Phase diagram in the imaginary chemical potential reigion
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Transcript Phase diagram in the imaginary chemical potential reigion
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
Our papers on imaginary chemical potential
2008-2009
•
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•
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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.
Prediction of QCD phase diagram
First-principle lattice calculation is difficult at finite real chemical potential,
because of sign problem.
Lattice calculation is done with some approximation.
Where is the
critical end point?
Where is it ?
Imaginary chemical potential
• Motivation
T
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).
Real μ
?
O.K.
0
μ2
effective model
Imaginary μ
Roberge-Weiss periodicity
Nucl. Phys. B275(1986)
Dimensionless imaginary chemical potential:
iT
T
Temperature:
QCD partition function
Z ( ) DqD q DA exp[ S ]
F
S d x q (D iT 4 m0 )q
4
2
4
Z3 transformation
q Uq,
i
1
A UAU U U ,
g
1
where
U ( x, )
is an element of SU(3) with the boundary condition
U ( x,1 / T ) e
for any integer k
i 2k / 3
U ( x,0)
RW periodicity and extended Z3 transformation
Z ( ) Z ( 2k / 3)
under Z3 transformation.
Z ( ) Z ( 2k / 3)
Roberge-Weiss Periodicity
Invariant under the extended Z3 transformation
i
1
q Uq, A UAU U U ,
g
1
2k / 3
for integer
k
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
Polyakov-loop Nambu-Jona-Lasinio (PNJL) model
Two-flavor
Fukushima; PLB591
quark part (Nambu-Jona-Lasinio type)
, Ratti, Weise; PRD75
gluon potential
1
Trc e iA4 / T
3
It reproduces the lattice data in
the pure gauge limit.
Mean-field Lagrangian in Euclidean space-time
LMF q( D 4 M )q GS 2 U
for
M m0 2GS ,
Performing the path integration of the PNJL partition function
the thermodynamic potential
logZ / V
Thermodynamic potential (1)
GS 2 U
where
1
, E ( p)
T
p2 M 2 ,
E ( P) E( p) E( p) iT ,
invariant under the extended Z3 transformation
Thermodynamic potential (2)
Polyakov-loop is not invariant under the extended Z3 transformation;
Modified Polyakov-loop
Thermodynamic potential
GS 2 U
Extended Z3 invariant
RW periodicity:
Θ-evenness
GS 2 U
Stationary condition
/ 0
Invariant under charge conjugation
, *
Θ-even
( ) ( )
Model parameters
Gs
3
d
p
0
, Ratti, Weise; PRD75
This model reproduces
the lattice data at μ=0.
Thermodynamic Potential
low T=Tc
high T=1.1Tc
Kratochvila, Forcrand; PRD73
low T
RW transition
high T
Polyakov-loop susceptibility
PNJL
Lattice data:
Wu, Luo, Chen, PRD76(07).
Phase of Polyakov loop
PNJL
Lattice data:
Forcrand, Philipsen, NP B642(02),
Wu, Luo, Chen, PRD76(07)
Phase diagram for deconfinement phase trans.
PNJL
RW
RW periodicity
Lattice data:
Wu, Luo, Chen, PRD76(07)
Chiral condensate and quark number density
Quark Number
Chiral Condensate
Θ-even
Low T
High T
Lattice
D’Elia, Lombardo(03)
Θ-odd
Phase diagram for chiral phase transition
PNJL
RW line
Chiral
Deconfinement
Forcrand,Philipsen,NP
B642
Chiral
Deconfinement
Θ-even higher-order interaction
Zero chemical potential
PNJL
Lattice data:
Karsch et al. (02)
Higher order correction
8-quark
PNJL
＋
Θ-even in next-to-leading order
＋
Power counting rule
based on mass dimension
Lattice
Karsch, et. al.(02)
Θ-even in next-to-leading order
PNJL
8-quark
＋
RW
Chiral
Forcrand,Philipsen,
NP B642
Deconfinement
Another correction
8-quark (Θ-even)
PNJL
＋
RW
difference
Chiral
Forcrand,PhilipsenN
PB642
Deconfinement
Vector-type interaction
8-quark (Θ-even)
PNJL
Vector-type (Θ-odd)
＋
＋
RW
Chiral
Forcrand,PhilipsenN
PB642
Deconfinement
Phase diagram at real μ
8-quark
PNJL
Vector-type
＋
＋
RW
Chiral
Deconfinement
confined
de-confined
C
E
CEP
P
1’ st order
Critical End Point
Stephanov Lattice2006
Our result
(784, 125)
Lattice
Taylor Exp.(LTE)
Reweighting(LR)
Model
Meson mass
Mesonic correlation function
Random phase approximation
(Ring diagram approximation)
1 2Gs ss 0
・・・
1
x 2iGs 2iGs x 2iGs ･･･ x
i
2iGs
x
x
1 2Gs x
( x)
eff
One-loop polarization function
1 2Gs pp 0
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 with RW periodicity
T=160 MeV
oscillation
I / T
Extrapolation
T=160 MeV
PNJL
8
M ck
2k
k 0
T
1
1
T
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
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.
Lattice
Karsch, et. al.(02)