Locating the CEP via the Locating the CEP via the Dyson-Schwing Equation Dyson-Schwing Equation

Approach of QCDApproach of QCD Yuxin Liu

Dept. Phys.,Peking Univ., China

CPOD 2014, Bielefeld University, Germany, Nov. 20, 2014

OutlineOutline I. I. IntroductionIntroduction ⅡⅡ. . The DS Eq. Approach ⅢⅢ. . Locating the CEPLocating the CEP Ⅳ Ⅳ. . RemarksRemarks

1

FD problems are sorted to QCD FD problems are sorted to QCD PTsPTs QCD Phase Diagram: Phase Boundary, Specific States, e.g., CEP, sQGP, Quakyonic,

Items Influencing the Items Influencing the Phase Transitions: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. – FSBChiral SymmetricQuark deconfined

SB, Quark confined

sQGP

？？

？

2

Theoretical ApproachesTheoretical Approaches ：： Two kindsTwo kinds--Continuum & Discrete Continuum & Discrete (lattice)(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 ,DS equations , AdS/CFT, HD(T)LpQCD ,

The approach should manifest simultaneously: (1) DCSB & its Restoration , (2) Confinement & Deconfinement . 3

For the location of the CEP, different For the location of the CEP, different

approaches give quite distinct approaches give quite distinct results.results. (p)NJL model & others give quite large E

q/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 E

q/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); 4

Slavnov-Taylor Identity

Dyson-Schwinger Equations

axial gauges BBZ

covariant gauges QCD

ⅡⅡ. . The Dyson-Schwinger Equation ApproachThe 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; .

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Algorithms Algorithms of Solving the DSEs of of Solving the DSEs of QCD 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

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Expression of the quark gap Expression of the quark gap

equationequation Truncation ： Preserving Symm. Quark Eq.

Decomposition of the Lorentz Structure

Quark Eq. in Vacuum :

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Quark Eq. in MediumQuark Eq. in MediumMatsubara Formalism

Temperature T : Matsubara Frequency

Density ： Chemical Potential

Decomposition of the Lorentz Structure

Tnn )12(

S

S

S

S

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Models of the eff. gluon propagatorModels 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

1/ 2 t

Derivation and analysis in PRD 87, 085039 (2013) show that the one in 4-D should be infrared constant. 9

Models of quark-gluon interaction Models of quark-gluon interaction vertexvertex

(1) Bare Ansatz

(2) Ball-Chiu Ansatz

(3) Curtis-Pennington Ansatz

),( pq (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.

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In DSE approach

)(

)(22

2

)(pA

pBpM

Dynamical chiral symmetry breakingDynamical chiral symmetry breaking

Increasing the interaction

strength induces the

dynamical mass generation

0qq

K.L. Wang, YXL, et al., PRD 86,114001(‘12); K.L. Wang, YXL, et al., PRD 86,114001(‘12);

Numerical results

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with D = 16 GeV2, 0.4 GeV

DCSB still exists beyond chiral DCSB still exists beyond chiral limit limit

Solutions of the DSE with

With = 0.4 GeV

16 0.4

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.

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

H. Chen, YXL, et al., Phys. Rev. D 78, H. Chen, YXL, et al., Phys. Rev. D 78, 116015 (2008)116015 (2008)

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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.

SS Mr /1 mdS rr

Hadron properties provide signals for not only the chiral phase transt. but also the confinement-deconfnmt. phase transition.

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Dyson-Schwinger Equations

QCD

A comment on the DSE approach of A comment on the DSE approach of QCDQCD

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; .

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III. III. Locating the CEP via the Locating the CEP via the DSEDSE

! condensatequark chiral : O qqparameterrder

Quantity to identify the phase transition Traditionally

Criterion in Dynamics: Equating Effective TPs

With fully Nonperturbative approach, one could not have one could not have the ETPs.the ETPs. New Criterion must be established! 16

Chiral Susceptibility as a Chiral Susceptibility as a CriterionCriterion

S.X. Qin,S.X. Qin, L. Chang, H. Chen, Y.X. Liu, C.D. Roberts, L. Chang, H. Chen, Y.X. Liu, C.D. Roberts, PRL 106, 172301(‘11)PRL 106, 172301(‘11)

For 2nd 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.

