1. Imaginary Chemical potential and Determination of QCD phase diagram
From the effective theory
M. Yahiro (Kyushu Univ.)
Collaborators:
H. Kouno (Saga Univ.),
K. Kashiwa, Y. Sakai(Kyushu Univ.)
2009/08/3 XQCD 2009

2. Our papers on imaginary chemical potential
Polyakov loop extended NJL model with imaginary chemical potential,　Phys. Rev. D77 (2008),
Phase diagram in the imaginary chemical potential region and extended Z(3) symmetry,　Phys. Rev. D78(2008),
Vector-type four-quark interaction and its impact on QCD phase structure, Phys. Rev. D78(2008),
Meson mass at real and imaginary chemical potential, Phys. Rev. D 79, (2009).
Determination of QCD phase diagram from imaginary chemical potential region, Phys. Rev. D 79, (2009).
Correlations among discontinuities in QCD phase diagram, J. Phys. G to be published.

3. Prediction of QCD phase diagram
First-principle lattice calculation is difficult at finite real chemical potential,
because of sign problem.
Where is it ?
Lattice calculation is done with some approximation.
Sign problem
If we obtain the phase structure by using the first principle calculation, that is the Lattice QCD,
can not solve at finite chemical potential exactly because of sign problem.
Therefore, some approximations and approaches are suggested,
but these are far from perfection and only can obtained at T>m.
It means that the reliability is lost if m leave from m=0 point.
Therefore, the study of phase diagram is done by using effective models.
For example the NJL type model.
Where is the
critical end point?

4. Imaginary chemical potential
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, (2003);
Phys. Rev. D 70, (2004);
M. D’Elia, F. D. Renzo, and M. P. Lombardo, Phys. Rev. D 76, (2007);
H. S. Chen and X. Q. Luo, Phys. Rev. D72, (2005);
arXiv:hep-lat/ (2007).
S. Kratochvila and P. de Forcrand, Phys. Rev. D 73, (2006)
L. K. Wu, X. Q. Luo, and H. S. Chen, Phys. Rev. D76, (2007). ? T μ2
O.K.
effective model
Real μ
Imaginary μ

5. Roberge-Weiss periodicity
Nucl. Phys. B275(1986)
Dimensionless imaginary chemical potential:
Temperature:
QCD partition function

6. Z3 transformation
where
is an element of SU(3) with the boundary condition
for any integer

7. RW periodicity and extended Z3 transformation
under Z3 transformation.
Roberge-Weiss Periodicity
Invariant under the extended Z3 transformation
for integer

8. QCD has the extended Z3 symmetry in addition to the chiral symmetry
This is important to construct an effective model.
The Polyakov-extended Nambu-Jona-Lasinio (PNJL) model
Fukushima; PLB591

9. Polyakov-loop Nambu-Jona-Lasinio (PNJL) model
Two-flavor
Fukushima; PLB591
quark part (Nambu-Jona-Lasinio type)
gluon potential
, Ratti, Weise; PRD75
The PNJL model can treat both the chiral and the deconfinement phase transitions.
This lagragian is described as this. D is the covariant derivative, m is the current quark mass.
The interaction is the same as the NJL model. The U is the gluon potential.
This gluon potential is determined to reproduce the energy density, the entropy density and the pressure in the pure gauge limit.
The quark sector has the model parameters, current quark mass, coupling constant of 4-quark interaction and cutoff of the momentum integration.
These are determined from the experimental data at vacuum, but it can reproduce the lattice data at finite temperature and zero chemical potential.
For example, this figure shows the Polyakov-loop as the function of temperature at zero chemical potential.
The PNJL model reproduces the lattice data at zero chemical potential.
It reproduces the lattice data in the pure gauge limit.

10. Mean-field Lagrangian in Euclidean space-time
for
Performing the path integration of the PNJL partition function
the thermodynamic potential

11. Thermodynamic potential (1)
where
invariant under the extended Z3 transformation

12. Thermodynamic potential (2)
Polyakov-loop is not invariant under the extended Z3 transformation;
Modified Polyakov-loop
Thermodynamic potential
Extended Z3 invariant
RW periodicity:

13. Invariant under charge conjugation
Θ-evenness
Stationary condition
Invariant under charge conjugation
Θ-even

14. Gs Model parameters This model reproduces the lattice data at μ=0.
, Ratti, Weise; PRD75
This model reproduces the lattice data at μ=0.

