1. QCD Phase Transition in the Dyson-Schwinger Equation Approach
Yu-xin Liu
Department of Physics, Peking University,
Beijing , China
Collaborators: Fan Wang, Xiao-fu Lü, Hong-shi Zong,
Lei Chang, Zhao Zhang, Bin Wang, Jian-fa Gu,
Yan-Jun Chen, Wei-jie Fu, Guo-yun Shao,
Wei Yuan, Huan Chen, C.D. Roberts, R. Alkofer, 
International Workshop on Heavy Ion Physics at LHC,
CCNU, Wuhan, May 21-24, 2008

2. I. Introduction
QCD Phase Diagram
Chiral Phase Transition
Confinement-Deconfinement
QCD phase transition plays significant role in the evolution of
universe and hadron physics.

3. Aspects Affecting the QCD Phase Transition
Temperature
Density (Chemical Potential)
Finite Size Effect (Surface, Curvature, •••••• )
Current Mass
Running Coupling Strength (Function)
Color-Flavor Structure
••• ••• ••• •••
Hadron Structure Approaches:
Bag Models (MIT , Cloud )
Soliton Models (chiral , GCM )

4. Approaches： Expt.: (1) High Energy Heavy-Ion Collision Thin Pancakes
Lorentz g=100
Nuclei pass thru each other < 1 fm/c
Huge Stretch
Transverse Expansion
High Temperature (?!)
The Last Epoch:
Final Freezeout--
Large Volume
We measure the “final” state,
we are most interested in the “intermediate” state,
we need to understand the “initial” state…

5. Phase & P-T of Nuclear Collective Motion
Phases of nuclear collective motion and the symmetries
 Shape of Nucleus
 Sphere
 Deforemation
quadrupole
octupole
hexadecupole
 Modes of Nuclear Collective Motion
vibration
axial rotation ( prolate, oblate)
-soft rotation
triaxial rotation ••• ••• ••• •••

6. 背景简介 Exp.: (2) Astronomical Observation(Compact Stars)
( F.Weber, J.Phys.G 25, R195 (1999) )

7. Theoretical Approaches：
Lattice QCD
Finite Temperature Field Theory, RG, LT
with dynamics approach:
QHD, NJL, DSE, GCM, QSR, INST, ···
Key Points:
Representing the two main features of QCD:
Chiral Symmetry & its Breaking
Confinenment

8. Debiting Problems: Whether there exists quark phase in NS ?
Y. Aoki, et al., Nature 443, 675 (2006) : Lattice QCD: T Crossover, Evolve smoothly. F. Wilczek, Nature 443, 637 (2006):
Mechanism of hadronization ?
S. M. Ransom, et al., Science 307, 892 (2005): Large mass neutron stars were observed. F.Oezel (Nature 441, 1115 (2006) ) exclude the EOS with condensate and quark. M.Alford (Nature 445, E7 (2007) ) : quark phase is possible and important.

9. Hot Topic: Mechanism of Type II SN Explosion
 Many explosions have been observed. Theor. simulation
has not yet been successful.
 The explosion depends on the behavior of the neutrino.
The Neutrinos come from
J.M. Lattimer & M. Prakash,
Science, 304, 536 (2004)
A. Burrows,
B.L. Dessart,
C.D. Ott,
E. Livne,
Phys. Rep.
442, 23 (‘07)
New
Mechanism:
Acoustic
Oscillation
Key point is the EoS !

10.  Key Point to Study Phase Transition:
Existence of Multi-solutions
e.g., Pagels’s Result (Phys. Rev. D 19, 3080 (1979) )
Whether there exist two even more solutions
in low energy region ?
 Key Point to the Composition of CS:
EoS with solid QCD Foundation

11. II. The Formalism of DSEs
1. The Main Point
 General Point of View
D-S equation is a set of coupled integral
eqs. among quark, gluon, ghost and
vertex functions,
where the n-point function depends on
the (n+1)-and higher point functions.
C. D. Roberts, et al, Prog. Part. Nucl. Phys. 33 (1994), 477; S1, 1 (2000);
EPJ-ST 140(2007,53; R. Alkofer, et. al, Phys. Rept. 353, 281 (2001);  .

12. 2. Rain-Bow Approximation
 Quark equation at zero chemical potential
where is the effective gluon propagator,
can be conventionally decomposed as
 Quark equation in medium
with

13. 3. Effective Gluon Propagators
(1) MN Model
(2)
(3)
(2) (Q4 +)1 Model
(3) More Realistic model
(4) An Analytical Expression of the Realistic Model:
Maris-Tandy Model
(5) Point Interaction: (P) NJL Model

14. III. Some Numerical Results of the P.T.
 Chiral Symmetry Restoration in a Model
Independent Approach
( L. Chang, Y. X. Liu, H. Guo,
Phys. Rev. D 72, (2005) )

15. Effect of the running coupling strength on the chiral phase transition
(W. Yuan, H. Chen, Y.X. Liu,
Phys. Lett. B 637, 69 (2006))
parameters are taken From Phys. Rev. D 65,
(1997), with fitted as
Lattice QCD result
PRD 72, (2005)

16.  Effect of Current Mass on PT
L. Chang, Y. X. Liu, C. D. Roberts, et al, Phys. Rev. C 75, (2007)
(nucl-th/ )
Solutions of the DSE with
Mass function
With =0.4 GeV
with D = 16 GeV2,   0.4 GeV

17. The above result has been confirmed soon after it was released by Pennington’s group (Phys. Lett. B 645, 167 (2007) (hep-ph/ ) )

