1. QCD thermodynamic on the lattice and the hadron resonance gas Péter Petreczky Physics Department and RIKEN-BNL ECT*/LOEWE/NIKHEF/CATHIE workshop, Trento, September 14-18, 2009 Lattice artifacts in staggered fermion formulation Thermodynamics at high temperaure : EoS, fluctuations HotQCD, RBC-Bielefeld, MILC Thermodynamics at low temperatures and the hadron resonance gas (HRG) Parametrization of the EoS based on lattice+HRG and its effect on elliptic flow Deconfinement and chiral aspects of the QCD transition : Budapest-Wuppertal, HotQCD results and highly improved staggered quark (HISQ) action

2. 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 freedomquark degrees of freedom quark dispersion relation p4, asqtad, HISQ stout for #flavors < 4 rooting trick

3. Lattice results on trace anomaly (2+1)-flavor calculations with p4 and asqtad and nearly physical lattice spacing from the heavy quark potential : arXiv:0903.4379 Bernard et al, (MILC) PRD 75 (07) 094505, Cheng et al, (RBC-Bielefeld) PRD 77 (08) 014511 Bazavov et al, (HotQCD), arXiv:0903.4379

4. Pressure, energy density and speed of sound rapid rise in number of d.o.f at T=185-195 MeV about 10% deviation from the ideal gas limit lattice discretization errors are small For energy densities relevant for RHIC the speed of sound is smaller than the ideal gas value The softest point corresponds to Bazavov et al, (HotQCD Coll.), arXiv:0903.4379

5. Equation of State for physical quark masses Thermodynamics quantities are quark mass independent for T>200MeV The quark mass effect is also very small at low temperatures (T<170MeV) because cutoff effects dominate, no agreement with hadron resonance gas In the transition region thermodynamic quantities are larger for the smaller quark mass, the enhancement of thermodynamic quantities is consistent with 5MeV shift of the transition region towards lower temperatures

6. QCD thermodynamics at non-zero chemical potential Taylor expansion : hadronic quark Fluctuation of conserved quantum numbers at zero baryon density : probe of deconfinement and chiral aspects of the QCD transitions at zero baryon density and also related to event-by-event fluctuations in RHIC Physics at non-zero baryon density: Isentropic EoS radius of convergence, critical end-point

7. Deconfinement : fluctuations of conserved charges baryon number electric charge strange quark number Ideal gas of quarks : conserved charges are carried by massive hadrons conserved charges carried by light quarks

8. Deconfinement : fluctuations of conserved charges baryon number electric charge strange quark number Ideal gas of quarks : conserved charges are carried by massive hadrons conserved charges carried by light quarks

9. 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 no constant non-perturbative term is present in the entropy density 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 and are in contrrast with AdS/CFT expectations Teaney, PRD 74 (06) 045025

10. Fluctuations in the hadron resonance gas model Cheng et al., arXiv:0811.1006 Kurtosis : ratio of the quartic fluctuations to quadratic fluctuations, can be studied also experimentally, see e.g. Schuster, arXiv:0903.2911 reasonable agreement with HRG for certain ratios at low T Hadron resonance gas (HRG) can be used as a reference at low temperatures

11. 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:0909.xxxx

12. 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 at T=800MeV Huovinen, P.P. arXiv:0909.xxxx

13. Speed of sound and elliptic flow softest point : c s 2 ~0.15 @ ε~1GeV/fm 3 significant enhancement of v 2 compared to the Bag EoS (see Huovinen, NPA761 (2005) 296 for similar results ) Huovinen, P.P. arXiv:0909.xxxx

14. Deconfinement and chiral transition stout : Budapest-Wuppertal Group, Aoki et al., PLB 643 (06) 46; arXiv:0903.4155 no qualitative change, but significant shift of the transition region toward smaller T Renormalized Polyakov loopRenormalized chiral condensate stout action is optimized to reduce the effect of flavor symmetry breaking, but not the quark the quark dispersion relation 5MeV, quark mass 6MeV, continuum extrapolation

15. Deconfinement and chiral transition stout : Budapest-Wuppertal Group, Aoki et al., PLB 643 (06) 46; arXiv:0903.4155 agreement between HotQCD results and Budapest-Wuppertal results at high T Strangeness fluctuations:Renormalized chiral condensate stout action is optimized to reduce the effect of flavor symmetry breaking, but not the quark the quark dispersion relation => significant discretization effects at T>200MeV

16. Deconfinement and chiral transition stout : Budapest-Wuppertal Group, Aoki et al., PLB 643 (06) 46; arXiv:0903.4155 Strangeness fluctuations agree quite well with HRG for T<200MeV !

17. Deconfinement and chiral transition for HISQ action The highly improved staggered fermion action (HISQ) improves the quark dispersion and is the most efficient in the improvement of flavor symmetry breaking (smallest pion splitting) Renormalized chiral condensate Renormalized Polyakov loop Preliminary results for N τ =6 and 8 HISQ calculations (Bazavov, P.P., Lattice 2009) significant shift of the transition region for the chiral condensate (much closer to stout) no apparent shift in the renormalized Polyakov loop (HISQ results differ significantly from the stout results)

18. Summary and outlook Rapid increase in thermodynamic quantities at T=185-195 MeV for p4 and asqtad action and N τ =8 ; things maybe different as continuum limit is approach 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 to be used in hydrodynamic modeling. Significant effect on the proton elliptic flow was observed in ideal hydro compared to bag EoS Comparison with HRG indicate significant cutoff effects in the low temperature region for p4 and asqtad actions. These discretization effects maybe responisble for discrepancy between HotQCD and stout results. The ongoing calculations with HISQ on N τ =8 and asqtad calculations with N τ =12 should clarify this problem

20. 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

21. Backup: weak coupling results versus lattice data