PoS(CPOD 2013)059 http://pos.sissa.it/ 4 , 2 , 1 sed in a wide range of ce. finement phase transitions dron resonance multiplicities in alogy to PNJL models for higher s of freedom and takes into account deconfinement phase transition. At ss resonance states in this region and ing matter and the QCD equation of , Horst Stöcker ermal model results. ckstr. 1, 64291 Darmstadt, 1 , USA confinement ng phase diagram of QCD matter at small fang-Str. 1, 60438 Frankfurt am x-von-Laue Str. 1, 60438 Frankfurt a hadron resonance gas is described using a SU(3)- B ρ , Stefan Schramm ive Commons Attribution-NonCommercial-ShareAlike Licen 3 and baryonic densities T , Jan Steinheimer 2 , . In this way, the correct asymptotic degrees of freedom are u . Here, results of this model concerning the chiral and decon 1 B B ∗ ρ ρ [email protected] and and chemical potentials is in line with latest lattice QCD and th their impact on the chiral transition behavior. The resulti and thermodynamic model propertiesthe are transition presented. region emphasize the Large importance ha of heavy-ma T Speaker. flavor sigma-omega model and a quark phaseT is introduced in an low temperatures state (EoS). The model includes both hadronthe and quark transition degree of chiral symmetry restoration as well as the This work presents an effective model for strongly interact ∗ Copyright owned by the author(s) under the terms of the Creat GSI Helmholtzzentrum für Schwerionenforschung GmbH, Plan Frankfurt Institute for Advanced Studies (FIAS), Ruth-Mou Institut für Theoretische Physik, Goethe - Universität, Ma Lawrence Berkeley National Laboratory, Berkeley, CA 94720 c March 11-15 Napa, California, USA 8th International Workshop on Critical Point and Onset of De Philip Rau QCD Equation of State From aModel Chiral Including Hadronic Quark Degrees of Freedom

4 Germany E-mail: 1 Main, Germany 2 am Main, Germany 3 PoS(CPOD 2013)059 v r − = , s ω int r ω i L g 0, lattice Philip Rau − i . To obtain > or chemical µ φ , B B ω µ , including the . Furthermore, = ρ N Φ ∗ , and all known i g Pol s · µ U , v r d − ) meson fields , plane. = 6 GeV). The effective φ round-state properties, V u . , / φ 2 µ , th ω – ω fective chiral flavor SU(3) ≤ B Ω T g which includes the particles’ ) by the coupling to the scalar m i + ting matter is still scarce and re gas of quarks and gluons is . All thermodynamic quantities decreases, the effective masses m scalar meson self interactions as 160 MeV. For all nd solutions in this region. To and mes δ mes 1. The resonance vector coupling σ ≈ 0, lattice QCD predicts a smooth strain free quarks from the ground ESB L s ζ ting phase diagram. For reasonable and baryon density d in a PNJL-like approach. , properties. All quark couplings are , approach for the quark phase. The ≈ -model for the hadronic phase using L he baryon octet are fixed such as to T σ a clear picture whether the transition B s with sion methods can be used. However, = + s −L g and lively scientific debate. T hadrons and point-like quarks. Due to ω N s, nuclear matter is known to exhibit r ρ let) are scaled by the coefficients + - B g The model includes all known hadronic ) and vector ( int int 0 s. Therefore, the phase structure of QCD · σ µ ζ L s L ry at , r and −L + + σ = T = . At kin ζ vec , T L V L σ / 2 B = g = Ω runs over the quark flavors L i defining the quark dynamics in the model [2]. The shift mes L and their effective chemical potentials Pol i U m δ . Index i + ψ ζ ] ∗ i ζ i m g 0, the scalar coupling is chosen + )+ = σ of hadrons and quarks which couple to the Polyakov loop 0 σ µ φ i th g φ i Ω g = ∗ i + 0 m -flavor chiral effective model [1] unifies a ) ω attains increasing interest. In addition to the well known g 3 ω ( i 1, the chiral transition is a smooth cross-over in the whole B g ( µ ≈ SU 0 γ v [ r i ¯ are generated dynamically (except for small explicit masse ψ Detailed knowledge on the phase diagram of strongly interac The last term of the model Lagrangian includes the vector and This work presents results on the QCD phase diagram from an ef The i φ has a large impact on the transition behavior and on the resul includes the Polyakov loop potential ∑ φ i v kinetic energy, the interaction of baryons with scalar ( r the study of QCD matter, in particular at finite temperature QCD EoS From a Chiral Hadronic Model Including Quarks 1. Introduction particle masses a non-linear realization ofmodel chiral is symmetry based and on a a PNJL-like mean field Lagrangian decline, and chiral symmetry isreproduce restored. well-known vacuum masses The and couplingschosen nuclear of according saturation t to the additivestate. quark model The and baryon resonance such couplings as (including to the re decup a smooth cross-over at QCD suffers from theobtain fermion information sign-problem on which QCD matter hindersuntil at to today finite neither fi potentials, from expan experimentpossibly nor changes from to theory first there order and is a critical end point exist model and contrasts them to recentdegrees results from of the freedom lattice. as well as a quark-gluon phase implemente 2. Chiral Effective Model g well as explicit chiral symmetry breaking matter under extreme conditions remains object of an ongoin values to the respective couplings of the nucleons via potential − cross-over transition for the restoration of chiral symmet baryons (octet, decuplet, and heavy-mass baryon resonance of degrees of freedom fromrealized a by hadron excluded volume resonance effects gas assuming (HRG) finite-volume to a pu and the vector fields respectively. With increasing are derived from the grand canonical potential from heavy-ion experiments withproperties highest of collision a energie nearly perfect fluid at very high thermal contribution Ω PoS(CPOD 2013)059 0. 350 = 0 as a = B µ 0 µ σ / at Philip Rau at least up σ 3 ) 300 T T ( σ 250 in the int. HRG+q sce- T [MeV] 164 MeV is determined. (II) σ within a small temperature flattens significantly. Com- = c ) σ ground state value 200 T T ( e transition, Fig. 2 (I) shows the ge extent. Due to quarks lacking uark phase with either no interac- σ lattice QCD (II). with quarks divided by arks are abundant above the critical I), the rapid increase of quark multi- ractions (int. HRG+q). While in the l. A pure interacting HRG without r (gray band) and newest continuum lattice data PNJL: PRD75, 034007 (2007) PNJL: PRD75, HRG+q int. HRG+q fective model (analogously to PNJL 3]. The illustration contrasts different and the number of quarks exceeds the llustrates the total quark density. ns define the slope of cline of 150 tly smaller quark scalar coupling causes c

