カイラル相転移・カラー超伝導の臨界温度近傍におけるクォークの準粒子描像

Masakiyo KitazawaKyoto Univ.

M.K., T.Koide, T.Kunihiro and Y.Nemoto, PRD70,056003 (2004),M.K., T.Koide, T.Kunihiro and Y.Nemoto, hep-ph/0502035,

M.K., T.Kunihiro and Y.Nemoto, in preparation× ２ .

第６回　関西ＱＮＰセミナー　於：京大基研

2SC pairingat low energy:

150~170MeV

Phase Diagram of QCDPhase Diagram of QCD

Color SuperconductivityHadrons

T

Chiral Symm.Broken

0

attractive channel in one-gluon exchange interaction.

quark (fermion) systemColor Superconductivity

Cooper instability at sufficiently low TSU (3)c color-gauge symmetry is broken!

RHIC

Compact Stars

GSI,J-PARC

[ ３ ]c×[ ３ ]c ＝ [ ３ ]c ＋ [ ６ ]c

u d

150~170MeV

Phase Diagram of QCDPhase Diagram of QCD

Color Superconductivity(CSC)Hadrons

T

Chiral Symm.Broken

0

~100MeV

Hadronic excitations in QGP phase•soft mode of chiral transition - Hatsuda, Kunihiro.•qq bound state - Shuryak, Zahed; Brown, Lee, Rho, Shuryak.•Lattice simulations – Asakawa, Hatsuda; etc.

Pre-critical region of CSC

•large pair fluctuations precursory phenomena of CSC

M.K., et al., 2002,2004

The pseudogap survives up to =0.05~0.1 ( 5~10% above TC ).

Numerical Result : Density of StateNumerical Result : Density of State

( )( )free

NN

Spectral Function of QuarksSpectral Function of Quarks

[MeV]k [MeV]

= 0 MeV = 0.05

0 0 0 0 0( , ) ( , ) ( , )A p p p p p pquark part

-(,k)

sharp peak withnegative dispersion

k [MeV]

[MeV]

quasiparticle peak ~ k

TABLE OF CONTENTS

1, Introduction2, Quarks above CSC phase transition 3, Quarks above chiral phase transition4, Summary

2,2, Quarks above CSC

phase transitionT

T

0

Nature of CSCNature of CSC

strong coupling! weak coupling

~ 100MeV/ EF ~ 0.1 / EF ~ 0.0001

in electric SC

Large pair fluctuations can

Short coherence length .

Mean field approx.works well.

Matsuzaki, PRD62,017501 (2000) Abuki, Hatsuda, Itakura, PRD 65, 074014 (2002) cf.) Bosonization of Cooper pairs

invalidate MFA.cause precursory phenomena of CSC.

There exist large fluctuations of pair field.

ペア場のゆらぎペア場のゆらぎ

二次相転移点では、秩序変数のゆらぎが発散。

ペア場 (x) for CSC

F()

Tc で原点に到達　　　　ソフトモード

T

カラー超伝導

1

ペア場のゆらぎは、集団モードを形成する。

極

クォーク対

Pair Fluctuations in SuperconductorsPair Fluctuations in Superconductorselectric conductivity

~10-3

enhancementabove Tc

Precursory Phenomena in Alloys•Electric Conductivity•Specific Heat•etc…

Thouless, 1960Aslamasov, Larkin, 1968Maki, 1968, …

High-Tc Superconductor(HTSC)

large fluctuations induced by strong coupling and quasi-two dimensionality

pseudogap

1986~in quasi-two-dimensional cuprates

C

C

T TT

k

2 2sgn( ) ( )k k

2 2( )

d kdk k

Quasi-particle energy:

2( ) dkN kd

( )N

2

Density of State:

Quarks in BCS TheoryQuarks in BCS Theory

The origin of the pseudogap in HTSC is still controversial.

:Anomalous depression of the density of state near the Fermi surface in the normal phase.

Pseudogap

Conceptual phase diagram

Renner et al.(‘96)

2 25

5 2 5 2

( ) ( )

( )( )

S

CCC A A

L i G i

G i i

τ

Nambu-Jona-Lasinio model (2-flavor,chiral limit) ：

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

A AIH

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

25.01GeV650MeV

/ 0.62

S

C S

G

G G

Parameters:

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

M.K. et al., (2002)

2SC is realized at low and near Tc.We neglect the gluon degree of freedom.

