第６回 関西ＱＮＰセミナー 於：京大基研
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 Kitazawa
Kyoto 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.
Phase Diagram of QCD
Color Superconductivity
quark (fermion) system
attractive channel in
one-gluon exchange interaction.
T
RHIC
150~170MeV
[３]c×[３]c＝[３]c＋[６]c
Cooper instability at sufficiently low T
SU (3)c color-gauge symmetry is broken!
Chiral Symm.
GSI,J-PARC
Broken
Color Superconductivity
Hadrons
0
Compact Stars
2SC pairing u
at low energy:
m
d
Phase Diagram of QCD
T
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.
150~170MeV
Pre-critical region of CSC
M.K., et al., 2002,2004
•large pair fluctuations
 precursory phenomena of CSC
Chiral Symm.
Broken
Hadrons
0
~100MeV
Color Superconductivity(CSC)
m
Numerical Result : Density of State
N ( )
N free ( )
The pseudogap survives up to e =0.05~0.1 ( 5~10% above TC ).
m = 0 MeV
e = 0.05
Spectral Function of Quarks
A(p, p )   (p, p )   (p, p )
0
0
0
0
0
quasiparticle
peak  ~ k
quark part
-(,k)
sharp peak with
negative dispersion
 [MeV]
k [MeV]
k [MeV]
 [MeV]
TABLE OF CONTENTS
1, Introduction
2, Quarks above CSC phase transition
3, Quarks above chiral phase transition
4, Summary
2, Quarks above CSC
phase transition
T
m
Nature of CSC
T
m
weak coupling
0
strong coupling!
D ~ 100MeV
D / EF ~ 0.1
in electric SC
D / EF ~ 0.0001
Short coherence length x.
Mean field approx.
works well.
There exist large fluctuations of pair field.
Large pair fluctuations can invalidate MFA.
cause precursory phenomena of CSC.
cf.) Bosonization of Cooper pairs
Matsuzaki, PRD62,017501 (2000)
Abuki, Hatsuda, Itakura, PRD 65, 074014 (2002)
ペア場のゆらぎ
F(D)
二次相転移点では、秩序変数のゆらぎが発散。
ペア場D(x) for CSC
D
クォーク対
ペア場のゆらぎは、
集団モードを形成する。
 1

極


T
カラー超伝導
m
Tcで原点に到達
ソフトモード
Pair Fluctuations in Superconductors
electric conductivity
Precursory Phenomena in Alloys
•Electric Conductivity
•Specific Heat
•etc…
e
T  TC
TC
enhancement
above Tc
e ~10-3
Thouless, 1960
Aslamasov, Larkin, 1968
Maki, 1968, …
High-Tc Superconductor(HTSC)
in quasi-two-dimensional cuprates 1986~
large fluctuations induced by
strong coupling and
quasi-two dimensionality
pseudogap
e
Quarks in BCS Theory
Quasi-particle energy:

D
  sgn(k  m ) ( k  m ) 2  D 2
D
m
k
Density of State:
N ( )
de

dk
dk
k
d
k m
2
N ( )
2D
(k  m ) 2  D 2
m

Pseudogap
:Anomalous depression of the density of state
near the Fermi surface in the normal phase.
Conceptual phase diagram
Renner et al.(‘96)
The origin of the pseudogap in HTSC is still controversial.
NJL model
Nambu-Jona-Lasinio model (2-flavor,chiral limit)：
L   i   GS ( )2  ( i 5 τ )2 
C
GC ( i 5 2A )( i 5 2A )
C
Parameters:
H I  A
A
：SU(2)F Pauli matrices
：SU(3)C Gell-Mann matrices
C :charge conjugation operator
GS  5.01GeV 2
  650MeV
GC / GS  0.62
so as to reproduce
  (250MeV)3 , f  93MeV
Klevansky(1992), T.M.Schwarz et al.(1999)
Notice:
2SC is realized at low m and near Tc.
We neglect the gluon degree of freedom.
M.K. et al., (2002)
Response Function of Pair Field
Linear Response
external field: H ex   dx  D†ex C i 5 22  h.c.
expectation value of induced pair field:
Dind   ( x)i 5 22 ( x)
Dind ( x)  2GC  ( x)i 5 22 C ( x)
 i  ds  Hex (s), O(x, t )
t
C
ex
ex
t0
  dt ' dxD R ( x, x ')D ex ( x ')
D R (x, t )  2GC  ( x)i 5 22 C ( x), (0)i 5 22 C (0)   (t )
Retarded Green function
Fourier Transformation
Dind (k )  D(k, )Dex (k )
T-matrix
Dtot (k )  Dind (k)  Dex (k)  GC1(k, )Dex (k )
Rondom Phase Approx. (RPA)
(k,  )  GC 
1
   1
GC  Q(k ,  )

