the hotqcd equation of state

The HotQCD Equation of State

Implications for

Hydrodynamic Models

03-APR-2009 1R. Soltz, LLNL-PRES-xxxxxx

for TC see presentation by P. Petreczky or poster by M. Cheng arxiv.org:0903.4379

(backup slides)

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:

gluon links

R. Soltz, LLNL-PRES-xxxxxx 3

€

Uμ (n) = e igaAμ (n )

€

U−μ (n + μ + ν ) = eiga −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 ⏐ → ⏐ 1

4d3x Tr Fμν Fμν[ ]

0

1/T

∫ = SgluonV

∫

fermion

field

fermion

field03-APR-2009

trouble with (discrete) fermions

• 1D Dirac Eq. has• degenerate fermion states


€

E = ±sin(ka)

a

€

∂ψ∂t

= −i

2aγ 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

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 5

LHC

orand

not a

nym

ore!

R. Soltz, LLNL-PRES-xxxxxx

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

€

Δ X = X0

− Xτ

notation used to express T=0 subtraction on next slide

Analysis

• Apply thermalization cut, remove autocorrelations• Construct Trace Anomaly (deviation from massless ideal gas)

• Temperature Scale Setting

7

€

ε−3p

T 4=

ΘFμμ (T)

T 4+

ΘGμμ (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 +1

4βΔ Tr (2Dl

−1 + Ds−1)

dM

du0

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

asqtad terms

€

Rβ (β ) = Tdβ

dT= −a

dβ

da

€

r2dV

q q(r)

dr

⎛

⎝ ⎜

⎞

⎠ ⎟r= 0.469(7)

€

=1.65

heavy 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

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 8

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


€

p(T)

T 4=

p(T0)

T04 + d ′ T

Θμμ ( ′ T )′ T 5

T0

T

∫

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


€

ε−3p

T 4=

3

4b0g

4 +d2

T 2+

d4

T 4

HRG mres<1.5, 2.5 (GeV)

Energy, Pressure, Entropy

• systematic error bars from interpolation p(T0=0)=0 MeV

• shaded offset uses p(T0=100 MeV)=HRG03-APR-2009 R. Soltz, LLNL-PRES-xxxxxx 11€

p(T)

T 4=

p(T0)

T04 + d ′ T

Θμμ ( ′ T )′ T 5

T0

T

∫

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)

12

€

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

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-2009 R. Soltz, LLNL-PRES-xxxxxx

Speed of Sound in Hydro

1. ready for hydro: smooth approx. to HotQCD EoS w/HRG2. able to propagate systematic variation through models


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 OrderPhase

Transition

Results with VH2 (viscous 2D+1)


• 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

HotQCD Collaboration


with help from S. Pratt, P. Huovinen, D. Molnar, S. Bass, P. Romatschke, A. Glenn, J. Newby

Evaluating Polyakov loop : the Movie


P. Vranas (now at LLNL) and IBM colleagues

M. Cheng QM2009 Poster


Backup slides

• Trace Anomaly (no fit)


Transition Temperature

• Deconfinement & Chiral – refer to poster


€

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=

1

VT

∂ 2 log(Z)

∂μ(l ,s)2

Scale Setting (more detail, p4)


M. Cheng, et al, PRD, 77:014511, 2008

Cutoff dependence


F. Karsch, Lect. Notes., 583:209, 2001

Strange Quark No. Susceptibility


Y. Aoki, et al, arxiv:0903.4155, 2009

€

χ(l ,s)

T 2=

1

VT

∂ 2 log(Z)

∂μ(l ,s)2

A. Bazavov, et al, arxiv.org:0903.4379, 2009

trouble with (discrete) fermions

• 1D Dirac Eq. has• degenerate fermion states


€

E = ±sin(ka)

a

€

∂ψ∂t

= −i

2aγ 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

the hotqcd equation of state

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