1. Locating the CEP via the Dyson- Schwing Equation Approach of QCD Yuxin Li u Dept. Phys.,Peking Univ., China CPOD 2014, Bielefeld University, Germany, Nov. 20, 2014 Outline Outline I. Introduction Ⅱ. Ⅱ. The DS Eq. Approach Ⅲ. Locating the CEP Ⅲ. Locating the CEP Ⅳ. Remarks Ⅳ. Remarks 1

2. FD problems are sorted to QCD PTs  QCD Phase Diagram: Phase Boundary, 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. 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; . 5

6.  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. Fischer’s talk 6

7.  Expression of the quark gap equation  Truncation ： Preserving Symm.  Quark Eq.  Decomposition of the Lorentz Structure   Quark Eq. in Vacuum : 7

8.  Quark Eq. in Medium Matsubara Formalism Temperature T :  Matsubara Frequency Density  ：  Chemical Potential Decomposition of the Lorentz Structure S S S S 8

9.  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. 9

10.  Models of quark-gluon interaction vertex (1) Bare Ansatz (2) Ball-Chiu Ansatz (3) Curtis-Pennington Ansatz (Rainbow-Ladder Approx.) (4) BC+ACM (Chang, Liu, etc, PRL 106, 072001, Qin, etc, PLB 722) Satisfying W-T Identity, L-C. restricted Satisfying Prod. Ren. 10

11.   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 11

12. 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. 12

13.  Analyzing the spectral density function indicate 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) 13

14. Kun-lun Wang, Yu-xin Liu, et al., Phys. Rev. D 87, 074038 (2013)  T-dependence of the screening masses of some hadrons in contact interaction model, when, the color gets deconfined. Screening mass Degenerate  Chiral symmetry restored. Hadron properties provide signals for not only the chiral phase transt. but also the confinement-deconfnmt. phase transition. 14

15. 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; . 15

16. III. Locating the CEP via the DSE   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! 16

17.  Chiral Susceptibility as a Criterion S.X. Qin, L. Chang, H. Chen, Y.X. Liu, C.D. Roberts, PRL 106, 172301(‘11) S.X. Qin, L. Chang, H. Chen, Y.X. Liu, C.D. Roberts, PRL 106, 172301(‘11) For 2 nd order PT & Crossover,  s of the two phases diverge at the same states. 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. 17

18.  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 18

19. 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 19

20. The 2 nd, 3 rd, 4 th order fluctuations where   Locating the CEP with q. n. fluctuations Generalized Susceptibility Correlation Quark number: 20

21.  Quark Number Density Fluctuations vs T in the DSE X.Y. Xin, S.X. Qin, YXL, PRD 90, 076006 (2014) 21

22.  Quark Number Density Fluctuations vs μ in the DSE X.Y. Xin, S.X. Qin, YXL, PRD 90, 076006 (2014) 22

23.  Locating the CEP with the Critical Behavior Fei Gao, Yu-xin Liu, et al., to be published 23

24.  Some thermal properties Fei Gao, Yu-xin Liu, et al., to be published 24

25.  Dynamical Mass is generated by DCSB;  Confinement can be described with the positivity violation of the spectral function. Thanks !!  Far from well established ! Ⅳ. Ⅳ. Summary & Remarks  QCD phase transitions are investigated via DSE  DSE, a npQCD approach, is described  CEP of the QCD Phase Transition is discussed  chiral susceptibility criterion;  quark number fluctuations;  critical exponents of the & the C v, etc. 25

26. ♠ 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) ) 26

27.  目前关注的核心问题 (1): 相变级次与相变机制 Y. Aoki, et al., Nature 443, 675 (2006) : Lattice QCD: T Crossover, Evolve smoothly. F. Wilczek, Nature 443, 637 (2006) ； Nature Phys. 3, 375 (2007); 研究包含有限流夸克质量效应的强相互作用物质的 相结构和相变、以及相变的级次和机制十分必要， 并且迫切 ! 密度效应 ?! 27

28.  目前关注的核心问题 (2): 手征相变与禁闭相变的关系 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 any hierarchy between the two kind PTs ??  Inconsistent with Coleman-Witten Theorem (PRL 45, 100 (‘80)) : Confinement coincides with DCSB !! quarkyonic 28

29.  目前关注的核心问题 (3): 临界终点的存在性及其位置和观测信号  (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); Gupta, arXiv:~0909.4630[nucl-ex]; Li, et al., PRD 84, 071503 (2011); Gupta & Xu, et al., Science 332, 1525 (2011);   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 (2011); PLB 718, 1036 (2013); arXiv:1405.4762;  各大装置都将之列为最重要的研究目标，国际上有每年 一次的专题研讨会，足见对该问题的重视程度。 R.A. Lacey, et al., nucl-ex/0708.3512;  但到现在仍未正式发表。 29

30.  目前关注的核心问题 (4): 手征相变赝临界温度之上物质的性质 HTL Cal. (Pisarski, PRL 63, 1129(‘89); Blaizot, PTP S168, 330(’07) ), Lattice QCD ( Karsch, et al., NPA 830, 223 (‘09); PRD 80, 056001 (’09) ) NJL (Wambach, et al., PRD 81, 094022(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, 094506 (2010) ) claims: No No qualitative difference between the quark propagators in the deconfined and confined phases near the T c. RHIC experiments ( Gyulassy, et al., NPA 750, 30 (2005); Shuryak, PPNP 62, 48 (2009); Song, et al., JPG 36, 064033 (2009); … … ) indicate: the matter is in sQGP state.  What’s the nature of the matter in npQCD?? 30

31.  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) 。 31

32.  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) ) 32

33. 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) 33

34. ( Wei-jie Fu, and Yu-xin Liu, Phys. Rev. D 79, 074011 (2009) )   T-dependence of some properties of  &  -mesons and  -  S-L. in the model with contact interaction 34

35. Taking into account the DCSB effect 35

36.  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. 36

37. Radio Pulses =  “ Neutron ” Stars Compositions and Phase Structure of Compact Stars and their Identification Composition & Structure of NS are Still Under Study ! 37

38. 背景简介 ( F.Weber, PPNP 54, 193 (2005) ) Conjecture of the Composition of Compact Stars 38