1. QCD Phase Transitions in Dyson-Schwinger Equation Approach
Yuxin Liu
Department of Physics, Peking University, China
Outline
I. Introduction
Ⅱ. The Approach
Ⅲ. P.Ts. in Intrinsic Space & in Medium
Ⅳ. Possible Observables
V . Summary
9th Workshop on QCD Phase Transitions & Heavy Ion Phys.，Hangzhou Normal University, – 20, 2011

2. I. Introduction  The Evolution Process of the Universe nuclei
Quarks
and
gluons
get
confined 
Hadrons
Non-confined
quarks and gluons
and leptons
(F.S.) (F.S.B.)
Nuclear
synthesis 
nuclei

3.  The Phase Transitions and Schematic PD
Phase Transitions involved：
Deconfinement–confinement
DCS – DCSB
Flavor Sym. – FSB
Chiral Symmetric
Quark deconfined
sQGP
Items Influencing the Phase Transitions:
Medium：Temperature T ,
Density  ( or  )
Size
Intrinsic：Current mass,
Coupling Strength,
Color-flavor structure,
••• •••
SB, Quark confined

4.  Theoretical Approaches： Two kinds-Continuum & Discrete (lattice)
 Lattice QCD：
Running coupling behavior，
Vacuum Structure，
Temperature effect，
“Small chemical potential”；
  
 Continuum：
(1)Phenomenological models
NJL、QMC、QMF、
(2)Field Theoretical
Chiral perturbation,
Renormalization Group,
QCD sum rules,
Instanton(liquid) model,
DS equations ,
AdS/CFT,
HD(T)LpQCD ,
The approach should manifest simultaneously:
(1) DCSB & its Restoration ,
(2) Confinement & Deconfinement .

5.  A comment on the DSE approach in QCD
Dyson-Schwinger Equations
Slavnov-Taylor Identity
axial gauges BBZ
covariant gauges
QCD
C. D. Roberts, et al, PPNP 33 (1994), 477; 45-S1, 1 (2000); EPJ-ST 140(2007), 53;
R. Alkofer, et. al, Phys. Rep. 353, 281 (2001); C.S. Fischer, JPG 32(2006), R253;  .

6. Ⅱ. The Dyson-Schwinger Equation Approach
Dyson-Schwinger Equations
Slavnov-Taylor Identity
axial gauges BBZ
covariant gauges
QCD
C. D. Roberts, et al, PPNP 33 (1994), 477; 45-S1, 1 (2000); EPJ-ST 140(2007), 53;
R. Alkofer, et. al, Phys. Rep. 353, 281 (2001); C.S. Fischer, JPG 32(2006), R253;  .

7.  Practical Algorithm at Present Stage
 Quark equation at zero chemical potential
where is the effective gluon propagator,
can be conventionally decomposed as
 Quark equation in medium
with
Meeting the requirements!

8.  Models of the Vertex (1) Bare Vertex (2) Ball-Chiu Vertex
(Rainbow-Ladder Approx.)
(2) Ball-Chiu Vertex
(3) Curtis-Pennington Vertex
(4) BC+ACM Vertex (Chang, Liu, Roberts, PRL 106, (‘11) )

9.  Effective Gluon Propagators
(1) MN Model
(2)
(3)
(2) Model
Cuchieri, et al, PRD, 2008
(3) More Realistic model
(4) An Analytical Expression of the Realistic Model:
Maris-Tandy Model
(5) Point Interaction: (P) NJL Model

10. Ⅲ. The Phase Transitions in I.-S. & in Medium
Chiral Susceptibility (S & SB phases simultaneously): Signature of the Chiral Phsae Transition
Point Interaction 
S Phase
Y. Zhao, L. Chang, W. Yuan, Y.X. Liu, Eur. Phys. J. C 56, 483 (2008)

11.  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)
Bare vertex
CS phase
CSB phase
(BC Vertex: L. Chang, Y.X. Liu, R.D. Roberts, et al., Phys. Rev. C 79, (2009))

12.  Effect of the Current Quark Mass on the Chiral Phase Transition
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

13. Euro. Phys. J. C 60, 47 (2009) gives the 5th solution .
Hep-ph/
confirms the existence of the 3rd solution,
and give the 4th solution .
Euro. Phys. J. C 60, 47 (2009) gives the 5th solution .

14.  Part of the QCD Phase Diagram in terms of the Current Mass and Coupling Strength
 The one with multi-node solutions is more complicated and more interesting.

