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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
 Effective theory of CS
 Gap equation
The inhomogeneous phase (LOFF): phase
diagram and crystalline structure
 Phonons
 LOFF phase in compact stellar objects
 Outlook
Introduction
 mu, md, ms << m: CFL phase

aL

bL

aR

bR
0  0  0 
0  
C
abC
 mu, md << m << ms : 2SC phase

aL

bL
3
0   0   ab
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 inhomogeneous 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
p – pF >> 
Quasi-particles (dressed fermions
as electrons in metals). Decoupling
of antiparticles (Hong 2000)  >>
pF + 
LHDET
p – pF >>      p
F
pF + 
Decoupling of gapped quasiparticles. Only light modes as
Goldstones, etc. (R.C. & Gatto; Hong,
Rho & Zahed 1999)
LGold
p – pF << 
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)
4-fermi attractive interaction is
marginal (relevant at 1-loop)


 

(p1  p 2 , p3  p 4 )
  


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)
SM gives rise di-fermion condensation producing a
Majorana mass term. 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
  10  100 MeV
Evaluation 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
  10  100 MeV
The inhomogeneous
phase (LOFF)
In many different situations the “would be” pairing fermions
belong to Fermi surfaces with different radii:
• Quarks with different masses
• Requiring electrical neutrality and/or weak equilibrium
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
 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
According LOFF, close to first order line, possible
condensation with non zero total momentum
 
   
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
 4  6
          
2
3
2
(for regular crystalline structures all the q are equal)
The coefficients can be determined microscopically for
the different structures (Bowers and Rajagopal (2002))
 Gap equation
 Propagator expansion
 Insert in the gap equation
We get the equation
          0
3
Which is the same as
 
 
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  .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
Single plane wave
Critical line from

 0,

Along the critical line
(at T  0, q  1.2m2 )

0
q
General
analysis
(Bowers and
Rajagopal (2002))
Preferred
structure:
face-centered
cube
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
Introduce:
( 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  
Using the symmetry group of the cube one gets:
 
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 also evaluate the effective lagrangian for the
gluons in the 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 ~ .75 BCS
14  m(MeV)  70
Orders of magnitude from a crude model: 3 free quarks
M u  Md  0, Ms  0
 n.m.is the saturation nuclear density ~ .15x1015 g/cm3
 At the core of the neutron star B ~ 1015 g/cm3
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 explanations require: metallic crust +
superfluide inside (neutrons)
 LOFF region inside the star might provide a
crystalline structure + superfluid CFL phase
 New possibilities for strange stars
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. Possibility of observing the LOFF
crystal?