1. QCD thermodynamic on the lattice and the hadron resonance gas Péter Petreczky Physics Department and RIKEN-BNL Winter Workshop on Nuclear Dynamics, Ocho Rios, Jamaica, January 2-9, 2010 Thermodynamics at high temperaure : EoS, fluctuations BI-RBC, MILC, HotQCD : p4, asqtad, N τ =4, 6 and 8 Quark Gluon Gas Thermodynamics at low temperatures and the Hadron Resonance Gas (HRG) Parametrization of the EoS based on lattice+HRG and its effect on flow

2. Thermodynamics at high temperature good agreement between lattice and resummed perturbative (NLA) calculations of the entropy Rebhan, arXiv:hep-ph/0301130; Blaizot et al, PRL 83 (99) 2906 The quark number susceptibilities for T>300MeV agree with resummed petrurbative predictions A. Rebhan, arXiv:hep-ph/0301130 Blaizot et al, PLB 523 (01) 143 Deviations from ideal gas limit at T=800MeV is only 5-10% The cutoff effects (estimated from N τ =6 and N τ =8) are about 5% similar results for HISQ and stout actions, see talks by Bazavov and Fodor

3. Lattice results for physical quark masses Thermodynamics quantities are quark mass independent for T>200MeV The quark mass effect is small at low temperature and is similar to cutoff effects dominate Lattice results are significantly below the Hadron Resonance Gas Lattice calculations at the physical quark mass and N τ =8, Cheng et al, arXiv:0911.2215

4. Improved staggered calculations at finite temperature high-T region T>200MeV low T region T<200 MeV cutoff effects are different in : a>0.125fm a<0.125fm improvement of the flavor symmetry is important hadronic degrees of freedom quark degrees of freedom quark dispersion relation p4, asqtad, HISQ stout for #flavors < 4 rooting trick

5. Quark mass and lattice spacing dependence of hadron masses Hadron specturm has been calculated with improved staggered (asqtad) quarks for several values of quark masses and a=0.18, 0.15, 0.12, 0.09 and 0.06 fm For range of the lattice spacing used in T>0 calculations cutoff effects on the hadron mass could be as large as 15-20% Fit lattice results with: Huovinen, P.P. arXiv:0912.2541

6. Lattice results vs. hadron resonance gas model Include all resonances up to 2.5GeV Use ground state hadron masses modified according to know lattice corrections Modify the masses of baryon resonances up to threshold 1.8GeV and 2.5GeV in the same way as the ground state baryons Baryon number fluctuations Strangeness fluctuations discretization effects result in “effective shift” of T-scale Huovinen, P.P. arXiv:0912.2541

7. Interpolating between HRG and lattice results Use interpolation of lattice data above 200MeV and match it to HRG at lower temperature with constrain that a s=0.95s SB or s=0.90s SB at T=800MeV Huovinen, P.P. arXiv:0912.2541 fit the lattice data

8. EoS parametrization EoS is never softer than HRG EoS Large transition region : 170MeV < T < 220MeV,where the system is neither hadronic nor partonic Huovinen, P.P. arXiv:0912.2541

9. EoS and hydrodynamic flow Huovinen, P.P. arXiv:0912.2541 Momentum anisotropy: ε p is sensitive to EoS, though the difference between different lattice parameterization is small About half of the momentum anisotropy is produced in the partonic state, half in the transition region and only negligible fraction in hadronic stage (T < 170 MeV) The sensitivity of flow to EoS is studied in ideal hydrodynamics Au+Au, √s =200 GeV, b=7 fm

10. EoS and hydrodynamic flow adjusting the freezout temperature to reproduce the spectra gives significantly larger proton v 2 compared to the EoS with 1 st order transition (EoSQ) (see Huovinen, NPA761 (2005) 296 for similar results ) Huovinen, P.P. arXiv:0912.2541 p T –differential v 2 is not sensitive to EoS, but the spectra are Within ideal hydrodynamics it is not possible to describe both the proton spectrum and v 2, i.e. ideal hydrodynamics does not work !

11. Summary and outlook In the high T region cutoff effects are under control and thermodynamics can be understood in terms of quark gluon gas In the low temperature region (T<200MeV) there are potentially large cutoff effects which are responsible for significant discrepancy between HRG model and lattice results Taking into account the lattice spacing dependence of hadron masses it is possible to get agreement between the HRG and lattice QCD Interpolating between HRG at low T and lattice QCD at high T it is possible construct realistic equation of state that be used in hydrodynamic modeling. Significant effect on the proton elliptic flow was observed in ideal hydro compared to bag model type EoS => ideal hydrodynamic model does not work if realistic EoS are used !

12. Comparison of EoS

13. HPQCD, UKQCD, MILC and Fermilab, PRL 92 (04) 022001 Fermilab, HPQCD, MILC PRL 94 (05) 011601 (hep-ph/0408306 ) Exp.: Belle, hep-ex/0510003 Back-up:Results from improved staggered calculations at T=0 a=0.125fm, 0.09fm, 0.06fm, chiral and continuum extrapolations LQCD : Fermilab, HPQCD, UKQCD PRL 94 (05) 172001 [hep-lat/0411027] Exp: CDF, PRL 96 (06) 082002 [hep-exp/0505076] Bernard et al (MILC), PoSLAT2007 (07) 137; Aoki et al, arXiv:0903.4155v1 [hep-lat] To obtain these results it was necessary to implement : 1) improvement of quark dispersion relation 2) reduce the flavor symmetry breaking in the staggered fermion formulation