Importance of imaginary chemical potential for QCD phase diagram in the PNJL model

Importance of imaginary chemical potential

for QCD phase diagram in the PNJL modelKouji Kashiwa

H. Kouno A, Y. Sakai, T. Matsumoto and M. Yahiro

Recent studies (2008 – 2010)

Kyushu Univ., Saga Univ.A

1/20

Y. Sakai, K. K., H. Kouno and M. Yahiro, Phys. Rev. D 77 (2008) 051901(R).　　　　　　Y. Sakai, K. K., H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Rev. D 78 (2008) 036001.　　　　　　

Y. Sakai, K. K, H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Rev. D 79 (2009) 096001.　　　　　　

K. K., M. Matsuzaki, H. Kouno, Y. Sakai and M. Yahiro, Phys. Rev. D 79 (2009) 076008.

K. K., M. Yahiro, H. Kouno, M. Matsuzaki and Y. Sakai, J. Phys. G. 36 (2009) 105001.

K. K, H. Kouno and M. Yahiro, Phys. Rev. D 80 (2009) 117901.

K. K., H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B 662 (2008) 26.　　　　　　

H. Kouno, Y. Sakai, K. K., and M. Yahiro, J. Phys. G. 36 (2009) 115010.

Introduction

Quantitative understanding of the QCD phase diagram at finite R is quite poor.

In recent theoretical studies, novel scenarios for QCD phase diagram are suggested.

Quarkyonic phase

Multi critical-endpoints generationex.)

M. Kitazawa, T. Koide, T. Kunihiro and Y. Nemoto, Prog. Theor. Phys. 108 (2002) 929.T. Hatsuda, M. Tachibana, N. Yamamoto and G. Baym, Phys. Rev. Lett. 97 (2006) 122001.

Z. Zhang, K. Fukushima and T. Kunihiro, Phys. Rev. D 79 (2009) 014004.

M. Harada, C. Sasaki and S. Takemoto, arXiv:0908.1361.

Qualitative understanding is now running well !!

However,

and more …

Lifshitz-point induced by the inhomogeneous phase

D. Nickel, Phys. Rev. Lett. 103 (2009) 072301; Phys. Rev. D 80 (2009) 074025.

L. McLerran and R. D. Pisarski, Nucl. Phys. A 796 (2007) 83.

Y. Hidaka, L. McLerran and R. D. Pisarski, Nucl. Phys. A 808 (2008) 117.K. Miura, T. Z. Nakano and A. Ohnishi. Prog. Theor. Phys. 122 (2009) 1045.

L. McLerran, K. Redlich and C. Sasaki, hep-ph/0812.3585

Introduction

A schematic viewLHC

RHIC

GSI

JPARC

Ear

ly u

niv

erse

Compact star

ρ0

AGS

SPS

KEK-PS

Quantitative understanding of 　　　　 the QCD phase diagram is important.

To investigate these properties quantitatively, our present understanding of the QCD phase diagram is not enough.

Problem

A schematic viewHowever, first principle lattice QCD simulation have the sign problem at real chemical potential.

K. K, H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B 662 (2008) 26. 　　

Effective models have some ambiguities.

Imaginary chemical potential

Reason 1 :

Strategy

LQCD simulation can exactly calculated in the imaginary chemical potential,because there is no sign problem.

Imaginary chemical potential region has almost all information of real one.

Reason 2:

In usual studies using information in the I region, we can not reach the moderate and high R region.

To exploit it more,

K. K, M. Matsuzaki, H. Kouno, Y. Sakai and M. Yahiro, Phys. Rev. D 79 (2009) 076008.M. P. Lombardo, PoSCPOD2006 (2006) 003, hep-lat/0612017.

We pay an attention to the imaginary chemical potential.

Strategy QCD phase diagram at finite

0 4/3

Imaginary chemical potential matching approach

11. We determine the strengths of interactions from LQCD data at I .

22. We then apply the model to the R region.

(Usual effective models have some ambiguityies caused by chemical potential effects)

(This extended model can well describe (real) chemical potential effects because the effects are taken into account through the comparison between model result and LQCD data at finite I.)

Through this method, we can explore the moderate and high chemical potential !

The imaginary chemical potential region have several and important information of real chemical potential.

Roberge Weiss (RW) phase transition line

Imaginary chemical potential

Phase diagram at imaginary chemical potential

Properties:

Thermodynamical quantities have the RW periodicity.

