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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

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の１コメント

北沢正清（京大）,国広悌二（京大基研),根本幸雄(RIKEN-BNL)

CONTENTS

T

1, Introduction

2, Collective Mode in CSC

3, Effective Equation for

Collective Mode

4, Numerical Simulation

5, Summary and Outlooks

critical endpoint

Chiral

symmetry

breaking

Color superconductivity

(CSC)

m

0

Main New Results given in This Talk

derive equation for the collective mode

of the pair field above CSC,

position of pole:

(determine A, B microscopically)

in linear response theory.

D (x)

T

x

D (x)

x

m

0

l=2p/k

1,Introduction

Phase Transitions and Fluctuations in QCD

At the critical point of the second order phase transitions,

fluctuation diverges.

W(D)

fluctuation of order parameter field

chiral transition

critical end point (CEP)

CSC transition

RHIC

D

SPS

might be responsible

for various observables.

T

AGS

?

susceptibilities

GSI,J-PARC

baryon number, chiral, etc…

??

light sigma meson

transport coefficient

2SC

CFL

???

Stephanov, Rajagopal, Shuryak / Berdnikov, Rajagopal /

Hatta, Ikeda / Fukushima / Fujii / Hatta, Stephanov

m

0

another CEP?

Fluctuation of pair field in CSC

M.K., T.Koide, T.Kunihiro, Y.Nemoto,

PRD 65, 091504 (2002)

Spectral Function of Pair Field

ε→０

（Ｔ→ＴＣ）

＋

＋・・・

As T is lowered toward TC,

The peak of r becomes sharp.

The peak survives up toe ~ 0.2

Thouless criterion

r(k=0,w =0)diverges at T=TC.

electric SC：e ~ 0.005

provided the second order transition,

mDependence of Pseudogap

Pseudogap in quark DOS!

Depth of the pseudogap hardly changes with m.

2,Collective Mode in CSC

Model

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

t：SU(2)F Pauli matrices

l：SU(3)C Gell-Mann matrices

C :charge conjugation operator

Parameters:

so as to reproduce

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

M.K. et al., (2002)

Response Function of Pair Field

Linear Response

external field:

expectation value of induced pair field:

Retarded Green function

Fourier transformation

with Matsubara formalism

RPA approx.:

where,

Analytic Properties ofQ(k,w)

Im Q(k,w)

1st term:

pair

creation

3,4th term:

scattering

Collective mode

w~0 k~0 near Tc

2nd term:

cf.) in the chiral phase transition

H.Fujii, PRD67 (2003) 094018

Cutoff Scheme

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,

(with 3-momentum cutoff)

Notice:

which ensures

Collective Mode in CSC

M.K., T.Koide, T.Kunihiro, Y.Nemoto,

PRD 65, 091504 (2002)

Spectral Function of Pair Field

ε→０

（Ｔ→ＴＣ）

＋

＋・・・

As T is lowered toward TC,

The peak of r becomes sharp.

The peak survives up toe ~ 0.2

Thouless criterion

r(k=0,w =0)diverges at T=TC.

electric SC：e ~ 0.005

provided the second order transition,

k

z

Pole of Collective Mode

Collective Mode

pair field Dind(k,w(k)) can be created with an infinitesimal Dex

pole of the response function

Notice:

pole locates in the lower half plane

k

z

first

sheet

second

sheet

Numerical Results

m=400MeV

k=0 ,50,100,…

e =0 ,0.2 ,… ,0.8

for k=0

k=0MeV

k=200MeV

e=0

e=0.2

e=0.4

k=100MeV

k=300MeV

Our calculation shows,

Poles locate in one direction

in the complex plane.

It is not pure imaginary.

linear

quadratic

damped oscillation

3,Effective Equation for Collective Mode

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

Notice

Thouless criterion:

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

here,

: real

: complex

: real

: complex

TDGL equation

second time derivative term can appear

when particle-hole symmetry is broken

pure imaginary

Notice:

pure imaginary in sigma mode of cSB

H. Fujii

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

It decreases as m increases.

Numerical Check

for m=400MeV (Tc=40.04MeV)

Im w(k)

:Full calculation

:Lowest expansion

k

Lowest expansion reproduces the full calculations well.

up to

covers the region where valid collective mode appear.

4,Numerical Simulation

Time Evolution of Pair Field

As T is lowered toward Tc,

e =0.01

lifetime of the collective mode

becomes longer.

large momentum mode

is not affected at all

near Tc.

e =0.05

Damped oscillation,

but heavy damping

e =0.1

k =0 MeV

k =50 MeV

e =0.5

k =100 MeV

k =150 MeV

k =200 MeV

Fluctuations in Coordinate Space

in infinite matter

initial condition:

200fm

e =0.1

e =0.01

e =0.5

Dt=200fm

t

Long wave length (low momentum) fluctuations survives.

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

Summary

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.

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

collective mode:

(w,k)

cf, in cSB: