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The HotQCD Equation of State

The HotQCD Equation of State. Implications for Hydrodynamic Models. for T C see presentation by P. Petreczky or poster by M. Cheng. arxiv.org:0903.4379. (backup slides). Evaluating Z (partition) on the lattice.

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The HotQCD Equation of State

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

  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” K. Wilson, Phys. Rev. D, 10:2445, 1974 ...see also M. Creutz, Phys. Rev. D, 21:2308, 1980 gluons fermions following slides draw on these texts: R. Soltz, LLNL-PRES-xxxxxx

  3. gluon links 1st Taylor series 2nd Taylor series fermion field R. Soltz, LLNL-PRES-xxxxxx

  4. trouble with (discrete) fermions • 1D Dirac Eq. has • degenerate fermion states 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(a2)+fat link smearing] • asqtad [O(a2)+tadpole coefficients] • B-W [stout link smearing] • all have Symanzik gauge improvements O(a2) • all should converge as a0 Wilson continuum dispersion M. Cheng, et al, PRD, 77:014511, 2008 improved staggered C. Bernard et al, PRD, 75:094505, 2007 naive lattice fermion Y. Aoki, et al, PLB643:46, 2006 Staggering Dirac spinor states along 4-corners thins degeneracy by 4 R. Soltz, LLNL-PRES-xxxxxx

  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 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 • 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 R. Soltz, LLNL-PRES-xxxxxx

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

  8. Θ fermionic/gluonic contributions • trace anomaly 85% gluonic (+ fermion interactions) • larger cutoff effects for p4 fermions from LCP Rm R. Soltz, LLNL-PRES-xxxxxx

  9. Θμμ interpolation and continuum • quadratic spline interpolations (needed to integrate pressure) • 5 MeV shift Nτ=68 shift by establishes continuum expectation • similar shift expected from approach to physical quark mass R. Soltz, LLNL-PRES-xxxxxx

  10. Θμμ low/high-T contact HRG/SB • T<180 MeV, Nτ=8 closer to, but below HRG • T>250,300 MeV fit to • perturbative term g4 not constrained; (d4)¼=175-225 MeV HRG mres<1.5, 2.5 (GeV) R. Soltz, LLNL-PRES-xxxxxx

  11. Energy, Pressure, Entropy • systematic error bars from interpolation p(T0=0)=0 MeV • shaded offset uses p(T0=100 MeV)=HRG R. Soltz, LLNL-PRES-xxxxxx

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

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

  14. Speed of Sound in Hydro • ready for hydro: smooth approx. to HotQCD EoS w/HRG • able to propagate systematic variation through models R. Soltz, LLNL-PRES-xxxxxx

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

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

  17. HotQCD Collaboration • MILC + RBC-Bielefeld ... with Arizona, Riken-BNL, Columbia, Indiana, LANL, LLNL, UC Santa Barbara, Utah with help from S. Pratt, P. Huovinen, D. Molnar, S. Bass, P. Romatschke, A. Glenn, J. Newby R. Soltz, LLNL-PRES-xxxxxx

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

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

  20. Backup slides • Trace Anomaly (no fit) R. Soltz, LLNL-PRES-xxxxxx

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

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

  23. Cutoff dependence F. Karsch, Lect. Notes., 583:209, 2001 R. Soltz, LLNL-PRES-xxxxxx

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

  25. trouble with (discrete) fermions • 1D Dirac Eq. has • degenerate fermion states 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(a2)+fat link smearing] • asqtad [O(a2)+tadpole coefficients] • B-W [stout link smearing] • all have Symanzik gauge improvements O(a2) • all should converge as a0 M. Cheng, et al, PRD, 77:014511, 2008 C. Bernard et al, PRD, 75:094505, 2007 DWF actions exponentially bind chiral states to opposing walls in 5th dimension preserve chiral symmetry at cpu cost Y. Aoki, et al, PLB643:46, 2006 P. Chen, et al, PRD, 64:014503, 2001 R. Soltz, LLNL-PRES-xxxxxx

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