QCD 相転移の臨界点近傍における

非平衡ダイナミクスについて北沢正清（京大） , 国広悌二（京大基研 ), 根本幸雄 (RIKEN-BNL)

0

T

の１コメント

Chiral symmetry breaking Color superconductivity

(CSC)

critical endpoint

CONTENTS

1, Introduction

2, Collective Mode in CSC

3, Effective Equation for Collective Mode4, Numerical Simulation

5, Summary and Outlooks

2( , ) c

c

A BT T

TT

k k

Main New Results given in This Talk

derive equation for the collective mode of the pair field above CSC,

position of pole:

0

T

(determine A, B microscopically)

(x)

x

(x)

x

=2/k

in linear response theory.

Phase Transitions and Fluctuations in QCD

fluctuation diverges. CT T

CT T

(

)

CT T

At the critical point of the second order phase transitions,

fluctuation of order parameter field

might be responsiblefor various observables.

chiral transitioncritical end point (CEP)CSC transition

ˆ ( ) ( ) ( )O x x x

5ˆ ( ) ( ) ( )C

A AO x x i x

ˆ ( )O x

0

2SC

T

CFL

RHIC

AGSSPS

GSI,J-PARC

another CEP?

1,1, Introduction

?

??light sigma meson

susceptibilitiesbaryon number, chiral, etc…

transport coefficient???Stephanov, Rajagopal, Shuryak / Berdnikov, Rajagopal /

Hatta, Ikeda / Fukushima / Fujii / Hatta, Stephanov

Spectral Function of Pair Field

C

C

T T

T

ε→ ０（Ｔ→ＴＣ）

＋ ＋・・・1( ) Im

k

As T is lowered toward TC,The peak of becomes sharp.

M.K., T.Koide, T.Kunihiro, Y.Nemoto, PRD 65, 091504 (2002)

The peak survives up to ~ 0.2

electric SC ： ~ 0.005(k=0,=0) diverges at T=TC.

Thouless criterion

provided the second order transition,

Fluctuation of pair field in CSC

Dependence of Pseudogap

Depth of the pseudogap hardly changes with .

Pseudogap in quark DOS!

Model

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

25.01GeV

650MeV

/ 0.62

S

C S

G

G G

Parameters:

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

M.K. et al., (2002)

2,2, Collective Mode in CSC

Response Function of Pair Field

5 2 2† h.c.ex

CexH d i x

( , )nD k

expectation value of induced pair field:

external field:

0

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

exex tx i x i ds H s O t x

† †(( ) , ) ( )ind n ex nnD kk k

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

C x

Rexe

G x i x Dd x x xtx d x

Linear Response

Fourier transformationwith Matsubara formalism

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

Retarded Green function

( , )nQ k

RPA approx.: ( , )

1 ( , )C n

C n

G Q

G Q

k

k

where,

( , )nQ k

, 'n mi i k k

, 'mi k

3

5 2 2 5 2 23

1 12 Tr

(2 ) ( ') 'm

dT i i

k k k

k'

3 2 2

3

' ( 2 )Im ( , ) ( 1)

(2 )

1 ( ) ( ) ( 2 )

1 ( ) ( ) ( 2 )

( ) ( ) ( 2 )

( ) ( ) ( 2 )

f c

d kQ N N

E E

f E f E E E

f E f E E E

f E f E E E

f E f E E E

k' k-k'

k' k-k' k' k-k'

k' k-k' k' k-k'

k' k-k' k' k-k'

k' k-k' k' k-k'

kk

Analytic Properties of Q(k,)

( , )nD k 1

CG

3 2 2

3

( 2 ) ( 2 )

( 2 ) (

1 ( ) ( ) 1 ( ) ( )

( ) ( )

' ( 2 )( 1)

(

( ) ( )

2 )

2 )

f c

f E f E f EE E E E

E E E

d kN

f E

f E f E f E f

NE

E

E

E

k' k-k' k' k-k'

k' k-k' k' k-k

k' k-k' k' k-k'

k' k

k' k-k' k' k

k'

'

-

-k'

k

1 ( ) 1 ( ) ( )

1

( )

( ) ( )

f E f E f E f E

f E f E

k' k-k

k' k-k' k' k-k'

'

, E k'k'

, E k-k'k - k'

,k , E k'k'

, E k-k'k - k'

,k

1st term:

( ) 1 ( )

( ) ( )

1 ( ) ( )f E f E f

E

f E

E

E

f f

k' k-k' k' k-k

k k'

'

' k-

3,4th term:

Im Q(k,)

k

2

2

paircreation

scattering

0

~0 k~0 near Tc

Collective mode

2nd term:

H.Fujii, PRD67 (2003) 094018cf.) in the chiral phase transition

( , 0) 0 k

2 2

( ) / 2

( ) / 2

( 2 )Im ( , ) ( 1)

2

cosh ( ) / 4log 2

cosh ( ) / 4

1log 2

1

f c

k

k

kQ N N

k

kk

k

ek

e

k

Since ImQ is free from UV divergence, we calculate it without cutoff. Then, we obtain a simple form,

Re Q is calculated from the dispersion relation,

2 2

2 2

P Im ( , )Re ( , ) ' ,

'

QQ d

k

k

Notice:which ensures

Im ( , 0) 0Q k

(with 3-momentum cutoff)

Cutoff Scheme

k

2

20

Spectral Function of Pair Field

C

C

T T

T

ε→ ０（Ｔ→ＴＣ）

＋ ＋・・・1( ) Im

k

As T is lowered toward TC,The peak of becomes sharp.

