QCD Phase Transitions QCD Phase Transitions in an Improved PNJL in an Improved PNJL

ModelModel 刘玉鑫

北京大学 物理系

The 10th Workshop on QCD Phase Transitions & RHIC Physics, Sichuan University, Chengdu, Aug. 8-10, 2013

OutlineOutline I. IntroductionI. Introduction Ⅱ. The Improved PNJL ModelThe Improved PNJL Model Ⅲ. Numerical Results Ⅳ. Remarks

宇宙的形成、组分及演化涉及的基本问题宇宙的形成、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

现 在 宇宙

“ 已知”明亮物质：原子、分子 ( 强子 ) 物质

高度对称 ,mq = 0 .

mq > 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！？

Mu,d >= 100mu,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？

对基本问题归结为相变，研究正如火如荼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：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 SymmetricQuark deconfined

SB, Quark confined

sQGP

研究方法：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

？？

？？

？

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 .

Special Topic: Special Topic: Relation Between Relation Between the DCSB & the the DCSB & the Confinement ?Confinement ?

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);

== 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

Lagrangian (three flavors, traditional ) NJL Model

Effective Thermodynamical Potential

ⅡⅡ. The Improved PNJL Model. The Improved PNJL Modelcan describe DCS-DCSB PT well !

Intuitive View : PNJL Model

PNJL Effective Thermodynamical Potential

ⅡⅡ. . The Improved PNJL ModelThe Improved PNJL Model

can describe conf.-deconf. phl.

●●

●

♣ SUc(3) has center Z3 , color appears if Z3 broken ! ♣ Pure gauge effective thermal potential

ΩΩPNJLPNJL = = ΩΩNJLNJL + + U U P-loopP-loop

Improvement on the PNJL ModelPNJL Model

OriginallyOriginally U U P-loopP-loop == TT44 f f ((TT00, , TT, , ΦΦ, , ΦΦ*) *) , , One commonly used form of One commonly used form of f f : : ff = RR = RR aa((TT,,TT00))

++ bb((TT,,TT00)) lnln[1 RR 6[1 RR 6ΦΦ**ΦΦ + 4(+ 4(ΦΦ*3*3++ΦΦ33) RR 3() RR 3(ΦΦ**ΦΦ))22] ]

withwith

Result in 2+1 flavor DSE Result in 2+1 flavor DSE (arXiv:1306.6022)(arXiv:1306.6022) hints: hints:

TT44 P P SBSB = = TT44 ++ TT22μμ22 ++ μμ4 4

21

3619 2

23

243

Improvement on the PNJL Model

Our ImprovementOur Improvement

(2) In the functions (2) In the functions aa((TT,,TT00) and ) and bb((TT,,TT00) ,) ,

with with AA, , BB, and , and ηη being parameters . being parameters .

(1) (1) TT44 TT44 ++ AA TT22μμ22 ++ BB μμ4 4

~~ rescaled- rescaled-P P SBSB = = TT44 ++ 0.288 0.288 TT22μμ22 ++ 0.0146 0.0146 μμ4 4

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 !

ⅢⅢ. . Numerical ResultsNumerical Results

Detailed data for: T induces crossover, μ drives first order phase transition !

ⅢⅢ. . Numerical ResultsNumerical Results

Phase Diagram

ⅢⅢ. . Numerical ResultsNumerical Results

(TEc,µE

c) = (135MeV, 363MeV)

(TEc,µE

c) = (147MeV, 193MeV)Quarkyonic phaseStable!

Quarkyonic phaseStable!

Quarkyonic phaseMetastable!

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 !! Thanks !!

IVIV. Summary & Remarks. 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 .

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; .

Dyson-Schwinger Equations

QCD

Dyson-Schwinger Equations (DSEs) in Dyson-Schwinger Equations (DSEs) in QCD 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; .

？？

Practical Algorithm at Present Practical Algorithm at Present Truncation ： Preserving Symm. Quark

Eq.

Decomposition of the Lorentz Structure

Quark Eq. in Vacuum :

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

Models of the effective gluon propagatorModels 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

1/ 2 t

Derivation and analysis in arXiv:1209.1974 show that the one in 4-D should be infrared constant.

Models of quark-gluon interaction vertexModels of quark-gluon interaction vertex

(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.

SB

< qq >0 ~ - (240 MeV)3

DSE approach meets the requirements!

Dynamical Mass

In DSE approach )(

)(22

2

)(pA

pBpM

III. QCD Phase Transitions in DSEIII. QCD Phase Transitions in DSE

! condensatequark chiral : O qqparameterrder

Quantity to identify the phase transition Traditionally

Criterion in Dynamics: Equating Effective TPs

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

New Criterion: Chiral New Criterion: Chiral SusceptibilitySusceptibility

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, PRL 106, 172301(‘11)172301(‘11)

Phase diagram in bare vertex Phase diagram in BC vertex

For 2nd 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.

