The HotQCD Equation of State

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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

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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

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