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Phase diagram is given, CEP is Phase diagram is given, CEP is proposedproposed

S.X. Qin, L. Chang, H. Chen, Y.X. Liu, & C.D. Roberts, S.X. Qin, L. Chang, H. Chen, Y.X. Liu, & C.D. Roberts, Phys. Rev. Lett. 106, 172301 (2011)Phys. Rev. Lett. 106, 172301 (2011)

Phase diagram in bare vertex Phase diagram in BC vertex

},{},{

DeconfinedCSConfinedDCSB

regionexistenceCo

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Small ωω long range in coordinate space

Different methods give distinct Different methods give distinct locations locations of the CEP arises from diff. Conf. of the CEP arises from diff. Conf. Length Length

MN model infinite range in r-spaceNJL 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 19

The 2nd, 3rd, 4th order fluctuations

where

Locating the CEP with q. n. fluctuations

Generalized Susceptibility

. XXX NNN

Correlation

Quark number:

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Quark Number Density Fluctuations vs T in the DSE

X.Y. Xin, S.X. Qin, YXL, PRD 90, 076006 (2014)21

Quark Number Density Fluctuations vs μ in the DSE

X.Y. Xin, S.X. Qin, YXL, PRD 90, 076006 (2014)22

Locating the CEP with the Critical Behavior

Fei Gao, Yu-xin Liu, et al., to be published23

Some thermal properties

Fei Gao, Yu-xin Liu, et al., to be published24

Dynamical Mass is generated by DCSB; Confinement can be described with the

positivity violation of the spectral function.

Thanks !! Thanks !! Far from well established !

ⅣⅣ. . 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 Cv , etc .

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♠ ♠ Positivity Violation of the Spectral FunctionPositivity 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 Maximum Entropy Method

Result in DSEResult in DSEcTT 8.0

( Asakawa, et al., PPNP 46,459 (2001); Nickel, Ann. Phys. 322, 1949 (2007) )

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目前关注的核心问题目前关注的核心问题 (1): (1): 相变级次与相变机制相变级次与相变机制

Y. Aoki, et al., Y. Aoki, et al., Nature 443, 675 (2006) : Nature 443, 675 (2006) : Lattice QCD: T Crossover, Lattice QCD: T Crossover, Evolve smoothly. Evolve smoothly. F. Wilczek, Nature 443, 637 (2006)F. Wilczek, Nature 443, 637 (2006) ；； Nature Phys. 3, 375 (2007); 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 密度效应 ?!?!27

目前关注的核心问题目前关注的核心问题 (2): (2): 手征相变与禁闭相变的关手征相变与禁闭相变的关系系

claim that there exists a quarkyonic phase.

and General (large-Nc) 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

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目前关注的核心问题目前关注的核心问题 (3): (3): 临界终点的存在性及其位置和观测信临界终点的存在性及其位置和观测信号号 (p)NJL model & others give quite large E

q/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); 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 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 (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。

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目前关注的核心问题目前关注的核心问题 (4): (4): 手征相变赝临界温度之上物质的性手征相变赝临界温度之上物质的性质质 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: NoNo qualitative difference between the quark propagators

in the deconfined and confined phases near the Tc. 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

Approach 1: Soliton bag modelⅣⅣ. . Hadrons via DSEHadrons 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) 。

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Effect of the F.-S.-B. (Effect of the F.-S.-B. (m0) on Meson) on Meson’’s Masss 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

Electromagnetic Property & PDF of Electromagnetic Property & PDF of hadronshadrons

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

( 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

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Taking into account the DCSB effectTaking into account the DCSB effect

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Analytic Continuation from Euclidean Space Analytic Continuation from Euclidean Space

to Minkowskian Space to Minkowskian Space

( W. Yuan, S.X. Qin, H. Chen, & YXL, PRD 81, 114022 (2010) )

= 0, ei=1, ==> E.S. = , ei=1, ==> M.S.

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Radio Pulses = “Neutron” Stars

Compositions and Phase Structure Compositions and Phase Structure of Compact Stars and their of Compact Stars and their IdentificationIdentification

Composition & Structure of NS are Still Under Study !

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背景简介背景简介( F.Weber, PPNP 54, 193 (2005) )

Conjecture of the Composition of Compact Stars

38