1. Phase Diagram and Thermal Properties of Strong Interaction Matter Yuxin Liu Dept. Phys., Peking Univ., China XQCD 2015, CCNU, Sept. 22, 2015 Outline Outline I. Introduction Ⅱ. Ⅱ. The DS Eq. Approach Ⅲ. The Phase Diagram Ⅲ. The Phase Diagram Ⅳ. Thermal Properties Ⅳ. Thermal Properties V. Remarks V. Remarks 1

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

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. 3

4.  For the location of the CEP, different approaches give quite distinct results.  (p)NJL model & others give quite large  E q /T E (> 3.0) Sasaki, et al., PRD 77, 034024 (2008); 82, 076003 (2010); 82, 116004 (1010); 0). Costa, et al., PRD 77, 096001 (‘08); EPL 86, 31001 (‘09); PRD 81, 016007(‘10); Fu & Liu, PRD 77, 014006 (2008); Ciminale, et al., PRD 77, 054023 (2008); Fukushima, PRD 77, 114028 (2008); Kashiwa, et al., PLB 662, 26 (2008); Abuki, et al., PRD 78, 034034 (2008); Schaefer, et al., PRD 79, 014018 (2009); Hatta, et al., PRD 67, 014028 (2003); Cavacs, et al., PRD 77, 065016(2008); Bratovic, et al., PLB 719, 131(‘13); Bhattacharyya, et al., PRD 87,054009(‘13); Jiang, et al., PRD 88, 016008 (2013); Ke, et al., PRD 89, 074041 (2014);   Lattice QCD gives smaller  E q /T E ( 0.4 ~ 1.1) Fodor, et al., JHEP 4, 050 (2004); Gavai, et al., PRD 71, 114014 (2005); Gavai, et al., PRD 78, 114503 (2008); Schmidt et al., JPG 35, 104013 (2008); Li, et al., PRD 84, 071503 (2011); Gupta, et al., PRD 90, 034001 (2014);   DSE Calculations with different techniques generate different results for the  E q /T E (0.0, 1.1 ~ 1.3, 1.4 ~ 1.6, ) Blaschke, et al, PLB 425, 232 (1998); He, et al., PRD 79, 036001 (2009); Qin, et al., PRL 106, 172301 (2011); Fischer, et al., PLB 702, 438 (‘11); PLB 718, 1036 (‘13); PRD 90, 034022 (‘14);  4

5.  Relation between the chiral PT and the deconfinement PT claim that there exists a quarkyonic phase. and General (large-N c ) Analysis McLerran, et al., NPA 796, 83 (‘07); NPA 808, 117 (‘08); NPA 824, 86 (‘09),   Lattice QCD Calculation de Forcrand, et al., Nucl. Phys. B Proc. Suppl. 153, 62 (2006) ;   Is there any hierarchy between the two PTs ?  Coleman-Witten Conjecture (PRL 45, 100 (‘80)) : Confinement coincides with DCSB !! Inconsistent with each other ?! quarkyonic

6. Slavnov-Taylor Identity Dyson-Schwinger Equations axial gauges BBZ covariant gauges QCDQCD Ⅱ. The Dyson-Schwinger Equation Approach 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; . 6

7.  Algorithms of Solving the DSEs of QCD ？ ？ (1) 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. 7

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

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

10.  Models of the eff. gluon propagator (3) Commonly Used: Maris-Tandy Model (PRC 56, 3369) Cuchieri, et al, PRD, 2008 A.C. Aguilar, et al., JHEP 1007-002 Recently Proposed: Infrared Constant Model ( Qin, Chang, Liu, Roberts, Wilson, PRC 84, 042202(R), (2011). ) Taking in the coefficient of the above expression Derivation and analysis in PRD 87, 085039 (2013) show that the one in 4-D should be infrared constant. 10

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

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

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

14. with D = 16 GeV 2,   0.4 GeV DCSB still exists beyond chiral limit  DCSB still exists beyond chiral limit Solutions of the DSE with With  = 0.4 GeV L. Chang, Y. X. Liu, C. D. Roberts, et al, arXiv: nucl-th/0605058; R. Williams, C.S. Fischer, M.R. Pennington, arXiv: hep-ph/0612061. 14

15.  Analyzing the spectral density function indicates that the quarks are confined at low temperature and low density S.X. Qin, D. Rischke, Phys. Rev. D 88, 056007 (2013) H. Chen, YXL, et al., Phys. Rev. D 78, 116015 (2008) 15 T=0.8Tc

16. K.L. Wang, Y.X. Liu, C.D. Roberts, Phys. Rev. D 87, 074038 (2013)  Screening masses of hadrons can identify the phase transitions, when, the color gets deconfined. GT Relation  M   M  can be a signal of the DCS. Hadron properties provide signals for not only the chiral phase transt. but also the confinement-deconfnmt. phase transition. 16

17. Dyson-Schwinger Equations QCDQCD  A comment on 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; . 17

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

19.  Chiral Susceptibility as a Criterion S.X. Qin, L. Chang, H. Chen, Y.X. Liu, C.D. Roberts, PRL 106, 172301(‘11) 19 In the chiral limit Beyond the chiral limit For 2 nd 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.

