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QCD@Work 2003
International Workshop on
Quantum Chromodynamics
Theory and Experiment
Conversano (Bari, Italy)
June 14-18 2003
Inhomogeneous color
superconductivity
Roberto Casalbuoni
Department of Physics and INFN – Florence
&
CERN TH Division - Geneva
Summary
 Introduction to color superconductivity
 Effective theory of CS
 Gap equation
The anisotropic phase (LOFF): phase diagram
and crystalline structure
 Phonons
 LOFF phase in compact stellar objects
 Outlook
Literature
 Reviews of color superconductivity:
 T. Schaefer, hep-ph/0304281
 K. Rajagopal and F. Wilczek, hep-ph/0011333
 G. Nardulli, hep-ph/0202037
 Original LOFF papers:
 A.J. Larkin and Y. N. Ovchinnikov, Zh. Exsp. Teor. Fiz.
47 (1964) 1136
 P. Fulde and R.A. Ferrel, Phys. Rev. 135 (1964) A550
 Review of the LOFF phase:
 R. Casalbuoni and G. Nardulli, hep-ph/0305069
Introduction
Study of CS back to 1977 (Barrois 1977, Frautschi 1978,
Bailin and Love 1984) based on Cooper instability:
At T ~ 0 a degenerate fermion gas is unstable
Any weak attractive interaction leads to
Cooper pair formation
 Hard for electrons (Coulomb vs. phonons)
 Easy in QCD for di-quark formation (attractive
channel 3 )
(3  3  3  6)
CS can be treated perturbatively for large m
due to asymptotic freedom
At high m, ms, md, mu ~ 0, 3 colors and 3 flavors
Possible pairings:
0 iaa bjb 0
 Antisymmetry in color (a, b) for attraction
 Antisymmetry in spin (a,b) for better use of the
Fermi surface
 Antisymmetry in flavor (i, j) for Pauli principle
 
s s

p
Only possible pairings
LL and RR

p
Favorite state CFL (color-flavor locking)
(Alford, Rajagopal & Wilczek 1999)
a
aL
b
bL
a
aR
b
bR
0  0  0 
0  
abC
abC
Symmetry breaking pattern
SU(3)c  SU(3) L  SU(3)R  SU(3)cL R
What happens going down with m? If m << ms we get
3 colors and 2 flavors (2SC)
a
aL
b
bL
ab3
0   0   ab
SU(3)c  SU(2) L  SU(2)R  SU(2)c  SU(2) L  SU(2)R
In this situation strange quark decouples. But what
happens in the intermediate region of m? The interesting
region is for (see later)
m ~ ms2/
Possible new anisotropic phase of QCD
Effective theory of
Color
Superconductivity
Relevant scales in CS
Fermi momentum defined by

E( p F )  m

(cutoff)

The cutoff is of order wD in
superconductivity and > LQCD
in QCD
    p F
(gap)
pF
Hierarchies of effective lagrangians
LQCD
Microscopic description
pF + 
Quasi-particles (dressed fermions
as electrons in metals). Decoupling
of antiparticles (Hong 2000)
LHDET
p – pF >> 
    pF
pF + 
Decoupling of gapped quasiparticles. Only light modes as
Goldstones, etc. (R.C. & Gatto;
Hong, Rho & Zahed 1999)
p – pF << 
LGold
p

Physics near the Fermi surface
(    p F )
Relevant terms in the effective description
(see: Polchinski, TASI 1992, also Hong 2000; Beane, Bedaque &
Savage 2000, also R.C., Gatto & Nardulli 2001)


d 3p


SR  
dt
i




(
E
(
p
)

m
)


t
3
(2)


 

Marginal term in the effective description (p1  p 2 , p3  p 4 )
and attractive interaction
  


G 4 d 3p k
3 
 
 
SM   
dt  (p1  p 2  p3  p 4 ) (p1 ) (p3 ) (p 2 ) (p 4 )
3
2 k 1 (2)
The marginal term becomes relevant at 1 – loop
BCS instability solved by condensation and
formation of Cooper pairs




1 d 3p
 
*
* T 
SM   
dt  (p)C (p)    (p)C (p)  Sres
3
2 (2)
 T
 2
3 
Sres   d x  ( x ) C( x ) 
G

  
 * 2* 
  ( x ) C( x ) 

G 

Sres is neglected in the mean field approximation
The first term in SM behaves as a Majorana mass term
and it is convenient to work in the Nambu-Gorkov basis:

1   ( p) 
   



C

(

p
)
2

Near the Fermi surface


 
  
E(p)

 p  E ( p)  m  
 (p  p F )  v F  (p  p F )
p p p F


p F  mv F



 
p  mv F  

E



p
1
S 
*
 
 


E  p 
 
E   p
S  2

*
2 

E  p 
E   p    

2
2
Dispersion relation
(p)    p  
1
At fixed vF only energy and
momentum along vF are relevant
Infinite copies of 2-d physics
v1
v2
Gap equation
4
d p
1
1  G
 2 2
2
4
(2) p 4  | p |   BCS
d 3p 
1
1  GT 
 2
3 
2
(2) n  (( 2n  1)T)  (p, )
G d 3p 1  n u  n d
1 