New transition line appears.

Period ： 2/3

RW transition line

These properties become strong constraint for the extended model.

2/3

0

Remains of the Z3 symmetry

in pure gauge limit.

A. Roberge and N. Weiss, Nucl. Phys. B 275 (1986) 735.

4/3

These properties are directly obtained from QCD.

= I/T

Roberge Weiss (RW) phase transition line

Dimensionless chemical potential

PNJL model

The Lagrangian density of the PNJL model

2 20 5( ) ( ) ( , )sq i D m q G qq qi q U

L

Chiral phase transition

（ Approximately) 　 Deconfinement phase transition

K. Fukushima, Phys. Lett. B 591 (2004) 277.

Which model should be used in the imaginary chemical potential matching approach?

Our model must have the RW periodicity.

We already know the model!

Effective gluon propagator

Y. Sakai, K. K, H. Kouno and M. Yahiro, Phys. Rev. D 77 (2008) 051901(R). 　　　　　　

Y. Sakai, K. K, H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Rev. D 78 (2008) 036001. 　　　　　　

Intuitively,

ex.) C. Ratti, M. A. Thaler and W. Weise, Phys. Rev. D 73 (2006) 014019.

The PNJL model can reproduce QCD properties at finite T and zero .

S. Ro¨ßner, C. Ratti, and W. Weise, Phys. Rev. D 75 (2007) 034007.

PNJL model

Importance of multi-leg interaction

Usual NJL-type model ・・・ scalar-type four-quark interaction only.

Other interactions are neglected.(vector-type four-quark, scalar-type eight-quark interactions…)

However, there are no reason that these are neglected.

Mass term

Same order （ 1/Nc expansion ）　　

The scalar-type eight-quark interaction leads the T and dependence to the effective coupling constant.

m0

G G4

Gs8

K. K., H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B662 (2008) 26. 　　

K. K., H. Kouno, T. Sakaguchi, M. Matsuzaki and M. Yahiro, Phys. Lett. B 647 (2007) 446. 　　

K. K., M. Matsuzaki, H. Kouno, and M. Yahiro, Phys. Lett. B 657 (2007) 143. 　　

4-leg 8-leg

(in Lagrangian density)

22 2

8 5sG qq qi q

2

vG q q+

+

Our model(Vector-type)

,ie

.ie

K.K., Y. Sakai, H. Kouno, M. Matsuzaki and M. Yahiro, J. Phys. G36 (2009) .

RW periodicity in chiral limit

Lattice data:

H. S. Chen and X. Q. Luo, Phys. Rev. D 72 (2005) 034504

M. D’ Elia and M. P. Lambard, Phys. Rev. D 67 (2003) 145005.

In this region, different –order discontinuities can co-exist .

This fact can be proofed by model independent analysis.

Modified Polyakov-loop

These are RW periodic quantities.

Parameter set obtained by our approach and results

PNJL model in realistic case

0

0.4 0.8 1.0 P. de Forcrand and O. Philipsen, Nucl. Phys. B 642 (2002) 290.

L. K. Wu, X. Q. Luo and H. S. Chen, Phys. Rev. D 76 (2007) 034505.

Lattice data:

(Test fitting)


Set C

PNJL model

Phase diagram


In our extended PNJL model has one critical endpoint !!

Meson mass

Meson masses in PNJL model

Meson masses are good quantities

to determined strengths and kinds of interactions in the PNJL model.

The 2 flavor case: and meson

The 2+1 flavor case: a0, f0 and ’ meson

Meson masses do not depend on the renormalization point.

Model parameters (NJL part) largely affect meson masses.

Meson mass formula

Meson masses in PNJL model

1 2 0s ssG

1 2 0s ppG 4 2

4 2 2 2 2

( / 2) 1

(2 ) [( / 2) ] [( / 2) ]ss

d k p q Mi

p q M i p q M i

4 2

4 2 2 2 2

( / 2) 1

(2 ) [( / 2) ] [( / 2) ]pp

d k p q Mi

p q M i p q M i

・・・

( ) 12 2 2

2

1 2

xeff x s s x s x

sx x

s x

iG iG iGi

iG

G

･･･

　　

Random phase approximation

At finite chemical potential in the PNJL model :

H. Hansen, W. M. Alberico, A. Beraudo, A. Molinari, M. Nardi and C. Ratti, Phys. Rev. D 75 (2007) 065004.

p4 p4 +(– iA4)

We consider the scalar and pseudo-scalar meson masses.

