1. The HotQCD Equation of State Implications for Hydrodynamic Models 03-APR-20091R. Soltz, LLNL-PRES-xxxxxx for T C see presentation by P. Petreczky or poster by M. Cheng arxiv.org:0903.4379 (backup slides)

2. Evaluating Z (partition) on the lattice “... consider a continuum action, substitute finite-difference approximations for derivatives, and replace the space- time integral by a sum over the lattice sites” 03-APR-20092R. Soltz, LLNL-PRES-xxxxxx K. Wilson, Phys. Rev. D, 10:2445, 1974...see also M. Creutz, Phys. Rev. D, 21:2308, 1980 gluonsfermions following slides draw on these texts:

3. gluon links R. Soltz, LLNL-PRES-xxxxxx3 1 st Taylor series 2 nd Taylor series fermio n field fermio n field 03-APR-2009

4. trouble with (discrete) fermions 1D Dirac Eq. has degenerate fermion states 03-APR-20094R. Soltz, LLNL-PRES-xxxxxx Wilson action lifts degenerate states, breaks chiral symmetry, not widely used in thermodynamics preserves a discrete chiral symmetry additional terms improve cutoff effects p4 [O(a 2 )+fat link smearing] asqtad [O(a 2 )+tadpole coefficients] B-W [stout link smearing] all have Symanzik gauge improvements O(a 2 ) all should converge as a  0 M. Cheng, et al, PRD, 77:014511, 2008 C. Bernard et al, PRD, 75:094505, 2007 Y. Aoki, et al, PLB643:46, 2006 continuum dispersion naive lattice fermion improved staggered Wilson Staggering Dirac spinor states along 4-corners thins degeneracy by 4

5. Aside to junior experimentalists Where to work? Because they have superb physics programs and... – your RHIC colleagues will assume you’re at CERN – your LHC colleagues will assume you’re at BNL – while you submit LQCD EoS jobs to your local BG/L 03-APR-20095 LHC or and not anymore! R. Soltz, LLNL-PRES-xxxxxx

6. Data Sets (≈1/4 shown below) > 100M cpu-hrs on LLNL,NYBlue, SDSC BG/L systems – as outlined in ~40 TF-yr proposal to DOE/NNSA 03-APR-20096R. Soltz, LLNL-PRES-xxxxxx table for 23 p4 Beta runs also 17 astad Beta runs and an equal number of T=0 runs for both notation used to express T=0 subtraction on next slide

7. Analysis Apply thermalization cut, remove autocorrelations Construct Trace Anomaly (deviation from massless ideal gas) Temperature Scale Setting 7 asqtad terms heavy quark potential ϒ (2S-1S) M. Cheng, et al, PRD, 77:014511, 2008 A. Gray, et al, PRD, 72:094507, 2005 (plaquette histories) Lines of Constant Physics 03-APR-2009R. Soltz, LLNL-PRES-xxxxxx

8. Θ fermionic/gluonic contributions trace anomaly 85% gluonic (+ fermion interactions) larger cutoff effects for p4 fermions from LCP R m 03-APR-2009R. Soltz, LLNL-PRES-xxxxxx8

9. Θ μμ interpolation and continuum quadratic spline interpolations (needed to integrate pressure) 5 MeV shift N τ =6  8 shift by establishes continuum expectation similar shift expected from approach to physical quark mass 03-APR-2009R. Soltz, LLNL-PRES-xxxxxx9

10. Θ μμ low/high-T contact HRG/SB T<180 MeV, N τ =8 closer to, but below HRG T>250,300 MeV fit to – perturbative term g 4 not constrained; (d 4 ) ¼ =175-225 MeV 03-APR-2009R. Soltz, LLNL-PRES-xxxxxx10 HRG m res <1.5, 2.5 (GeV)

11. Energy, Pressure, Entropy systematic error bars from interpolation p(T 0 =0)=0 MeV shaded offset uses p(T 0 =100 MeV)=HRG 03-APR-2009R. Soltz, LLNL-PRES-xxxxxx11

