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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
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
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).
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).