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, 15-11-16

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

 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 .

 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, 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 Eq/TE ( 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 Eq/TE (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); 

 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 ?

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;  .

？ ？  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. ？ ？

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

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

 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, 042202(R), (2011). ) A.C. Aguilar, et al., JHEP 1007-002 Taking in the coefficient of the above expression Derivation and analysis in PRD 87, 085039 (2013) show that the one in 4-Dimmession should be infrared constant.

 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))

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

 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); 

 DCSB still exists beyond chiral limit 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. Solutions of the DSE with With  = 0.4 GeV with D = 16 GeV2,   0.4 GeV

 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, 116015 (2008) S.X. Qin, D. Rischke, Phys. Rev. D 88, 056007 (2013)

 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, 074038 (2013)

 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;  .

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!

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

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, 172301 (2011)

Phase diagram is given, CEP is proposed  Beyond Chiral Limit Fei Gao, Y.X. Liu, C.D. Roberts, et al., arXiv: 1507.00875.

Phase diagram in intrinsic space K.L. Wang, S.X. Qin, YXL, & CDR, et al., Phys. Rev. D 86, 114001 (2012).

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

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

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

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

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

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

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

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

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

 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, 076006 (2014)

 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: 1510.07543

 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:1510.07543.

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

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

 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

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 !!

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

 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, 114022 (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, 014017(2011)

 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, 014017(2011); F. Gao, S.X. Qin, Y.X. Liu, et al., Phys. Rev. D 89, 076009 (2014).

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)。

 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, 045203 (2007) )

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

 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: 0810.1084, Phys. Rev. Lett. 101， 181102 (2008)

Taking into account the DCSB effect

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, 114022 (2010) )