Effective Thermal Potential

New Crirerion: m2 of mesons

Phase diagram

K.L. Wang, L. Chang, Y.X. Liu, C.D. Roberts, et al., Phys. Rev. D 86, K.L. Wang, L. Chang, Y.X. Liu, C.D. Roberts, et al., Phys. Rev. D 86, 114001 (2012)114001 (2012)

parameters are taken from Phys. Rev. D 65, 094026 (1997), with fitted as

Effect of the Running Coupling Strength Effect of the Running Coupling Strength on the Chiral Phase Transition on the Chiral Phase Transition

f MeVf 93

(W. Yuan, H. Chen, Y.X. Liu, Phys. Lett. B 637, 69 (2006))(W. Yuan, H. Chen, Y.X. Liu, Phys. Lett. B 637, 69 (2006))

Lattice QCD result Lattice QCD result PRD 72, 014507 (2005)PRD 72, 014507 (2005)

(BC Vertex: L. Chang, Y.X. Liu, R.D. Roberts, et al., Phys. Rev. C 79, 035209 (BC Vertex: L. Chang, Y.X. Liu, R.D. Roberts, et al., Phys. Rev. C 79, 035209 (2009))(2009))

Bare vertexBare vertexCS phaseCS phase

CSB CSB phasephase

with D = 16 GeV2, 0.4 GeV

DCSB still exists beyond chiral limitDCSB still exists beyond chiral 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.

Intuitive picture of Mass Intuitive picture of Mass GenerationGeneration

K.L. Wang, L. Chang, Y.X. Liu, C.D. Roberts, et al., Phys. Rev. D 86, K.L. Wang, L. Chang, Y.X. Liu, C.D. Roberts, et al., Phys. Rev. D 86, 114001 (2012); 114001 (2012);

Chiral Symmetry Breaking generates the Chiral Symmetry Breaking generates the anomalous Magnetic Moment of Quarkanomalous Magnetic Moment of Quark

L. Chang, Y.X. Liu, & C.D. Roberts, PRL 106, 072001 (‘11)

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

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

Small σ long range in coordinate space

Diff. of CEP comes from diff. Conf. Length Diff. of CEP comes from diff. Conf. Length

MN model infinite range in r-spaceNJL model “ zero” range in r-space Longer range Int. Smaller E/TE 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)

Solving quark’s DSE Quark’s Propagator

Property of the matter above Property of the matter above but but near the T near the Tcc

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 Qin, Chang, Liu, et al., PRD 84, 014017(2011)84, 014017(2011)

T = 3.0Tc

Disperse Relation and Momentum Dependence of the Residues of the Quasi-particles’ poles

T = 1.1Tc

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

(<7.0Tc),

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.

( 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

“QCD” Phase Transitions may Happen

ⅣⅣ. . Possible ObservablesPossible 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., v2, Viscosity, , CC Fluct. & Correl., Hadron

Prop.,···

In Astron. Observ. M-R Rel., Rad. Sp., Inst. R. Oscil., Freq. G-M. Oscil., ···

QCD Phase TransitionQCD Phase Transitions

,c PTR ,c PTR

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)

Density & Temperature Dependence of some Properties of Nucleon in DSE Soliton Model

0/ BB 0/ RR0/MM (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) )

( S.X. Qin, L. Chang, Y.X. Liu, C.D. Roberts, et al., Phys. Rev. C 84, 042202(R) (2011) )

Some properties of mesons in DSE-BSE

( L. Chang, & C.D. Roberts, Phys. Rev. C 85, 052201(R) (2012) )

Present work

( 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

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

Only the star matter including deconfined Only the star matter including deconfined quarks can give stars with M quarks can give stars with M 2.0M 2.0Msunsun

Hadron-star’s mass can not be larger than 1.8Msun

G.Y. Shao, Y. X. Liu, Phys. Rev. D 82, 055801 (2010).

Quark-star’s mass can be larger than 2.0Msun

（ 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;

Gravitational Mode Pulsation Frequency Gravitational Mode Pulsation Frequency can be an Excellent Astronomical Signal 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 Frequency of g-mode oscillationg-mode oscillation

Ott et al. have found that these g-modepulsation 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).

Dynamical Mass is generated by DCSB; Phase Diagram is given; CEP is fixed & Coexisting Phase is discussed;

sQGP above but near the Tc is discussed.

Thanks !! Thanks !! Far from well established !

VV. Summary & Remarks. Summary & Remarks

QCD phase transitions are investigated via DSE

DSE, a npQCD approach, is described

Some possible observables are discussed .

Thin PancakesLorentz =100

Nuclei pass thru each other

< 1 fm/c

Huge StretchTransverse ExpansionHigh 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)

（ 2 ） Compact Star Observations

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/TE (> 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/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., NPA 830, 633c (2009);

Gupta & Xu, et al., Science 332, 1525 (2011); RHIC Exp. Estimate hints quite small E/TE ( 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);

Special topic (2): Special topic (2): Coexistence region Coexistence region (Quarkyonic ? ) (Quarkyonic ? )

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);

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

Special Topic (3): Special Topic (3): Quark Matter at T Quark Matter at T above but near T above but near Tc 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: 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?

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) 。

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) )

( 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

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)

Taking into account the Taking into account the SB effectSB effect

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.

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 !

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

Conjecture of the Composition of Compact Stars