Q(k, n ) 
p, im
k  p, in  im
Thouless Criterion
- for second order phase transitions
 2 W(D  0)
  1  GCQ(0,0) 
D 2
1
Thermodynamic
Potential
r.h.s. is equal to zero at Tc due to the critical conditon.
The fluctuation diverges at Tc.
W(D)
DR(0,0) diverges at TC
D.J. Thouless, AoP 10,553(1960)
T  TC
D
 2 W(D  0)
D 2
Softening of Pair Fluctuations
Dynamical Structure Factor
S (k )  
1
e

1
1 
Im  (k )
T =1.05Tc
m= 400 MeV
T  Tc
e
Tc
The peak grows from e ~ 0.2
electric SC：e ~ 0.005
Softening of Pair Fluctuations
Dynamical Structure Factor
S (k )  
1
e

1
1 
Im  (k )
T =1.05Tc
m= 400 MeV
Pole of Collective Mode
Dtot (k )  GC1(k, )Dex (k )
pole: 1  GC1  Q(k, )  0

T  Tc
e
Tc
The peak grows from e ~ 0.2
electric SC：e ~ 0.005
The pole approaches the origin
as T is lowered toward Tc.
(the soft-mode of the CSC)
T-matrix Approximation
1
G(k , in )  0
G (k , in )  (k , in )
Quark Green function :
q, im
(k, n ) 
k  q, in  im




d 3q
0
 T

(
k

q
,



)
G
(q, m )
n
m
3
(2 )
m
Decomposition of G:
quark part
0
0






G(p, p0 ) 


p  m 0   p0  m  p   p0  m  p  
1
1   0  p
 
2
:projection op.
Spectral Function of Quarks
Spectral Function
1
A(k ,  )   Im G R (k ,  )
vanishes in
the chiral limit

from parity and rotational invariance
A(k, )   0 (k,  ) 0  V (k,  )kˆ     S (k, )
spectral function of baryon density
Density of State N()
d 3k 0
N ( )  
 (k ,  )
3
(2 )
N   d 3 x  0
 0 (k ,  ) 
1
Tr  0 Im G R (k ,  ) 
4
Numerical Result : Density of State
N ( )
N free ( )
The pseudogap survives up to e =0.05~0.1 ( 5~10% above TC ).
Spectral Function of Quarks
0(,k)
m= 400 MeV
e=0.01
Depression
at Fermi surface
quasi-particle peak,
 =k)~ km

k
0
k [MeV]
kF
Im  ,k=kF)
 [MeV]
kF
The peak in Im around =0
owing to the decaying process:
 [MeV]
Im  ,k)
e
|Im -|
T  TC
TC
Peak of |Im  |
w =m –k k
kF
k
-m

0
m

k, k  m : on-shell
|Im | has peaks around  =mk,
which is found to be the hole energy.
peak of ImS
quasi-particle peak
w =m–k
w =k–m
coincide at
fermi surface.
Re  ,k)
0, 0 : collective mode
Dispersion Relation of Quarks
 [MeV]
80
m= 400 MeV
e=0.01
 =(p)
40
0
rapid increase
around  =0
-40
-80
320
400
k [MeV]
Re G ( , p)   0
1
  p  m  Re  (, p)  0
480
cf.)

Super
Normal

D
0
kF
k
0
D
kF
k
Dispersion Relation of Quarks
 [MeV]
80
m= 400 MeV
e=0.01
 =(p)
40
0
-40
-80
320
400
k [MeV]
rapid increase
around  =0 However,
w.f. renormalization
1
Z p1m
 Re

(

,
p
)

0 0.7

 ( k ,  ) /  
480
Re  ,k=kF)
 [MeV]
still Fermi-liquid-like
Im  R (k ,  ')
Re  (k ,  )    d

   ' i
R
1
Diquark Coupling Dependence
stronger diquark coupling GC
GC
×1.3
m= 400 MeV
e=0.01
×1.5
Resonant Scattering of Quarks
GC=4.67GeV-2
Janko, Maly, Levin, PRB56,R11407 (1995)
  p  m  Re  (, p)  0
 p m
Re  ( , p)
Resonant Scattering of Quarks
GC=4.67GeV-2
Mixing between quarks and holes

kF
nf ()

k
Level Repulsion
D
D
f B ( ) Im (p,  )  D2 (2 )4  (3) (p) ( )
4
q0
q0   
d
q

0
Im  R (k ,  )  
Im

(
k

q
,


q
)
Im
G
(
q
,
q
)
tanh

coth
0
0 

(2 )4
2
T
2
T


Im R (p, )  2D2 (  | p | m )

