Imaginary Chemical potential and Determination of QCD phase diagram

M. Yahiro (Kyushu Univ.)

Collaborators: H. Kouno (Saga Univ.),

K. Kashiwa, Y. Sakai(Kyushu Univ.)

2009/08/3 XQCD 2009

From the effective theory

2008-2009

• Polyakov loop extended NJL model with imaginary chemical potential,　 Phys. Rev. D77 (2008), 051901.

• Phase diagram in the imaginary chemical potential region and extended Z(3) symmetry,　 Phys. Rev. D78(2008), 036001.

• Vector-type four-quark interaction and its impact on QCD phase structure, Phys. Rev. D78(2008), 076007.

• Meson mass at real and imaginary chemical potential, Phys. Rev. D 79, 076008 (2009).

• Determination of QCD phase diagram from imaginary chemical potential region, Phys. Rev. D 79, 096001 (2009).

• Correlations among discontinuities in QCD phase diagram, J. Phys. G to be published.

Our papers on imaginary chemical potential

First-principle lattice calculation is difficult at finite real chemical potential, because of sign problem.

Lattice calculation is done with some approximation.Where is it ?

Sign problem

Where is thecritical end point?

Prediction of QCD phase diagram

• Motivation Lattice QCD has no sigh problem.

• Lattice data P. de Forcrand and O. Philipsen, Nucl. Phys. B642, 290 (2002); P. de Forcrand and O. Philipsen, Nucl. Phys. B673, 170 (2003). M. D’Elia and M. P. Lombardo, Phys. Rev. D 67, 014505(2003); Phys. Rev. D 70, 074509 (2004); M. D’Elia, F. D. Renzo, and M. P. Lombardo, Phys. Rev. D 76,

114509(2007); H. S. Chen and X. Q. Luo, Phys. Rev. D72, 034504 (2005); arXiv:hep-lat/0702025 (2007). S. Kratochvila and P. de Forcrand, Phys. Rev. D 73, 114512

(2006) L. K. Wu, X. Q. Luo, and H. S. Chen, Phys. Rev. D76,

034505(2007).

Imaginary chemical potential

?

0

T

μ2

O.K.

effective model

Real μ Imaginary μ

Nucl. Phys. B275(1986)

QCD partition function

Dimensionless imaginary chemical potential:

Ti

TTemperature:

]exp[)( SDAqDqDZ

4

)(2

044 F

qmiTDqxdS

Roberge-Weiss periodicity

,Uqq ,11 UUg

iUAUA

),( xU

where

is an element of SU(3) with the boundary condition

)0,()/1,( 3/2 xUeTxU ki

for any integerk

Z3 transformation

under Z3 transformation.)3/2()( kZZ

Roberge-Weiss Periodicity

)3/2()( kZZ

,Uqq ,11 UUg

iUAUA

3/2 k for integer k

Invariant under the extended Z3 transformation

RW periodicity and extended Z3 transformation

QCD has the extended Z3 symmetry in addition to the chiral symmetry

The Polyakov-extended Nambu-Jona-Lasinio (PNJL) model

Fukushima; PLB591

This is important to construct an effective model.

gluon potentialquark part (Nambu-Jona-Lasinio type)

Polyakov-loop Nambu-Jona-Lasinio (PNJL) model

It reproduces the lattice data in the pure gauge limit.

, Ratti, Weise; PRD75

Fukushima; PLB591

TiAe /c

4Tr3

1

Two-flavor

UGqMDqL SMF 24 )(

for

Performing the path integration of the PNJL partition function

,20 SGmM

Mean-field Lagrangian in Euclidean space-time

VZ /logthe thermodynamic potential

UGS 2

,)( 22 MppE ,)()()( TipEpEPE

where

Thermodynamic potential (1)

,1

T

invariant under the extended Z3 transformation

Thermodynamic potential (2)

Modified Polyakov-loop

UGS 2

Thermodynamic potential

RW periodicity:

Polyakov-loop is not invariant under the extended Z3 transformation;

Extended Z3 invariant

Invariant under charge conjugation

Θ-evenness

Θ-even

*,

)()(

UGS 20/

Stationary condition

Gs

0

3 pd

This model reproduces the lattice data at μ=0.

, Ratti, Weise; PRD75

Model parameters

Thermodynamic Potential

Kratochvila, Forcrand; PRD73

low T

high T

low T=Tchigh T=1.1Tc

RW transition

Polyakov-loop susceptibility

Lattice data: Wu, Luo, Chen, PRD76(07).

PNJL

Phase of Polyakov loop

Lattice data:Forcrand, Philipsen, NP B642(02), Wu, Luo, Chen, PRD76(07)

PNJL

Phase diagram for deconfinement phase trans.

Lattice data:Wu, Luo, Chen, PRD76(07)PNJL RW

RW periodicity

Chiral condensate and quark number densityChiral Condensate Quark Number

LatticeD’Elia, Lombardo(03)

High T

Low T

Θ-even Θ-odd

Chiral

RW line

Deconfinement Forcrand,Philipsen,NP B642

Phase diagram for chiral phase transition

PNJL

ChiralDeconfinement

Θ-even higher-order interaction

Zero chemical potential

PNJL

Lattice data:Karsch et al. (02)

Higher order correction

＋

LatticeKarsch, et. al.(02)

＋PNJL

8-quark

Θ-even in next-to-leading order

Power counting rule based on mass dimension

RW

Chiral

＋PNJL 8-quark

Deconfinement

Forcrand,Philipsen,NP B642

Θ-even in next-to-leading order

Deconfinement

RW

Chiral

difference

＋PNJL

Forcrand,PhilipsenNPB642

Another correction8-quark (Θ-even)

Vector-type interaction

RW

Chiral

＋PNJL ＋

8-quark (Θ-even) Vector-type (Θ-odd)

Deconfinement

Forcrand,PhilipsenNPB642

CEP

confined

de-confinedChiral

Deconfinement

Lattice

1’ st order

RW

CEP

Phase diagram at real μ

＋PNJL ＋

8-quark Vector-type

(784, 125)Our result

LatticeTaylor Exp.(LTE)Reweighting(LR)Model

Critical End Point

Stephanov Lattice2006

1 2 0s ssG

1 2 0s ppG

・・・

( ) 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 (Ring diagram approximation)

H. Hansen, W. M. Alberico, A. Beraudo, A. Molinari, M. Nardi and C. Ratti, Phys. Rev. D 75 (2007) 065004.K. Kashiwa, M. Matsuzaki, H. Kouno, Y. Sakai and M. Yahiro1, Phys. Rev. D 79, 076008 (2009).

Meson mass

Mesonic correlation function

One-loop polarization function

/I T

oscillation

T=160 MeV

Meson mass with RW periodicity

T=160 MeV

Extrapolation

1T

8

0

2

k

kkcM

PNJL

1T

Conclusion

• QCD has a higher symmetry at imaginary μ, called the extended Z3 symmetry.

• PNJL has this property. • PNJL well reproduces lattice data at imaginary μ.• PNJL predicts that the CEP survives, even if the

vector interaction is taken into account. • Meson mass also has RW periodicity at

imaginary μ.

Thank you

Higher order correction

＋

LatticeKarsch, et. al.(02)

＋PNJL 8-quark

Mean field approx.1/N expansion

Kashiwa et al. PLB647(07),446;PLB662(08),26.

Θ-even in next-to-leading order