The HotQCD Equation of State
25
The HotQCD Equation of State Implications for Hydrodynamic Models 03-APR-2009 1 R. Soltz, LLNL-PRES-xxxxxx for T C see presentation by P. Petreczky or poster by M. Cheng arxiv.org:090 3.4379 (backup slides)
description
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. - PowerPoint PPT Presentation
Transcript of The HotQCD Equation of State
R. Soltz, LLNL-PRES-xxxxxx 1
The HotQCD Equation of StateImplications
for Hydrodynamic Models
03-APR-2009
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”
03-APR-2009
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
€
Uμ (n) = e igaAμ (n )
€
U−μ (n + μ + ν ) = e iga −Aμ (n )−a∂ν Aμ (n )+O(a 2 )[ ] 1st Taylor series
2nd Taylor series
€
=e iga 2 (∂ μ Aν −∂ν Aμ )−ig Aμ ,Aν[ ] = e ia 2gFμν
€
=1− a4g2
2Fμν Fμν + O(a2)( )
€
S = Tr Fμν Fμν[ ]n,μ <ν∑ + const.
€
a= 0 ⏐ → ⏐ 14
d3x Tr Fμν Fμν[ ]0
1/T
∫ = SgluonV∫
fermionfield03-APR-2009
R. Soltz, LLNL-PRES-xxxxxx 4
trouble with (discrete) fermions
• 1D Dirac Eq. has• degenerate fermion states
03-APR-2009
€
E = ± sin(ka)a
€
∂ψ∂t
= − i2a
γ 5 ψ (n +1) −ψ (n −1)[ ]
Wilson action lifts degenerate states, breaks chiral symmetry, not widely used in thermodynamics
€
2d n f( )
• 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 a0
M. Cheng, et al, PRD, 77:014511, 2008
C. Bernard et al, PRD, 75:094505, 2007
Y. Aoki, et al, PLB643:46, 2006
continuum dispersio
n
naivelattice fermion
improved staggered
Wilson
Staggering Dirac spinor states along4-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
03-APR-2009
LHC
orand
not a
nymore
!
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-2009
• table for 23 p4 Beta runs
• also 17 astad Beta runs
• and an equal number of T=0 runs for both
€
Δ X = X0
− Xτ
notation used to express T=0 subtraction on next slide
R. Soltz, LLNL-PRES-xxxxxx
Analysis
• Apply thermalization cut, remove autocorrelations• Construct Trace Anomaly (deviation from massless ideal gas)
• Temperature Scale Setting
7
€
ε−3pT 4 = ΘF
μμ (T)T 4 + ΘG
μμ (T)T 4 = Rβ (β )Nτ
4Δ s
€
ΘFμμ (T)T 4 = −Rβ RmNτ
4 2 ˆ m lΔ ψψl+ ˆ m sΔ ψψ
s( )
€
ΘGμμ (T)T 4 = Rβ Nτ
4 Δ sG − Ru 6 ′ β rtΔ R + 4 ′ β pgΔ C + 14β
Δ Tr (2Dl−1 + Ds
−1) dMdu0
⎛ ⎝ ⎜
⎞ ⎠ ⎟
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
asqtad terms
€
Rβ (β ) = T dβdT
= −a dβda
€
r2dVq q (r)
dr
⎛
⎝ ⎜
⎞
⎠ ⎟r= 0.469(7)
€
=1.65heavy quarkpotentialϒ(2S-1S) M. Cheng, et al, PRD,
77:014511, 2008A. Gray, et al, PRD, 72:094507, 2005
(plaquette histories)
Lines of Constant Physics
€
ml = 0.1ms(LCP)
03-APR-2009
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
03-APR-2009
R. Soltz, LLNL-PRES-xxxxxx 9
Θμμ interpolation and continuum
• quadratic spline interpolations (needed to integrate pressure) • 5 MeV shift Nτ=68 shift by establishes continuum expectation• similar shift expected from approach to physical quark mass
03-APR-2009
€
p(T)T 4 = p(T0)
T04 + d ′ T Θμμ ( ′ T )
′ T 5T0
T
∫
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
03-APR-2009
€
ε−3pT 4 = 3
4b0g
4 + d2
T 2 + d4
T 4
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
03-APR-2009 €
p(T)T 4 = p(T0)
T04 + d ′ T Θμμ ( ′ T )
′ T 5T0
T
∫
R. Soltz, LLNL-PRES-xxxxxx 12
Θμμreprise : 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)
€
1− 1
1+ e(T −c1 )/ c2[ ]2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟× d2
T 2 + d4
T 4
⎛ ⎝ ⎜
⎞ ⎠ ⎟
• 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-2009
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
03-APR-2009
R. Soltz, LLNL-PRES-xxxxxx 14
Speed of Sound in Hydro
1. ready for hydro: smooth approx. to HotQCD EoS w/HRG2. able to propagate systematic variation through models
03-APR-2009
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
03-APR-2009
First OrderPhase
Transition
R. Soltz, LLNL-PRES-xxxxxx 16
Results with VH2 (viscous 2D+1)
03-APR-2009
• 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
R. Soltz, LLNL-PRES-xxxxxx 17
HotQCD Collaboration
03-APR-2009
=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
03-APR-2009
P. Vranas (now at LLNL) and IBM colleagues
R. Soltz, LLNL-PRES-xxxxxx 19
M. Cheng QM2009 Poster
03-APR-2009
R. Soltz, LLNL-PRES-xxxxxx 20
Backup slides
• Trace Anomaly (no fit)
03-APR-2009
R. Soltz, LLNL-PRES-xxxxxx 21
Transition Temperature• Deconfinement & Chiral – refer to poster
03-APR-2009
€
Δ l,s(T) =ψ ψ
l ,T− ml
ms
ψ ψs,T
ψ ψl ,0
− ml
ms
ψ ψs,0
€
χ(l ,s)
T 2 = 1VT
∂ 2 log(Z)∂μ(l ,s)
2
R. Soltz, LLNL-PRES-xxxxxx 22
Scale Setting (more detail, p4)
03-APR-2009
M. Cheng, et al, PRD, 77:014511, 2008
R. Soltz, LLNL-PRES-xxxxxx 23
Cutoff dependence
03-APR-2009
F. Karsch, Lect. Notes., 583:209, 2001
R. Soltz, LLNL-PRES-xxxxxx 24
Strange Quark No. Susceptibility
03-APR-2009
Y. Aoki, et al, arxiv:0903.4155, 2009
€
χ(l ,s)
T 2 = 1VT
∂ 2 log(Z)∂μ(l,s)
2
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
03-APR-2009
€
E = ± sin(ka)a
€
∂ψ∂t
= − i2a
γ 5 ψ (n +1) −ψ (n −1)[ ]
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 a0
DWF actions exponentially bind chiral states to opposing walls in 5th dimensionpreserve chiral symmetry at cpu cost
€
2d n f( )
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
