nuclear matter properties XLVIII International Winter Meeting Bormio 2010 Phase structure of strongly interacting matter and results for heavy-ion collisions and neutron stars OUTLINE hadronic SU(3) model nuclear matter properties generating a critical endpoint in a hadronic model molecular / hydro simulation including quark degrees of freedom phase diagram excluded volume corrections J. Steinheimer, V. Dexheimer, H. Stöcker, SWS Goethe University, Frankfurt

hadronic model based on non-linear realization of chiral symmetry degrees of freedom SU(3) multiplets: baryons (n,Λ, Σ, Ξ) scalars (, , 0) vectors (ω, ρ, φ) , pseudoscalars, glueball field χ A) SU(3) interaction ~ Tr [ B, M ] B , ( Tr B B ) Tr M B) meson interactions ~ V(M) <> = 0  0 <> =  0  0 C) chiral symmetry  m = mK = 0 explicit breaking ~ Tr [ c  ] ( mq q q )  light pseudoscalars, breaking of SU(3) _ _ _ _ _ _ _  ~ <u u + d d> <  ~ <s s> 0 ~ < u u - d d> _

fit parameters to hadron masses  mesons *  *        K* ’ p,n    Model can reproduce hadron spectra via dynamical mass generation  K 

Lagrangian (in mean-field approximation) L = LBS + LBV + LV + LS + LSB baryon-scalars: _ LBS = -  Bi (gi  + gi  + gi  ) Bi baryon-vectors: _ LBV = -  Bi (gi  + gi  + gi  ) Bi / / / meson interactions: LBS = k1 (2 + 2 + 2 )2 + k2/2 (4 + 2 4 + 4 + 6 2 2 ) + k3  2  - k4 4 - 4 ln /0 +  4 ln [(2 - 2) / (020)] I LV = g4 (4 + 4 + 4 + β 22) I explicit symmetry breaking: LSB = c1  + c2 

important reality check nuclear matter properties at saturation density asymmetry energy E/A (p- n) equation of state E/A () binding energy E/A ~ -15.2 MeV saturation (B)0 ~ .16/fm3 compressibility ~ 223 MeV asymmetry energy ~ 31.9 MeV phenomenology: 200 - 250 MeV 30 - 35 MeV + good description of finite nuclei / hypernuclei SWS, Phys. Rev. C66, 064310

Task: self-consistent relativistic mean-field calculation coupled 7 meson/photon fields + equations for nucleons in 1 to 3 dimensions parameter fit to known nuclear binding energies and hadron masses 2d calculation of all measured (~ 800) even-even nuclei error in energy  (A  50) ~ 0.21 % (NL3: 0.25 %)  (A  100) ~ 0.14 % (NL3: 0.16 %) relativistic nuclear structure models good charge radii rch ~ 0.5 % (+ LS splittings) SWS, Phys. Rev. C66, 064310 (2002)

phase transition compared to lattice simulations heavy states/resonance spectrum is effectively described by single (degenerate) resonance with adjustable couplings Tc ~ 180 MeV µc ~ 110 MeV reproduction of LQCD phase diagram, especially Tc, μc + successful description of nuclear matter saturation phase transition becomes first-order for degenerate baryon octet ~ Nf = 3 with Tc ~ 185 MeV D. Zschiesche et al. JPhysG 34, 1665 (2007)

lines of constant entropy per baryon, i.e. perfect fluid expansion Isentropes, UrQMD and hydro evolution lines of constant entropy per baryon, i.e. perfect fluid expansion E/A = 5, 10, 40, 100, 160 GeV E/A = 160 GeV goes through endpoint J. Steinheimer et al. PRC77, 034901 (2008)

connect hadronic and quark degrees of freedom order parameter of the phase transition confined phase deconfined phase effective potential for Polyakov loop, fit to lattice data U = ½ a(T) ΦΦ* + b(T) ln[1 – 6 ΦΦ* + 4 (ΦΦ*)3 – 3 (ΦΦ*)2] a(T) = a0T4 + a1 µ4 + a2 µ2T2 baryonic and quark mass shift δ mB ~ f(Φ) δ mq ~ f(1-Φ) q quarks couple to mean fields via gσ, gω q V. Dexheimer, SWS, astro-ph0901.1748 Ratti et al. PRD 73 014019 (2006) Fukushima, PLB 591, 277 (2004) minimize grand canonical potential

