1. Phase Transitions of Strong Interaction System in Dyson-Schwinger Equation Approach Yu-xin Liu （刘玉鑫） Department of Physics, Peking University, China 第 13 届全国中高能核物理大会，中国科技大学，合肥， 2009 年 11 月 5-7 日 Outline I. Introduction II. The Approach III. Some Numerical Results of Our Group IV. Summary & Outlook

2. I. Introduction Schematic QCD Phase Diagram Items Affecting the PTs: Medium Effects ： Temperature, Density (Chem. Potent.) Finite size Intrinsic Effects ： Current mass, Run. Coupl. Strength, Color-Flavor Structure, Related Phase Transitions: Confinement(Hadron.) –– Decconfinement Chiral Symm. Breaking CS Restoration –– CS Restoration Flavor Symmetry –– Flavor Symm. Breaking Chiral Symmetric Quark deconfined  SB SB, Quark confined sQGP  How do the aspects influence the phase transitions ?  Why there exists partial restoration of dynamical  S in low density matter ?  How does matter emerge from vacuum ?

3. Theoretical Methods Theoretical Methods ： Lattice QCD Finite-T QFT, Renormal. Group, Landau T.,  Dynamical Approaches （ models ） : QHD, (p)NJL, QMC, QMF, QCD Sum Roles, Instanton models, Dyson-Schwinger Equations (DSEs),  General Requirements for the approaches: not only involving the chiral symmetry & its breaking, but also manifesting the confinement and deconfinement. AdS/CFT

4. Slavnov-Taylor Identity Dyson-Schwinger Equations axial gauges BBZ covariant gauges QCDQCD II. The DSE Approach of 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; .

5.  Practical Way 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!

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

7.  Models of Vertex (1) Bare Vertex (2) Ball-Chiu Vertex (3) Curtis-Pennington Vertex (Rainbow-Ladder Approx.)

8. For Hadron Structure (2) Soliton Model (non-local fields) (1) Bethe-Salpeter Equation approach

9. Examples of achievements of the DSE of QCD Generation of Dynamical Mass Taken from: The Frontiers of Nuclear Science – A Long Range Plan (DOE, US, Dec. 2007). Origin: MSB, CDR, PCT, et al., Phys. Rev. C 68, 015203 (03) Taken from: Tandy ’ s talk at Morelia-2009

10. More recent result for the mass splitting between ρ& a 1 mesons

11. Y. Zhao, L. Chang, W. Yuan, Y.X. Liu, Eur. Phys. J. C 56, 483 (2008) Point Interaction Point Interaction  S Phase  III. Some Numerical Results of Our Group  Chiral Susceptibility (  S &  SB phases simultaneously): Signature of the Chiral Phsae Transition

12. parameters are taken From Phys. Rev. D 65, 094026 (1997), with fitted as  Effect of the Running Coupling Strength on the Chiral Phsae Transition (W. Yuan, H. Chen, Y.X. Liu, Phys. Lett. B 637, 69 (2006)) Lattice QCD result PRD 72, 014507 (2005) (BC Vertex: L. Chang, YXL, RDR, Zong, et al., Phys. Rev. C 79, 035209 ( ‘ 09)) Bare vertex CS phase CSB phase

13. with D = 16 GeV 2,   0.4 GeV  Effect of the Current Quark Mass on the Chiral Phase Transition Solutions of the DSE with Mass function With  =0.4 GeV L. Chang, Y. X. Liu, C. D. Roberts, et al, Phys. Rev. C 75, 015201 (2007) (nucl-th/0605058)

14. Distinguishing the Dynamical Chiral Symmetry Breaking From the Explicit Chiral Symmetry Breaking ( L. Chang, Y. X. Liu, C. D. Roberts, et al, Phys. Rev. C 75, 015201 (2007) )

15. Phase Diagram in terms of the Current Mass and the Running Coupling Strength BC vertex gives qualitatively same results.

16. Euro. Phys. J. C 60, 47 (2009) gives the 4th solution. Hep-ph/0612061 confirms the existence of the 3rd solution, and give the 4th solution.

17.  Effect of the Chemical Potential on the Chiral Phase Transition Diquark channel: ( W. Yuan, H. Chen, Y.X. Liu, Phys. Lett. B 637, 69 (2006) ) Chiral channel: ( L. Chang, H. Chen, B. Wang, W. Yuan,Y.X. Liu, Phys. Lett. B 644, 315 ( L. Chang, H. Chen, B. Wang, W. Yuan, and Y.X. Liu, Phys. Lett. B 644, 315 (2007) ) Chiral Susceptibility of Wigner-Vacuum in DSE Some Refs. of DSE study on CSC 1. D. Nickel, et al., PRD 73, 114028 (2006); 2. D. Nickel, et al., PRD 74, 114015 (2006); 3. F. Marhauser, et al., PRD 75, 054022 (2007); 4. V. Klainhaus, et al., PRD 76, 074024 (2007); 5. D. Nickel, et al., PRD 77, 114010 (2008); 6. D. Nickel, et al., arXiv:0811.2400; …………

18.  Partial Restoration of Dynamical  S  & Matter Generation H. Chen, W. Yuan, L. Chang, YXL, TK, CDR, Phys. Rev. D 78, 116015 (2008); H. Chen, W. Yuan, YXL, JPG 36 (special issue for SQM2008), 064073 (2009) Bare vertex BC vertex CSB phase BC vertex CS phase NJL Model Alkofer ’ s Solution-2cc Alkofer ’ s Solution-BCFit1 Solution with BC vertex

