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

qq

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