20.  Phase diagram is given, CEP is proposed S.X. Qin, L. Chang, H. Chen, Y.X. Liu, & C.D. Roberts, Phys. Rev. Lett. 106, 172301 (2011) Phase diagram in bare vertexPhase diagram in BC vertex 20  In Chiral Limit

21.  Phase diagram is given, CEP is proposed Fei Gao, Y.X. Liu, et al., to be published  Beyond Chiral Limit 21

22. IV. Thermal Properties Pressure Sound Speed  Basic Formulae: 22 Heat capability & latent heat

23. IV. Thermal Properties Trace Anomaly at Zero Chemical Potential 23 T M = 140 MeV

24. IV. Thermal Properties Pressure & Trace Anomaly at Non-Zero Chemical Potential 24 T M = 140 MeV

25. IV. Thermal Properties Sound Speed squared 25 T M = 140 MeV

26. IV. Thermal Properties Specific heat capability & Latent heat 26 T M = 140 MeV

27. The 2 nd, 3 rd, 4 th order fluctuations where   Quark Number Fluctuations Quark number: 27

28.  Quark Number Density Fluctuations vs T in the DSE X.Y. Xin, S.X. Qin, YXL, PRD 90, 076006 (2014) 28

29.  Quark Number Density Fluctuations vs μ in the DSE X.Y. Xin, S.X. Qin, YXL, PRD 90, 076006 (2014) 29

30.  Quark Number Density Fluctuations vs μ in the DSE X.Y. Xin, S.X. Qin, YXL, Phys. Rev. D 90, 076006 (2014) 30 In crossover region, the fluctuations oscillate obviously; In 1 st transt., overlaps exist. At CEP, they diverge!

31.  Relating with Experiment Directly Jing Chen, Fei Gao, Yu-xin Liu, et al., to be published 31 Key issue: Taking the Finite size effect into account ! Chemical Freeze out Conditions

32.  Critical Behavior Fei Gao, Yu-xin Liu, et al., to be published 32

33.  Critical Behavior Fei Gao, Yu-xin Liu, et al., to be published 33

34. ω Small ω  long range in coordinate space  Different methods give distinct locations of the CEP arises from diff. Conf. Length MN model  infinite range in r-space NJL model  “zero” range in r-space Longer range Int.  Smaller  E /T E S.X. Qin, YXL, et al, PRL106,172301(‘11); X. Xin, S. Qin, YXL, PRD90,076006 34

35.  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.  Far from well established  promising ! V. V. Summary & Remarks  QCD phase transitions are investigated via DSE  DSE, a npQCD approach, is described  Some thermal properties are discussed  Trace anomaly, sound speed, etc;  quark number fluctuations;  critical exponents of the & the C v, etc. 35 Thanks !!

36. ♠ Positivity Violation of the Spectral Function   Criterion for Confinement S.X. Qin, and D.H. Rischke, Phys. Rev. D 88, 056007 (2013) S.X. Qin, and D.H. Rischke, Phys. Rev. D 88, 056007 (2013) Maximum Entropy Method Result in DSE ( Asakawa, et al., PPNP 46,459 (2001); Nickel, Ann. Phys. 322, 1949 (2007) ) 36

37. Solving quark’s DSE  Quark’s Propagator  Property of the matter above but near the T c Maximum Entropy Method ( Asakawa, et al., PPNP 46,459 (2001) ; Nickel, Ann. Phys. 322, 1949 (2007) )  Spectral Function In M-Space, only Yuan, Liu, etc, PRD 81, 114022 (2010). Usually in E-Space, Analytical continuation is required. Qin, Chang, Liu, et al., PRD 84, 014017(2011)

38. T = 3.0T c Disperse Relation and Momentum Dependence of the Residues of the Quasi-particles’ poles T = 1.1T c S.X. Qin, L. Chang, Y.X. Liu, et al., Phys. Rev. D 84, 014017(2011); S.X. Qin, L. Chang, Y.X. Liu, et al., Phys. Rev. D 84, 014017(2011); F. Gao, S.X. Qin, Y.X. Liu, et al., Phys. Rev. D 89, 076009 (2014). F. Gao, S.X. Qin, Y.X. Liu, et al., Phys. Rev. D 89, 076009 (2014). Normal T. Mode Plasmino M. Zero Mode  The zero mode exists at low momentum (<7.0T c ), 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.

39.  Approach 1: Soliton bag model Ⅳ. Hadrons via DSE  Approach 2: BSE + DSE  Mesons BSE with DSE solutions being the input  Baryons Fadeev Equation or Diquark model (BSE+BSE) Pressure difference provides the bag constant. L. Chang, et al., PRL 103, 081601 (2009) 。 39

40.  Effect of the F.-S.-B. () on Meson ’ s Mass  Effect of the F.-S.-B. (m 0 ) 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, 045203 (2007) ) 40

41. ( S.X. Qin, L. Chang, Y.X. Liu, C.D. Roberts, et al., Phys. Rev. C 84, 042202(R) (2011) )   S ome properties of mesons in DSE-BSE ( L. Chang, & C.D. Roberts, Phys. Rev. C 85, 052201(R) (2012) ) Present work

42. Electromagnetic Property & PDF of hadrons Proton electromagnetic forma factor L. Chang et al., AIP CP 1354, 110 (‘11) P. Maris & PCT, PRC 61, 045202 (‘00) PDF in pion PDF in kaon R.J. Holt & C.D. Roberts, RMP 82, 2991(2010); T. Nguyan, CDR, et al., PRC 83, 062201 (R) (2011) 42

43.  Gravitational Mode Pulsation Frequency can be an Excellent Astronomical Signal W.J. Fu, H.Q. Wei, and Y.X. Liu, arXiv: 0810.1084, Phys. Rev. Lett. 101 ， 181102 (2008) Neutron Star: RMF, Quark Star: Bag Model Frequency of g-mode oscillation

44. Taking into account the DCSB effect 44

45.  Analytic Continuation from Euclidean Space to Minkowskian Space ( W. Yuan, S.X. Qin, H. Chen, & YXL, PRD 81, 114022 (2010) )  = 0, e i  =1, ==> E.S.  = , e i  =  1, ==> M.S. 45