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, fluctuationsHotQCD , 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

Improved staggered calculations at finite temperature

high-T regionT>200MeV

low T region T<200 MeV

cutoff effects are different in :

a>0.125fm a<0.125fm

improvement of the flavor symmetry isimportant

hadronic degrees of freedom quark degrees of freedomquark dispersion relation

p4, asqtad, HISQstout

for #flavors < 4rooting trick

Lattice results on trace anomaly

(2+1)-flavor calculations with p4 and asqtad and nearly physical

lattice spacing from the heavyquark 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

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 relevantfor RHIC the speed of sound issmaller than the ideal gas value

The softest point corresponds to

Bazavov et al, (HotQCD Coll.) , arXiv:0903.4379

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

QCD thermodynamics at non-zero chemical potential

Taylor expansion :

hadronic

quark

Fluctuation of conserved quantum numbers at zero baryon density :

probe of deconfinement andchiral aspects of the QCD transitionsat zero baryon density and also relatedto event-by-event fluctuations in RHIC

Physics at non-zero baryondensity:

Isentropic EoSradius of convergence,critical end-point

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 carriedby light quarks

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 carriedby light quarks

Thermodynamics at high temperature

good agreement between latticeand resummed perturbative (NLA)calculations of the entropyRebhan, 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 susceptibilitiesfor T>300MeV agree with resummed petrurbativepredictions A. Rebhan, arXiv:hep-ph/0301130Blaizot et al, PLB 523 (01) 143and are in contrrast with AdS/CFT expectations Teaney, PRD 74 (06) 045025

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 alsoexperimentally, 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

Lattice results vs. hadron resonance gas model

Include all resonances up to 2.5GeV Use ground state hadron masses modified according to know lattice correctionsModify 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

Interpolating between HRG and lattice results

Use interpolation of lattice data above 200MeV and match it to HRG at lower temperaturewith constrain that a s=0.95sSB at T=800MeV

Huovinen, P.P. arXiv:0909.xxxx

Speed of sound and elliptic flow

softest point : cs2~0.15 @ ε~1GeV/fm3

significant enhancement of v2 compared to the Bag EoS

(see Huovinen, NPA761 (2005) 296 for similar results )

Huovinen, P.P. arXiv:0909.xxxx

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 loop Renormalized chiral condensate

stout action is optimized to reduce the effect of flavor symmetry breaking, but not the quarkthe quark dispersion relation

5MeV, quark mass6MeV, continuumextrapolation

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 quarkthe quark dispersion relation => significant discretization effects at T>200MeV

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 !

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 Nτ=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 thestout results)

Summary and outlook

• Rapid increase in thermodynamic quantities at T=185-195 MeV for p4 and asqtad action

and Nτ=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 Nτ=8 and asqtad calculations with Nτ=12 should clarify this problem

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, UKQCDPRL 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 relation2) reduce the flavor symmetry breaking in the staggered fermion formulation

Backup: weak coupling results versus lattice data