1

Reviews of color superconductivity:Reviews of color superconductivity:

T. Schaefer, hep-ph/0304281T. Schaefer, hep-ph/0304281

K. Rajagopal and F. Wilczek, hep-ph/0011333K. Rajagopal and F. Wilczek, hep-ph/0011333

G. Nardulli, hep-ph/0202037G. Nardulli, hep-ph/0202037

Original LOFF papers:Original LOFF papers:

A.J. Larkin and Y. N. Ovchinnikov, Zh. Exsp. Teor. Fiz. A.J. Larkin and Y. N. Ovchinnikov, Zh. Exsp. Teor. Fiz. 47 (1964) 113647 (1964) 1136

P. Fulde and R.A. Ferrel, Phys. Rev. 135 (1964) A550P. Fulde and R.A. Ferrel, Phys. Rev. 135 (1964) A550

Review of the LOFF phase:Review of the LOFF phase:

R. Casalbuoni and G. Nardulli, hep-ph/0305069R. Casalbuoni and G. Nardulli, hep-ph/0305069

2

ppF2 F2 = =

ppF1 F1 = = – M– M22/2/2

ppFFcc = = – M – M22/4/44

M2

4M2

EE11(p(pFFcc)) = =

4M2

4M2

EEF1F1== EEF2F2 = =

EE22(p(pFFcc) = ) = MM22/4/4

2

4cFiFi

cF 16

M))p()(pp(

3

4

5

BCS2 75.0 BCS9.0|q|

2.1|q|

6

NormalNormal

LOFFLOFF

BCSBCS

weak couplingweak coupling strong couplingstrong coupling

7

Tricritical point

General study by Combescot and Mora (2002). General study by Combescot and Mora (2002). Favored structureFavored structure 2 antipodal vectors2 antipodal vectors

At T = 0 the antipodal At T = 0 the antipodal vector leads to a second order vector leads to a second order phase transition. Another phase transition. Another tricritical point ? (Matsuo et tricritical point ? (Matsuo et al. 1998)al. 1998)

Change of crystalline Change of crystalline structure from tricritical to structure from tricritical to zero temperature?zero temperature?

8

Two-dimensional case (Two-dimensional case (Shimahara 1998Shimahara 1998))

2

c

caNa T

TTb

2

1

0qqq

)]rqcos(

)rqcos()rq[cos(2)r(

]eee[)r(

)]qycos()qx[cos(2)r(

)rqcos(2)r(

e)r(

321

3

21hexa

rqirqirqitra

sqa

FFLOa

rqiFFa

321

Analysis close to the critical lineAnalysis close to the critical line

9

Bowers & Rajagopal Bowers & Rajagopal (2002(2002))

10

0)x(

4)x(

4)x(

11

|q|cos q

12

s,d,ui

iqeqees,d,ui

ii NNQNNN

0Qe

2

s,d,ui

iF2B )p(

3

1

3

1

Weak equilibrium:Weak equilibrium:

2s

2s

sFes

ddFed

uuFeu

Mp,3

1

p,3

1

p,3

2

Electrical neutrality:Electrical neutrality:

13)GG(

4

1G),G3G(

4

1G MEBMEA

iM

0E

G

G

ME GG

Relativistic Relativistic casecase

14

RotationRotation

'nn

and translation byand translation by a

compensate locally atcompensate locally at PP

15

LOFF window for one massless and one massive quarksLOFF window for one massless and one massive quarks