15.  Special Topic (1) of the P.T. in Medium： Critical EndPoint (CEP)
The Position of CEP is a highly debated problem!
 (p)NJL model & others give quite large E/TE (> 3.0)
Sasaki, et al., PRD 77, (2008); Costa, et al., PRD 77, (2008);
Fu & Liu, PRD 77, (2008); Ciminale, et al., PRD 77, (2008);
Fukushima, PRD 77, (2008); Kashiwa, et al., PLB 662, 26 (2008);
Abuki, et al., PRD 78, (2008); Schaefer, et al., PRD 79, (2009);
Costa, et al., PRD 81, (2010); 
Hatta, et al., PRD 67, (2003); Cavacs, et al., PRD 77, (2008);

 Lattice QCD gives smaller E/TE ( 0.4 ~ 1.1)
Fodor, et al., JHEP 4, 050 (2004); Gavai, et al., PRD 71, (2005);
Gupta, arXiv:~ [nucl-ex]; Li, et al., NPA 830, 633c (2009);
Gupta & Xu, et al., Science 332, 1525 (2011); 
 RHIC Exp. Estimate hints quite small E/TE (  1)
R.A. Lacey, et al., nucl-ex/ ; 
 Simple DSE Calculations with Different Effective Gluon Propagators Generate Different Results (0.0, 1.3)
Blaschke, et al, PLB 425, 232 (1998); He, et al., PRD 79, (2009); 
What can sophisticated DSE calculation produce ?
Why different models give distinct results ?

16.  Special topic (2) of the P. T
 Special topic (2) of the P.T. in Medium： Coexistence region (Quarkyonic ? )
 Lattice QCD Calculation
de Forcrand, et al.,
Nucl. Phys. B Proc. Suppl. 153, 62 (2006); 
quarkyonic
and Generaal (large-Nc) Analysis
McLerran, et al., NPA 796, 83 (‘07);
NPA 808, 117 (‘08);
NPA 824, 86 (‘09), 
claim that there exists a quarkyonic phase.
 Inconsistent with Coleman-Witten Theorem !!
 Can sophisticated continuous field approach of QCD give the coexistence (quarkyonic) phase ?
 What can we know more for the coexistence phase?

17.  What is the nature of the matter in DSE?
 Special Topic (3) of the P.T. in Medium： Quark Matter at T above but near Tc
HTL Cal. (Pisarski, PRL 63, 1129(‘89); Blaizot, PTP S168, 330(’07)),
Lattice QCD (Karsch, et al., NPA 830, 223 (‘09); PRD 80, (’09))
NJL (Wambach, et al., PRD 81, (2010))
& Simple DSE Cal. (Fischer et al., EPJC 70, 1037 (2010) ) show:
there exists thermal & Plasmino excitations in hot QM.
Other Lattice QCD Simulations
(Hamada, et al., Phys. Rev. D 81, (2010)) claims:
No qualitative difference between the quark propagators in the deconfined and confined phases near the Tc.
RHIC experiments (Gyulassy, et al., NPA 750, 30 (2005); Shuryak,
PPNP 62, 48 (2009); Song, et al., JPG 36, (2009); … … ) indicate:
the matter is in sQGP state.
 What is the nature of the matter in DSE?

18. Phase Diagram of Strong Interaction Matter in Present DSE Approach of QCD
Phase diagram in bare vertex
Phase diagram in BC vertex
S.X. Qin, L. Chang, H. Chen, Y.X. Liu, C.D. Roberts, PRL 106, (‘11)

19.  Model Parameter Dependence of the CEP
Small σ  short range in momentum space
 long range in coordinate space
MN model  infinite range in r-space
NJL model “zero” range in r-space
Longer range Int.  Smaller E/TE

20.  Property of the matter above but near the Tc
Solving quark’s DSE  Quark’s Propagator
In M-Space, only Yuan, Liu, etc, PRD 81, (2010).
Usually in E-Space, Analytical continuation is required.
Maximum Entropy
Method
(Asakawa, et al.,
PPNP 46,459 (2001);
Nickel, Ann. Phys. 322,
1949 (2007))
Spectral
Function
Qin, et al., PRD 94, (2011)

21. Disperse Relation and Momentum Dependence of the Residues of the Quasi-particles’ poles at T=3.0Tc and T = 1.1Tc
T = 3.0Tc
T = 1.1Tc
S.X. Qin, L. Chang, Y.X. Liu, C.D. Roberts, PRD 94, (‘11)

