1. The QCD EoS from simulations on BlueGene L Supercomputer at LLNL and NYBlue Rajan Gupta T-8, Los Alamos National Lab Lattice 2008, College of William and Mary The HotQCD collaboration (All HotQCD data are preliminary)

2. 2 HotQCD Collaboration T. Battacharya (LANL) A. Bazavov (Arizona) M. Cheng (Columbia) N. Christ (Columbia) C. DeTar (Utah) S. Gottlieb (Indiana) R. Gupta (LANL) U. Heller (APS) K. Huebner (BNL) C. Jung (BNL) F. Karsch (BNL/Bielefeld) E. Laermann (Bielefeld) L. Levkova (Utah) T. Luu (LLNL) R. Mawhinney (Columbia) P. Petreczky (BNL) D. Renfrew (Columbia) C. Schmidt (BNL) R. Soltz (LLNL) W. Soeldner (BNL) R. Sugar (UCSB) D. Toussaint (Arizona) P. Vranas (LLNL) A unique US wide opportunity for simulating QCD at finite temperature using the BlueGene L at LLNL and NYBlue

3. 3 Equilibration Hadronization Thermal freeze-out EoS, viscosity RHIC probes QGP between 150-250MeV LHC will probe the range 150-700 MeV  = 15 GeV/fm 3  = 5.4 GeV/fm 3 Time after collision (fm/c) Probe QGP at RHIC and LHC

4. 4 HotQCD Collaboration: Goals Nature of the transition –Deconfinement and  S restoration? Transition temperature Tc Equation of State (EOS) at (  =0,  0) Spectral Functions Spatial and temporal correlators versus T Transport coefficients of the quark gluon plasma

5. HotQCD Collaboration: Precision LQCD (Controlling all systematic errors) Two improved staggered formulations (Asqtad, p4) (different O(a 2 ) errors) Continuum limit (N  = 4, 6, 8, …) Chiral extrapolation (m l /m s = 0.2, 0.1, 0.05) Dense sampling of the transition region High statistics N space ≥ 4 N  Testing domain wall fermions

6. HOTQCD data sets: Asqtad  NsNs M l /m s Number of  values Streams Total # trajectories per  (unit length) 6320.27118-20K 6320.17118-19K 8320.28113K 8320.123115-16K 8640.1113.7K 32 0.11815-6K

7. HOTQCD data sets: p4  NsNs M l /m s Number of  values Streams Total # trajectories per  (0.5 length) 6160.2910-60K 616,240.18 25-60K, 5-8K 8320.27214-15K 8320.1301-38-37K 32 0.12112-6K hep-lat/0608013

8. 8 Line of constant physics (LCP) using T=0 simulations Scale set using potential (r 0 and r 1 ) M ss r 0 = 1.58 (M  r 0  0.52  M   220 MeV) Quark mass m l /m s = 0.1 (real world ~0.04) Take a  0 along this LCP varying just the gauge coupling 

9. Equation of State Experiments at the LHC will probe QGP over 700-200 MeV. Provide realistic EoS to hydro models of evolution of the QGP.

10. Equation of State -- details Shift to smaller T by ~5 MeV with N  =6  8. Data lies below hadron resonance gas (purple curve) Peak above 200 MeV. 15-20% change with N  =6  8 Need to estimate finite volume uncertainty for T>400 MeV (point at 400 MeV with 8  64 3 ). Need more points to fix the shape of the curve Fit: c 0 + c 2 /T 2 + c 4 /T 4 Asymptotic g 4 behavior not evident in c 0

11. Entropy density At the LHC (~700 MeV) the degrees of freedom have a much cleaner interpretation as quarks and gluons Band in T: 185-195 MeV

12. Transition Temperature Need to extrapolate to m l =0 to define a transition temperature from thermodynamics quantities –Polyakov line and its susceptibility –Chiral condensate and its susceptibility –Quark number and its susceptibility Polyakov loop Quark number susceptibility Chiral condensate Chiral susceptibility Deconfinement  S Restoration

13. 13 Renormalized Polyakov Loop K. Petrov: arXiv:0710.4237 A. Bazavov: Unpublished Difference between Asqtad and p4 & N  = 6, 8  10% In all figures the band indicates the range T=185-195 MeV

14. 14 Quark Number Susceptibility States carrying such quantum numbers (  0   baryon number/strangeness) are heavy at low T and light at high T. Probe confinement Does not require renormalization Peak in fourth derivative!

15. 15 Quark Number Susceptibility Difference between Asqtad and p4 most pronounced between 200-300 MeV

16. Chiral Condensate:  S restoration Subtraction to eliminate additive renormalization Sharp decrease in  reflects  Symmetry restoration

17. 17 N  =6, 8: and for P4 and AsqTAD

18. 18 Chiral Susceptibility: N  = 4, 6, 8 One flavor susceptibility for light (l = u,d) and strange (s) quarks:   2  2 + - Connected Disconnected Note: Isosinglet  =  disconnected + 2  connected One flavor  =  disconnected + 4  connected

19. 19 Disconnected Chiral Susceptibility: N  = 8 ? Should one choose position of the peak or right edge for T c ? See talk by F. Karsch

20. 20 Conclusions Asqtad and p4 data are consistentAsqtad and p4 data are consistent –Differences at O(a 2 ) level –Maximum differences seen between 200-300 MeV Estimate of EoS over the range 150-700 MeV ready for input into hydrodynamics modelsEstimate of EoS over the range 150-700 MeV ready for input into hydrodynamics models Need (m l  0) limit to define transition temperature. (N  = 8, m l /m s = 0.1) estimates ~185-195 MeVNeed (m l  0) limit to define transition temperature. (N  = 8, m l /m s = 0.1) estimates ~185-195 MeV Complete N  = 6, 8 (m l /m s = 0.2, 0.1) runs and extend toComplete N  = 6, 8 (m l /m s = 0.2, 0.1) runs and extend to –m l /m s = 0.05 –N  = 12

21. Extras

22. 22 P4: Setting the lattice scale r 0 =0.469(7) fm, r 0  0.5 =1.104(5),r 0 /r 1 =1.463(7) Jan van der Heide at LAT2007