Notice:

NJL modelNJL model

5 2 2† h.c.ex

CexH d i x

expectation value of induced pair field:external field:

0

5 2 2( ) ( ) ( ), ( , )tC

exexind tx i x i ds H s O t x

5 2 22 ( )( ) ( '( ,( )) ' )'indC

C x

Rexe

G x i x Dd x x xtx d x

Linear Response

5 2 2 5 2 22 ( ) ( ), (0) (0) (, ) )( CR CCG x i xD t i t x

Retarded Green function

Fourier Transformation

( ) (( ) ),ind exD kk k

Response Function of Pair FieldResponse Function of Pair Field

1( ) ( ) ( ) (( , ) )tot ind e Cx exG kk k k k

T-matrix

Rondom Phase Approx. (RPA)

( , )nQ k, n mi i k p

, mip

CT T

(

)

2

2

( 0)

Thouless CriterionDR(0,0) diverges at TC

2

2

( 0)

1 1 ( ,0)CG Q 0

- for second order phase transitions

D.J. Thouless, AoP 10,553(1960)

r.h.s. is equal to zero at Tc due to the critical conditon.

The fluctuation diverges at Tc.

ThermodynamicPotential

( , ) CG k 1

1( , )CG Q

k

Softening of Pair FluctuationsSoftening of Pair Fluctuations

Dynamical Structure Factor

T =1.05Tc

The peak grows from ~ 0.2 electric SC ： ~ 0.005

1 1( ) I (m )1

Se

kk

= 400 MeVc

c

T TT

Softening of Pair FluctuationsSoftening of Pair Fluctuations

Dynamical Structure Factor

T =1.05Tc

= 400 MeV

The peak grows from ~ 0.2 electric SC ： ~ 0.005

1 1( ) I (m )1

Se

kk

Pole of Collective Mode

1 1 ( , ) 0CG Q k

1( ) ( , () )tot C exG kk k

pole:

c

c

T TT

The pole approaches the origin as T is lowered toward Tc.

(the soft-mode of the CSC)

0

1( , )( , ) ( , )n

n n

Gi i

iG

k

kk

T-matrix ApproximationT-matrix Approximation

Quark Green function :

( , )n k , n mi i k q

, miq

Decomposition of G:

0 00

0 0 0

1( , )G pp p p

pp p

quark part

012

p

:projection op.

30

3

q ( , )(2 )

( , ) mn mm

dT G

q qk

Spectral Function of QuarksSpectral Function of QuarksSpectral Function

1( , ) Im ( , )RA G

k k

30

3( ) ( , )(2 )dN

k k 0 01( , ) Tr Im ( , )

4RG k k

Density of State N() 3 0N d x

0 0 ˆ( , ) ( , ) ( , ) ( , )V SA k k k k kfrom parity and rotational invariance

vanishes in the chiral limit

spectral function of baryon density

The pseudogap survives up to =0.05~0.1 ( 5~10% above TC ).

Numerical Result : Density of StateNumerical Result : Density of State

( )( )free

NN

0(,k)= 400 MeV=0.01

Spectral Function of QuarksSpectral Function of Quarks

k

0[MeV]

quasi-particle peak,=k)~ k

Depressionat Fermi surface

Im ,k=kF)

[MeV]

The peak in Im around =0owing to the decaying process:

k [MeV]kF

kF

Im ,k)

quasi-particle peak=–kpeak of Im=k–

,00

,k k

: collective mode

: on-shell

|Im | has peaks around =k, which is found to be the hole energy.

|Im -|

k

coincide at fermi surface.

Re

,k)

=–k

-0

kPeak of |Im |

kF

C

C

T TT

Dispersion Relation of QuarksDispersion Relation of Quarks=(p)

rapid increase around =0

[

MeV

]

k [MeV]

40

80

0

-40

-80400320 480

0

kkF0

k

kF

Normal Supercf.) 1Re ( , ) 0G p

Re ( , ) 0 p p

= 400 MeV=0.01

Dispersion Relation of QuarksDispersion Relation of Quarks=(p)

rapid increase around =0

[

MeV

]

k [MeV]

40

80

0

-40

-80400320 480

ImRe ( , ) ( , '1'

)RR d

i

kkRe ,k=kF)

[MeV]

Re ( , ) 0 p p

= 400 MeV=0.01

w.f. renormalization

still Fermi-liquid-like

11 ( , ) / 0.7Z k

However,

stronger diquark coupling GC

Diquark Coupling DependenceDiquark Coupling Dependence

GC ×1.3 ×1.5

= 400 MeV=0.01

Resonant Scattering of QuarksResonant Scattering of QuarksGC=4.67GeV-2

Re ( , ) 0 p p

p

Re ( , ) p

Janko, Maly, Levin, PRB56,R11407 (1995)

Resonant Scattering of QuarksResonant Scattering of QuarksGC=4.67GeV-2

Mixing between quarks and holes

k

nf ()kF

Level RepulsionLevel Repulsion

40 0 0

0 04

qIm ( , ) Im ( , ) Im ( , ) tanh coth(2 ) 2 2

R q qd q G qT T

k k q q

2Im ( , ) 2 ( | | )R p p

2 4 (3)( ) Im ( , ) (2 ) ( ) ( )Bf p p

2

1 Im ( , ')Re ( , ) ''

12| |

RR P d

pp

p

Re ( , ) 0 p p 2 2 2( ) 2 0 p

ppF

Quarks at very high Quarks at very high TT•1-loop (g<<1)•Hard Thermal Loop ( p, , mq<<T )

