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高密度クォーク物質における
カイラル凝縮とカラー超伝導の競合
国広悌二(京大基研）
東大特別講義2005年12月5-7日
Ref. M. Kitazawa ,T. Koide,Y. Nemoto and T.K.
Prog. of Theor. Phys., 108, 929(2002)
1 Introduction
Color Superconductivity(CSC)
Cooper instability:
In sufficiently cold fermionic matter, any attractive
interaction leads to the instability to form infinite
Cooper pairs.
QCD at high density:
asymptotic freedom
Fermi surface
attractive channel in one-gluon exchange interaction
[３]C×[３]C＝[３]C ＋[６]C
D.Bailin and A.Love, Phys.Rep.107,325(’84)
Attractive!
Ｔ
Hadrons
CSC
0
m or r
Cold, dense quark matter is
color superconducting
Recent Progress in CSC (’98~)
The di-quark gap D can become ~100MeV.
M.Alford et al.,PLB422(’98)247 / R.Rapp et al.,PRL81(’98)53 / D.T.Son,PRD59(’99)094019
The possibility to observe the CSC
in neutron stars or heavy ion collisions
Another symmetry breaking pattern
Color-flavor locked (CFL) phase at high density(mq >>ms)
M.Alford ,K.Rajagopal, F.Wilczek,Nucl.Phys.B537(’98)443
CFL : SU (3)C  SU (3) L  SU (3) R  SU (3)CLR
< ms)
2SC : SU (3)C  SU (2)C (mq ~
2SC:
ｄｕｄ
ｕｄ
ｕ
CFL:
ｕｄ
ｄｓ
ｓ
ｕ
R.Pisarski,D.Rischke(’99)
T.Schaefer,F.Wilczek(’99)
K.Rajagopal,F.Wilczek(’00)
Phase Diagram of QCD
Ｔ
End point of the 1st order transition
170MeV
M.Asakawa, K.Yazaki (’89)
Chiral
Symmetry
Broken
0
40 80MeV
2SC
CFL
μ
K. Rajagopal and F. Wilczek (’02),
”At the Frontier of Particle Physics / Handbook of QCD” Chap.35
Various models lead to qualitatively the same results.
NJL-type 4-Fermi model J.Berges, K.Rajagopal(’98) / T.Schwarz et al.(’00)
Random matrix model B.Vanderheyden,A.Jackson(’01)
Schwinger-Dyson eq. with OGE M.Harada,S.Takagi(’02) / S.Takagi(’02)
However, almost all previous works have considered only the
scalar and pseudoscalar interaction in qq and qq channel.
2
2
2
2
GS     i 5  
GC  i 5 22 C   22 C 




Vector Interaction
GV   
m
2
density-density correlation
GV     GV  
0
2
0
2
 GV r 2
r   0
The importance of the vector interaction is well known :
Hadron spectroscopy Klimt,Luts,&Weise (’90)
Chiral restoration Asakawa,Yazaki (’89) / Buballa,Oertel(’96)
E
m
r
r
M
0 m
Vector interaction naturally appears in the effective theories.
Instanton-anti-instanton molecule model Shaefer,Shuryak (‘98)
1
 2
a m
2
a m
2 

  L8
L  G 2 ( a ) 2  ( ai 5 ) 2 
(




)

(





)
5
2
2 NC
 NC

GV / GS  1 / 4
Renormalization-group analysis
L0LL  Gll ( L 0 L )2  ( L m L )2 
N.Evans et al. (‘99)
Effects of GV on Chiral Restoration
GV→Large
First Order
Cross Over
As GV is increased,
Chiral restoration is shifted to higher densities.
The phase transition is weakened.
Asakawa,Yazaki ’89 /Klimt,Luts,&Weise ’90 / Buballa,Oertel ’96
Chiral Restoration at Finite m
E
E
q
Chiral condensate
( q-q condensate )
0
q
m
q
0
q
m:Small
Baryon density
suppresses the
formation of q-q
pairing.
m:Large
CSC
( q-q condensate )
g*  gN F
Small Fermi
sphere
Large Fermi sphere leads to
strong Cooper instability
2 Formulation
Nambu-Jona-Lasinio(NJL) model (2-flavors,3-colors):

