1. QCD Phase Transitions in an Improved PNJL Model 刘玉鑫 北京大学 物理系 The 10 th Workshop on QCD Phase Transitions & RHIC Physics, Sichuan University, Chengdu, Aug. 8-10, 2013 Outline Outline I. Introduction The Improved PNJL Model Ⅱ. The Improved PNJL Model Ⅲ. Numerical Results Ⅳ. Remarks

2. 宇宙的形成、组分及演化涉及的基本问题 不束缚的夸克、 胶子及电子等 夸克、 胶子 囚禁 强子 核 合 成 原 子 核 复 合 时 期 原 子 星系形成星系形成 现 在 宇宙现 在 宇宙 “ 已知 ” 明亮物质 ： 原子、分子 ( 强子 ) 物质 高度对称, m q = 0. m q > 0 ， 呈 6 味 3 代. 为何禁闭 ? 如何禁闭 ? 明亮物质 ~ 5%. > 95% ? ? 暗物质 ？ 暗能量？ 手征对称性硬破缺, Higgs 机制, Higgs 粒子的确认！ ？ M u,d >= 100m u,d, 如何产生 ？ 强作用非微扰效应 ！ ? “ 全新 ” 形态物质： QPG? sQGP? 强子结构 ？

3. 对基本问题归结为相变，研究正如火如荼 Schematic QCD Phase Diagram 影响 QCD 相变的因素 : 介质效应：温度, 密度 ( 或化学势 ) 有限尺度 内禀因素：流质量, 跑动耦合强度, 色味结构, 涉及相变： 禁闭（强子化） – 退禁闭 手征对称性破缺 – 恢复 味对称 – 味对称性破缺 Chiral Symmetric Quark deconfined  SB SB, Quark confined sQGP 研究方法： 实验： RHIC 、 Ast-Obs. 理论：离散场论、连续场论 计算：实现理论、模拟 ？ ？ ？ ？ ？

4.  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.

5. Special Topic: Relation Between the DCSB & the Confinement ? 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) ;  ==  DCSB does not coincide with confinement !  What practical approaches can give ?  Phase boundaries ? CEPs?  Coleman-Witten Theorem (conjecture) : They coincide with each other (PRL 1980) quarkyonic

6. Lagrangian (three flavors, traditional ) NJL Model Effective Thermodynamical Potential Ⅱ. The Improved PNJL Model can describe DCS-DCSB PT well !

7. Intuitive View : PNJL Model PNJL Effective Thermodynamical Potential Ⅱ. The Improved PNJL Model can describe conf.-deconf. phl. ●● ● ♣ SU c (3) has center Z 3,  color appears if Z 3 broken ! ♣ Pure gauge effective thermal potential

8. Ω PNJL = Ω NJL + U P-loop Ω PNJL = Ω NJL + U P-loop   Improvement on the PNJL Model PNJL Model Originally U P-loop = T 4 f (T 0, T, Φ, Φ*), U P-loop = T 4 f (T 0, T, Φ, Φ*), One commonly used form of f : f = ̶̶ a (T,T 0 ) f = ̶̶ a (T,T 0 ) + b(T,T 0 ) ln [1 ̶̶ 6Φ * Φ + 4(Φ *3 +Φ 3 ) ̶̶ 3(Φ*Φ) 2 ] + b(T,T 0 ) ln [1 ̶̶ 6Φ * Φ + 4(Φ *3 +Φ 3 ) ̶̶ 3(Φ*Φ) 2 ] with with Result in 2+1 flavor DSE (arXiv:1306.6022) hints: T 4  P SB = T 4 + T 2 μ 2 + μ 4 T 4  P SB = T 4 + T 2 μ 2 + μ 4

9.   Improvement on the PNJL Model Our Improvement (2) In the functions a(T,T 0 ) and b(T,T 0 ), with A, B, and η being parameters. with A, B, and η being parameters. (1) T 4  T 4 + A T 2 μ 2 + B μ 4 ~ rescaled- P SB = T 4 + 0.288 T 2 μ 2 + 0.0146 μ 4 ~ rescaled- P SB = T 4 + 0.288 T 2 μ 2 + 0.0146 μ 4