18. Phase Diagram in terms of the Current Mass and the Running Coupling Strength

19. Quark Condensate at Finite Current Mass Arising from the Dynamical Breaking
( L. Chang, Y. X. Liu, C. D. Roberts, et al, Phys. Rev. C 75, (2007) )

20. Effect of Current Quark Mass on Meson Mass
( L. Chang, Y. X. Liu, C. D. Roberts, et al., Phys. Rev. C 76, (2007) )
Solving the B-S equation with the kernel being fixed by the solution of DS equation
and flavor symmetry breaking, we obtain

21. Effect of the chemical potential dependence of the gluon propagator on PT
(L. Chang, H. Chen, B. Wang, W. Yuan, Y. X. Liu, Phys. Lett. B 644, 315 (2007) ) in

22. Components of the vacuum of the system with
finite isospin chemical potential
Case ， , ， ；
Case , , ， ；
Case ， ， , ；
Case ， , ，No Solution.
(Z. Zhang, Y.X. Liu, Phys. Rev. C 75, (07))

23. *Ratio of the pion mixedq-G Cond. and the pion cond. in the
system with mq = 0.
*Ratio of the pion mixedq-G Cond. and the pion cond. in the
system with mq  0.

24. Chiral Susceptibility of the Wigner-Vacuum in DSE
No competition between SB phase and CSC phase
( W. Yuan, H. Chen, Y.X. Liu, Phys. Lett. B 637, 69 (2006) )
Chiral Susceptibility of the Wigner-Vacuum in DSE

25. Phase transition from vacuum to hadron matter
Phase with SB & Confinement is stable hadron matter appears
H. Chen, Y.X. Liu, et al., to be published

26. (L. Chang, Y. X. Liu, H. Guo, Nucl. Phys. A 750, 324 (2005))
Variation of Nucleon Properties with Respect to the Density of the Matter
(L. Chang, Y. X. Liu, H. Guo, Nucl. Phys. A 750, 324 (2005))
- relation nucleon properties

27. Phase Diagram of Quark Matter in P-NJL Model
(W.J. Fu, Z. Zhao, Y.X. Liu, Phys. Rev. D 77, (2008) (2+1 flavor)
Simple case: 2-flavor, Z. Zhang, Y.X. Liu, Phys. Rev. C 75, (2007) )
- relation nucleon properties

28. Order of the QCD Phase Transitions
( W.J. Fu, Z. Zhao, Y.X. Liu, Phys. Rev. D 77, (2008) (2+1 flavor) )

29.  Finite Size Effect on Isospin Asymmetry and
Chiral Symmetry Restoration
( G.Y. Shao, L. Chang, Y.X. Liu, X.L. Wang, Phys. Rev. D 73, (2006) )
Possible Applications: Star quake, Glitch of pulsars, Effective mass splitting of N

30. IV. Composition of Compact Stars and Their Effects
condensate and hyperons soften the EOS
(J.F. Gu, H. Guo, Y.X. Liu, et al., Astrophys. J. 622, 549 (2005) )

31. Softened EOS MNS decreases
(J.F. Gu, H. Guo, Y.X. Liu, et al., Astrophys. J. 622, 549 (2005) )

32. Kaon condensation may exist in Large mass NS
(G.H. Wang, W.J. Fu, & Y.X. Liu, Phys. Rev. C 76, (2007) )

33. Effect of the isovector-scalar -meson on NS
( G.H. Wang, W.J. Fu, & Y.X. Liu, to be published. )

34. The Components of Neutron Stars

35. Quark Effect on “Neutron Star”
J.F. Gu, H, Guo, Y.X. Liu, et al.,
Phys. Rev. C73, (2006)

36. The CSC phase softens the EOS The mass of the hybrid star decreases
(J.F. Gu, H. Guo, Y.X. Liu,et al., Eur. Phys. J. A 30, 455 (2006) )

37. Distinguishing Quark Star From Neutron Star
(W.J. Fu, H.Q. Wei, and Y.X. Liu, to be published )

38. V. Processes in Compact Stars
Neutrino Scattering
(Y.J. Chen, H. Guo, Y.X. Liu, Phys. Rev. C 75, (2007) )

39. Mean free path of neutrinos
(Y.J. Chen, H. Guo, Y.X. Liu,
Phys. Rev. C 75, (2007) )

40. The density and temperature dependence of the EC reaction rate
( W.J. Fu, G.H. Wang, Y.X. Liu, Astrophys. J. 678, 1517 (2008) )

41. The Density and Temperature Dependence of the Mean Free Path of Neutrinos
(W.J. Fu, G.H. Wang, Y.X. Liu, Astrophys. J. 678, 1517 (2008) )

42. Effect of the nucleon-nucleon interaction on the Reactions
( W.J. Fu, G.H. Wang, Y.X. Liu, Astrophys. J. 678, 1517 (2008) ) IG IG

43. V. Summary
 The Phase structure of strongly interacting matter (or QCD) is presented.
 The effects of the running coupling strength, the current quark mass on the QCD Phase transition are given in the continuous field theory approach of QCD.
 The chemical potential, the finite size and other effects are also discussed.
 The phase structure of neutron star matter and its effects are also discussed.
 The electron capture, neutrino scattering and neutrino absorption processes in compact stars are discussed, too.

44. Thanks !

45. Three different solutions exist in chiral limit
M+ shifts upward too.

46. M+ shifts upward too.

49. 色超导（Color Superconductivity, 简称 CSC）