12. Θ μμ reprise : Hydro Parametrization Three fits each action (p4, asqtad) 1.lattice data (solid) 2.lattice data and HRG from 100-130 MeV (double-dot) 3.lattice-10 MeV shift to approx. chiral/continuum shifts (dash) 12 physically constrains high-T region reasonably describes peak, low-T single function avoids fluctuations few parameters (easy to transfer) see also poster by P. Huovinen 03-APR-2009R. Soltz, LLNL-PRES-xxxxxx

13. Energy, Pressure revisited new fits fall within previous sys. errors all curves below SB limit (inc. HRG merger) 13 trace anomaly numerically integrated starting 50 MeV 03-APR-2009R. Soltz, LLNL-PRES-xxxxxx

14. Speed of Sound in Hydro 1.ready for hydro: smooth approx. to HotQCD EoS w/HRG 2.able to propagate systematic variation through models 03-APR-2009R. Soltz, LLNL-PRES-xxxxxx14

15. Conclusions No one should use 1 st order bag EoS, unless μ>μ c HotQCD EoS parametrization now available to hydro community to be used and improved 03-APR-2009R. Soltz, LLNL-PRES-xxxxxx15 First Order Phase Transition

16. Results with VH2 (viscous 2D+1) 03-APR-2009R. Soltz, LLNL-PRES-xxxxxx16 Beginning to propagate EOS thru Hydro Preparing to add cascade afterburner->spectra/flow/HBT M. Cheng M. Luzum and P. Romatschke, PRC, 78:034915, 2008

17. HotQCD Collaboration 03-APR-2009R. Soltz, LLNL-PRES-xxxxxx17 with help from S. Pratt, P. Huovinen, D. Molnar, S. Bass, P. Romatschke, A. Glenn, J. Newby

18. Evaluating Polyakov loop : the Movie 03-APR-200918R. Soltz, LLNL-PRES-xxxxxx P. Vranas (now at LLNL) and IBM colleagues

19. M. Cheng QM2009 Poster 03-APR-200919R. Soltz, LLNL-PRES-xxxxxx

20. Backup slides Trace Anomaly (no fit) 03-APR-2009R. Soltz, LLNL-PRES-xxxxxx20

21. Transition Temperature Deconfinement & Chiral – refer to poster 03-APR-200921R. Soltz, LLNL-PRES-xxxxxx

22. Scale Setting (more detail, p4) 03-APR-2009R. Soltz, LLNL-PRES-xxxxxx22 M. Cheng, et al, PRD, 77:014511, 2008

23. Cutoff dependence 03-APR-2009R. Soltz, LLNL-PRES-xxxxxx23 F. Karsch, Lect. Notes., 583:209, 2001

24. Strange Quark No. Susceptibility 03-APR-2009R. Soltz, LLNL-PRES-xxxxxx24 Y. Aoki, et al, arxiv:0903.4155, 2009 A. Bazavov, et al, arxiv.org:0903.4379, 2009

25. trouble with (discrete) fermions 1D Dirac Eq. has degenerate fermion states 03-APR-200925R. Soltz, LLNL-PRES-xxxxxx lifts degenerate states, breaks chiral symmetry, not widely used in thermodynamics preserves a discrete chiral symmetry additional terms improve cutoff effects improved staggered fermion actions: p4 [O(a 2 )+fat link smearing] asqtad [O(a 2 )+tadpole coefficients] B-W [stout link smearing] all have Symanzik gauge improvements O(a 2 ) all should converge as a  0 DWF actions exponentially bind chiral states to opposing walls in 5 th dimension preserve chiral symmetry at cpu cost M. Cheng, et al, PRD, 77:014511, 2008 C. Bernard et al, PRD, 75:094505, 2007 Y. Aoki, et al, PLB643:46, 2006 P. Chen, et al, PRD, 64:014503, 2001