19. Collective Quantization: Nucl. Phys. A790, 593 (2007).  Properties of Nucleon in DSE Soliton Model B. Wang, H. Chen, L. Chang, & Y. X. Liu, Phys. Rev. C 76, 025201 (2007) Model of the effective gluon propagator

20.  Density Dependence of some Properties of Nucleon in DSE Soliton Model  -  relation nucleon properties (Y. X. Liu, et al., Nucl. Phys. A 695, 353 (2001); NPA 725, 127 (2003); NPA 750, 324 (2005) )

21.  Chemical Potential Dependence of   N L. Chang,Y. X. Liu, H. Guo, Phys. Rev. D 72, 094023 (2005) In BC vertex:   N = (   N = (60~80) MeV 。 Newly result (H. Chen, YXL, et al., to be published) :

22.  Temperature Dependence of the Propagators of Gluon and Ghost Lattice QCD Results (A. Maas, et al., EPJC 37, 335 (2004); A. Cucchieri, et al., PRD 75, 076003 (2007) ) : Previous DSE solutions in torus momentum space do not give the same results (C.S. Fischer, et al., Ann. Phys. 321, 1918 (2006); ··· ). with

23.  Our Newly Results in continuum momentum space Solving coupled equations of gluon and ghost:  (H. Chen, R. Alkofer, Y.X. Liu, to be published)

24.  Phase Diagram of Strong Interaction Matter  S.X. Qin, L. Chang, Y.X. Liu, to be published. S.X. Qin, L. Chang, Y.X. Liu, to be published. Result in bare vertex Result in Ball-Chiu vertex  (GeV)

25. Phase Diagram of the （ 2+1 ） Flavor System in P-NJL Model  -  relation nucleon properties Simple case: 2-flavor, Z. Zhang, Y.X. Liu, Phys. Rev. C 75, 064910 (2007) ) Simple case: 2-flavor, Z. Zhang, Y.X. Liu, Phys. Rev. C 75, 064910 (2007) ) (W.J. Fu, Z. Zhao, Y.X. Liu, Phys. Rev. D 77, 014006 (2008) (2+1 flavor)

26.  An Astronomical Signal Identifying the QCD Phase Transition W.J. Fu, H.Q. Wei, and Y.X. Liu, arXiv: 0810.1084,1084 Phys. Rev. Lett. 101 ， 181102 (2008) Neutron Star: RMF, Quark Star: Bag Model Frequency of g-mode oscillation

27. Taking into account the  SB effect

28. Ott et al. have found that these g-mode pulsation of supernova cores are very efficient as sources of g-waves (PRL 96, 201102 (2006) ) DS Cheng, R. Ouyed, T. Fischer, ····· The g-mode oscillation 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. IV. Summary & Discussion ：  We propose an astronomical signal manifesting the quark deconfinement phase transition in dense matter.  We develop the Polyakov-NJL model for (2+1) flavor system and study the phase transitions.  above a criticalμ,  S can be restored partially.  above a critical coupling strength and below a critical current mass, DCSB appears;  QCD Phase Transitions: With the DSE approach of QCD, we show that A mechanism is proposed ! ?!  Driving the Polyakov-loop from DSE ?! Being checked in sophisticated DSE approach !  Being checked in sophisticated DSE approach !  The finite-T effect on the pure gauge fields are given. Thanks !!! Thanks !!!  Phase diagram of strong interaction matter (T-μ) ?!

30. 背景简介 ( F.Weber, J.Phys.G 25, R195 (1999) ) Composition of Compact Stars

31. Calculations of the g-mode oscillation Oscillations of a nonrotating, unmagnetized and fluid star can be described by a vector field, and the Eulerian (or “local”) perturbations of the pressure, density, and the gravitational potential,,, and. Employing the Newtonian gravity, the nonradial oscillation equations read We adopt the Cowling approximation, i.e. neglecting the perturbations of the gravitational potential.

32. Factorizing the displacement vector as, one has the oscillation equations as where is the eigenfrequency of a oscillation mode; is the local gravitational acceleration.

33. The eigen-mode can be determined by the oscillation Eqns when complemented by proper boundary conditions at the center and the surface of the star The Lagrangian density for the RMF is given as Five parameters are fixed by fitting the properties of the symmetric nuclear matter at saturation density.

34. For a newly born SQS, we implement the MIT bag model for its equation of state. We choose, and a bag constant. The equilibrium sound speed can be fixed for an equilibrium configuration, with baryon density, entropy per baryon, and the lepton fraction being functions of the radius. ( taken from Dessart et al. ApJ,645,534,2006 ).

35. We calculate the properties of the g-mode oscillations of newly born NSs at the time t=100, 200 and 300ms after the core bounce, the mass inside the radius of 20km is 0.8, 0.95, and 1.05 M Sun, respectively. We assume that the variation behaviors of and for newly born SQSs are the same as for NSs.  As ω changes to 100.7, 105.9, 96.1 Hz, respectively.  When M SQS = 1.4M sun, ω changes to 100.2, 91.4, 73.0 Hz, respectively.  As M SQS = 1.68M sun, ω changes to 108.8, 100.9, 84.5 Hz, respectively.

36. The reason for the large difference in the g-mode oscillation eigenfrequencies between newly born NSs and SQSs, is due to The components of a SQS are all extremely relativistic and its EOS can be approximately parameterized as are highly suppressed.