1. Toward QCD Phase Transitions via Continuum NP-QCD
Yuxin Liu
Dept. Phys., Peking Univ., China
Outline
I. Introduction
Ⅱ. DS Eqs. — A C-QCD Approach
Ⅲ. The Phase Diagram
Ⅳ. Thermal Properties
V. Remarks
Sino-Americas Workshop on BSP-CQCD, CCNU,

2. I. FD problems are sorted to QCD PTs
 QCD Phase Diagram: Phase Boundary,
Thermal Property, Specific States, e.g., CEP, sQGP, Quakyonic,
Phase Transitions involved：
Deconfinement–confinement
DCS – DCSB
Flavor Sym. – FSB ？
sQGP
Chiral Symmetric
Quark deconfined ？
Items Influencing the Phase Transitions:
Medium：Temperature T ,
Density ρ ( or  )
Size
Intrinsic：Current mass,
Coupling Strength,
Color-flavor structure,
••• ••• ？
SB, Quark confined

3.  Theoretical Approaches： Two kinds-Continuum & Discrete (lattice)
 Lattice QCD：
Running coupling behavior，
Vacuum Structure，
Temperature effect，
“Small chemical potential”；
  
 Continuum：
(1)Phenomenological models
(p)NJL、(p)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 .

4.  For the location of the CEP, different approaches give quite distinct results.
 (p)NJL model & others give quite large Eq/TE (> 3.0)
Sasaki, et al., PRD 77, (2008); 82, (2010); 82, (1010); 0).
Costa, et al., PRD 77, (‘08); EPL 86, (‘09); PRD 81, (‘10);
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);
Hatta, et al., PRD 67, (2003); Cavacs, et al., PRD 77, (2008);
Bratovic, et al., PLB 719, 131(‘13); Bhattacharyya, et al., PRD 87,054009(‘13);
Jiang, et al., PRD 88, (2013); Ke, et al., PRD 89, (2014); 
 Lattice QCD gives smaller Eq/TE ( 0.4 ~ 1.1)
Fodor, et al., JHEP 4, 050 (2004); Gavai, et al., PRD 71, (2005);
Gavai, et al., PRD 78, (2008); Schmidt et al., JPG 35, (2008);
Li, et al., PRD 84, (2011); Gupta, et al., PRD 90, (2014); 
 DSE Calculations with different techniques generate different results for the Eq/TE (0.0, 1.1 ~ 1.3, 1.4 ~ 1.6, )
Blaschke, et al, PLB 425, 232 (1998); He, et al., PRD 79, (2009);
Qin, et al., PRL 106, (2011);
Fischer, et al., PLB 702, 438 (‘11); PLB 718, 1036 (‘13); PRD 90, (‘14); 

5.  Relation between the chiral PT and the deconfinement PT
 Lattice QCD Calculation
de Forcrand, et al.,
Nucl. Phys. B Proc. Suppl. 153, 62 (2006); 
quarkyonic
and General (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.
Coleman-Witten Conjecture (PRL 45, 100 (‘80)):
Confinement coincides with DCSB !!
Inconsistent with each other ?!
 Is there any hierarchy between the two PTs ?

6. Ⅱ. DS Eqs — A Continuum NP-QCD 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); LYX, Roberts, et al., CTP 58 (2012), 79;  .

7. ？ ？  Algorithms of Solving the DSEs of QCD
Solving the coupled quark, ghost and
gluon (parts of the diagrams) equations,
e.g.,
(2) Solving the truncated quark equation
with the symmetries being preserved. ？ ？

8.  Expression of the quark gap equation
 Truncation：Preserving Symm.
 Quark Eq.
 Decomposition of the Lorentz Structure
 Quark Eq. in Vacuum : 

9.  Quark Eq. in Medium S S Matsubara Formalism
Temperature T :  Matsubara Frequency
Density  ：  Chemical Potential S S
Decomposition of the Lorentz Structure S S

10.  Models of the Dressed Gluon Propagator
Commonly Used: Maris-Tandy Model (PRC 56, 3369)
Cuchieri, et al, PRD, 2008
(3)
Recently Proposed: Infrared Constant Model
( Qin, Chang, Liu, Roberts, Wilson,
PRC 84, (R), (2011). )
A.C. Aguilar, et al.,
JHEP
Taking in the coefficient
of the above expression
Derivation and analysis in PRD 87, (2013) show that the one in 4-Dimmession should be infrared constant.