M.K., T.Koide, T.Kunihiro, Y.Nemoto, PRD 65, 091504 (2002)

The peak survives up to ~ 0.2

electric SC ： ~ 0.005(k=0,=0) diverges at T=TC.

Thouless criterion

provided the second order transition,

Collective Mode in CSC

Collective Mode

pole of the response function ( , )( , )

1 ( , )C

C

G QD

G Q

kk

k

1 ( , ) 0CG Q k

pair field ind(k,(k)) can be created with an infinitesimal ex

( ) k

Notice:pole locates in the lower half plane

Pole of Collective Mode

kz

k

z

firstsheet

secondsheet

22

22

tanh8 2( , ) P2( )

tanh8 22( )

kTQ z k dk

z k

kTk dk

z k

0

2( , ) c

c

T TT A B

T

k k

Numerical Results

=0.4=0.2=0k=200MeVk=100MeVk=0MeV

k=300MeV

=400MeV

Our calculation shows,

linear quadratic

for k=0

=0 ,0.2 ,… ,0.8k=0 ,50,100,…

Poles locate in one directionin the complex plane.

It is not pure imaginary.damped oscillation

3,3, Effective Equation for Collective Mode

21 1( , ) ( , ) cC

c

aT T

bG cQT

k k k

Near the crtical temperature, -1=g-1+Q expands,

1( ,0) 0cT T

0

Thouless criterion:Notice

2 2 2 22 2 2

( 1) 'P '( ' 2 ) cosh

16 4f cN N

a dT T

22 2 3

2 22 2

( 1) ' ' ( ' 2 ) ' 'P tanh sinh cosh

4 ' 4 48 4 4f cN N d

bT T T T

2 22 2 2

2 2 2

( 1) ' ' ( ' 2 ) '( ) P ( ' 2 ) tanh cosh

2 ' 4 8 4 2f cN N d

c C iT T T T

The solution of collective mode (-1=0) reads,

2

2

2

2* *

c c

c c

a b aT T T T

T T

c bc

c c c c k k

here, : real

: complex

(0,0)

cT T

Qa T

T

2

(0,0)

cT T

Qb

k

(0,0)

cT T

Qc T

: real: complex

,a bc

TDGL equation

2 3 0'c

c

bT

at T

aT

c

2

2

2

2* *

c c

c c

a b aT T T T

T T

c bc

c c c c k k

2c

c

T T

T

a b

ic ic k

( ),

i kx tk e

pure imaginary

second time derivative term can appear when particle-hole symmetry is broken

21 1( , ) ( , ) 0cC

c

a bT

G QT

cT

k k k

Notice: pure imaginary in sigma mode of SB H. Fujii

Particle-hole asymmetry in CSC caused the real part of .

It decreases as increases.

Numerical Check for=400MeV (Tc=40.04MeV)

* *

2

2

2

2

4

0.614 1.2940.7 85.= 3

10c c

c c

ac bc ii

c c

T T T T

T T

k k

Lowest expansion reproduces the full calculations well.

1.7 , 200MeVcT T k up to

covers the region where valid collective mode appear.

0 0.2 0.4 0.6

Im

(k)

k:Lowest expansion

:Full calculation0.8

Time Evolution of Pair Field

=0.01

=0.05

=0.1

=0.5k =0 MeVk =50 MeVk =100 MeVk =150 MeVk =200 MeV

cT T

As T is lowered toward Tc,

lifetime of the collective modebecomes longer.

large momentum modeis not affected at allnear Tc.

Damped oscillation,but heavy damping

4,4, Numerical Simulation

2( , ) c

c

A BT T

TT

k k

Fluctuations in Coordinate Space in infinite matter

initial condition:

=0.5 =0.1 =0.01

200fm

t

200fm

Long wave length (low momentum) fluctuations survives.

t

Time scale of CSC is longer than the one of SB.

We calculated the collective mode of pair field in CSC.

We derived effective equation which describes non-equilibrium dynamics of the pair-field near Tc and low momentum, and confirmed that nature of the collective mode is damped oscillation.

Summary

The collective mode with pole near the origin might affect various observables

(,k)collective mode:

5 A ACi

cf, in SB: