. The symbols denote lattice data, wherethe open symbols represent older data with the asqtad andp4 action and the colored symbols are more recent results.The grey line represents the interaction measure for a hadronresonance gas equation of state which includes all hadronicresonances up to 2.2 GeV. This leads to an increase of the pressure of the pion gaswith decreasing scalar condensate, thus driving the phasetransition to lower temperatures.It is well known that at some temperature QCD ex-hibits a transition to a deconfined phase at which thequarks become the dominant degrees of freedom. Whenthis deconfinement will appear and what the order pa-rameter for this transition might be is still under heavydebate [28, 29]. Assuredly one can only say that it oc-curs in a temperature region of T dec ≈ −
400 MeV.Nevertheless at some point the hadronic parity doubletmodel will not be the appropriate effective description ofQCD and one needs to introduce a deconfinement mech-anism in the model. In this work we will apply a mech-anism that has been introduced in [30] to add a decon-finement transition in a chiral hadronic model. This isdone by adding an effective quark and gluon contribu-tion as done in the PNJL approach [31, 32]. This modeluses the Polyakov loop Φ as the order parameter for de-confinement. Φ is defined via Φ = Tr[exp ( i R dτ A )],where A = iA is the temporal component of the SU(3)gauge field, distinguishing Φ, and its conjugate Φ ∗ at fi-nite baryon densities [33–35]. In recent years the PNJLmodel has been widely used and extended to include non-local interactions as well as an imaginary chemical poten-tial (see also [36–65]).The effective masses of the quarks are generated bythe scalar mesons except for a small explicit mass term( δm q = 5 MeV and δm s = 150 MeV for the strange
50 150 300 450012350 150 300 4500123 T =220 MeV T =270 MeVblack : Q + Q green : Meson red : B + B T [MeV] / T FIG. 7: (Color online) Number densities of the different par-ticle species as a function of temperature at µ B = 0. Thesolid lines show results for T = 220 MeV and the dashedline for T = 270 MeV. The mesons are shown in green, thedensities for baryons plus anti-baryons in red and the quarksplus anti-quarks in black. quark) and m (explain): m ∗ q = g qσ σ + δm q + m q ,m ∗ s = g sζ ζ + δm s + m q , (8)with values of g qσ = g sζ = 4 .
0. As in the case of thebaryons we also introduced a mass parameter m q = 200MeV for the quarks. Again this additional mass termcan be due to a coupling of the quarks to the dilatonfield (gluon condensate). Fot this mass term the quarksdo not appear in the nuclear ground state which wouldbe an unphysical result. This allows to set the vectortype repulsive interaction strength of the quarks to zero.A non-zero vector interaction strength would lead to amassive deviation of the quark number susceptibilities tolattice data as has been indicated in different mean fieldstudies [66–68].A coupling of the quarks to the Polyakov loop is intro-duced in the thermal energy of the quarks. Their thermalcontribution to the grand canonical potential Ω, can bewritten as:Ω q = − T X i ∈ Q γ i (2 π ) Z d k ln (cid:18) E ∗ i − µ i T (cid:19) (9)andΩ q = − T X i ∈ Q γ i (2 π ) Z d k ln (cid:18) ∗ exp E ∗ i + µ i T (cid:19) (10)The sums run over all quark flavors, where γ i is thecorresponding degeneracy factor, E ∗ i = p m ∗ i + p theenergy and µ ∗ i the chemical potential of the quark.All thermodynamical quantities, energy density e , en-tropy density s as well as the densities of the differentparticle species ρ i , are derived from the grand canonicalpotential. It includes the effective potential U (Φ , Φ ∗ , T ),which controls the dynamics of the Polyakov-loop. In ourapproach we adopt the ansatz proposed in [32]: U = − a ( T )ΦΦ ∗ + b ( T ) ln [1 − ∗ + 4(Φ Φ ∗ ) − ∗ ) ] (11)with a ( T ) = a T + a T T + a T T , b ( T ) = b T T .The parameters a , a , a and b are fixed, as in [32],by demanding a first order phase transition in thepure gauge sector at T = 270 MeV, and that theStefan-Boltzmann limit of a gas of gluons is reached for T → ∞ . In general of course the presence of quarks mayhave a significant influence on the Polyakov potential[42] so one should not regard the parameters to beabsolutely fixed.In the following we introduce excluded volumesfor the hadrons in the system. As a onsequence thehadronic contributions from the equation of state at hightemperatures and densities will be suppressed. Includingeffects of finite-volume particles in a thermodynamicmodel for hadronic matter, was proposed long ago[69–78]. In recent publications [30, 68] we adopted thisansatz to successfully describe a smooth transition froma hadronic to a quark dominated system (see also [79]).In particular we introduce the quantity v i which isthe volume excluded of a particle of species i where weonly distinguish between hadronic baryons, mesons andquarks. Consequently v i can assume three values: v Quark = 0 v Baryon = vv Meson = v/a Where a is a number larger than one. In our cal-culations we choose a value of a = 8, which assumesthat the radius r of a meson is half of the radius of abaryon. Note that at this point we neglect any possibledensity-dependent and Lorentz contraction effects on theexcluded volumes as introduced in [76, 77].The modified chemical potential e µ i , which is connectedto the real chemical potential µ i of the i -th particlespecies, is obtained by the following relation: e µ i = µ i − v i P (12)where P is the sum over all partial pressures. To bethermodynamically consistent, all densities ( e e i , e ρ i and e s i ) have to be multiplied by a volume correction factor f ,which is the ratio of the total volume V and the reducedvolume V ′ , not being occupied: f = V ′ V = (1 + X i v i ρ i ) − (13)
300 400 500 6000.000.030.060.09300 400 500 6000.000.030.060.09 / q [MeV] (n+p) (u+d) (N*) FIG. 8: (Color online) Densities of the different particlespecies at T = 0 and as a function of µ q = µ B /
3. The blacksolid line depicts the density of protons plus neutrons, thered dashed line for up plus down quarks and the green dashdotted line for the chiral partners of the nucleons. e = X i f e e i ρ i = f e ρ i s = X i f e s i (14)As a consequence, the chemical potentials of thehadrons are decreased by the quarks, but not viceversa. In other words as the quarks start appearingthey effectively suppress the hadrons by changing theirchemical potential, while the quarks are only affectedthrough the volume correction factor f . IV. THERMODYNAMIC PROPERTIES
A surprising result from recent lattice studies of the2+1 flavor QCD equation of state at finite temperatureis the apparent decoupling of the chiral phase transitionand the increase of the Polyakov loop which was longthought to be a good order parameter for deconfinement.In particular one observes that the steepest change in thechiral condensate occurs at a low temperature of roughly150 −
160 MeV, depending on the choice of the lattice ac-tion as well as the scale which is used to translate latticequantities into the physical temperature. On the otherhand a considerable increase in the value of the PolyakovLoop is only observed above T ≈
210 MeV, where thisresult seems quite independent of the lattice action thatis applied.In the following we will compare results on the or-der parameters and thermodynamic quantities calculatedwithin our model to recent lattice results from the differ-ent collaborations [1, 2, 9, 10, 80–84] Figure 4 displaysthe results for the temperature dependence of the expec-tation value of the σ field, normalized to its ground state T [ M e V ] q [MeV] T =220 MeV T [ M e V ] q [MeV] T =270 MeV FIG. 9: (Color online) Phase diagrams for our model with two different values of T . Within the orange region the normalizedvalue of the chiral condensate lies between 0 . < σ/σ < .