10. Preliminary calculations are carried out with A = 0.30 ( ~ 0.288, the one in rescaled P SB ), {PNJL parameters} = { commonly used }, and B and η as free parameters. The Improvement gives: T induces crossover, μ drives first order phase transition ! Ⅲ. Numerical Results

11. Detailed data for: T induces crossover, μ drives first order phase transition ! Ⅲ. Numerical Results

12. Phase Diagram Ⅲ. Numerical Results (T E c,µ E c ) = (135MeV, 363MeV) (T E c,µ E c ) = (147MeV, 193MeV) Quarkyonic phase Stable! Quarkyonic phase Stable! Quarkyonic phase Metastable!

13.  Phase Diagrams of the two PTs are given;  The two PTs separate from each other;  The CEPs of the two PTs may coincide;  Quarkyonic phase may be a metastable phase. Thanks !! IV. Summary & Remarks  QCD phase transitions are investigated  An improved PNJL model is proposed with the quark chemical potential effect being involved explicitly in the Polyakov-loop potential.  Thorough investigations are needed.

14. 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; .

15. Dyson-Schwinger Equations QCDQCD  Dyson-Schwinger Equations (DSEs) in 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; . ？ ？

16.  Practical Algorithm at Present  Truncation ： Preserving Symm.  Quark Eq.  Decomposition of the Lorentz Structure   Quark Eq. in Vacuum :

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

18.  Models of the effective 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 arXiv:1209.1974 show that the one in 4-D should be infrared constant.

19.  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.

20. DSE approach meets the requirements to describe the evolution process of early universe matter  SB 0 ~ - (240 MeV) 3 DSE approach meets the requirements! Dynamical Mass In DSE approach

21. III. QCD Phase Transitions in 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!

22.  New Criterion: Chiral Susceptibility 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) Phase diagram in bare vertexPhase diagram in BC vertex For 2 nd order PT & Crossover,  s diverge at 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.

23. Effective Thermal Potential New Crirerion: m 2 of mesons Phase diagram K.L. Wang, L. Chang, Y.X. Liu, C.D. Roberts, et al., Phys. Rev. D 86, 114001 (2012)

24. parameters are taken from Phys. Rev. D 65, 094026 (1997), with fitted as  Effect of the Running Coupling Strength on the Chiral Phase 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, Y.X. Liu, R.D. Roberts, et al., Phys. Rev. C 79, 035209 (2009)) Bare vertex CS phase CSB phase

25. with D = 16 GeV 2,   0.4 GeV  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.

26.  Intuitive picture of Mass Generation K.L. Wang, L. Chang, Y.X. Liu, C.D. Roberts, et al., Phys. Rev. D 86, 114001 (2012); 

27. Chiral Symmetry Breaking generates the anomalous Magnetic Moment of Quark L. Chang, Y.X. Liu, & C.D. Roberts, PRL 106, 072001 (‘11)

28.  Phase diagram is given, CEP is fixed. 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

29. Small σ  long range in coordinate space  Diff. of CEP comes 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, L. Chang, H. Chen, Y.X. Liu, et al, PRL 106, 172301(‘11) S.X. Qin, L. Chang, H. Chen, Y.X. Liu, et al, PRL 106, 172301(‘11)

30. 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)

31. 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, C.D. Roberts, PRD 84, 014017(‘11) S.X. Qin, L. Chang, Y.X. Liu, C.D. Roberts, PRD 84, 014017(‘11) 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.