mics consistency. utions from low- and intermediate-mass

T Φ emperature from PNJL models [2] and lattice QCD [3] than predicted by lattice QCD (compiled in ≥ Φ T Φ is shown in Fig. 1 (II). The chiral model results 1 0 0.8 0.6 0.4 0.2 Φ 3 350 = 0 B µ hadrons vanish from the system and the QGP-phase prevails. lattice data int. HRG HRG+q int. HRG+q HISQ, cont. stout, cont. µ 300 and T 175 MeV (Fig. 1 (II)), the slope of 250 = 0 stays at its ground state value, the decrease of T (I) σ 200 generates a distinctly faster increase of . The Polyakov loop order parameter c T T T [MeV]

150

0

σ σ / and, thus, the chiral transition is driven by hadrons to a lar Other model scenarios additionally include the PNJL-like q Fig. 1 (I) shows the chiral order parameter normalized by its Quantifying the impact of heavy-mass resonances at the phas c T to decrease much slower. However, as illustrated in Fig. 1 (I 0 1 0.2 0.8 0.6 0.4 to an eigenvolume, they are preferentially populated at number of hadrons (Fig. 2 (a)). In this region, the significan tions between meson fields and particlesnon-interacting (HRG+q) case or full inte σ The total densities (gray line) are broken down into contrib particles as well as heavy-mass resonances and the red line i QCD EoS From a Chiral Hadronic Model Including Quarks function of range. In line with most recent lattice results, a critical t Figure 1: Normalized chiral condensateextrapolated (I) lattice compared data to [3]. olde models The [2]) Polyakov shows a loop more from rapid the shift ef to deconfined quarks as in the excluded volume, at high nario is rather steep inPolyakov the temperature beginning. However, as soonparing as the qu full model results to pure HRG, apparently, hadro Introducing volume correction factors ensures thermodyna 3. Results the gray band). A fundamental discrepancy between plicities at baryon number density of baryons (a) and mesons (b) together in the transition region is apparent. are compared to lattice QCD withparameter different fermion sets actions and [ particlequarks compositions (int. of HRG) the exhibits chiral the steepest mode (and continuous) in PoS(CPOD 2013)059 at at ∼ 350 4 3 2 = 0 T T B ∗ mes =8 µ / m e =8 Philip Rau 300 asqtad, N p4, N stout, cont. HRG int. HRG int. HRG+q -plane. uark density is B . For both quan- s scaling µ 4 – 250 T T sing temperature, the . This dip in c nt the largest share of T T [MeV] (II) 200 from different model sce- 4 and slightly above underlines T nfluence, heavy-mass hadron greement up to slightly higher of the resonances decline due around c . Regarding the mesons (Fig. 2 T ∗ 4 T (b), and quarks divided by l as a perceivable dip in the total 150 (d) divided by ensity shares of different particle m T ition region. Notwithstanding this e / crease in the vector meson densities r hadron contribution in this region. e

ty over

ich become the most abundant mesons e/T

p/T olume effects suppress hadrons at high at all

4 4 eracting model including quarks yields in nuum extrapolated lattice QCD results (d) (c) al transition. The effect of resonances is 100 3 2 1 0 8 6 4 2 0 14 12 10 3.5 2.5 1.5 0.5 , pseudoscalar meson multiplicities are dominant . c 4 T T and rises sharply again thereafter. c T 350 = 0 µ , 3 (c) and the energy density total octet decuplet resonances quarks 300 , heavy-mass resonances are the most abundant baryons in the p int. HRG+q ρ ρ ρ ρ ρ pseudoscalar vector resonances quarks total c ρ ρ ρ ρ ρ T =2.7 fm ex V ≥ 250 T 0, this conclusion should also hold true in the whole = -resonances reach rather large multiplicities with increa 200 T [MeV] ∆ µ (I) , the fast rising quark number and the pseudoscalar meson mas c -field. At T MeV is caused by the hard-core repulsion of the hadrons. The q σ 150 ) , low-mass baryons from the octet, mainly nucleons, represe c 15 T ± 100

