Determination of QCD phase diagram

from regions with no sign problem

Yahiro（ Kyushu University）

Collaborators:Y. Sakai, T. Sasaki and H. Kouno

Sign problem of LQCD at finite chemical potential

Phase factor

In the two-flavor case, the average of the phase factor isZ in the 1+1 system Z in the 1+1* system

0.4

Prediction of PNJL model

When ,LQCD is feasible onlyfor the deconfinement phase

2/ mq

4.02 ie

CEP

Y. Sakai, T. Sasaki, H. Kouno, and M. Yahiro, arXiv: 1005.0993[hep-ph](2010).

Fermion determinant

1. Purpose and StrategyPurpose is to determine QCD phase diagram at real chemical potential

1. LQCD has the sign problem at real chemical potential2. Regions with no sign problemI) Imaginary quark-number chemical potentialII) Real and imaginary isospin chemical potentials

3. Our strategy 1) construct reliable effective models in I)-II)

2) apply the model to real quark-number chemical potential.

2. Imaginary chemical potential

Roberge-Weiss Periodicity

confined

de-confined

de-confined

de-confined

Roberge, Weiss NPB275(1986)

RW phase transition

QCD partition function )3/2()( kZZ

,Uqq ,11 UUg

iUAUA

),( xU

where

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

Z3 transformation

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

for any integer k

Z3 transformation

)(Z

,Uqq ,11 UUg

iUAUA

3/2 k for integer k

Invariant under the extended Z3 transformation

)3/2()( ZZ

Extended Z3 transformation

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

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

Fukushima; PLB591

This is important to construct an effective model.

Y. Sakai, K. Kashiwa, H. Kouno and M. Yahiro,

Phys. Rev. D77 (2008), 051901; Phys. Rev. D78(2008), 036001.

Symmetries of QCD

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

Gs

0

3 pd

This model reproduces the lattice data at μ=0.

, Ratti, Weise; PRD75

Model parameters

Phase diagram for deconfinement phase trans.

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

PNJL RW

RW periodicity

Chiral

RW line

Deconfinement Forcrand,Philipsen,NP B642

Phase diagram for chiral phase transition

PNJL

ChiralDeconfinement

Model building

A. PNJL with higher-order interactions

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

B. PNJL with an effective four-quark vertex depending on Polyakov loop

Y. Sakai, T. Sasaki, H. Kouno, and M. Yahiro, arXiv:1006.3648[Hep-ph](2010).

Chiral

RW line

Deconfinement Forcrand,Philipsen,NP B642

Phase diagram for chiral phase transition

PNJL

ChiralDeconfinement

Θ-even higher-order interaction

Deconfinement

RW

Chiral

difference

＋PNJL

Forcrand,PhilipsenNPB642

Another correction

8-quark (Θ-even)

Vector-type interaction

RW

Chiral

＋PNJL ＋

8-quark (Θ-even) Vector-type (Θ-odd)

Deconfinement

Forcrand,PhilipsenNPB642

Sakai

Model building (B)

s

PNJL

sG

extended Z3 symmetry

333000

aaaaaa aAA

880

330 AiAi

eTr

888000

aaaaaa aAA More precise discussionby K. Kondo, in Hep-th:1005.0314and His poster.

Entanglement interactions such as 2)( sG

B. PNJL with an effective four-quark vertex depending on Polyakov loop Y. Sakai, T. Sasaki, H. Kouno, and M. Yahiro, arXiv:1006.3648[Hep-ph](2010).

Phase Diagram by original PNJL Phase Diagram by entanglement- PNJL

Entanglement PNJL

Isospin chemical potential

Entanglement PNJL

Real quark-number chemical potential

CEP

Summary

1) We proposed two types of PNJL models. One has higher-order interactions and the other

has an entanglement vertex. 2) Both the models give the same quality of agreement

with LQCD data.3) In both the models, CEP can survive.

Last question: Which model is better ?PNJL with the entanglement vertex is more reliable.

RW

E

M. D’Elia and F. Sanfilippo, Phys. Rev. D 80, 11501 (2009).P. de Forcrand and O. Philipsen, arXiv:1004.3144 [heplat](2010).

The order is first order for small and large quark masses, but second order for intermediate masses.

Order of RW phase transition

Entanglement PNJL

List of papers1. Y. Sakai, K. Kashiwa, H. Kouno and M. Yahiro. Phys. Rev. D77 (2008), 051901.2. Y. Sakai, K. Kashiwa, H. Kouno and M. Yahiro. Phys. Rev. D78(2008), 036001.3. Y. Sakai, K. Kashiwa, H. Kouno, M. Matsuzaki and M. Yahiro. Phys. Rev. D78(2008), 076007. 4. Y. Sakai, K. Kashiwa, H. Kouno, M. Matsuzaki, and M. Yahiro, Phys. Rev. D 79, 096001 (2009).5. Y. Sakai, H. Kouno, and M. Yahiro, arXiv: 0908.3088(2009).6. T. Sasaki, Y. Sakai, H. Kouno, and M. Yahiro, arXiv:hepph/ 1005.0910 [hep-ph] (2010).7. Y. Sakai, T. Sasaki, H. Kouno, and M. Yahiro, arXiv: 1005.0993 [hep-ph](2010).8. Y. Sakai, T. Sasaki, H. Kouno, and M. Yahiro, arXiv:1006.3648 [Hep-ph](2010).