The QCD equation of state for two flavor QCD at non-zero chemical potential

Shinji Ejiri (University of Tokyo)

Collaborators: C. Allton, S. Hands (Swansea),

M. Döring, O.Kaczmarek, F.Karsch, E.Laermann (Bielefeld),

K.Redlich (Bielefeld & Wroclaw)

(Phys. Rev. D71, 054508 (2005) +)

Quark Matter 2005, August 4-9, Budapest

Numerical Simulations of QCD at finite Baryon Density

• Boltzmann weight is complex for non-zero .– Monte-Carlo simulations: Configurations are generated with t

he probability of the Boltzmann weight.– Monte-Carlo method is not applicable directly.

Reweighting method Sign problem

1, Perform simulations at =0. for large

2, Modify the weight for non-zero .

Studies at low density• Taylor expansion at =0.

– Calculations of Taylor expansion coefficients: free from the sign problem.

– Interesting regime for heavy-ion collisions is low density. (q/T~0.1 for RHIC, q/T~0.5 for SPS)

• Calculation of thermodynamic quantities.– The derivatives of lnZ: basic information in lattice simulations.

6

q6

4

q4

2

q2034

ln1

Tc

Tc

TccZ

VTT

p

du,du,du,

ln

pZ

V

Tn

2q

2

dudu

q

p

nn

Quark number density:

Quark number susceptibility:

m

Z

V

T

ln

Chiral condensate: Higher order terms: natural extension.

Equation of State via Taylor Expansion

Equation of state at low density

• ; quark-gluon gas is expected.Compare to perturbation theory

• Near ; singularity at non-zero (critical endpoint).Prediction from the sigma model

• ; comparison to the models of free hadron resonance gas.

QGP

color super-conductor?

hadron

T

cT T

cT

cT T

Simulations We perform simulations for =2 at ma=0.1 (m/m0.70 at T

c) and investigate T dependence of Taylor expansion coefficients.

Moreover, Taylor expansion coefficients of chiral condensate and static quark-antiquark free energy are calculated.• Symanzik improved gauge action and p4-improved staggered fermion action

• Lattice size: 41633site NNN

6

q6

4

q4

2

q244

0T

cT

cT

cT

p

T

p

2q

2I

3

3I

q3

3

)()(

ln

! ,

)(

ln

!

n

n

nn

n

n TT

Z

Nn

Nc

T

Z

Nn

Nc

4

qI6

2

qI4

I22

I 30122T

cT

ccT

4

q6

2

q422

30122T

cT

ccT

q

2 ,2 duIduq

Quark number susceptibility:

Isospin susceptibility:

Pressure:

fN

Derivatives of pressure and susceptibilities

• Difference between and is small at =0.– Perturbation theory: The difference is

• Large spike for , the spike is milder for iso-vector.• at

– Consistent with the perturbative prediction in .

0at q0 cTT

4

qI6

2

qI4

I22

I 30122T

cT

ccT

4

q6

2

q422

30122T

cT

ccT

q

6 0c cT T4c

qI

3( )O g

3( )O g

Difference of pressure for >0 from =0

Chemical potential effect is small. cf. pSB/T4~4 at =0.

RHIC : only ~1% for p.

The effect from O(6) term is small.

8q

6

q6

4

q4

2

q244

0

O

Tc

Tc

Tc

T

p

T

p 6q

4

q4

2

q244

0

O

Tc

Tc

T

p

T

p

( / 0.1)q T

Quark number susceptibility and Isospin susceptibility

• Pronounced peak for around Critical endpoint in the (T,) ?• No peak for Consistent with the prediction from the sigma model.

6q

4

q6

2

q42q4

q 30122

O

Tc

Tcc

T 6

q

4

qI6

2

qI4

I2q4

I 30122

O

Tc

Tcc

T

q

/ 1q T qI

Chiral susceptibility

• Peak height increases as

increases.

Consistent with the prediction from the sigma model.

6q

4

q4

2

q20

O

Tc

Tcc cscscs

m

Z

V

T

ln

(disconnected part only)

q

Comparison to hadron resonance gas model

• At , consistent with hadron resonance gas model.

• At , approaches the value of a free quark-gluon gas.

Hadron resonance gas

Free QG gas ,

3cosh92

42

2

q

TTF

T

Tp

Tq

q

,3

cosh 4

TTFTG

T

p q

,103 ,43 4624 cccc

Hadron resonance gas prediction

6

q6

4

q4

2

q204 T

cT

cT

ccT

p

cT T

cT T

Hadron resonance gas model for Isospin susceptibility and chiral condensate

• At , consistent with hadron resonance gas model.

4

q4

2

q203 T

cT

ccT

Hadron resonance gas

4

qI6

2

qI4

I22

I 30122T

cT

ccT

Free QG gas

Hadron resonance gas

T

TFTGT

qII 3cosh

2I

TTFTG

Tq3

cosh3

cT T

Debye screening mass• QQ free energy from Polyakov loop correlation

Singlet free energy (Coulomb gauge) Averaged free energy

where : Polyakov loop

• Assumption at T>Tc

Color-electric screening mass:

rTFTFr

Tm QQQQr,,,,ln

1lim, )1()1(

av1

2

1mm

perturbative prediction (T. Toimela, Phys.Lett.B124(1983)407)

0at

rTmQQQQ e

r

TTFrTF

,,

3

4,,,,

63

,2

31, 0

2

q

20fc

fc

f NNTTAgm

TNN

NmTm

)(Tr

1)0(Tr

1Reln,, ,)()0(

1Reln,, †av†1 rL

NL

NTrTFrLLTr

NTrTF

ccQQ

cQQ

N

t

txUxL1

4 ,)(

O.Kaczmarek and F.Zantow, Phys.Rev.D71 (2005) 114510

Taylor expansion coefficients of screening mass

Consistent with perturbative prediction

perturbative prediction

2av2

12 mm

02 ,0 av4

14 mm

6

6

4

4

2

20,T

TmT

TmT

TmTmTm

2 , av6

16 mm

0at 1 Tm

Summary • Derivatives of pressure with respect to q up to 6th order are computed.

• The hadron resonance gas model explains the behavior of pressure and susceptibilities very well at .– Approximation of free hadron gas is good in the wide range.

• Quark number density fluctuations: A pronounced peak appears for .

• Iso-spin fluctuations: No peak for . • Chiral susceptibility: peak height becomes larger as q increases.

This suggests the critical endpoint in plane?

• Debye screening mass at non-zero q is consistent with the perturbative result for .

• To find the critical endpoint, further studies for higher order terms and small quark mass are required.

cT T

0/ 1q T 0/ 1q T

2 cT T

( , )T