3
2 (2) (p, )
n u  nd 
1
e

 ( p , ) / T
1
For T T 0
G d 3p
1 
2 (2)3
1

 (p)  2BCS
2
At weak coupling
G p
2
1 2
log
2 v F
 BCS
2
F
 BCS  2e
2

G
(  cutoff )
2
F
p
 2
 vF
density of states
With G fixed by SB at T = 0, requiring
Mconst ~ 400 MeV
and for typical values of m ~ 400 – 500 MeV one gets
  100 MeV
Evaluationd from QCD first principles at asymptotic m
(Son 1999)
32

2g s
5
s
  bmg e
Notice the behavior exp(-c/g) and not exp(-c/g2) as one
would expect from four-fermi interaction
For m ~ 400 MeV one finds again
  100 MeV
The anisotropic phase
(LOFF)
In many different situations pairing may happen between
fermions belonging to Fermi surfaces with different radius,
for instance:
• Quarks with different masses
• Requiring electric neutrality
Consider 2 fermions with m1 = M, m2 = 0 at the same
chemical potential m. The Fermi momenta are
p F1  m  M
2
pF2  m
2
To form a BCS condensate one needs common momenta
of the pair pFcomm
p comm
F
M2
m
4m
Grand potential at T = 0
for a single fermion
pF
3

dp
(p)  m
  2
3
(2)
0
2
  2 m (p
2
i 1
comm
F
 p Fi )(i (p
Pairing energy
comm
F
)  m)  M
4
 m 
2 2
2
Pairing possible if
M

m
The problem may be simulated using massless fermions
with different chemical potentials (Alford, Bowers &
Rajagopal 2000)
Analogous problem studied by Larkin &
Ovchinnikov, Fulde & Ferrel 1964. Proposal of
a new way of pairing. LOFF phase
2
pF2 = m
M
4m
M2
4m
pFc = m – M2/4m
pF1 = m – M2/2m
4
M
(p cF  p Fi )(i (p cF )  m) 
16m 2
E1(pFc) = mM2/4m
EF1= EF2 = m
E2(pFc) = mM2/4m
M2
4m
M2
4m
 LOFF: ferromagnetic alloy with paramagnetic
impurities.
 The impurities produce a constant exchange
field acting upon the electron spins giving rise to
an effective difference in the chemical potentials
of the opposite spins.
 Very difficult experimentally but claims of
observations in heavy fermion superconductors
(Gloos & al 1993) and in quasi-two dimensional layered
organic superconductors (Nam & al. 1999, Manalo & Klein
2000)
m1  m 2
or paramagnetic impurities (m ~ H) give
rise to an energy additive term
H I  m3
Gap equation
4
d p
1
1  G
 2 2
4
2
(2) (p 4  im)  | p |  
Solution as for BCS   BCS, up to (for T = 0)
 BCS
m1 
 0.707 BCS
2
 2
2
 BCS   normal   ( BCS  2m )
4
 First order transition, since for m > m1,   0 T  0
 For m  0, usual BCS second order transition
at T= 0.5669 BCS
 Existence of a tricritical point in the plane (m, T)
According LOFF possible condensation with
non zero total momentum of the pair
 
   
p1  k  q p 2  k  q
More generally
(x)(x)  e

2iqx
 ( x ) ( x )    c m e
 
2iq m x
m
 

p1  p 2  2q

|q|
fixed variationally
 
q/|q|
chosen
spontaneously
Simple plane wave: energy shift
 

E(p)  m  E( k  q)  m  m    m
 
m  m  v F  q
g d p 1 nu  nd
1 

3
2 (2) (p, )
3
Gap equation:
nu  nd
For T T 0
n u ,d 
1
e

(  ( p, ) m ) / T
blocking region
g d 3p
1
1 
(1  (  m )  (  m ))

3
2 (2) (p, )
1
 | m |
The blocking region reduces the gap:
 LOFF   BCS
Possibility of a crystalline structure (Larkin &
Ovchinnikov 1964, Bowers & Rajagopal 2002)
 ( x ) ( x ) 


|q i | 1.2 m

qi
e
 
2iq i x
see later
The qi’s define the crystal pointing at its vertices.
The LOFF phase is studied via a Ginzburg-Landau
expansion of the grand potential
b 4  6
  a        
2
3
2
(for regular crystalline structures all the q are equal)
The coefficients can be determined microscopically for
the different structures.
 Gap equation
 Propagator expansion
 Insert in the gap equation
We get the equation
a  b        0
3
Which is the same as
a 
b 
3
 
5
5

0

with
The first coefficient has
universal structure,
independent on the crystal.
From its analysis one draws
the following results
 BCS   normal
 2
  ( BCS  2m2 )
4
 LOFF   normal  0.44(m  m2 ) 2
 LOFF  1.15 (m2  m)
m1   BCS / 2
m2  0.754 BCS
Small window. Opens
up in QCD? (Leibovich,
Rajagopal & Shuster
2001; Giannakis, Liu &
Ren 2002)
Results of Leibovich, Rajagopal &
Shuster (2001)
m(MeV)
m2//BCS
(m2  m1/BCS
(LOFF)
0.754
0.047
400
1.24
0.53
1000
3.63
2.92
Corrections for non weak coupling
Normal
LOFF
weak coupling
BCS
strong coupling
Single plane wave
Critical line from