11.  Models of quark-gluon interaction vertex
(1) Bare Ansatz
(Rainbow-Ladder Approx.)
(2) Ball-Chiu Ansatz
Satisfying W-T Identity, L-C. restricted
(3) Curtis-Pennington Ansatz
Satisfying Prod. Ren.
(4) BC+ACM (Chang, etc, PRL 106,072001(‘11), Qin, etc, PLB 722,384(‘13))

12. A regirous check on the ACM model for
the quark-gluon interaction vertex

13.  DSE meets the requirements for an approach to describe the QCD PTs
 Dynamical chiral symmetry breaking
DCSB
In DSE approach
Numerical results
Increasing the
interaction strength
induces the dynamical
mass generation
K.L. Wang, YXL, et al., PRD 86,114001(‘12); 

14.  DCSB still exists beyond chiral limit
L. Chang, Y. X. Liu, C. D. Roberts, et al, arXiv: nucl-th/ ;
R. Williams, C.S. Fischer, M.R. Pennington, arXiv: hep-ph/
Solutions of the DSE with
With  = 0.4 GeV
with D = 16 GeV2,   0.4 GeV

15.  Analyzing the spectral density function
indicates that the quarks are confined at
low temperature and low density
T=0.8Tc
H. Chen, YXL, et al., Phys. Rev. D 78, (2008)
S.X. Qin, D. Rischke, Phys. Rev.
D 88, (2013)

16.  Screening masses of hadrons can identify the phase transitions
GT Relation
 M  M
can be a
signal of the
DCS.
, when , the
color gets deconfined.
Hadron properties provide signals
for not only the chiral phase transt.
but also the confinement-deconfnmt.
phase transition.
K.L. Wang, Y.X. Liu, C.D. Roberts, Phys. Rev. D 87, (2013)

17.  A comment on the DSE approach of QCD
Dyson-Schwinger Equations
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;  .

18. III. The Phase Diagram  Quantity to identify the phase transition
 Traditionally
Criterion in Dynamics: Equating Effective TPs
With fully Nonperturbative approach, one could not have the ETPs. New Criterion must be established!

19.  Chiral Susceptibility as a Criterion
In the chiral limit
S.X. Qin, L. Chang, H. Chen, Y.X. Liu, C.D. Roberts, PRL 106, (‘11)
For 2nd order PT & Crossover,
s of the two phases diverge at the same state.
For 1st order PT, the s diverge at different states.
 the  criterion can not only give the phase boundary, but also determine the position of the CEP.
Beyond the chiral limit

20. Phase diagram is given, CEP is proposed
 In Chiral Limit
Phase diagram in bare vertex
Phase diagram in BC vertex
S.X. Qin, L. Chang, H. Chen, Y.X. Liu, & C.D. Roberts,
Phys. Rev. Lett. 106, (2011)

21. Phase diagram is given, CEP is proposed
 Beyond Chiral Limit
Fei Gao, Y.X. Liu, C.D. Roberts, et al., arXiv:

22. Phase diagram in intrinsic space
K.L. Wang, S.X. Qin, YXL, & CDR, et al., Phys. Rev. D 86, (2012).

23. IV. Thermal Properties  Basic Formulae: Pressure Sound Speed
Heat capability & latent heat

24.  Trace Anomaly at Zero Chemical Potential
IV. Thermal Properties
 Trace Anomaly at Zero Chemical Potential
TM = 140 MeV

25.  Pressure & Trace Anomaly at Non-Zero
IV. Thermal Properties
 Pressure & Trace Anomaly at Non-Zero
Chemical Potential
TM = 140 MeV

26. IV. Thermal Properties
 Sound Speed squared
TM = 140 MeV

27.  Specific heat capability & Latent heat
IV. Thermal Properties
 Specific heat capability & Latent heat
TM = 140 MeV

28.  Entropy IV. Thermal Properties Only the Uniform Part
Including the surface Part
Fei Gao, Y.X. Liu, et al., to be published.