K. K, M. Matsuzaki, H. Kouno, Y. Sakai and M. Yahiro, Phys. Rev. D 79 (2009) 076008.

pp

Meson masses have the RW periodicity!

/I T

Opposite oscillation

T=160 MeV

PNJL resultsK. K, M. Matsuzaki, H. Kouno, Y. Sakai and M. Yahiro, Phys. Rev. D79 (2009) 076008.

M

T Matsumoto, K. K, H. Kouno and M. Yahiro, in preparation.

PNJL results

We use the 2+1-flavor PNJL model.

To consider the UA(1) anomaly effect, following determinant interaction is introduced.

This K can have the T and dependence because of the variation of instanton density.

However, quantitative behavior is not known.

Therefore, the 2+1-flavor system is very ambiguous at finite real chemical potential.

ex.) J. -W. Chen, K. Fukushima, Hi. Kohyama, K.Ohnishi, U. Raha, Phys. Rev. D80 (2009) 054012.

Meson masses also have the RW periodicity!T=300 MeV


PNJL results

Chiral symmetry re-brokenChiral symmetry re-brokenQualitatively behavior is same as the 2-flavor results.

Result

Near =/3, the chiral symmetry is broken again.

To investigate the UA(1) anomaly effects,we will vary the strength of K.


T=300 MeV T=300 MeV

PNJL results

Chiral symmetry re-brokenChiral symmetry re-broken

Qualitative difference arises near =/3. T=300 MeV


PNJL results

Qualitative difference arises near =/3.

meson mass ’ meson mass

T=300 MeV

Summary

We investigate properties of the imaginary chemical potential by using the PNJL model.

At the imaginary chemical potential, the strengths of the vector-type interaction and also determinant interaction can be determined.

To quantitatively investigate the phase structure at finite real chemical potential,we propose the imaginary chemical potential matching approach.

If we can refer LQCD data for several meson masses and thermodynamical quantities, we can quantitatively investigate the phase structure at finite R.

The imaginary chemical potential matching approach can be applied to the 2+1 flavor system.

END

Rw periodicity

Strange worldImaginary chemical potential

2/3 periodicity

Phase diagram at imaginary chemical potential

Partition function of SU(N) gauged theory with imaginary chemical potential

4 20 4

1( ) exp ( ) ( )

4Z D D DA d x D m F i

,

,D iA 2

a

aA gA

is the Gell-Mann matrix, g is the gauge coupling constanta

2/3


4 20

1( ) exp ( ) ( )

4Z D D DA d x D m F

( , ) exp( / ) ( , )x i x

Boundary condition : ( , ) exp( ) ( ,0)x i x

;U 1 1( )i

A UAU U Ug

2

( , ) exp ( ,0)ik

U x U xN

Boundary condition : ( , ) exp(2 / ) exp( ) ( ,0)x ik N i x

2( )

kZ Z

N

Ｚ（ θ ） has the periodicity of 2πk/N !!

ZN transformation(Gauge)

Relation between imaginary and real chemical potential


This strange world is useful space because there are many important information about real chemical potential.

M. P. Lombardo, PoSCPOD2006 (2006) 003, hep-lat/0612017.

We assume the smooth connection at 2=0.

Those results suggest that the physics at real is deeply related to imaginary one.

(4-flavor)

0 1

0 1

[ , ]( )M

i M ii N

i N i

a a aP N M

b b b

P[0,M] is corresponding the to Taylor N-th partial sum.

Numerical extrapolation Good method?

M. P. Lombardo, PoSCPOD2006 (2006) 003, hep-lat/0612017.

ex.) Gross-Neveu model

Tri-critical point

H. Kohyama, Phys. Rev. D 77 (2008) 045016.

0 0/r if the CSC is take into account,the phase structure is modified.

In this simple case, we can not reach the structure at high chemical potential region by using the extrapolation.

But,

More simpler model than the NJL (QCD motivated) model

Phase diagram

F. Karbstein and M. Thies, Phys. Rev. D 75 (2007) 025003.

In QCD

Thermodynamical value become -even or -odd function at imaginary chemical potential.

Point of extrapolation

ex.)

chiral condensate, real part of modified Polyakov-loop-even ・・・ :

-odd: ・・・quark number density, imaginary part of modified Polyakov-loop

CP invariance Quantities are function of 2.