22. (e.g., T  1.4Tc), there exists a zero mode,
Numerical Results:
 At high temperature (e.g., T = 3.0Tc), there exists
Normal thermal mode & Plasmino mode excitations.
At high momentum,
the N.T. mode plays the main role
& behaves like free particle.
 At the temperature near but above the Tc
(e.g., T  1.4Tc), there exists a zero mode,
besides the N.T. mode and the P. mode.
 The zero mode exists at low momentum (<7.0Tc),
and is long-range correlation ( ~ 1 >FP) .
 The quark at the T where S is restored involves
still rich phases. And the matter is sQGP.
S.X. Qin, L. Chang, Y.X. Liu, C.D. Roberts, PRD 94, (‘11)

23. Ⅳ. Possible Observables
“QCD” Phase Transitions may Happen
General idea，
Phenom. Calc.,
Sophist. Calc.,
quark may get
deconfined(QCD PT)
at high T and/or 
QCD Phase Transitions
Signals for QCD Phase Transitions:
In Lab. Expt.
Jet Q., v2, Viscosity, , CC Fluct. & Correl., Hadron Prop.,···
In Astron. Observ.
M-R Rel., R.S., Rad., Inst. R. Oscil., Freq. G. Oscil., ···

24. Density & Temperature Dependence of some Properties of Nucleon in DSE Soliton Model
(Y. X. Liu, et al.,
NP A 695,
353 (2001);
NPA 725,
127 (2003);
NPA 750,
324 (2005) )
( Y. Mo, S.X. Qin, and Y.X. Liu, Phys. Rev. C 82, (2010) )

25.  Fluctuation & Correlation of Conserved Charges
Temperature dependence of some properties of  & -mesons in the model with contact interaction
( Wei-jie Fu, and Yu-xin Liu, Phys. Rev. D 79, (2009) )
 Fluctuation & Correlation of Conserved Charges
( W.J. Fu, Y.X. Liu, & Y.L. Wu, Phys. Rev. D 79, (2010) )

26.  Distinguishing Newly Born Strange Quark Stars from Neutron Stars
Neutron Star: RMF, Quark Star: Bag Model Frequency of g-mode oscillation
W.J. Fu, H.Q. Wei, and Y.X. Liu, arXiv: ,
Phys. Rev. Lett. 101， (2008)

27. Taking into account the SB effect

28. pulsation of supernova cores are very efficient as sources of g-waves
Ott et al. have found
that these g-mode
pulsation of supernova
cores are very efficient
as sources of g-waves
(PRL 96, (2006) )
DS Cheng, R. Ouyed,
T. Fischer, ·····
The g-mode pulsation frequency can be a signal to
distinguish the newly born strange quark stars from
neutron stars,
i.e, an astronomical signal of QCD phase transition.

29. Thanks !! V. Summary & Remarks  Far from Well Established !
 Discussed some aspects of QCD phase transitions
in the DS equation approach of QCD
 Coexistence Phase；  CEP (E/TE comparable with Lattice D & EE)  Effects of the C.-Strength & Current Mass
 Far from Well Established !
 Observables ?!
 Mechanism ?! Process ?!   
Thanks !!

30. In DSE approach  SB Dynamical Mass
Dynamical chiral symmetry breaking (SB) generates the mass of Fermion  SB
Dynamical Mass
< qq >0 ~ - (240 MeV)3
In DSE approach
Maris & Roberts

31. Analytic Continuation from Euclidean Space to Minkowskian Space
= 0, ei=1,
==> E.S.
= , ei=1,
==> M.S.
( W. Yuan, S.X. Qin, H. Chen, & YXL, PRD 81, (2010) )

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

33. Chang, Liu, et al., Phys. Rev C 75, 015201 (2007)
M+ shifts upward too.

34. Chang, Liu, et al., Phys. Rev C 75, 015201 (2007)

35. Chang, Liu, et al., Phys. Rev C 75, 015201 (2007)

36. Hadron Structure

37. Some Numerical Results

38.  Effect of the F.-S.-B. (m0) on Meson’s Mass
Solving the 4-dimenssional covariant B-S equation with the kernel being fixed by
the solution of DS equation and flavor symmetry breaking, we obtain
( L. Chang, Y. X. Liu, C. D. Roberts, et al., Phys. Rev. C 76, (2007) )

39. DSE Soliton Description of Nucleon
B. Wang, H. Chen, L. Chang, & Y. X. Liu, Phys. Rev. C 76, (2007)
Collective Quantization: Nucl. Phys. A790, 593 (2007).

40. Compositions and Phase Structure of Compact Stars and their Identification
Radio Pulses = “Neutron” Stars
Composition & Structure of NS are Still Under Study !

41. 背景简介 Conjecture of the Composition of Compact Stars
( F.Weber, PPNP 54, 193 (2005) )