Lattice QCD at finite density

Shinji Ejiri (University of Tokyo)

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

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

K.Redlich (U. Bielefeld & U. Wroclaw)

(hep-lat/0501030)

RIKEN, February 2005

phase

hadron phase

RH

ICearly universe

SP

S

AG

S

mN/3~300MeV

color flavor locking?

color super conductor?

nuclear matter

Numerical simulationsNumerical simulations

q

Tquark-gluon plasma

Introduction• High temperature and density QCD

• Low density region– Heavy-ion collisions

Comparison with different density

– Critical endpoint?

Simulation parameter: q/T

• High density region Y. Nishida

Tc~

170M

eV

Chemical freeze out parameter

• Statistical thermal model– Well explains the particle pr

oduction rates(P. Braun-Munzinger et al., n

ucl-th/0304013)

• Relation to the chiral/ confinement phase transition

• Relation to (,p,S,n)

Lattice calculations

qB 3

Lattice(10% error)

Critical endpoint• Various model calculations (M.A. Stephanov, Prog.Theor.Phys.Suppl.153 (2004)139)

qB 3

Baryon fluctuations becomes bigger as large.

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

• Reweighting method only at small .– Not very serious for small lattice. (~ Nsite)– Interesting regime for heavy-ion collisions is low density. (q

/T~0.1 for RHIC, q/T~0.5 for SPS)

• Taylor expansion at =0.– Taylor expansion coefficients are free from the sign problem.

(The partition function is a function of q/T)

6

q6

4

q4

2

q244

0T

cT

cT

cT

p

T

p

• Quark number density:

• Quark (Baryon) number susceptibility: diverges at E.

• Iso-vector susceptibility: does not diverge at E.

• Charge susceptibility: important for experiments.

• Chiral susceptibility: order parameter of the chiral phase transition

• We compute the Taylor expansion coefficients of these susceptibilities.

Fluctuations near critical endpoint E

du,du,du,

ln

pZ

V

Tn

2q

2

dudu

q

p

nn

2I

2

dudu

I

p

nn

dudu

C 3

1

3

2

3

1

3

2nn

qdu 2I

2

q

2C

4

1

36

1

TTT

For the case:

2duI

2duq

Z

V

Tp ln

Equation of State via Taylor Expansion

Equation of state at low density

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

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

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

Simulations• We perform simulations for Nf=2 at ma=0.1 (m/m0.70 at Tc)

and investigate T dependence of Taylor expansion coefficients.

• 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

6

q

6

3

3

64q

4

3

3

42q

2

3

3

2 )(

ln

!6 ,

)(

ln

!4 ,

)(

ln

2 T

Z

N

Nc

T

Z

N

Nc

T

Z

N

Nc

.)()(

ln

!6 ,

)()(

ln

!4 ,

)(

ln

2 4q

2I

6

3

3I62

q2

I

4

3

3I42

I

2

3

3I2 TT

Z

N

Nc

TT

Z

N

Nc

T

Z

N

Nc

4

qI6

2

qI4

I22

I 30122T

cT

ccT

4

q6

2

q422

30122T

cT

ccT

q

Derivatives of pressure and susceptibilities

• Difference between q and I is small at =0.Perturbation theory: The difference is O(g3)

• Large spike for c4, the spike is milder for iso-vector.

0at q0 cTT

Shifting the peak of d2/d2

• c6 changes the sign at Tc.• The peak of d2/d2 moves left, corresponding

to the shift of Tc.

increases

• c6 < 0 at T > Tc.Consistent with the perturbative pr

ediction in O(g3).

Difference of pressure for >0 from =0

Chemical potential effect is small. cf. pSB/T4~4. RHIC (q/T0.1): 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 6

q

4

q4

2

q244

0

O

Tc

Tc

T

p

T

p

Quark number susceptibility

• We find a pronounced peak for q/T~ 1. Critical endpoint in the (T,)?

• Peak position moves left as increases, corresponds to the shift of Tc()

6q

4

q6

2

q42q4

q 30122

O

Tc

Tcc

T 4

q

2

q42q4

q 122

O

Tcc

T

Iso-vector susceptibility

• No peak is observed.

Consistent with the prediction from the sigma model.

dudu

I nn

6

q

4

qI6

2

qI4

I2q4

I 30122

O

Tc

Tcc

T

(disconnected) chiral susceptibility

• Peak height increases as q increases.

Consistent with the prediction from the sigma model.

133

tr 1

)(

detln1

M

NNma

M

NN

6q

4

q4

2

q20

21213

tr tr 1

OT

cT

cc

MMNN

cscscs

Comparison to the hadron resonance gas• Non-interacting hadron gas: dependence must be

• Taylor expansion:

we get

,06

q6

4

q4

2

q244

Tc

Tc

Tc

T

p

T

p

,3

cosh92

q

T

TFT

q ,3

cosh 4

TTFTG

T

p q

T

TFTGT

qII 3cosh

2I

4

qI6

2

qI4

I22

I 30122T

cT

ccT

,103 ,103 ,43 I4

I64624 cccccc

4

q6

2

q422

30122T

cT

ccT

q

Hadron resonance gas or quark-gluon gas

• At T<Tc, consistent with hadron resonance gas model.

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

4

q6

2

q422

q 30122T

cT

ccT

4

qI6

2

qI4

I22

I 30122T

cT

ccT

Hadron resonance gas

Free QG gasFree QG gas

Hadron resonance gas

Hadron resonance gas for chiral condensate

• At T<Tc, consistent with hadron resonance gas model.

4

q4

2

q20 T

cT

cc Hadron resonance gas

Singular point at finite densityRadius of convergence

•

• We define the radius of convergence

• The SB limit of n for n>4 is

• At high T, n is large and 4 > 2 > 0

– No singular point at high T.

nnnn

nn

n ccTTcTc

2 2

2 qqq

nn

lim

,06

q6

4

q4

2

q244

Tc

Tc

Tc

T

p

T

p

Radius of convergence• The hadron resonance gas prediction

• The radius of convergence should be infinity at T<Tc.

• Near Tc, n is O(1)• It suggests a singular point around

Tc ~ O(1) ??

– However, still consistent with HRGM.– Too early to conclude.

n

nlim

9

)12)(22(

2

nn

nc

nc

n

Mechanical instability

• Unstable point

• We expect q to diverge at the critical endpoint.

Unstable point appears?

• There are no singular points.• Further studies are necessary.

0

q

q

Tq

n

n

p

T

n

n

p q

q

q

T

3tanh

3

1(resonance gas)

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

d.

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

• Quark number density fluctuations: A pronounced peak appears for /T0 ~ 1.0.

• Iso-spin fluctuations: No peak for /T0 <1.0.• Chiral susceptibility: peak height becomes larger as q increases.

This suggests the critical endpoint in (T,) plane?

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

• Also the extrapolation to the physical quark mass value and the continuum limit is important for experiments.