imaginary chemical potential and determination of qcd phase diagram m. yahiro (kyushu univ.)...
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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