QCD 相転移における秩序変数揺らぎとクォークスペクトル

根本幸雄 ( 名古屋大 ) with北沢正清 ( 基研 )国広悌二 ( 基研 )小出知威 (Rio de Janeiro Federal U.)

Phase Diagram of QCD

RHIC

TT

ETc

2Tc

chiral sym. broken (antiquark-quark condensate)confinement

chiral sym. restoreddeconfinement

quark-quark condensate

FAIR

compact stars

fluctuations of

fluctuations of

PRD65,091504,2002 (KKKN)

PRD70,056003,2004 (KKKN)

PTP.114,117,2005 (KKKN)

PLB631,157,2005 (KKN)

~170 MeVfrom Lattice QCD qq

qq PLB633,269,2006 (KKN)

T

0

QGP from high T to low TQGP from high T to low TQGP from high T to low TQGP from high T to low T

strong coupling weak coupling

HTL approximation

Hadronic

QGP

1T GeV~

1(10?)T GeV

• Lattice QCD at finite T (current status)

quenched approximationfull QCD with heavy quark mass

• our approachmodel calculationmassless quark limit (chiral limit)

genuine phase transition dynamics

~~weak coupling

Mean field approx.

exact chiral symmetry

CSC from high CSC from high to low to low CSC from high CSC from high to low to low

CSC

HDL appoximation

Matsuura et al. 2004

Quark spectrum aboveQuark spectrum above

T

CSC phase transitionCSC phase transition

fluctuations of qq

2 25

5 2 5 2

( ) ( )

( )( )

S

CCC A A

L i G i

G i i

τ

NJL-like model (w/ diquark-correlation)(2-flavor,chiral limit)

： SU(2)F Pauli matrices： SU(3)C Gell-Mann matricesC :charge conjugation operator

A A

3( 250MeV) , 93MeVf so as to reproduce

25.01GeV

650MeV

/ 0.62

S

C S

G

G G

Parameters:

Klevansky(1992), T.M.Schwarz et al.(1999)

2SC is realized at low and near Tc.

Nambu-Jona-Lasinio modelNambu-Jona-Lasinio modelNambu-Jona-Lasinio modelNambu-Jona-Lasinio model

2nd order transition from Wigner-to-CSC, even in the finite current quark mass.

Wigner phase

Description of fluctuations

Linear response theoryResponse of quark plasma to the perturbation caused byan external 　　　 pair field: ),(ext xtqq

A pair field is induced in the neighborhood of the external field:

qqGxt C2),(ind

)','()','(''),( extind xtxxttDdxdtxt R Linear response

),( xtDR :Response function=Retarded Green function

( , ) F.T. ( ), (0) ( )RD p qq x qq t We use RPA: ),( kDR

Collective ModesCollective mode is an elementary excitation of the system induced spontaneously.

),(),(),( extind kkDk R 0),(ext k

For the infinitesimally small external field, is non-zero if the denominator of is zero.

indRD

)(0),( 1 kkD

Dispersion relation of the collective mode

Spectral function: Strength of the response of the system to the external field.

In general, is complex.

),(Im1

),( kDkA R

Spectrum of diquark-fluctuations

Dynamical Structure Factor

T =1.1Tc T =1.05Tc

for= 400 MeV

),(1

1),( kA

ekS

Peaks of the collective modes survive up to T=1.2 Tc. (cf. 1.005 Tc in Metal)Large fluctuations

)(,0 CTT soft modes

ReIm diffusion-like

Pole position in the complex plane

( , )ni k

Quark self-energy (T-approximation)

Spectral Function of quark0 0( , ) ( , ) ( , )A p p p

quark

Spectrum of a single-quark

anti-quark

= 400 MeV=0.01 =(p)

[

MeV

]

k [MeV]

40

80

0

-40

-80400320 480

0

kF

kF

Normal Super

Disp. Rel.

stronger diquark coupling GC

Stronger diquark couplingsStronger diquark couplingsStronger diquark couplingsStronger diquark couplings

GC ×1.3 ×1.5

= 400 MeV=0.01

Resonant ScatteringResonant ScatteringResonant ScatteringResonant Scattering

GC=4.67GeV-2Mixing between quarks and holes

k

nf ()

kF

Quark spectrum aboveQuark spectrum above

T

chiral phase transitionchiral phase transition

fluctuations of qq

Recent topics near Tc

TT

ETc

2Tc

RHIC experimentsrobust collective flow

• good agreement with rel. hydro models• almost perfect fluid

(quenched) Lattice QCD

charmonium states up to 1.6-2.0 Tc(Asakawa et al., Datta et al., Matsufuru et al. 2004)

Strongly coupled plasma rather than weakly interacting gas

Description of fluctuations

Linear response theoryResponse of quark plasma to the perturbation caused byan external 　　　 pair field: ),(ext xtqq

