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
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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
M. Kitazawa ,T. Koide,Y. Nemoto and T.K.Prog. of Theor. Phys., 108, 929(2002)
国広 悌二 (京大基研)東大特別講義 2005年 12月 5-7日
Ref.
11 IntroductionColor Superconductivity(CSC)
asymptotic freedom Fermi surfaceattractive channel in one-gluon exchange interaction
Cooper instability: In sufficiently cold fermionic matter, any attractive interaction leads to the instability to form infinite Cooper pairs.
QCD at high density:
[ 3 ]C×[ 3 ]C = [ 3 ]C + [ 6 ]C
Attractive!
Cold, dense quark matter is color superconducting
D.Bailin and A.Love, Phys.Rep.107,325(’84)
T
or 0
CSC
Hadrons
Recent Progress in CSC (’98~)
The di-quark gap can become ~100MeV.
The possibility to observe the CSC in neutron stars or heavy ion collisions
Another symmetry breaking patternColor-flavor locked (CFL) phase at high density(q >>ms)
M.Alford et al.,PLB422(’98)247 / R.Rapp et al.,PRL81(’98)53 / D.T.Son,PRD59(’99)094019
udd duu
2SC : (3) (2)C CSU SU
CFL : (3) (3) (3) (3)C L R C L RSU SU SU SU
(q < ms)~
M.Alford ,K.Rajagopal, F.Wilczek,Nucl.Phys.B537(’98)443
uddssu
2SC: CFL:R.Pisarski,D.Rischke(’99)T.Schaefer,F.Wilczek(’99)
K.Rajagopal,F.Wilczek(’00)
T
μ0
ChiralSymmetryBroken 2SC
Phase Diagram of QCD
NJL-type 4-Fermi modelRandom matrix modelSchwinger-Dyson eq. with OGE
CFL
End point of the 1st order transitionM.Asakawa, K.Yazaki (’89)
170MeV
K. Rajagopal and F. Wilczek (’02),”At the Frontier of Particle Physics / Handbook of QCD” Chap.35
40 80MeV
Various models lead to qualitatively the same results.J.Berges, K.Rajagopal(’98) / T.Schwarz et al.(’00)
B.Vanderheyden,A.Jackson(’01)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
5 2 2 2 2C C
CG i 2 25SG i
Instanton-anti-instanton molecule model Shaefer,Shuryak (‘98)
82
52
22
52
2 )()(2
1)()(2 LN
iN
GL aa
C
aa
C
Renormalization-group analysis N.Evans et al. (‘99)
2200 )()( LLLLllLL GL
2VG
The importance of the vector interaction is well known :
Vector interaction naturally appears in the effective theories.
Hadron spectroscopy Klimt,Luts,&Weise (’90)
Chiral restoration Asakawa,Yazaki (’89) / Buballa,Oertel(’96)
Vector Interaction
density-density correlation
2 20 0 2V V VG G G 0
M
E
0
4/1/ SV GG
m
Effects of GV on Chiral Restoration
Chiral restoration is shifted to higher densities.The phase transition is weakened.
As GV is increased,
First OrderCross Over
GV→Large
Asakawa,Yazaki ’89 /Klimt,Luts,&Weise ’90 / Buballa,Oertel ’96
Chiral Restoration at Finite
:Small :Large
Chiral condensate( q-q condensate )
CSC( q-q condensate )
E
0
E
0
Small Fermi sphere
Large Fermi sphere leads to strong Cooper instability
q
q
Baryon density suppresses the formation of q-q pairing.