8. In the green area the value of the Polyakov loop is in between0 . < φ < .
7. The black lines indicate first order phase transitions, where the points indicate the critical endpoints. value, from the SU(3) parity model including quarks.The dashed line depicts the results when the parameterof the Polyakov potential is unchanged ( T = 270 MeV)and the solid line when T is changed to 220 MeV. Notethat such readjustments of T are commonly used in anumber of PNJL studies and T can even depend on µ B [61, 85].Our results are compared to calculations from chiralperturbation theory (grey short dashed line) [26] andrecent lattice results (colored symbols depict the morerecent results while the open symbols refer to previouslyused lattice actions).At temperatures below 160 MeV the decrease of the σ is dominantly caused by the pseudoscalar-scalarcoupling. Compared to the chiral perturbation theoryresults, which depict the pion contribution to the chiralcondensate, our model still shows a slower decrease ofthe chiral condensate with temperature, while latticedata are rather well described by the chiral perturbationtheory up to temperatures of 150 MeV. Above thattemperature the baryon scalar interaction (and at evenhigher temperature the quark scalar interaction) startsto contribute to the value of the σ -field. In this regimeour model gives a good description of the lattice data.The chiral critical temperature T χc only weakly dependson the value of T because chiral symmetry restorationis mainly driven by the hadronic interactions with thefields. We obtain the values of T χc from the maximum in ∂σ/∂T as T χc = 172 and 165 MeV respectively. While the temperature dependence of the latticeresults on the chiral condensate strongly depends onthe action and lattice spacing that is applied, thisdependence is not so strong for the Polyakov loop as canbe seen in figure 5. Here again we compare our modelresults with lattice data. Because T directly influencesthe dynamics of the Polyakov loop we see a strongdependence of T P Lc on the change in T . We obtain T P Lc = 210 and 173 MeV. the general shape of the curveof φ ( T ) is not changed considerably with T and weobserve that we cannot accommodate the almost linearincrease of the Polyakov loop at lower temperatures, asseen on the lattice, by simply adjusting T .Recent lattice studies start to agree on the tempera-ture dependence of the order parameters. However thisis not the case when the interaction measures, closelyrelated to the thermodynamic properties of the matter,are compared. In figure 6 we compare our model resultson the interaction measure, for the two values of T , withdifferent lattice data sets. First we have to note thatthe results from the HotQCD collaboration (red squaresand green triangles) differ considerably from thoseof the Wuppertal-Budapest group (black diamonds).Furthermore we see that the interaction measure forour calculation with T = 220 MeV gives a drasticallybetter agreement with the HotQCD lattice results thanfor 270 MeV. in any case our model underestimatesthe contribution to the interaction measure at lowtemperatures. However, this could be understood by
100 200 300 4000.00.20.40.60.81.0100 200 300 4000.00.20.40.60.81.0
Lattice data: HISQ N =6 HISQ N =8 asqtad N =12 stout cont.
T [MeV] T =270 MeV T =210 MeV a =8.47 FIG. 10: (Color online) Value of the Polyakov loop as a func-tion of temperature at µ B = 0. This time the solid line isthe model result for T = 210 MeV and a modifies parameter a = − .
47. The dashed line stands for the previous resultwith T = 270 MeV and a = − .
47. The symbols denote lat-tice data, where the open symbols represent older data withthe asqtad and p4 action and the colored symbols are morerecent results. missing contributions from hadronic resonances as isseen from the grey line which depicts ( e − p ) /T fora hadron resonance gas which has already been shownto give good results for the interaction measure attemperatures below T χc .The transition from a hadron to quark dominated systemis depicted in figure 7 with the different particle numberdensities at µ B = 0. While the mesons dominate the lowtemperature region (green lines) one can clearly see thatthe hadrons are slowly removed from the system as thequark density increases (black lines) and consequentlythe hadrons play less of a role for the thermodynamicquantities. A. Finite µ B An advantage of our effective model is that, unlikelattice studies, we can easily extend our studies to fi-nite baryon densities and explore the phase behavior at µ B >
0. Figure 8 shows again the number densities ofdifferent hadrons and quarks, this time at T = 0, andas a function of µ B . at the liquid-gas phase transitionthe nucleon density exhibits a jump as expected. Ateven higher chemical potentials we can observe a sec-ond jump in the densities. at this point the chiral part-ner of the nucleon is activated, as well as the quark de-grees of freedom. Both steps in the densities correspondto jumps in the chiral condensate as can be seen moreclearly in figure 9. In this figure we depict the phasediagrams obtained from our model for the two values
100 200 300 40002468100 200 300 40002468 T =270 MeV T =210 MeV, a =-8.47 HRG ( e - ) / T T [MeV]
FIG. 11: (Color online) The interaction measure (defined as( e − p ) /T ) as a function of temperature at µ B = 0. Thistime the solid line is the model result for T = 210 MeV anda modifies parameter a = − .
47. The dashed line stands forthe previous result with T = 270 MeV and a = − .
47. Thesymbols denote lattice data, where the open symbols repre-sent older data with the asqtad and p4 action and the coloredsymbols are more recent results. The grey line represents theinteraction measure for a hadron resonance gas equation ofstate which includes all hadronic resonances up to 2.2 GeV. of T . The black lines with endpoints depict the regionwhere the change of the chiral condensate is of first or-der. Instead of drawing an ambiguous crossover ’line’ weshow the regions in which the value of the chiral con-densate changes from 0 . < σ/σ < . . < φ < .
8, which we willrefer to as the deconfinement crossover. Please note thatwithin our model the Polyakov loop generally changes ina smooth crossover and only exhibits a first order phasetransition at the chiral phase transition, where it jumpsfrom φ = 0 to 0 .
