A method of finding the critical point in finite density QCD

Shinji Ejiri(Brookhaven National Laboratory)

Existence of the critical point in finite density lattice QCDPhysical Review D77 (2008) 014508 [arXiv:0706.3549]

Canonical partition function and finite density phase transition in lattice QCDPhysical Review D 78 (2008) 074507[arXiv:0804.3227]

Tools for finite density QCD, Bielefeld, November 19-21, 2008

QCD thermodynamics at 0

• Interesting properties of QCDMeasurable in heavy-ion collisions

Critical point at finite density

• Location of the critical point ?

– Distribution function of plaquette value

– Distribution function of quark number density

• Simulations: – Bielefeld-Swansea Collab., PRD71,054508(2005).

– 2-flavor p4-improved staggered quarks with m770MeV

– 163x4 lattice

– ln det M: Taylor expansion up to O(6)

quark-gluon plasma phase

hadron phase

RH

ICS

PS

AG

S

color super conductor?

nuclear matter

q

T

Effective potential of plaquette

Physical Review D77 (2008) 014508 [arXiv:0706.3549]

Effective potential of plaquette Veff(P) Plaquette distribution function (histogram)

• First order phase transition Two phases coexists at Tc

e.g. SU(3) Pure gauge theory

• Gauge action• Partition function

,, , PWdPZ

PNS siteg 6

PWPV ln)(eff

'e det ,' f PPMDUPW gSN Effective potential

SU(3) Pure gauge theoryQCDPAX, PRD46, 4657 (1992)

histogram

histogram

)6( 2g

Problem of complex quark determinant at 0

Problem of Complex Determinant at 0

Boltzmann weight: complex at 0 Monte-Carlo method is not applicable. Configurations cannot be generated.

55 )()( MM (5-conjugate)

)(det)(det)(det * MMM

)0,(

)0,(

f

f

f 0detdet

0det

det ,

PP-

MM

PP-

MPP-DU

MPP-DUPR

N

N

N

Distribution function and Effective potential at 0 (S.E., Phys.Rev.D77, 014508(2008))

• Distributions of plaquette P (1x1 Wilson loop for the standard action)

, , PWPRdPZ PNS siteg 6

gSNMPP-DUPW e 0det , f

Effective potential:

(Weight factor at

R(P,): independent of , R(P,) can be measured at any .

, ,ln)(eff PWPRPV

1st order phase transition?

,ln PR ,ln PW

(Reweight factor

+ = ??

=0 crossovernon-singular

Reweighting for and curvature of –lnW(P)

Change: 1(T) 2(T)

Weight:

Potential:

PNSS gg 12site12 6)()(

12112 WeWW gg SS

+ =

212site1 ln6ln WPNW

Curvature of –lnW(P) does not change.

Curvature of –lnW(P,) : independent of .

, , PWPRdPZ 'e 0det ,' f PPMDUPW gSN

-dependence of the effective potential

T

hadron

QGP

CSC

+ =

1st order phase transition

,ln PR ,ln PW

Critical point

,ln PR ,ln PW ,ln PW

+ =

Crossover

=0 reweighting

=0 reweighting

Curvature: Zero

Curvature: Negative

Sign problem and phase fluctuations• Complex phase of detM

– Taylor expansion: odd terms of ln det M (Bielefeld-Swansea, PRD66, 014507 (2002))

• || > /2: Sign problem happens. changes its sign.

• Gaussian distribution– Results for p4-improved staggered

– Taylor expansion up to O(5)

– Dashed line: fit by a Gaussian function

Well approximated

CTT histogram of

)(detlnImf MN

: NOT in the range of [-, ]

T

M

TT

M

TT

M

TN

5

55

3

33

f d

detlnd

!5

1

d

detlnd

!3

1

d

detlndIm

ie

error) al(statistic )0(det

)(det

FeM

M i

N f

FeFe Fi 22

Effective potential at (S.E., Phys.Rev.D77, 014508(2008))

Results of Nf=2 p4-staggared, m/m0.7

[data in PRD71,054508(2005)]

• detM: Taylor expansion up to O(6)

• The peak position of W(P) moves left as increases at =0.

,ln,ln,,eff PRPWPV

Rln

Wlnat =0

Solid lines: reweighting factor at finite /T, R(P,)

Dashed lines: reweighting factor without complex phase factor.

Truncation error of Taylor expansion

• Solid line: O(4)

• Dashed line: O(6)

• The effect from 5th and 6th order term is small for q/T 2.5.

N

nn

n

T

M

TnNMN

0

n

ffd

detlnd

!

1)(detln

Curvature of the effective potential

• First order transition for q/T 2.5• Existence of the critical point: suggested

– Quark mass dependence: large– Study near the physical point is important.

0

,ln, ln,,2

2

2

2

2eff

2

dP

PRd

dP

PWd

dP

PVd

at q=0

Critical point:

2

2 ln

dP

Wd

+ = ?

Nf=2 p4-staggared, m/m0.7

Canonical approach

Physical Review D 78 (2008) 074507[arXiv:0804.3227]

Canonical approach• Canonical partition function

• Effective potential as a function of the quark number N.

• At the minimum,

• First order phase transition: Two phases coexist.

NN

CGC NWTNNTZTZ exp ,,

TNNTZNWNV C ),(ln)(ln)(eff

0),(ln)(ln)(eff

TN

NTZ

N

NW

N

NV C

NN

N

NZC

)(ln

)(eff NV

T

First order phase transition line

• Mixed state First order transition

• Inverse Laplace transformation by Glasgow method Kratochvila, de Forcrand, PoS (LAT2005) 167 (2005)

Nf=4 staggered fermions, lattice– Nf=4: First order for all .