15. Thermodynamic Potential
low T=Tc
high T=1.1Tc
Kratochvila, Forcrand; PRD73
low T
RW transition
high T
This figure shows the theta dependence of the PNJL thermodynamic potential.
The blue line shows the low temperature case and the red line shows the high temperature case.
The thermodynamic potentials are the RW periodic and the theta even.
At the low temperature, Omega is the smooth everywhere.
At the high temperature, Omega has the cusp at the theta equal to plus minus pi over 3.
This feature is consistent with the lattice data qualitatively.
As the result, the PNJL model can reproduce the lattice data at imaginary chemical potential.

16. Polyakov-loop susceptibility
PNJL
Lattice data:
Wu, Luo, Chen, PRD76(07).

17. Phase of Polyakov loop PNJL Lattice data:
Forcrand, Philipsen, NP B642(02),
Wu, Luo, Chen, PRD76(07)

18. Phase diagram for deconfinement phase trans.
Lattice data:
Wu, Luo, Chen, PRD76(07)
PNJL RW
RW periodicity

19. Chiral condensate and quark number density
Θ-odd
Θ-even
Low T
High T
Lattice
D’Elia, Lombardo(03)

20. Phase diagram for chiral phase transition
PNJL
Chiral
RW line
Deconfinement
Forcrand,Philipsen,NP B642
Chiral
Deconfinement
Θ-even higher-order interaction

21. Zero chemical potential
PNJL
Lattice data:
Karsch et al. (02)
At zero chemical potential, PNJL reproduce the lattice data for the Polyakov-loop, but not for the chiral condensate.

22. Higher order correction
8-quark ＋
PNJL
Θ-even in next-to-leading order ＋
Power counting rule
based on mass dimension
To solve this problem, we introduce the scalar-type 8-quark interaction.
The blue lines are results with the 8-quark interaction.
This difference is much reduce by the 8-quark interaction.
Lattice
Karsch, et. al.(02)

23. PNJL Θ-even in next-to-leading order 8-quark RW Chiral Deconfinement ＋
Forcrand,Philipsen,
NP B642
As the result, the chiral line approaches to the lattice data particularly at small theta.
Deconfinement

24. Another correction PNJL difference 8-quark (Θ-even) RW Chiral＋
PNJL RW
difference
Chiral
Forcrand,PhilipsenNPB642
But, the sizeable difference remains at large theta.
Deconfinement

25. Vector-type interaction
8-quark (Θ-even)
Vector-type (Θ-odd) ＋
PNJL RW
Chiral
Forcrand,PhilipsenNPB642
In order to solve this problem, we introduce the vector-type interaction.
The phase diagram at real chemical potential is quite sensitive to the strength, and it is important to determine the strength.
By tuning the strength of the vector-type interaction, the chiral transition line becomes consistent with the lattice data.
Therefore, by introducing the two interactions, the PNJL model can reproduce the lattice data quantitatively.
Deconfinement

26. Phase diagram at real μ PNJL Lattice 8-quark Vector-type confined＋
PNJL
CEP
confined
de-confined
Chiral
Deconfinement
Lattice
1’ st order RW
Finally, we have succeeded in determining the model parameters at imaginary chemical potential.
This figure shows the phase diagram at real and imaginary chemical potential by the PNJL calculation in which parameters are fixed at imaginary chemical potential.
The horizontal axis is the square of mu. The minus region is the imaginary chemical potential and the plus region is the real chemical potential.
The green dotted line is the chiral crossover line. The blue solid line is the confinement crossover line.
The red line is the chiral and deconfinement phase transition line of first-order.

27. Our result Critical End Point (784, 125) Stephanov Lattice2006 Lattice
Taylor Exp.(LTE)
Reweighting(LR)
Model
This is the accumulation of predictions on the location of critical end points presented by Stephanov.
Our result is located here.

28. Meson mass One-loop polarization function Mesonic correlation function
Random phase approximation
(Ring diagram approximation)
One-loop polarization function
H. Hansen, W. M. Alberico, A. Beraudo, A. Molinari, M. Nardi and C. Ratti, Phys. Rev. D 75 (2007)
K. Kashiwa, M. Matsuzaki, H. Kouno, Y. Sakai and M. Yahiro1, Phys. Rev. D 79, (2009).

29. Meson mass with RW periodicity
T=160 MeV
oscillation

30. Extrapolation
T=160 MeV
PNJL

31. PNJL well reproduces lattice data at imaginary μ.
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.

32. Thank you

33. Higher order correction
Θ-even in next-to-leading order ＋
PNJL
8-quark ＋
Mean field approx.
1/N expansion
Kashiwa et al. PLB647(07),446;
PLB662(08),26.
To solve this problem, we introduce the scalar-type 8-quark interaction.
The blue lines are results with the 8-quark interaction.
This difference is much reduce by the 8-quark interaction.
Lattice
Karsch, et. al.(02)