1. The position of the chiral criticalendpoint is T cep = 58 MeV and µ cepB ≈ T = 270 MeV and for T = 220 MeV only the chemicalpotential changes to µ cepB ≈ B. Varying Polyakov Loop parameters
As had been mention in the previous section, our modeldoes not yield a good description of the temperature be-
T=56 MeV S = B /3f s =0.5f s =-0.2 S [ M e V ] B [MeV] f s =0 T=150 MeV S [ M e V ] f s =-0.2f s =0 S = B /3f s =0.5 B [MeV] / FIG. 12: (Color online) Contour plots of the normalized chiral condensates as a function of the chemical potentials µ B and µ S for fixed temperature. The red lines correspond to different values of a fixed strangeness to baryon fraction f s . havior of the Polyakov loop together with a good descrip-tion of the interaction measure. Within the PNJL modelit has also been pointed out that a concurrent descriptionof both order parameters and the interaction measure isusually not achieved when simply T is varied (see e.g.[86]).In our model in particular the linear increase of thePolyakov loop is ill described. Usually the parametersof the Polyakov potential are fitted to pure gauge latticeresult. However it has been shown that the presence ofquarks may have an influence on these parameters [42]and in general there is no reason that only T should beaffected by such a quark coupling. To investigate andillustrate what effects a different parameter set of thePolyakov potential has on the interaction measure wewill adjust one parameter ( a ) to a = − .
47 (insteadof a = − . T to T = 210MeV this gives an improved description of the latticedata for the Polyakov loop behavior as can bee seen infigure 10.Figure 11 shows the resulting interaction measure for ouradjusted parameter. One can see that, due to the slowerrise of the Polyakov loop, the peak in the interaction mea-sure is considerably lowered. Taking into account miss-ing contributions from resonances one can even concludethat this parametrization compares more favorably withthe Wuppertal-Budapest results than with the HotQCDlattice data.Consequently, to understand the interplay between theorder parameters and the interaction measure from lat-tice calculations, one has to resolve the still existing dis-crepancies. Only then conclusions regarding the role ofthe degrees of freedom in the chiral and deconfinementphase transition can be drawn. V. THE STRANGE PHASE DIAGRAM
Until now all calculations where restricted to the limitof vanishing net strange density. Because the strong in-teraction conserves the net strange particle number, theequation of state used for the description of heavy ioncollisions is usually considered to be net strange free.However, there are several issues that make the studyof the strange EoS interesting. Some of these issues are:1. As has been shown in [87] the net strangeness dis-tribution in coordinate and momentum space of aheavy ion collision can fluctuate, although the to-tal net strangeness is zero. To dynamically treatsuch a system, and calculate observables that arisefrom such a strangeness fluctuation, the equationof state for ρ s = 0 needs to be evaluated.2. Compact stars are very dense and long lived ob-jects. Due to a β -equilibrium inside the star, net-strange conservation can be violated by the weakinteraction.3. Lattice QCD results at finite µ B are often evaluatedthrough a Taylor expansion in µ B at µ B = µ S =0. A vanishing strange number chemical potentialusually induces a non-vanishing net strangeness,which means that the equation of state of net-strange matter is calculated.First investigations on the strange equation of statewere done in [88], where one usually considered a firstorder transition from a hadron to a quark phase. Inour model we are able to discuss the strange EoS in thecontext of a smooth transition from a confined hadronphase to a deconfined quark phase.Figure 12 presents our results on the order parameterof the chiral phase transition as a function of µ B and0
150 200 250 300 350 4000.000.050.100.150.200.25150 200 250 300 350 4000.000.050.100.150.200.25
Exact c and c only c / T T [MeV] B /T=1, S =0 FIG. 13: (Color online) Baryon number density divided by T as a function of temperature for µ B /T = 1. Shown is theexact solution from the model (black solid line) and Taylorexpansions of the density, taking into account only the second(green dashed line) and forth (red solid line) coefficient. µ S at fixed temperature. The red lines indicate pathsof constant values for f s = ρ s /ρ B , the strangeness perbaryon fraction. note that f s = 0 corresponds to ourresults in section IV (with T = 220 MeV). At thetemperature T = 56 MeV, the critical endpoint of thechiral phase transition was located at µ cepB ≈ f s , the change in theorder parameter becomes steeper and the value of T CEP increases slightly to T CEP = 68 MeV for f s = 0 .
5. Atthe larger temperature we also observe a slight changein the phase structure. Here, for increasing f s , thecrossover moves closer to the µ B,S = 0-line.For a gas of deconfined quarks there is a strong corre-lation between the baryon number and strangeness. In ahadronic medium such a correlation is usually not triv-ial as strangeness can be found in mesons and baryons.These considerations led to the idea that the so calledstrangeness-baryon correlation factor c BS is sensitive tothe deconfinement and/or chiral phase transition [89].On the other hand the strangeness to baryon ratio f s should also be sensitive on any phase transition at finitebaryon densities.On the lattice such quantities are usually calculated asfunctions of the expansion coefficients, it is defined as[89]: c BS = − h N B N S i − h N B i h N S ih N S i − h N S i (15)The question is how many coefficients are needed toevaluate the baryon and/or strange densities at finite µ B /T to a given accuracy [90]. As an example figure 14shows the baryon density as a function of temperatureat fixed µ B /T = 1 and µ S = 0 for our exact modelcalculation. Alternatively we can also numerically
100 200 300 400 50001234100 200 300 400 50001234 B /T=3, S =0: c BSS /T=1, B =0: f S /3 T [MeV]
FIG. 14: (Color online) Shown are the strangeness to baryoncorrelation coefficient c BS (red short dashed line) comparedto the quark-gluon fraction λ = e Quarks + Gluons /e Tot (blacksolid line) as a function of temperature for µ B /T = 3 and µ S = 0. The plot also shows the strangeness per baryonfraction f s (green dashed line) and the quark-gluon fraction λ (grey solid line) as a function of temperature for µ S /T = 1and µ B = 0. extract the expansion coefficients for our model andexpand the density in powers of µ/T . One can see thatalready the result, taking into account only the firstnon vanishing coefficient c , gives a quite reasonabledescription of the exact result. Taking into account the2nd and 4th order coefficient already allows to describethe exact result to high accuracy, except at the point ofthe crossover transition. This means that, in order tocalculate f s at finite µ S /T , it is sufficient to extract thecoefficients up to 4th order ( c B,S , ) from the lattice.The information that can be extracted from thesequantities is exemplified in figure 14. Here we show theexact solution for c BS as a function of temperature for µ B /T = 3 and µ S = 0. One can observe a distinctpeak at T ≈
150 MeV ⇒ µ B = 450 MeV. Comparingwith figures 9 and 12 one can identify this peak withthe crossover transition of the chiral condensate. Sucha behavior of c BS has been predicted and also has beenshown to exist in lattice data [91]. At higher temper-atures the strangeness to baryon correlation approachesunity which resembles closely the behavior of the quarkand gluon fraction λ = e Quarks + Gluons /e T ot of the sys-tem. In comparison figure 14 also shows the tempera-ture dependence of f s at µ S /T = 1 and µ B = 0. Thisquantity is even more sensitive in the quark-gluon frac-tion as is c BS , while it seems to be not very sensitive tothe chiral phase transition. The peak in f s can ratherbe understood as a consequence of our excluded volumetreatment, where mesons have a smaller excluded volumethan baryons. Hence mesons, that can carry strangeness,are less suppressed than baryons.1 VI. DISCUSSION
We presented results on the phase structure of a SU(3)parity-doublet description of hot and dense hadronicmatter. With appropriate parameters we could generatea quantitatively acceptable nuclear ground state. Thephase diagram in temperature and baryochemical poten-tial exhibits a liquid-gas first-order phase transition aswell as a chiral phase transition that is connected to thepopulation of the parity partners and the onset of theirdegeneracy with the normal baryon states. Dependingon the mass gap between the baryons and their paritypartners this transition is first-order at high densitiesand low temperatures and a crossover otherwise, or asmooth crossover for all values of
T, µ for smaller massgaps. In order not to be overwhelmed by too many newparameters some simplifications of the parameter choicehave been made, assuming an equal mass gap between allpositive and negative parity baryons. These restrictionsshould be relaxed in further studies to explore the modelin more detail.In the second part of this paper we extended the SU(3)parity doublet model to incorporate a deconfinementphase transition. When comparing our results to latticedata at µ B = 0 we find that the low temperaturebehavior of the chiral condensate is dominated byhadronic interactions. In such a scenario a decoupling ofthe Polyakov loop and the chiral condensate, as is seenin recent lattice studies, can be easily understood. Afeature which is common to PNJL-type models is thata simultaneous description of the interaction measureand the Polyakov loop cannot be achieved simply byadjusting the parameter T . If we loosen also theconstraint on other parameters of the Polyakov looppotential we obtain an improved description of thePolyakov loop dynamics. It would be interesting toinvestigate if, e.g. in PNJL models, the slower increasealso shows to have drastic effects on the interactionmeasure as has the parameter change presented in thiswork. As lattice results still differ strongly in theirresults on the interaction measure it is not possible tosay if such a reparametrization improves or weakensthe model. Consequently it is of utmost importanceto understand and settle the differences in the latticeresults to be able to understand the interplay betweenthe order parameters and the thermodynamics, i.e. theactive degrees of freedom.At finite baryon densities our model describes thedeconfinement transition as a continuous crossoverfor all values of µ B . Only the chiral order parameter exhibits two discontinuities. One is related to thenuclear liquid gas phase transition while the other canbe identified as the chiral phase transition and appearsat larger densities. We also observe that the criticalendpoint of the chiral phase transition has a rathersmall temperature T cep = 56 MeV. As the chiral phasetransition is driven mainly by hadronic interactionsand the deconfinement by quarks and the Polyakovpotential, we see a decoupling of the order parameters,which becomes stronger for large chemical potentials.We observe several different states of matter that canform, starting from a nucleon liquid which changes toa phase of chirally symmetric hadrons. Only at highertemperature these hadrons disappear and the quarksare the dominant degrees of freedom. Whether such achirally symmetric hadronic phase can be the N c = 3equivalent of the N c = ∞ quarkyonic phase [92] is stillunder extensive debate [93–95]. In any case the highdensity part of the QCD phase diagram could have arather rich phase structure to explore.In the last part of this paper we discuss the propertiesof our models phase diagram at finite net-strange den-sity. This aspect of QCD matter is not only interestingfor heavy ion collisions and compact stars, but also fora comparison with lattice results extrapolated to finite µ B,S . We find that the location of the critical endpointshifts to a slightly higher temperature for a finite netstrangeness (lattice results).We briefly discussed quantities that are sensitive on thechiral and/or deconfinement phase transition. In partic-ular these are the strangeness baryon correlation factor c BS and the strangeness per baryon fraction f s . Bothshow to be sensitive to the deconfined fraction on thesystem while c BS also shows a distinct peak at the chi-ral crossover at finite chemical potential. The advantageof extracting f s from the lattice is that it can be evalu-ated e.g. in a Taylor expansion, using only the first twonon-zero expansion coefficients for the strange and lightquark number susceptibilities. Acknowledgments
This work was supported by BMBF, HGS-hire and theHessian LOEWE initiative through the Helmholtz Inter-national center for FAIR (HIC for FAIR). The authorsthank P. Petreczky for fruitful discussions. The computa-tional resources were provided by the LOEWE FrankfurtCenter for Scientific Computing (LOEWE-CSC). [1] S. Borsanyi et al. , JHEP , 077 (2010)[2] A. Bazavov and P. Petreczky [HotQCD collaboration], J.Phys. Conf. Ser. , 012014 (2010) [3] Z. Fodor and S. D. Katz, JHEP , 014 (2002)[4] Z. Fodor and S. D. Katz, JHEP , 050 (2004)[5] P. de Forcrand and O. Philipsen, JHEP , 012 (2008) [6] G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo, JHEP , 001 (2011)[7] P. Huovinen and P. Petreczky, Nucl. Phys. A , 26(2010)[8] P. Huovinen and P. Petreczky, arXiv:1106.6227 [nucl-th].[9] Y. Aoki, S. Borsanyi, S. Durr, Z. Fodor, S. D. Katz,S. Krieg and K. K. Szabo, JHEP , 088 (2009)[10] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg,C. Ratti and K. K. Szabo [Wuppertal-Budapest Collab-oration], JHEP , 073 (2010)[11] J. Boguta, Phys. Lett. B , 34 (1983).[12] N. K. Glendenning, Annals Phys. , 246 (1986).[13] I. Mishustin, J. Bondorf and M. Rho, Nucl. Phys. A ,215 (1993).[14] E. K. Heide, S. Rudaz and P. J. Ellis, Nucl. Phys. A ,713 (1994)[15] P. Papazoglou, S. Schramm, J. Schaffner-Bielich,H. Stoecker and W. Greiner, Phys. Rev. C , 2576(1998)[16] P. Papazoglou, D. Zschiesche, S. Schramm, J. Schaffner-Bielich, H. Stoecker and W. Greiner, Phys. Rev. C ,411 (1999)[17] K. Tsubakihara, H. Maekawa, H. Matsumiya andA. Ohnishi, Phys. Rev. C , 065206 (2010)[18] C. E. Detar and T. Kunihiro, Phys. Rev. D , 2805(1989).[19] D. Zschiesche, L. Tolos, J. Schaffner-Bielich andR. D. Pisarski, Phys. Rev. C , 055202 (2007)[20] V. Dexheimer, S. Schramm and D. Zschiesche, Phys. Rev.C , 025803 (2008)[21] V. Dexheimer, G. Pagliara, L. Tolos, J. Schaffner-Bielichand S. Schramm, Eur. Phys. J. A , 105 (2008)[22] S. Gallas, F. Giacosa and D. H. Rischke, Phys. Rev. D , 014004 (2010)[23] C. Sasaki and I. Mishustin, Phys. Rev. C , 035204(2010)[24] Y. Nemoto, D. Jido, M. Oka and A. Hosaka, Phys. Rev.D , 4124 (1998)[25] J. R. Ellis, J. I. Kapusta and K. A. Olive, Phys. Lett. B , 123 (1991).[26] P. Gerber and H. Leutwyler, Nucl. Phys. B , 387(1989).[27] A. Mishra, S. Schramm and W. Greiner, Phys. Rev. C , 024901 (2008)[28] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz and K. K. Sz-abo, Nature , 675 (2006)[29] A. Bazavov and P. Petreczky [HotQCD Collaboration],arXiv:1009.4914 [hep-lat].[30] J. Steinheimer, S. Schramm and H. Stocker, J. Phys. G , 035001 (2011)[31] K. Fukushima, Phys. Lett. B , 277 (2004)[32] C. Ratti, M. A. Thaler and W. Weise, Phys. Rev. D ,014019 (2006)[33] K. Fukushima and Y. Hidaka, Phys. Rev. D , 036002(2007)[34] C. R. Allton et al. , Phys. Rev. D , 074507 (2002)[35] A. Dumitru, R. D. Pisarski and D. Zschiesche, Phys. Rev.D , 065008 (2005)[36] C. Ratti and W. Weise, Phys. Rev. D , 054013 (2004)[37] S. Roessner, C. Ratti and W. Weise, Phys. Rev. D ,034007 (2007)[38] C. Sasaki, B. Friman and K. Redlich, Phys. Rev. D ,074013 (2007)[39] C. Ratti, S. Roessner and W. Weise, Phys. Lett. B , 57 (2007)[40] S. Roessner, T. Hell, C. Ratti and W. Weise, Nucl. Phys.A , 118 (2008)[41] M. Ciminale, R. Gatto, N. D. Ippolito, G. Nardulli andM. Ruggieri, Phys. Rev. D , 054023 (2008)[42] B. J. Schaefer, J. M. Pawlowski and J. Wambach, Phys.Rev. D , 074023 (2007)[43] W. j. Fu, Z. Zhang and Y. x. Liu, Phys. Rev. D ,014006 (2008)[44] T. Hell, S. Roessner, M. Cristoforetti and W. Weise,Phys. Rev. D , 014022 (2009)[45] H. Abuki, R. Anglani, R. Gatto, G. Nardulli and M. Rug-gieri, Phys. Rev. D , 034034 (2008)[46] K. Fukushima, Phys. Rev. D , 114028 (2008)[47] K. Fukushima, Phys. Rev. D , 114019 (2008)[48] P. Costa, C. A. de Sousa, M. C. Ruivo and H. Hansen,Europhys. Lett. , 31001 (2009)[49] P. Costa, M. C. Ruivo, C. A. de Sousa, H. Hansen andW. M. Alberico, Phys. Rev. D , 116003 (2009)[50] H. Hansen, W. M. Alberico, A. Beraudo, A. Molinari,M. Nardi and C. Ratti, Phys. Rev. D , 065004 (2007)[51] S. Mukherjee, M. G. Mustafa and R. Ray, Phys. Rev. D , 094015 (2007)[52] H. Abuki, M. Ciminale, R. Gatto, N. D. Ippolito, G. Nar-dulli and M. Ruggieri, Phys. Rev. D , 014002 (2008)[53] H. Abuki, M. Ciminale, R. Gatto, G. Nardulli andM. Ruggieri, Phys. Rev. D , 074018 (2008)[54] K. Fukushima, Phys. Rev. D , 074015 (2009)[55] H. Mao, J. Jin and M. Huang, J. Phys. G , 035001(2010)[56] B. J. Schaefer, M. Wagner and J. Wambach, Phys. Rev.D , 074013 (2010)[57] T. Hell, S. Rossner, M. Cristoforetti and W. Weise, Phys.Rev. D , 074034 (2010)[58] G. A. Contrera, D. G. Dumm and N. N. Scoccola, Phys.Rev. D , 054005 (2010)[59] A. E. Radzhabov, D. Blaschke, M. Buballa andM. K. Volkov, Phys. Rev. D , 116004 (2011)[60] G. A. Contrera, M. Orsaria and N. N. Scoccola, Phys.Rev. D , 054026 (2010)[61] T. K. Herbst, J. M. Pawlowski and B. J. Schaefer, Phys.Lett. B , 58 (2011)[62] V. Pagura, D. G. Dumm and N. N. Scoccola,arXiv:1105.1739 [hep-ph].[63] K. Kashiwa, T. Hell and W. Weise, arXiv:1106.5025 [hep-ph].[64] W. Weise, Prog. Theor. Phys. Suppl. , 390 (2010)[65] D. Blaschke, M. Buballa, A. E. Radzhabov andM. K. Volkov, Phys. Part. Nucl. , 921 (2010).[66] T. Kunihiro, Phys. Lett. B , 395 (1991).[67] L. Ferroni and V. Koch, Phys. Rev. C , 045205 (2011)[68] J. Steinheimer and S. Schramm, Phys. Lett. B , 257(2011)[69] R. Hagedorn and J. Rafelski, Phys. Lett. B , 136(1980).[70] J. Baacke, Acta Phys. Polon. B , 625 (1977).[71] M. I. Gorenstein, V. K. Petrov and G. M. Zinovev, Phys.Lett. B , 327 (1981).[72] R. Hagedorn, Z. Phys. C , 265 (1983).[73] D. H. Rischke, M. I. Gorenstein, H. Stoecker andW. Greiner, Z. Phys. C , 485 (1991).[74] J. Cleymans, M. I. Gorenstein, J. Stalnacke and E. Suho-nen, Phys. Scripta , 277 (1993).[75] J. I. Kapusta and K. A. Olive, Nucl. Phys. A , 478 (1983).[76] K. A. Bugaev, M. I. Gorenstein, H. Stoecker andW. Greiner, Phys. Lett. B , 121 (2000)[77] K. A. Bugaev, Nucl. Phys. A , 251 (2008).[78] L. M. Satarov, M. N. Dmitriev and I. N. Mishustin, Phys.Atom. Nucl. , 1390 (2009)[79] Y. Sakai, T. Sasaki, H. Kouno and M. Yahiro,arXiv:1104.2394 [hep-ph].[80] M. Cheng et al. , Phys. Rev. D , 054504 (2010)[81] A. Bazavov et al. , Phys. Rev. D , 014504 (2009)[82] A. Bazavov and P. Petreczky, PoS LATTICE2010 , 169(2010)[83] P. Petreczky, Nucl. Phys. A , 11C (2009)[84] O. Kaczmarek, F. Karsch, P. Petreczky and F. Zantow,Phys. Lett. B , 41 (2002)[85] K. Fukushima, arXiv:1008.4322 [hep-ph].[86] T. Hell, K. Kashiwa and W. Weise, Phys. Rev. D , 114008 (2011)[87] J. Steinheimer, M. Mitrovski, T. Schuster, H. Petersen,M. Bleicher and H. Stoecker, Phys. Lett. B , 126(2009)[88] K. S. Lee and U. W. Heinz, Phys. Rev. D , 2068 (1993).[89] V. Koch, A. Majumder and J. Randrup, Phys. Rev. Lett. , 182301 (2005)[90] F. Karsch, B. J. Schaefer, M. Wagner and J. Wambach,Phys. Lett. B , 256 (2011)[91] C. Schmidt, PoS C POD2009 , 024 (2009)[92] L. McLerran and R. D. Pisarski, Nucl. Phys. A , 83(2007)[93] S. Lottini and G. Torrieri, arXiv:1103.4824 [nucl-th].[94] L. Bonanno and F. Giacosa, Nucl. Phys. A859