32. ( Kun-lun Wang, Yu-xin Liu, Lei Chang, C.D. Roberts, & S.M. Schmidt, Phys. Rev. D 87, 074038 (2013) )   T-dependence of some hadrons’ properties in DSE

33. “QCD” Phase Transitions may Happen Ⅳ. Possible Observables General idea ， Phenom. Calc., Sophist. Calc., quark may get deconfined (QCD PT) at high T and/or  Signals for QCD Phase Transitions: In Lab. Expt. Jet Q., v 2, Viscosity, , CC Fluct. & Correl., Hadron Prop.,··· In Astron. Observ.. M-R Rel., Rad. Sp., Inst. R. Oscil., Freq. G-M. Oscil., ··· QCD Phase Transitions s

34. Collective Quantization: Nucl. Phys. A790, 593 (2007). DSE Soliton Description of Nucleon B. Wang, H. Chen, L. Chang, & Y. X. Liu, Phys. Rev. C 76, 025201 (2007)

35.   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, 025206 (2010) )

36. ( 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

37. ( Wei-jie Fu, and Yu-xin Liu, Phys. Rev. D 79, 074011 (2009) ; Kun-lun Wang, Yu-xin Liu, & C.D. Roberts, PRD 87, 074038)   T-dependence of some properties of mesons in the model with contact interaction

38.  Fluctuation & Correlation of Conserved Charges Recent Lattice QCD results agree with ours excellently! W.J. Fu, Y.X. Liu, & Y.L. Wu, PRD 81, 014028 (2010) 。 HotQCD Collaboration, 1203.0784

39.  Only the star matter including deconfined quarks can give stars with M  2.0M sun  Hadron-star’s mass can not be larger than 1.8M sun G.Y. Shao, Y. X. Liu, Phys. Rev. D 82, 055801 (2010).  Quark-star’s mass can be larger than 2.0M sun （ Nature 467, 1081 (2010) reported ） H.S. Zong, et al., PRD 82, 065017 (2010); MPL 25, 47 (2010); PRD 83, 025012 (2011); PRD 85, 045009 (2012); etc;

40.  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

41. 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, ····· g-mode oscillation frequency can be a signal identifying the QCD phase transitions in high density matter ( compact star matter).

42.  Dynamical Mass is generated by DCSB;  Phase Diagram is given;  CEP is fixed & Coexisting Phase is discussed;  sQGP above but near the T c  is discussed. Thanks !!  Far from well established ! V. Summary & Remarks  QCD phase transitions are investigated via DSE  DSE, a npQCD approach, is described  Some possible observables are discussed.

43. Thin Pancakes Lorentz  =100 Nuclei pass thru each other < 1 fm/c Huge Stretch Transverse Expansion High Temperature (?!) The Last Epoch: Final Freezeout-- Large Volume We measure the “final” state, we are most interested in the “intermediate” state, we need to understand the “initial” state…  Experimental Aspects ： （ 1 ） Relativistic Heavy Ion Collisions (u23u) (u23u) （ 2 ） Compact Star ObservationsObservations

44.  Special Topic (1) :  Special Topic (1) : Critical EndPoint (CEP) The Existence & Position of CEP is highly debated ! What sophisticated DSE calculation can give? Why different models give distinct results ?  (p)NJL model & others give quite large  E /T E (> 3.0) Sasaki, et al., PRD 77, 034024 (2008); Costa, et al., PRD 77, 096001 (2008); 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); Costa, et al., PRD 81, 016007 (2010);  Hatta, et al., PRD 67, 014028 (2003); Cavacs, et al., PRD 77, 065016(2008);   Lattice QCD gives smaller  E /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., NPA 830, 633c (2009); Gupta & Xu, et al., Science 332, 1525 (2011);   RHIC Exp. Estimate hints quite small  E /T E (  1) R.A. Lacey, et al., nucl-ex/0708.3512;   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, 036001 (2009);  

45.  Special topic (2): Coexistence region (Quarkyonic ? ) 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) ;   Can sophisticated continuous field approach of QCD give the coexistence (quarkyonic) phase ?  What can we know more for the coexistence phase?  Inconsistent with Coleman-Witten Theorem !! quarkyonic

46.  Special Topic (3): Quark Matter at T above but near T c 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??

47.  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) 。

48.  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) )

49. ( 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

50. 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)

51. Taking into account the  SB effect

52.  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.

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

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