29.  Quark Number Fluctuations
The 2nd, 3rd, 4th order fluctuations
where

30.  Quark Number Density Fluctuations vs T in the DSE
X.Y. Xin, S.X. Qin, YXL, PRD 90, (2014)

31.  Quark Number Density Fluctuations vs μ in the DSE
X.Y. Xin, S.X. Qin, YXL, PRD 90, (2014)

32.  Quark Number Density Fluctuations vs μ in the DSE
In crossover
region, the
fluctuations
oscillate
obviously;
In 1st transt.,
overlaps
exist.
At CEP,
they
diverge!
X.Y. Xin, S.X. Qin, YXL, Phys. Rev. D 90, (2014)

33.  Relating with Experiment Directly
Key issue:
Taking the Finite size effect into account !
Chem. Freeze-out Conditions
c=676.8 MeV,
d=0.168 GeV-1 ;
Jing Chen, Fei Gao, Yu-xin Liu, arXiv:

34.  Relating with Experiment Directly
Taking the Finite size effect into account !
In case of R=2.2fm:
Jing Chen, Fei Gao, Yu-xin Liu, arXiv:

35.  Critical Behavior
Fei Gao, Yu-xin Liu, C.D. Roberts, et al., to be published

36.  Critical Behavior
Fei Gao, Yu-xin Liu, C.D. Roberts, et al., to be published

37.  Different methods give distinct locations of the CEP arises from diff. Conf. Length
Small ω  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
S.X. Qin, YXL, et al, PRL106,172301(‘11); X. Xin, S. Qin, YXL, PRD90,076006

38. Thanks !! V. Summary & Remarks
DSE, a npQCD approach, is described
QCD phase transitions are investigated via DSE
 Dynamical Mass is generated by DCSB;  Confinement can be described with the positivity violation of the spectral function.  The phase diagram and CEP are given.
Some thermal properties are discussed
 Trace anomaly, sound speed, etc;
 quark number fluctuations;  critical exponents of the & the Cv , etc .
 Far from well established  promising !
Thanks !!

39.  Criterion for Confinement
♠ Positivity Violation of the Spectral Function
S.X. Qin, and D.H. Rischke, Phys. Rev. D 88, (2013)
Maximum Entropy Method
( Asakawa, et al., PPNP 46,459 (2001);
Nickel, Ann. Phys. 322, 1949 (2007) )
Result in DSE

40.  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, Chang, Liu, et al., PRD 84, (2011)

41.  The zero mode exists at low momentum (<7.0Tc),
Disperse Relation and Momentum Dependence of the Residues of the Quasi-particles’ poles
T = 3.0Tc
T = 1.1Tc
Normal
T. Mode
Zero Mode
Plasmino M.
 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, et al., Phys. Rev. D 84, (2011);
F. Gao, S.X. Qin, Y.X. Liu, et al., Phys. Rev. D 89, (2014).

42. Ⅳ. Hadrons via DSE  Approach 1: Soliton bag model
Pressure difference provides the bag constant.
Approach 2: BSE + DSE
 Mesons
BSE with DSE solutions being the input
 Baryons
Fadeev Equation or Diquark model (BSE+BSE)
L. Chang,
et al.,
PRL 103,
081601
(2009)。

43.  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) )

44. Some properties of mesons in DSE-BSE
Present work
( L. Chang, & C.D. Roberts, Phys. Rev. C 85, (R) (2012) )
( S.X. Qin, L. Chang, Y.X. Liu, C.D. Roberts,
et al., Phys. Rev. C 84, (R) (2011) )

45.  Gravitational Mode Pulsation Frequency can be an Excellent Astronomical Signal
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)

46. Taking into account the DCSB effect

47. 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) )