Phase Diagram for HQM model µc = 360 MeV Tc = 166 MeV µc. = 1370 MeV ρc ~ 4 ρo hybrid hadron-quark model critical endpoint tuned to lattice results V. Dexheimer, S. Schramm, astro-ph0901.1748

Mass-radius relation using Maxwell/Gibbs construction Gibbs construction allows for quarks in the neutron star mixed phase in the inner 2 km core of the star V. Dexheimer, SWS, astro-ph0901.1748

isentropic expansion overlap initial conditions Elab = 5, 10, 40, 100, 160 AGeV Csph ~ ¼ Cs,ideal averaged Cs significantly higher than 0.2

Include modified distribution functions for quarks/antiquarks different approach – hadrons, quarks, Polyakov loop and excluded volume Include modified distribution functions for quarks/antiquarks Following the parametrization used in PNJL calculations U = - ½ a(T) ΦΦ* + b(T) ln[1 – 6 ΦΦ* + 4 (ΦΦ*)3 – 3 (ΦΦ*)2] a(T) = a0T4 + a1 T0T3 + a2 T02T2 , b(T) = b3 T03 T The switch between the degrees of freedom is triggered by excluded volume corrections thermodynamically consistent - Vq = 0 Vh = v Vm = v / 8 ~ ~ ~ µi = µ i – vi P e = e / (1+ Σ vi ρi ) Steinheimer,SWS,Stöcker hep-ph/0909.4421 D. H. Rischke et al., Z. Phys. C 51, 485 (1991) J. Cleymans et al., Phys. Scripta 84, 277 (1993)

Temperature dependence of chiral condensate and Polaykov loop at µ = 0 Interaction measure e – 3p lattice data taken from Bazavov et al. hep-lat 0903.4379 speed of sound shows a pronounced dip around Tc !

Lattice comparison of expansion coefficients as function of T lattice results lattice data from Cheng et al., PRD 79, 074505 (2009) Steinheimer,SWS,Stöcker hep-ph/0909.4421 suppression factor peaks

Φ Dependence of Polyakov loop and chiral condensate on µ, T Lines mark maximum in T derivative µ σ Φ

quark, meson, baryon densities at µ = 0 densities of baryon, mesons and quarks natural mixed phase, quarks dominate beyond 1.5 Tc ρ Energy density and pressure compared to lattice simulations

hybrid simulation of a Pb-Pb collision at 40 GeV/A red regions show the areas dominated by quarks

SUMMARY / OUTLOOK general hadronic model as starting point works well with basic vacuum properties, nuclear matter, nuclei, … phase diagram with critical end point via resonances implement EOS in combined molecular dynamics/ hydro simulations quarks included using effective deconfinement field realistic phase transition line implementing excluded volume term, natural switch of d.o.f. strangeness distillation / modify model towards metastable hyperon matter explore µS dependence resonances – 1st order transition?

Temperature distribution from UrQMD simulation as initial state for (3d+1) hydro calculation Use smeared-out particle distributions as starting point for hydro simulation

simple time evolution including π, K evaporation (E/A = 40 GeV) with evaporation C. Greiner et al., PRD38, 2797 (1988) E/A-mN 300 g2 = 0 If you want it exotic … g2 = 2 200 additional coupling g2 of hyperons to strange scalar field 100 g2 = 4 follow star calcs by J. Schaffner et al., PRL89, 171101 (2002) g2 = 6 0 0.5 1 1.5 barrier at fs ~ 0.4 – 0.6 fs

Hypernuclei -  single-particle energies Nuclear matter Model and experiment agree well

Elab≈ 5-10 AGeV sufficient to overshoot Evolution of the collision system Elab≈ 5-10 AGeV sufficient to overshoot phase border, 100-160 AGeV around endpoint

amount of volume scanning the critical endpoint (lattice)

Comparison of gross properties of initial conditions Overlap of projectile and target

time integrated volume around the critical end point

effective volume sampling the critical end point window T = Tc ± 10 MeV µ = µc ± 10 MeV maximum shifts in time