QCD at very high density
Roberto Casalbuoni
Department of Physics and INFN - Florence
http://theory.fi.infn.it/casalbuoni/barcellona.pdf
http://theory.fi.infn.it/casalbuoni/loff_rev.pdf
[email protected]
Perugia, January 22-23, 2007
1
Summary
 Introduction and basics in Superconductivity
 Effective theory
 Color Superconductivity: CFL and 2SC phases
 Effective theories and perturbative
calculations
 LOFF phase
2
Introduction
 Important to explore the entire QCD
phase diagram: Understanding of
Hadrons
QCD-vacuum
Understanding of its modifications
 Extreme Conditions in the Universe:
Neutron Stars, Big Bang
 QCD simplifies in extreme conditions:
Study QCD when quarks and gluons are the
relevant degrees of freedom
3
Studying the QCD vacuum under different and
extreme conditions may help our understanding
Neutron star
Heavy ion collision
Big Bang
4
5
Limiting case   
q
q
 R  0
Free quarks
R
Asymptotic freedom:
When nB >> 1 fm-3
free quarks expected
 1 fm
6
Free Fermi gas and BCS
(high-density QCD)
For T  0 (β = 1/kT  )
f(E)
E
EF  μ
7
pF
 High density means high pF
 Typical scattering at
momenta of order of pF
For p F   QCD
 No chiral breaking
 No confinement
 No generation of masses
Trivial
theory ?
8
Grand potential unchanged:
(F  E  μN)
• Adding a particle to the Fermi surface
 Taking out a particle (creating a hole)
9
For an arbitrary attractive interaction it is
convenient to form pairs particle-particle or
hole-hole (Cooper pairs)
E + (±2E F - E B ) - μ(N ± 2) = F - E B
In matter SC only under particular conditions (phonon
interaction should overcome the Coulomb force)
0
Tc (electr.)
1  10 K
3
4
 4

10

10
E(electr.) 10  105 0K
In QCD attractive interaction
(antitriplet channel)
SC much more efficient in QCD
10
Effective theory
 Field theory at the Fermi surface
 The free fermion gas
 One-loop corrections
11
Field theory at the Fermi surface
(Polchinski, TASI 1992, hep-th/9210046)
Renormalization group analysis a la Wilson
How do fields behave scaling down the energies
toward eF by a factor s<1?



k
  
pk
Scaling:
E

sE


 s
kk
12
Using the invariance under phase transformations,
construction of the most general action for the
effective degrees of freedom: particles and holes
close to the Fermi surface (non-relativistic
description)
†
†


dtd
p
iψ
p

ψ
p

ε
p

ε
ψ
p
ψ
p












σ
t
σ
F
σ
σ



3
Expanding around eF:
ε  p 
ε  p  εF 
p
 O
2
v
F

0
13
S   dtd kd iψ  p   t ψσ  p   v Fψ  p  ψ σ  p 
2dψ 1
2
†
σ
Scaling:
  s
1
dt  s dt


dk  dk


d   sd 
 t  s t
†
σ
Ss
S
requiring the
action S to be
invariant
ψs
1/2
ψ
14
The result of the analysis is that all possible
interaction terms are irrelevant (go to zero going
toward the Fermi surface) except a marginal
(independent on s) quartic interaction of the
form:
V  dtd p1d p2 ψ (p1 )ψσ (p2 )ψ (-p1 )ψσ' (-p2 )
3
3
†
σ
 ,'
†
σ'
corresponding to a Cooper-like interaction
p1
p2
 p1
 p2
15
s-1+4
Quartic








2
2
2
2
 dtd k1d 1d k 2d 2d k 3d 3d k 4d 4
   
  

 
V( k1 , k 2 , k 3 , k 4 )ψ σ (p1 )ψ σ (p3 )ψ σ' (p2 )ψ σ' (p4 )
  
3 
δ (p1  p2  p3  p4 )
sd ??
s-4x1/2
Scales as s1+d
16
Scattering:
 
 
p1  p2  p3  p4


 
p3  p1  δk 3  δ 3




p 4  p 2  δk 4  δ  4





3
δ δk3  δk4  δ 3  δ 4

17
irrelevant
marginal

 
d

3

d 4 p1
  p

2

d 4 p1
δk 3
δk 3


δk 4

d 3
δk 4
0


s
p2  p1






3
δ δk3  δk4  δ 3  δ 4




2
-1
δ (δk3  δk 4 )d(δ 3  δ 4 )
s
18
Higher order interactions
irrelevant
Free theory BUT check quantum corrections
to the marginal interactions among the
Cooper pairs
19
The free fermion gas
Eq. of motion:
Propagator:
(i t  v F ) (p, t)  0

(i t  v F )G ,' ( p, t )  d,'d( t )
G σ,σ' (p, t) = δσ,σ'G(p, t) =
= -iδσ,σ' θ(t)θ( ) - θ(-t)θ(- )  e-i v Ft
Using:
i
e  it
(t) 
d

2
  ie
20
1
( )
(  ) 
 it 
G(p, t)  lim
de 



e 0 2


v

i
e


v

i
e
F
F


or:


1
ip0 t
G( p, t ) 
dp0 e G( p0 , p)

2
1
G ( p) 
(1  ie)p0  v F
0
Fermi field decomposition
0


  ipx
ipx
  ( x )   b ( p, t )e   b ( p)e

p

p



x  ( t, x ), p  (v F , p)