 0,

Along the critical line
(at T  0, q  1.2m2 )

0
q
Preferred
structure:
face-centered
cube
Tricritical point
General study by Combescot and Mora (2002).
Favored structure 2 antipodal vectors
 At T = 0 the antipodal
vector leads to a second order
phase transition. Another
tricritical point ? (Matsuo et
al. 1998)
 Change of crystalline structure from
tricritical to zero temperature?
Two-dimensional case (Shimahara 1998)
Analysis close to the critical line
 Tc  T 
1

 a   N   b a 
2
 Tc 
2


iq r
 a ( r )   FFe

 
 a ( r )  2 FFLO cos( q  r )

 a ( r )  2 sq [cos( qx )  cos( qy)]
 
 
 

iq 3  r
iq1  r
iq 2  r
 a ( r )   tr [e  e  e ]

 
 
 a ( r )  2 hex [cos( q1  r )  cos( q 2  r )
 
 cos( q 3  r )]
  
q1  q 2  q 3  0
Phonons
In the LOFF phase translations and rotations are broken
phonons
Phonon field through the phase of the condensate (R.C.,
Gatto, Mannarelli & Nardulli 2002):
 
(x)(x)  e
 e
(x)  2q  x
 
1
introducing
( x )  ( x )  2q  x
f

2iqx
i ( x )
2
2
2
1  2








2
2
L phonon     v   2  2   v|| 2 
y 
z 
 x
2
Coupling phonons to fermions (quasi-particles) trough
the gap term
(x) C  e
T
i ( x )
 C
T
It is possible to evaluate the parameters of Lphonon
(R.C., Gatto, Mannarelli & Nardulli 2002)
+
2




1
m
v 2  1       0.153
2 | q | 


2
 m 
v      0.694
| q |
2
||
Cubic structure
8
( x )    e
 
2 iq k x

k 1
e

2 i|q| i x i
i 1, 2 , 3; i  

 (x)  2 | q | x i
(i )

1 (i )
(i )
 (x)   (x)  2 | q | x i
f

e
i i  ( i ) ( x )
i 1, 2 , 3; i  

( x )  0

( x )  4

( x )  4
(i)(x) transforms under the group Oh of the cube.
Its e.v. ~ xi breaks O(3)xOh ~ Ohdiag. Therefore we get
 
1
L phonon   
2 i 1, 2,3  t
(i )


2
 (i ) 2
 a
   |  |
 2 i 1, 2,3

b
(i ) 2
  i
 c   i  ( i )  j  ( j)
2 i 1, 2,3
i  j1, 2 , 3

Coupling phonons to fermions (quasi-particles) trough
the gap term
( x ) C  
T
e
i i  ( i ) ( x )
i 1, 2, 3; i  
 C
T
we get for the coefficients

1   m 
c
3    1

12   | q | 


2
1
a
12
b0
One can evaluate the effective lagrangian for the gluons in
tha anisotropic medium. For the cube one finds
Isotropic propagation
This because the second order invariant for the cube
and for the rotation group are the same!
LOFF phase in CSO
Why the interest
in the LOFF
phase in QCD?
In neutron stars CS can be studied at T = 0
Tns
6
7
 10  10
 BCS
20   BCS (MeV )  100
(1MeV  10 K )
10
For LOFF state from pF ~ 0.75 BCS
14  m(MeV)  70
Orders of magnitude from a crude model: 3 free quarks
M u  Md  0, Ms  0
2
m u  m  m e , p uF  m u
3
1
d
Weak equilibrium:
md  m  me , pF  md
3
1
m s  m  m e , p sF  m s2  M s2
3
 mi N i  me N e  m N q  meQ N q   Ni
i  u , d ,s
i  u , d ,s
Electric neutrality:

Q
0
m e
1 
1
B  
 2
3 m 3
i 2
(
p
 F)
i  u , d ,s
 n.m.is the saturation nuclear density ~ .15x1015 g/cm
 At the core of the neutron star B ~ 1015 g/cm
Choosing m ~ 400 MeV
Ms = 200 pF = 25
Ms = 300 pF = 50
B
 56
 n .m.
Right ballpark
(14 - 70 MeV)
Glitches: discontinuity in the period of the pulsars.
 Standard explanation: metallic crust + neutron
superfluide inside
 LOFF region inside the star providing the crystalline
structure + superfluid CFL phase
6
(Ω/Ω  10 )
Outlook
 Theoretical problems: Is the cube the optimal
structure at T=0? Which is the size of the LOFF
window?
 Phenomenological problems: Better discussion
of the glitches (treatment of the vortex lines)
 New possibilities: Recent achieving of degenerate
ultracold Fermi gases opens up new fascinating
possibilities of reaching the onset of Cooper pairing of
hyperfine doublets. However reaching equal populations
is a big technical problem (Combescot 2001). LOFF phase?