/Canonical Grand Canonical( , ) IiB TIZ T B d e Z

T

/Grand Canonical Canonical( , ) ( , )RB T

RB

Z T e Z T B

Fugacity expansion:

Relation between imaginary and real chemical potential

Fourier representation:

Numerical extrapolation

Current quark mass dependence

Meson mass is good indicator of the consistency about current mass.

T=160 MeV

Result 3-3

QCD to GCM to NJL model

Derivation of the NJL model from QCD 1

Lagrangian density of QCD

0

1( )

4q i m q F F gq A q

L

Gluon Field strength : a a abc b cF A A gf A A

2

aaj q q

4expaQCD QCDZ DqDq DA i d x L

40exp ( ) [ ]DqDq i d x q i m q iW j

4 41[ ] ln exp ,

4a aa aiW j DA d x F F ig d x A j


1 2 1 1 2

1

1 1 1

1

(1) 41 1 1

2,(2) 4 4

1 2 , 1 2 1 2

...( ) 4 41 ... 1 1

[ ] [0] ( ) ( )

( , ) ( ) ( ) ...2

( ... ) ( )... ( ) ...!

n n n

n

a

a a a a

na a aan

n n n

iW j iW g W x j x d x

gW x x j x j x d x d x

gW x x j x j x d x d x

n

...

Expansion in powers of quark current.

( )1Cofficients ( ... ) are the connected n-point function of gluons without quark loop.n

nW x x

The GCM consists in keeping only W(2).

It can reproduce properties of the QCD such as the confinement, asymptotic freedom.


Lagrangian density of GCM

2(2)

0( ) ( ) ( , ) ( )2

a babGCM

gq i m q j x W x y j y

L

Effective gluon propagator

2 28

0 50

( )2 2

i i

NJL sa

q i m q G q q qi q

L

Fierz transformation

Local approximation

Other method is using the field strength method. In this method, quadratic expansion around auxiliary field and limit of small momentaare used.

Interaction of the NJL model 1

1 gluon exchange interaction 4 quark interaction

gluon

3 3 1 8 Color-singlet( )qqAttractive

Gluon degree of freedom is integrated out

Local approximation

1 1

2 2ab cd ac bd ab cd ac bd

N N

N N

aT aT

a

b

c

d

, , ,a b c d Color…Color anti-symmetric Color symmetric

Attractive Repulsive

3 3 3 6

( )qq

In the ordinary NJL model, interactions are constructed by four-quark interaction only.

Parameter set

Formalism

Polyakov-loop potential

3 3 232 44

( )( , ; )( ) ( )

2 6 4

bb T bU T

T

RTW05:

4

( , ; ) 1( )

2

U Ta T

T

RRW06:

3 3 2( ) ln 1 6 4( ) 3( )b T

2 3

0 0 02 0 1 2 3( )

T T Tb T a a a a

T T T

2

0 00 1 2( )

T Ta T a a a

T T 3

03( )T

b T aT

C. Ratti, M. A. Thaler and W. Weise, Phys. Rev. D73, 014019 (2006).

S. Ro¨ßner, C. Ratti and W. Weise. Phys. Rev. D75, 034007 (2007).

PNJL model

Parameter set

K. K., H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B662 (2008) 26. 　　

K. K., H. Kouno, T. Sakaguchi, M. Matsuzaki and M. Yahiro, Phys. Lett. B 647 (2007) 446. 　　

K. K., M. Matsuzaki, H. Kouno, and M. Yahiro, Phys. Lett. B 657 (2007) 143. 　　

138

650

93.3

M MeV

M MeV

f MeV

Gs 　 (scalar type 4-quark interaction) Gs8 (scalar type 8-quark interaction) Λ 　 (3 dimensional momentum cutoff ).

Parameters: Gs, Gv, Gs8

138

600

93.3

M MeV

M MeV

f MeV

or

Our usual procedure

Our new procedureParameters: Gs, Gv, Gs8

Gv is free parameter.

All parameters are fitted in imaginary chemical potential region (and =0 region).

Y. Sakai, K. K, H. Kouno and M. Yahiro, Phys. Rev. D 77 (2008) 051901(R). 　　　　　　

Y. Sakai, K. K, H. Kouno and M. Yahiro, Phys. Rev. D 78 (2008) 036001. 　　　　　　


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Importance of imaginary chemical potential for QCD phase diagram in the PNJL model