R
Im

(p,  ')
Re  R (p,  )   P  d '

  '
1
2
 2D
  | p | m
pF
1
  p  m  Re  (, p)  0
p
 2  ( p  m )2  2D2  0
Quarks at very high T
•1-loop (g<<1)
•Hard Thermal Loop ( p, , mq<<T )
0
0






G(p, p0 ) 

D ( , p) D ( , p)
(, p) 
m 2f 
dispersion relations
1 2 2
gT
6
Ep
Eh
Re[D ( , p)]  0
  E p , Eh1
Re[D ( , p)]  0
  E p , Eh1
plasmino
Eh
 Ep
plasmino
Quarks at very high T
•1-loop(g<<1)
•Hard Thermal Loop approximation( p, , mq<<T )
 0
 0
G(p, p ) 

D ( , p) D ( , p)
0
(, p) 
m 2f 
dispersion relations
1 2 2
gT
8
Eh
Ep
Re[D ( , p)]  0
  E p , Eh1
Re[D ( , p)]  0
  E p , Eh1
 Ep
Eh
3, Quarks above chiral
phase transition
T
m
Soft Mode of Chiral Transition
Response Fucntion D(k,)
D(k ,  ) 

Hatsuda, Kunihiro (’85)

scalar and pseudoscalar parts
fluctuations of the chiral order parameter
Spectral Function
A(k )  
1

Im D(k )
T
m
for k=0
ε→０
（Ｔ→ＴＣ）
Sigma Mode above Tc
Spectral Function
Hatsuda, Kunihiro (’85)
sharp peak
in time-like region

s -mode
s  k 2  ms2
k
soft mode of CSC
sharp peak
around  = k =0

k
Quark Self-enrgy
1
Quark Green function : G(k , in )  0
G (k , in )  (k , in )
Self-energy:
(k, in ) 




d 3q
0
 T
D
(
k

q
,



)
G
(q, m )
n
m
3
(2 )
m
D(k, in )  1 
G0 (k, in ) 


:free quark progagator
in the chiral limit
m = 0 MeV
e = 0.05
Spectral Function of Quarks
A(p, p )   (p, p )   (p, p )
0
0
0
0
0
quasiparticle
peak  ~ k
quark part
-(,k)
sharp peak with
negative dispersion
 [MeV]
k [MeV]
k [MeV]
 [MeV]
Self Energy
 [MeV]
k [MeV]
Two peaks in Im produces five solutions
of the dispersion relation.
m = 0 MeV
e = 0.05
Spectral Function of Quarks
A(p, p )   (p, p )   (p, p )
0
0
0
0
0
positive energy part
 [MeV]
-(,k)
+(,k)
k [MeV]
k [MeV]
k [MeV]
-(,k)
 [MeV]
Resonant Scatterings of Quarks
q q
h
q q
( q q)soft
q
( q q)soft
q, qhole q, qhole
( q q)soft
q
h
q
( q q)soft
q , qhole q , qhole
These resonant scatterings affect the peaks of the
spectral functions in a non-trivial way.
q
Level Repulsion
D
m,-m
D
m,-m
for the CSC
f B ( ) Im (p,  )
2
4 (3)
f B (2 ) Im 
(
p
,

)

D
(2

)
 (p) ( )
4 (3)
 D (2 )  (p)  (  m)   (  m)
Im R (p,)  2D2  (  | p | m  m)   (  | p | m  m) 
dispersion relation
m>D
m=D
Self Energy
( q q)soft
q
( q q)soft
q
q
qhole
T dependence
 [MeV]
e = 0.05
+(,k)
k [MeV]
-(,k)
k [MeV]
T dependence
 [MeV]
e = 0.1
+(,k)
k [MeV]
-(,k)
k [MeV]
T dependence
 [MeV]
e = 0.15
+(,k)
k [MeV]
-(,k)
k [MeV]
T dependence
 [MeV]
e = 0.2
+(,k)
k [MeV]
-(,k)
k [MeV]
T dependence
 [MeV]
e = 0.25
+(,k)
k [MeV]
-(,k)
k [MeV]
T dependence
 [MeV]
e = 0.3
+(,k)
k [MeV]
-(,k)
k [MeV]
T dependence
 [MeV]
e = 0.35
+(,k)
k [MeV]
-(,k)
k [MeV]
T dependence
 [MeV]
e = 0.4
+(,k)
k [MeV]
-(,k)
k [MeV]
T dependence
 [MeV]
e = 0.5
+(,k)
k [MeV]
-(,k)
k [MeV]
Summary
The 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
above chiral transition:
Three peak structure appears!
two resonant scatterings of
quarks and anti-quarks
Future: finite quark mass, finite density,
phenomenological applications

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