Baryons Mesons . Also the non-interacting HRG yields a qualitatively good a

210 MeV. This finding of large resonance multiplicities at

c

i

i ] [ ] [ ρ ) MeV 1/(fm /T

ρ

) MeV 1/(fm /T

3 3 3 155 3 3 3 T (a) (b) cause pseudoscalar meson multiplicities to decrease as wel 0. Illustrated are the total densities (gray lines) and the d ≈ Fig. 2 (II) shows the pressure Below σ T = = ( 0 0 , and should be even larger in the pure HRG. Due to their large i / 0.5 0.4 0.3 0.2 0.1 0.6 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.08 µ just starting to rise sharplyTherefore, and the energy can density not flattens compensate up the to lowe tities the interacting pure HRGup to perfectly reproduces conti temperatures. In the whole temperaturereasonable range, results the with fully only int minor deviations from lattice QCD EoS From a Chiral Hadronic Model Including Quarks resonances must be considered whenstudy only studying considers the phase trans meson density. Higher temperatures are accompaniedand by a distinctive an appearance in of heavier mesonat resonances wh Figure 2: Total baryon number density of baryons (a), mesons T in this region. At the substantial impact of heavy-mass resonances onsubstantial the chir even in the presence of quarks,T where excluded v species. Panels (c) and (d) depict pressure and energy densi system. Even though narios contrasted to lattice QCD as functions of combined density contribution of the decuplet stays lowest 1 (b)), due to the abundant light-mass pions below the total baryon density (Fig. 2 (a)). This changes as soon as to a decreasing PoS(CPOD 2013)059 , , et et D85 C57 , 1665 in GeV). , 034007 Philip Rau G34 NN s , 077 (2010); D75 , 088 (2009); t curves from l model fits of [STAR], Phys. √

, Phys. Rev.

1011 T [MeV] T , Phys. Rev. 0906 , J. Phys. et al. et al. et al. JHEP

et al. , JHEP 300

, Phys. Rev. 17.3 , arXiv:0909.4421; P. Rau al transition is a smooth cross- et al. et al. ch not only provides the correct et al. , 044921 (2012); A. Andronic et al. 250 r properties in heavy-ion collisions. C85 [MeV] SPS to LHC energies ( d) and from the chiral effective model s. B CD results. An extracted hadron-quark µ e chiral model (Fig. 3) yields a plausible rom a confined HRG to a deconfined quark- 200 , 142 (2009); B. I. Abelev , Phys. Rev. 5 , 289 (1983); P. Papazoglou 150 B673 et al. , 014504 (2011); A. Bazavov B120 , 014504 (2009); Y. Aoki , 034905 (2006); D. Zschiesche D83 , 054503 (2012). ): PRD83, 014504 (2011) , 213 (2007); S. Roessner 100 B , 054504 (2010); S. Borsanyi µ

(

D80 C73 c 62.4 D85 , 411 (1999); J. Steinheimer C49 D81 , Phys. Lett. , 535c (2013).

C59 50

130 lattice T Cleymans: PRC73, 034905 (2006) Cleymans: PRC73, 034905 (2009) Andronic: PLB673, 142 chiral model (2009) STAR: PRC79, 034909 (2012) NA49: PRC85, 044921 (2013) ALICE: NPA904, 535c

200 et al.

, Phys. Rev. 2760 , Phys. Rev. , Phys. Rev. , Phys. Rev. 0 . The estimates for the transition are compared to freeze-ou . In comparison to lattice QCD extrapolations [4] and therma A904-905 , Phys. Rev. et al. B B , Eur. Phys. J. µ µ et al. et al. et al. et al. 140 130 120 110 100 170 160 150 , 034909 (2009); F. Becattini et al. C79 , Nucl. Phys. , arXiv:1302.3836. al. Rev. (2007); A. Andronic 054503 (2012). M. Cheng al. (2007). 2576 (1998); Phys. Rev. A. Bazavov Due to the large number of hadron degrees of freedom, the chir In summary, this work presented a unified effective model whi [1] J. Boguta and H. Stöcker, Phys. Lett. [5] J. Cleymans [2] C. Ratti [3] A. Bazavov [4] O. Kaczmarek over even at large Figure 3: Chiral transition from lattice QCD [4] (yellow ban (red line) at small QCD EoS From a Chiral Hadronic Model Including Quarks statistical models and to data from thermal model fits [5] for experimental results [5], the phaseestimate transition located line at from the th upper boundary of lattice QCD result ground state properties but also describes the transitiongluon f phase reasonably well and inEoS line is with to latest be lattice used Q within dynamic models to study nuclear matte References