),( p

2 2 216fm g T

pE

Re[ ( , )] 0D p

1,p hE E hE

Re[ ( , )] 0D p

1,p hE E pE

hEdispersion relations plasmino

plasmino

0 00

( , ))

,,

( )(

Dp

DG

pp

p

pE

pE

Quarks at very high Quarks at very high TT•1-loop(g<<1)•Hard Thermal Loop approximation( p, , mq<<T )

),( p

2 2 218fm g T

Re[ ( , )] 0D p

1,p hE E

hERe[ ( , )] 0D p

1,p hE E

hEdispersion relations

0 00

( , ))

,,

( )(

Dp

DG

pp

p

3,3, Quarks above chiral

phase transitionT

Soft Mode of Chiral TransitionSoft Mode of Chiral Transition

Response Fucntion D(k,)

( , )D k

fluctuations of the chiral order parameter

(1( ) Im )DA kk

Spectral Functionε→ ０（Ｔ→ＴＣ）

for k=0

T

Hatsuda, Kunihiro (’85)

scalar and pseudoscalar parts

Sigma Mode above Sigma Mode above TTcc Hatsuda, Kunihiro (’85)

sharp peak in time-like region

-mode

Spectral Function

soft mode of CSC

k

2 2k m

k

sharp peak around = k =0

0

1( , )( , ) ( , )n

n n

Gi i

iG

k

kk

( , )ni k

30

3

q ( , )(2 )

( , ) mm

n mdT GD

q qk

( , )nD i k 1

Quark Self-enrgyQuark Self-enrgy

Quark Green function :

0 ( , )nG i k :free quark progagator

Self-energy:

in the chiral limit

Spectral Function of QuarksSpectral Function of Quarks

[MeV]k [MeV]

= 0 MeV = 0.05

0 0 0 0 0( , ) ( , ) ( , )A p p p p p pquark part

-(,k)

sharp peak withnegative dispersion

k [MeV]

[MeV]

quasiparticle peak ~ k

Self EnergySelf Energy

k [MeV]

[MeV]

Two peaks in Im produces five solutionsof the dispersion relation.

Spectral Function of QuarksSpectral Function of Quarks

[MeV]k [MeV]

= 0 MeV = 0.05

0 0 0 0 0( , ) ( , ) ( , )A p p p p p ppositive energy part

-(,k)

k [MeV]

[MeV]

-(,k)+(,k)

k [MeV]

Resonant Scatterings of QuarksResonant Scatterings of Quarks

q hq q hq

hole,qqq

soft)( qq

hole,qqq

soft)( qq

hole, qq q

soft)( qq

hole,qq q

soft)( qq

These resonant scatterings affect the peaks of the spectral functions in a non-trivial way.

Level RepulsionLevel Repulsion

2Im ( , ) 2 ( | | ) ( | | )R m m p p p

2 4 (3)( ) Im ( , ) (2 ) ( ) ( )Bf p p

2 4 (3)

( ) Im ( , )

(2 ) ( ) ( ) ( )Bf

m m

pp

m> m=

dispersion relation

m,-m m,-m

for the CSC

Self EnergySelf Energy

holeqq

soft)( qq

q q

soft)( qq

k [MeV]

[MeV]

-(,k)+(,k)

k [MeV]

T dependenceT dependence

= 0.05

k [MeV]

[MeV]

-(,k)+(,k)

k [MeV]

T dependenceT dependence

= 0.1

k [MeV]

[MeV]

-(,k)+(,k)

k [MeV]

T dependenceT dependence

= 0.15

k [MeV]

[MeV]

-(,k)+(,k)

k [MeV]

T dependenceT dependence

= 0.2

k [MeV]

[MeV]

-(,k)+(,k)

k [MeV]

T dependenceT dependence

= 0.25

k [MeV]

[MeV]

-(,k)+(,k)

k [MeV]

T dependenceT dependence

= 0.3

k [MeV]

[MeV]

-(,k)+(,k)

k [MeV]

T dependenceT dependence

= 0.35

k [MeV]

[MeV]

-(,k)+(,k)

k [MeV]

T dependenceT dependence

= 0.4

k [MeV]

[MeV]

-(,k)+(,k)

k [MeV]

T dependenceT dependence

= 0.5

SummarySummaryThe soft mode associated with the chiral and color-superconducting phase transitions drastically modifies the property of quarks near Tc.

above CSC phase:Gap-like structure manifests itself! resonant scattering of quarks

Future: finite quark mass, finite density,phenomenological applications

above chiral transition:Three peak structure appears! two resonant scatterings of quarks and anti-quarks