Lattice QCD thermodynamics Kalman Szabo Bergische Universitat, - - PowerPoint PPT Presentation

Lattice QCD thermodynamics

Kalman Szabo Bergische Universitat, Wuppertal

Wuppertal-Budapest collaboration

Outline

Equation of State [1309.5258] Fluctuations [1305.5161] T > 0 with Wilson-type fermions [1205.0440]

Equation of state

As of 2010 height and position are different:

Crossover

[Wuppertal-Budapest,WB,’06]

volume dependence of chiral susceptibility:

Tc

Tc =

• ∼ 190 MeV

[BNL-Bielefeld-RIKEN-Columbia,’06]

∼ 150 MeV

[WB,’06]

= ?

0.2 0.4 0.6 0.8 1 120 140 160 180 200 T [MeV]

∆l,s

fK scale asqtad: Nτ=8 Nτ=12 HISQ/tree: Nτ=6 Nτ=8 Nτ=12 Nτ=8, ml=0.037ms stout cont.

[WB’06’09’10] Tc = 147(2)(3)MeV [hotQCD ’12] Tc = 154(9)MeV

Tc

[hotQCD ’12]

150 160 170 180 190 200 210 220 0.02 0.04 0.06 0.08 0.1 1/Nτ

2

Tc [MeV]

Nt a[fm] 4 0.30 6 0.20 8 0.15 Nt = 4 results are unreliable

Tc: scaling regime

quark number susceptibility [WB,’11] Nt a[fm] 6 0.20 8 0.15 10 0.12 12 0.10 16 0.075 a2-scaling for Nt 10 or a 0.12 fm

Equation of state

As of 2012 only the height is different:

Equation of state

A couple of improvements made:

[WB,arXiv:1309.5258]

Equation of state

Finer lattices: Nt = 6, 8, 10 → 6, 8, 10, 12, 16

0.005 0.01 0.015 0.02 0.025 0.03

1/Nt

2 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

p/T

4 6

• 2

12

• 2 10
• 2

8

• 2

tree level correction no correction

pcorr = pLAT ·

• pcont

pLAT

free tree-level improvement is as good as Naik,p4, but much cheaper not the slope matters, but the beginning of the a2-scaling-regime

Equation of state

Thermodynamics is more expensive system size increase → N4

t

autocorrelation → N2

t

Hybrid-Monte Carlo stepsize → Nt fermion inversion → Nt OT − OT=0 remove UV divergence of N4

t

→ N8

t

volume contributes to statistics → N−4

t

Total cost ∼ N12

t

Equation of state

Extreme fine lattice spacings downto a ≈ 0.02 fm

• T > 0 is easier than T = 0 (smaller box needed, faster inverter,

no tunneling problem) → use T/2 to cancel the divergence: OT − OT/2

• L = 1fm is easier, than L = 6fm → use Wilson-flow based

step-scaling to determine running coupling and mass

0.05 0.1 0.15 0.2

a[fm]

3.4 3.6 3.8 4 4.2 4.4 4.6

β

fit to fK scale fit to w0 scale scale from w0 based step scaling scale from w0 along LCP

Equation of state

Trace anomaly is unchanged

200 300 400 500

T[MeV]

1 2 3 4 5 6

(ε-3p)/T

4 hotQCD HISQ Nt=6 8 10 12 HRG 2 stout continuum WB 2010 4 stout crosscheck s95p-v1

extended the estimation of systematic error (896 methods)

Equation of state

Independent check from a different action

0.005 0.01 0.015 0.02 0.025 0.03

1/Nt

2 4 5 6 12

• 2

10

• 2

8

• 2

6

• 2

tree level correction no correction 0.005 0.01 0.015 0.02

1/Nt

2 4 5 6 16

• 2

12

• 2

10

• 2

8

• 2

4-stout smearing with charm quark has different βLAT ǫ − 3p T 4 ∼ dβLAT d log a · (OT − OT=0)

Equation of state

As of 2013 discrepancy remains ...

200 300 400 500

T[MeV]

1 2 3 4 5 6

(ε-3p)/T

4 hotQCD HISQ Nt=6 8 10 12 HRG 2 stout continuum WB 2010 4 stout crosscheck s95p-v1

WB is “full-result”: continuum with physical mass

Equation of state

Zero point of the pressure p(T∗) p(T) T 4 − p(T∗) T 4

∗

= T

T∗

dT ′ dp(T ′) dT ′ = T

T∗

dT ′ ǫ − 3p T ′5 Previously: set p(T∗) to “0” or to HRG at T∗ = 100 MeV Now: integrate in mass instead of T, divergence is only N2

t

p(T) T 4 = m=∞

mphys

dm′ dp(m′) dm′

1 2 3 4 5 6

log(mq/mphys)

• 0.2

0.2 0.4 0.6 0.8

• Nt

4 lnZ

ln(ma)

Nt=12 Nt=16 light strange

Equation of state

Pressure

200 300 400 500

T[MeV]

1 2 3 4 5 6

p/T

4 HRG HTL NNLO lattice continuum limit SB

at low T agreement with the HRG model (result, not an input!)

Fluctuations

Fluctuations

B,Q,S fluctuations ≡ u,d,s susceptibilities χBQS

ijk

= 1 VT 3

• ∂i

∂(µB/T)i ∂j ∂(µQ/T)j ∂k ∂(µS/T)k

• log Z

χ1 = Tr M−1∂µM χ2 = Tr . . .2 − Tr . . . 2 + ∂µTr . . .

Expensive!

higher orders bring volume penalty (disconnected) err(χ1) ∼ 1 √ N · VT 3 err(χ2) ∼ 1 √ N err(χ4) ∼ VT 3 √ N traces are calculated with noisy estimators (multigrid, TSA, GPU)

Fluctuations

HRG at low T, rapid rise, upto 90%SB, s “melts” at higher T

Uses of fluctuations

1 dominant degrees of freedom 2 thermodynamics at finite µB (Tc and EoS) 3 compare with experiment → freezout parameters

Low temperature

diagnose the breakdown of hadronic description, eg. uncorrelated hadron gas: p(µB, µS) =

• S

pmes,S cosh(SµS) +

• S=0...3

pbar,S cosh(µB − SµS) use fluct-combinations, which are zero

High temperature

How does strangeness behave?

1 bound with other partons (eg. sg,sq) 2 do not couple with other partons (quasi-quark)

CBS = −3δBδS δS2

High temperature

How does strangeness behave?

1 bound with other partons (eg. sg,sq) 2 do not couple with other partons (quasi-quark)

CBS = −3δBδS δS2

200 300 400 500

T[MeV]

1 2 3 4 5 6

p/T

4 HRG HTL NNLO lattice continuum limit SB

Uses of fluctuations

1 dominant degrees of freedom 2 thermodynamics at finite µB (Tc and EoS)

OµB = O+µ2

B

2 ·

• OχB

2 − OχB 2 + 2O′χB 1 + O′′

• +. . .

3 compare with experiment → freezout parameters

The transition line

from change of chiral-condensate with µB (only LO) is there an endpoint along this line? higher orders, sign problem

The freezout line

extract T, µB, µS, µQ, V of a heavy-ion collision from abundancies Can a hadronic approach be applied at such high temperatures? → Freezout parameters from the lattice

Event-by-event

What fluctuates in a heavy-ion collision, if you start with a fixed number of conserved charges (Z=82, A=207)? Consider particles coming only from a small part of the whole system, defined by imposing kinematical constraints. Charges from subvolumes will fluctuate from one event to the

• ther.

Experiment vs. lattice

fluct.exp

?

= fluct.latt,T,µB,µQ,µS

1 use 4 fluctuations to determine 4 freezout T, µB, µQ, µS 2 do they originate from thermal & chemical equilibrium? 3 how is the freezout curve (latt+exp) related to QCD

transition line (latt)? is the freezout curve close enough to the QCD endpoint?

Freezeout parameters

[BNL-Bielefeld ’12]

How to get freezout T, µB, µQ, µS? in a lead-lead collision (Z=82, A=207): S = 0, Q = 82 207B can be used to obtain µQ = µQ(T, µB) and µS = µS(T, µB) choose two fluctuations ( 1. thermometer, 2. baryometer ) and find T and µB, where . . . exp = . . . latt,T,µB to cancel Vol of the subsystem work with fluctuation ratios

Thermometer

experimentally charge is the cleanest δQ3/Q Tf 157 MeV

Baryometer

Q/δQ2 ∼ µB

Freezout vs. transition line

[BNL-Bielefeld,’13]

ab-initio determinations of freezout T, µ: STAR electric charge + lattice PHENIX electric charge + lattice STAR proton number + lattice baryon number good agreement with QCD transition line

Wilson/overlap at T > 0

Problems with staggered quarks

rooting is not necessarily universal

→ probably correct

finite µ at finite “a” is not well-defined

(det M)1/4 =

• k

jkλ1/4

k

, where jk = {+1, −1, +i, −i}

chiral limit at finite “a” is not well-defined

→ confusion about nf = 2 order of the transition

correlators are just awful

(−1)t

Check staggered

rooting is not necessarily universal

→ check it by Wilson/overlap check only meaningful in continuum limit for fully renormalized nite quantities for the check only, physical quark masses not essential (heavier quarks also universal)

Wilson setup

fixed β approach: change Nt with β fixed three pion-masses, strange mass is physical four lattice spacings (Nt = 8, 12, 16, 20 at Tc)

Wilson comparison

renormalized chiral condensate mR ¯ ψRψRT,0 = 2m2

PCACZ 2 A

• x

P(x)P(0)T,0 agreement between Wilson and staggered also for Polyakov-loop and strange number fluctuation

Overlap setup

2HEX smeared links Zolotarev-approximation project low lying modes by Krylov-Schur fixed topology by determinant suppression w0 for scale setting mπ = 350 MeV Nt = 6, 8, 10, 12 Ns/Nt = 2

Overlap comparison

• 0.08
• 0.07
• 0.06
• 0.05
• 0.04
• 0.03
• 0.02
• 0.01

0.01 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 120 140 160 180 200 220 240 260 mR

–

ψψR/m4

π

Tw0 T [MeV]

Preliminary

staggered 6×123 8×163 10×203 12×243

agreement between overlap and staggered

Summary

Equation of State → full result Fluctuations → ab-initio freeze-out parameters Wilson/overlap at T > 0 → staggered is OK