21
with:
b (p) 0  0
for
| p | pF
b (p) 0  0 for
| p | p F
†

[b (p), b† (p)]  dp,p'd, '
[ (x, t),  (x ', t)]  d, 'd (x  x ')
†

3
The following representation holds:
G , ' (x)  id, '  0 T(b (p, t)b† (p,0) 0 e ipx  d, '  G(p, t)
In fact, using
p
p
0 b† (p)b (p) 0  (p F  p)  (  )
0 b (p)b (p) 0  1  (p F  p)  (p  p F )  ( )
†

 iv F t


i

(

)
e
, t0

G( p, t )  
iv F t
i

(


)
e
, t0

22
The following property is useful:
lim G , (0, d)  i lim 0 T( (0, d)† (0) 0 
d0
d0
 i 0   0  i F
†

0
0
d 4 p ip0d
1
 F  2i lim  G , (0, d)  2i lim 
e
4
d0
d0
(2)
(1  ie)p0  v F


3
dp
dp
p3F
F  2
( )  2
( p F  p)  2
3
3
( 2 )
( 2 )
3
3
23
One-loop corrections
1
(1  ie)p0  v F
2
dE'd
kd
1
2
iG(E) = iG - G 
(2π)4 ((E + E')(1+ iε) - v F )((E - E')(1+ iε) - v F )
Closing in the
upper plane
we get
24
1 2
3
G(E)  G  G log( d/E)  O(G )
2

2
dk 1

  2
3
2  v F (k)
d, UV cutoff
on v F
From RG equations:
dG(E) 1

G(E) 2
dE
2E
25
G
G(E) 
G
1
log( d/E)
2
E0
BCS
instability
Attractive, stronger
for E  0
26
Functional approach
G † 2

†
S ,     d x  (i t  e(|  |)  )  ( ) 
2


†
4
Fierzing (C = i2)
†a a †b b  †a †ba  b 
1
1 † * T
† †c
d
  eabeabc d    C  C
4
2
G † * T


†
S ,     d x  (i t  e(|  |)  )  ( C )( C) 
4


†
4
Quantum theory
Z   D(,  )e
†
iS ,† 


27
const.   D( ,  )e
*

i 4  G T
G


d x   (  C )  *  ( †C* ) 
G
2
 2



Z
1
†
*

D(

,

)D(

,

)e

Z0 Z0
 ||2 1

1
iSo [  , ] i d x  
  ( †C* )  * ( TC ) 
2
 G 2

†

4
1   

 * 
2  C 
2


|

|
4
† 1
S0     d x   S  

G 

 p 0  p
S ( p)  
*



1
 
p0  p 
28
Since * appears already in  we are doublecounting. Solution: integrate over the
fermions with the “replica trick”:
||2
i d x
1/ 2
G
Z
1

*
1



D
(

,

)
det(
S
)
e


Z0 Z0 

4
e
iSeff

i
1
4 ||
Seff ( ,  )   Tr log( S0S )   d x
2
G
*
2
Evaluating the saddle point:
d4p

  iG 
(2)4 p02  2p  |  |2
G d 3p
 
2 ( 2 ) 3

2p  |  |2
29
At T not 0, introducing the Matsubara frequencies
n  (2n  1)T
dp

  GT  
3
2
2
2
n   ( 2 ) n  p  |  |

3
and using (f(E) is the Fermi particle density)

1
1

(1  2f ( E p ))

2
2
2
2 E pT
n   n  p  |  |
G d 3p
 
2 ( 2 ) 3

2p  |  |2
tanh( E p / 2T )
30
By saddle point:
Z
1
†
*

D(

,

)D(

,

)e

Z0 Z0
 ||2 1

1
iSo [  , ]i d x  
  ( †C* )  * ( TC ) 
2
 G 2

†

4
G T

 C
2
Introducing the em interaction in S0 we see
that Z is gauge invariant under
  e
i( x )
,   e
2 i( x )
Therefore also Seff must be gauge invariant
and it will depend on the space-time
derivatives of  through
31
D    2ieA 
In fact, evaluating the diagrams (Gor’kov 1959):
32
got the result (with a convenient
renormalization of the fields):
1
 1 *
2
2
4
H  d r 
 (r) | (  i2eA) | (r)   | (r) |   | (r) | 
2
 4m

3
charge of the pair
This result gave full justification to the
Landau treatment of superconductivity
33
The critical temperature
By definition at Tc the gap vanishes. One
can perform a GL expansion of the grand
potential
1 2 1 4
    
2
4
with extrema:
  3  0
 and  from the

d 3p

expansion of the   GT  
3
2
2
2
(
2

)




|

|
n  
n
p
gap equation up to
normalization
34
To get the normalization remember (in the
weak coupling and relatively to the normal
state):
1 2
H 0     
4
1
2d
Starting from the gap equation:    G log
0
2

Integrating over  and using
G 2


the gap equation one finds:
8
Rule to get the effective potential from the gap
equation: Integrate the gap equation over  and multiply
35
by 2/G
Expanding the gap equation in :
(n  (2n  1)T)
 d



3
  2GT Re   d  2
 2
     0
2
2 2
n 0 0
 (n   ) (n   )

One gets:
 d
2
d 

  1  2GT Re   2
2 
G
n 0 0 ( n   ) 
 d
d
  4T Re   2
2 2
(



)
n 0 0
n
Integrating over 
and summing over
n up to N
d
N  d  N 
2T
36
T
(T)   log
 0
Requiring (Tc) = 0
Also
γ = eC , C = 0.577...

Tc   0  0.56693  0

7ρ
β(T) = 2 2 ζ(3)
8π T
and, from the gap equation


T 
 (T)    1   
 Tc  

1/ 2
(T)
2 2 Tc 
T
 (T)  
 (T) 
1  
(T)
7(3)  Tc 
2
1/ 2

T
 3.06Tc 1  
 Tc 
37