* Fg gN
22 Formulation
Parameters: , , , ,S C Vm G G G
To reproduce the pion decay constant the chiral condensate
5.5MeVm-25.50GeV
631MeVSG
VG : is varied in the moderate range.
current quark mass
93MeV,f 3250MeV
Hatsuda,Kunihiro(’94)
Nambu-Jona-Lasinio(NJL) model (2-flavors,3-colors):
2 ψγψG μ
V
25
2 )( ψiγψψψGψmiγψL S τ
222
2225 CC
C ψψψiψG
6.0/ SC GG0 0.5VG
)1)(1log(2
)1()1log()2(
4
44),;,(
)~()~(3
3
222
eeT
eeTEpd
GGG
MTM
pp EEp
VCS
DD
Thermodynamic Potential in mean field approximation
2 2
22
,
( )p
p
E p M
E
5 2 2
22
D SC
C
M GG i
:chiral condensate:di-quark condensate
Quasi-particle energies: 2 VG 0 /
cf.) - model
Gap EquationsThe absolute minimum of gives the equilibrium state.
3
34 1 ( ) ( )(2 )
1 2 ( ) 1 2 ( )2
p pp
p p
S
d p M n E n EE
E E Mn nG
3
34 1 2 ( ) 1 2 ( ) 2(2 )d p n n G
0M
0
If there are several solutions, one must choose the absolute minimum for the equilibrium state.
T=0MeV, =314MeV GV=0
Gap equations ( the stationary condition):
Effect of Vector Interaction on
Vector interaction delays the chiral restoration toward larger .
large M small
small m large
V SG /G = 0 V SG /G = 0.2= Contour map of in MD- plane =
T=0 MeV=314 MeV
M
E
0m
33 Numerical Results0/ SV GG
0,0:isting)coex.(coex
0,0:mal)Wigner(nor
0,0:cting)SuperconduCSC(Color
0,0:Broken)Symmetry χSB(chiral
Phase Diagram
Order Parameters
MD :Chiral Condensate:Diquark Gap
The existence of the coex. phase
Berges,Rajagopal(’98):×Rapp et al.(’00) : ○
First OrderSecond OrderCross Over
2.0/ SV GG As GV is increased…
0/ SV GG
/ 0V SG G
(1) The critical temperatures of the SB and CSC hardly changes.
It does not change at all in the T- plane.
35.0/ SV GG
5.0/ SV GG
Another end point appears from lower temperature, and hence there can exist two end points in some range of GV ! 38.0~~33.0 VG
(3)
(4)
The region of the coexisting phase becomes broader.
(2)
Appearance of the coexisting phase becomes robust.
The first order transition between SB and CSC phases is weakened and eventually disappears.
Order Parameters at T=0 (in the case of chiral limit)
[MeV]300 400
/ 0V SG G
/ 0.5V SG G
/ 0.75V SG G
DM
DM
DM
Chiral restoration is delayed toward larger .
SB survives with larger Fermi surface.
Stronger Cooper instability is stimulated with SB.
The region of the coexisting phase becomes broader.
Con
tour
of
with
GV/G
S=0.
35
5M
eV
12M
eV
15M
eV
T= 22M
eV
Lar
ge fl
uctu
atio
n ow
ing
to th
e in
terp
lay
betw
een
SB
and
CSC
is e
nhan
ced
by G
V.
End Point at Lower Temperature
pFp
T( )n p
pFp
( )n pSB CSC
This effect plays a role similar to the temperature, and new end point appears from lower T.
As GV is increased,
Coexisting phase becomes broader .
becomes larger at the phase boundary between CSC and SB. The Fermi surface becomes obscure.
Phase Diagram in 2-color Lattice Simulation
J.B.Kogut et al. hep-lat/0205019
Summary
The vector interaction enhances the interplay between SB and CSC.
More deep understanding about the appearance of the 2 endpoints
Future Problems
The phase structure is largely affected by the vector interaction especially near the border between SB and CSC phases.
Coexistence of SB and CSC,2 endpoints phase structure, Large fluctuation near the border between SB and CSC
The calculation including the electric and color charge neutrality.
Phase Diagram in the T plane
0/ SV GG
35.0/ SV GG 5.0/ SV GG
2.0/ SV GG