A pair field is induced in the neighborhood of the external field:

qqGxt C2),(ind

)','()','(''),( extind xtxxttDdxdtxt R Linear response

),( xtDR :Response function=Retarded Green function

( , ) F.T. ( ), (0) ( )RD p qq x qq t We use RPA: ),( kDR

qq

ind ( , ) 2 St x G qq

( , ) F.T. ( ), (0) ( )RD p qq x qq t

Hatsuda, Kunihiro (’85)

sharp peak in time-like region

-mode

Spectral Function

k

2 2 ( )k m T

propagating mode

T = 1.1Tcm = 0

T

m

m softm

Tc

Spectrum of quark-antiquark fluctuations

|p|

( , )ni k

Quark self-energy

Spectral Function0 0( , ) ( , ) ( , )A p p p

0,05.1 CTT

| | Re 0 p

quark

3 peaks in also 3 peaks in

|p|

Spectrum of a single-quark

Resonant Scatterings of Quark for Resonant Scatterings of Quark for CHIRALCHIRAL Fluctuations Fluctuations

( , ) :p = + + …

E

E

0,08.1 CTT

dispersion law| | Re 0 p

0,05.1 CTT

Im

Re

Landau damping processes

p [MeV]p

[MeV]

+(,k)-(,k)

Resonant Scatterings of Quark for Resonant Scatterings of Quark for CHIRALCHIRAL Fluctuations Fluctuations

E

E

“quark hole”: annihilation mode of a thermally excited quark

“antiquark hole”: annihilation mode of a thermally excited antiquark(Weldon, 1989)

lead to quark-”antiquark hole” mixing

( )m T

pp

pp

cf: hot QCD (HTL approximation)

(Klimov, 1981)

0,05.1 CTT

( )m T

1.4 Tc1.2 Tc

1.1 Tc1.05 Tc

Spectral Contour and Dispersion RelationSpectral Contour and Dispersion Relation

p p

p p

p

p

p

p

+ (,k)

+ (,k)

+ (,k)

+ (,k)

-(,k)

-(,k)

-(,k)

-(,k)

Soft modes vs. massive scalar boson

the collective (soft) modes above Tc

propagating mode

2 2 ( )k m T

The widths are smaller as CT T

The soft modes can be approximately replaced by anelementary massive scalar boson.

The interaction of a quark and the soft modes are expressedby that of a fermion (quark) and a massive scalar boson.

Yukawa model at finite T

Fermion Spectrum in the Yukawa TheoryFermion Spectrum in the Yukawa TheoryFermion Spectrum in the Yukawa TheoryFermion Spectrum in the Yukawa Theory

quark + scalar boson ””

mq=0 m>0

cf.)mq>0 m=0

Baym, Blaizot, Svetitsky(’92)

( , )ni p

2 21( )

2L i i g m

One-loop Self-energy

Parameters: g, m, T

(on-shell renormalization for the T=0 part.)

The fermion (quark) spectral function

g=1

T/m=0.8 T/m=1.2 T/m=1.6

0 2 2 2 2

Im1 Im( , )

2 ( Re ) (Im ) ( Re ) (Im )p

p p

( , )p ( , )p

w w wp p p

Imaginary part of Imaginary part of Imaginary part of Imaginary part of

(a) (b)

Im

(,0

) g=1m=1T=2

0

{ ( )

( )

(

( )

( )

( ) )

( ) }

1

1( )

p k kp k

p p k k

p k k

p k k

k

p k

p k

k E

k

n f

n f

n f

n f

E

k E

k E

k

k

k

k

Parameters: g=1,m=1,T=2

for p=0

(a)

(b)

32

3

1Im ( , )

(2 ) 2 p

d ki g

E

p

k

Ek

E

(a)

(b)

Two Landau damping processes make two peak structure of Im.

Landau dampings:

2 2( )p kE p k m

energy of scalar boson

Dispersion RelationDispersion RelationDispersion RelationDispersion Relation

Im

(,0

)

Parameters: g=1,m=1,T=2

Re (

,0)

There appear five dispersions for k=0.

p

Three-peak structure in the quark spectrum also appears.

Two Landau damping processesform two peaks of the decay process.

Summary of quark spectrum in Yukawa modelSummary of quark spectrum in Yukawa modelSummary of quark spectrum in Yukawa modelSummary of quark spectrum in Yukawa model

From the analysis of the self-energy, we have found that

Yukawa NJL near Tc

Two resonant scatterings three peaks in the spectral function

E

w

0p massive bosonic mode

massless fermion

Im ( , 0)p

Summary 1Summary 1Summary 1Summary 1

Around Tc of the CSC and chiral phase transitions,existence of large fluctuations of the order parameters.

They affect a single-quark spectrum

CSC: mixing between a quark and a hole at the Fermi surfaceChiral: mixing between a quark and an antiquark-hole,

mixing between a antiquark and a quark-hole,

CSC chiral

Summary 2Summary 2Summary 2Summary 2

Similarity of the quark spectrum near chiral transition and the fermion spectrum in the Yukawa model.

interaction of a massless fermion and a massive boson at finite temperature

Fluctuations of qqare propagating modesmassive boson-like

chiral Yukawa

Outlook• finite quark mass effect (WIP)

2nd order crossover

• explicit gluon degrees of freedom (WIP)

• effects of observables on the fluctuations

The next talk (Mitsutani)

with S.Yoshimoto and M.Harada.

based on the Schwinger-Dyson approach

• improvement of approximationself-consistent T-approximation

quark-antiquark loop (cf. Braaten, Pisarski, Yuan 1990)paraconductivity,

dilepton production through