 G  ψi   ψ
 G  ψγ ψ  
L  ψ (iγ    m)ψ  GS ψψ   ψiγ5 τψ 
2
C
5 2
μ
2
2
C 2

 ψ 2 2ψ
C 2

2
V
Parameters: m, , GS , GC , GV
m  5.5MeV current quark mass
To reproduce
GS  5.50GeV -2
the pion decay constant f  93MeV,
3
  631MeV
the chiral condensate    250MeV 
Hatsuda,Kunihiro(’94)
GC / GS  0.6
GV : is varied in the moderate range. 0  GV  0.5
Thermodynamic Potential in mean field approximation

:chiral condensate
 M D  2GS 

C
:di-quark condensate
D

2
G

i




C
5 2 2


D
M D2
 ( M D , D; T , m ) 

 GV r 2
4GS 4GC
2

d3p
  ( E p  m~ )
  ( E p  m~ )
 4
E p  T log(1  e
)(1  e
)
3
(2 )
       2T log(1  e    )(1  e    )
m  m  2GV r
r      / m
0
Quasi-particle energies:
Ep 
p2  M 2 ,
   (Ep  m )  D
2
cf.) s- model
2
Gap Equations
The absolute minimum of  gives the equilibrium state.
Gap equations ( the stationary condition):

0
M

0
D
d3 p M
4
1  n( E p  m )  n( E p  m )

3
(2 ) E p
Ep  m
Ep  m

1  2n(  )  
1  2n(  )   M


2GS
d3 p  D
D 1  2n( )   D
4
1

2
n
(

)





 
3 


(2 )  

 2GD
T=0MeV, m=314MeV GV=0
If there are several solutions,
one must choose the absolute
minimum for the equilibrium
state.
Effect of Vector Interaction on 
= Contour map of  in MD-D plane =
GV /GS = 0
GV /GS = 0.2
E
m
T=0 MeV
m=314 MeV
r
m
r
M
0
large M
small r
small m
large r
Vector interaction delays the chiral restoration toward larger m.
3 Numerical Results
First Order
Second Order
Cross Over
GV / GS  0
Phase Diagram
χSB(chiralSymmetryBroken):   0,   0
CSC(Color Superconducting):   0,   0
Wigner(normal):   0,   0
coex.(coexisting):   0,   0
The existence of
the coex. phase
Berges,Rajagopal(’98):×
Rapp et al.(’00) :
○
Order Parameters
MD :Chiral Condensate
D : Diquark Gap
GV / GS  0
GV / GS  0
GV / GS  0.2
As GV is increased…
(1) The critical temperatures of the
cSB and CSC hardly changes.
It does not change at all
in the T-r plane.
GV / GS  0.35
(2) The first order transition
between cSB and CSC phases is
weakened and eventually
disappears.
(3) The region of the coexisting
phase becomes broader.
GV / GS  0.5
Appearance of the coexisting phase
becomes robust.
(4) Another end point appears from
lower temperature, and hence there
can exist two end points in some
range of GV !
0.33 ~ GV ~ 0.38
Order Parameters at T=0 (in the case of chiral limit)
GV / GS  0
MD
D
Chiral restoration is
delayed toward larger m.
cSB survives with
larger Fermi surface.
GV / GS  0.5
MD
D
Stronger Cooper instability
is stimulated with cSB.
GV / GS  0.75
MD
D
300
The region of the coexisting
phase becomes broader.
400 m [MeV]
Large fluctuation owing to the interplay
between cSB and CSC is enhanced by GV.
5MeV
12MeV
15MeV
T=
22MeV
Contour of  with GV/GS=0.35
m
End Point at Lower Temperature
As GV is increased,
n( p )
T
p
pF
cSB
CSC
Coexisting phase
becomes broader .
n( p )
D
pF
p
D becomes larger at the phase boundary between CSC and
cSB.
The Fermi surface becomes obscure.
This effect plays a role similar to the temperature,
and new end point appears from lower T.
Phase Diagram in 2-color Lattice Simulation
J.B.Kogut et al. hep-lat/0205019
Summary
The vector interaction enhances the interplay between cSB and CSC.
The phase structure is largely affected by the vector interaction
especially near the border between cSB and CSC phases.
Coexistence of cSB and CSC,
2 endpoints phase structure,
Large fluctuation near the border between cSB and CSC
Future Problems
More deep understanding about the appearance of the 2 endpoints
The calculation including the electric and color charge neutrality.
Phase Diagram in the Tr plane
GV / GS  0
GV / GS  0.2
GV / GS  0.35
GV / GS  0.5

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