In the thermodynamic limit, ,0)(eff

N

NV

N

NTZ

TC

),(ln*

cpTT

cpTT

463

3

*

sNTT

• Fugacity expansion (Laplace transformation)

canonical partition function

• Inverse Laplace transformation

– Note: periodicity

• Derivative of lnZ

Canonical partition function

N

CGC TNNTZTZ exp ,, VN /

IGCTiTN

IC iTZeTdNTZ I

0

3

3,

2

3, 0

0

f

f

)0(det

)(detdet

0

1

0)(

NSN

GCGC

GC

M

MeMDU

ZZ

Z g

R

I

0

Integral

N

NTZ

TC

),(ln*

,32, TZiTTZ GCGC

Arbitrary 0

Integral path, e.g. 1, imaginary axis2, Saddle point

• Inverse Laplace transformation

• Saddle point approximation (valid for large V, 1/V expansion)– Taylor expansion at the saddle point.

• At low density: The saddle point and the Taylor expansion coefficients can be estimated from data of Taylor expansion around =0.

Saddle point approximation(S.E., arXiv:0804.3227)

00 zT

f

I

I

N

ITiTNI

GC

IGCTiTN

IC

M

iMeTd

Z

iTZeTdNTZ

0det

det

2

)0(3

, 2

3,

03

3

0

3

3

0

0

0detln

0

f

zT

T

M

V

NSaddle point: 0z

R

I

0

Integral

Saddle point

0

tf0

n

ffd

detlnd

!

1)(detln

n

n

nn

n

n

TDNVN

T

M

TnNMN

3sNV

VN /

• Canonical partition function in a saddle point approximation

• Chemical potential

Saddle point approximation

)0,(

)0,(0

20

0f

exp2

3

1

)0(det

)(detlnexp

2

3

0,

,

T

T

i

GC

C

iF

zRVezV

M

zMN

TZ

TZ

ieR

T

M

V

N

TR 2

2f detln

Saddle point: 0z

)0,(

)0,(0*

exp

exp),(ln1)(

T

TC

iF

iFzTZ

VT

saddle point reweighting factor

Similar to the reweighting method(sign problem & overlap problem)

Saddle point in complex /T plane

• Find a saddle point z0 numerically for each conf.

• Two problems– Sign problem– Overlap problem

0detln

0

f

zT

T

M

V

N

Technical problem 1: Sign problem

• Complex phase of detM– Taylor expansion (Bielefeld-Swansea, PRD66, 014507 (2002))

• || > /2: Sign problem happens. changes its sign.• Gaussian distribution

– Results for p4-improved staggered– Taylor expansion up to O(5)

– Dashed line: fit by a Gaussian function

Well approximated

0.23 T

histogram of 2

eW

)(detlnIm(phase) f MN

: NOT in the range [-, ]

ie

2Im 0

10tf

zzDNNVn

n

FFi eeee F22

Technical problem 2: Overlap problemRole of the weight factor exp(F+i)

• The weight factor has the same effect as when (T) increased.• */T approaches the free quark gas value in the high density

limit for all temperature.

3

2f3

1

TTN

T

free quark gas

free quark gas

free quark gas

Technical problem 2: Overlap problem

• Density of state method W(P): plaquette distribution

dPPWiF

dPPWiFz

TP

P

)(exp

)(exp)( 0*

)( 2exp)(exp 2 PWFPWiFPPP

linear for small Plinear for small P

)( exp eff PWP for small PSame effect when changes.

Reweighting for =6g-2

Change: 1(T) 2(T)

Distribution:

Potential:

PNSS gg 12site12 6)()(

12112 WeWW gg SS

+ =

212site1 ln6ln WPNW

site

2

eff

1

2

1

NdP

d

dP

FdPP

(Data: Nf=2 p4-staggared, m/m0.7, =0)

Effective (temperature) for 0

( increases) ( (T) increases)

'e det ,' )(f PPMDUPW gSN

Overlap problem, Multi- reweightingFerrenberg-Swendsen, PRL63,1195(1989)

• When the density increases, the position of the importance sampling changes.

• Combine all data by multi- reweighting

Problem:• Configurations do not cover all

region of P.• Calculate only when <P> is near

the peaks of the distributions.

Plaquette value by multi-beta reweighting peak position of the distribution ○ <P> at each

P

)0,(

)0,(

exp

exp

T

T

iF

iFPP

Chemical potential vs density

• Approximations: – Taylor expansion: ln det M– Gaussian distribution: – Saddle point approximation

• Two states at the same q/T– First order transition at T/

Tc < 0.83, q/T >2.3

• */T approaches the free quark gas value in the high density limit for all T.

• Solid line: multi-b reweighting• Dashed line: spline interpolation• Dot-dashed line: the free gas limit

Nf=2 p4-staggered, lattice4163

Number density

Summary• Effective potentials as functions of the plaquette value and the quark

number density are discussed.

• Approximation:– Taylor expansion of ln det M: up to O(6)– Distribution function of =Nf Im[ ln det M] : Gaussian type.– Saddle point approximation (1/V expansion)

• Simulations: 2-flavor p4-improved staggered quarks with m/m 0.7 on 163x4 lattice– Existence of the critical point: suggested.– High limit: /T approaches the free gas value for all T.– First order phase transition for T/Tc < 0.83, q/T >2.3.

• Studies near physical quark mass: important.– Location of the critical point: sensitive to quark mass