Transcript Document

Color Superconductivity:
CFL and 2SC phases

Introduction

Hierarchies of effective lagrangians

Effective theory at the Fermi surface (HDET)

Symmetries of the superconductive phases
1
Introduction
 Ideas about CS back in 1975 (Collins & Perry-1975,
Barrois-1977, Frautschi-1978).
 Only in 1998 (Alford, Rajagopal & Wilczek; Rapp,
Schafer, Schuryak & Velkovsky) a real progress.
 The phase structure of QCD at high-density
depends on the number of flavors with mass m < m.
 Two most interesting cases: Nf = 2, 3.
 Due to asymptotic freedom quarks are almost free
at high density and we expect difermion
condensation in the color channel 3*.
2
Consider the possible pairings at very high density
α
ia
β
jb
0ψ ψ 0
a, b color; i, j flavor; a,b spin
 Antisymmetry in spin (a,b) for better use of
the Fermi surface
 Antisymmetry in color (a, b) for attraction
 Antisymmetry in flavor (i,j) for Pauli principle
3
 
s s

p
Only possible pairings

p
LL and RR
For m >> mu, md, ms
Favorite state for Nf = 3, CFL (color-flavor
locking) (Alford, Rajagopal & Wilczek 1999)
0 ψαiLψβjL 0 = - 0 ψαiR ψβjR 0  ΔεαβCεijC
Symmetry breaking pattern
SU(3)c  SU(3) L  SU(3) R  SU(3)c+L+R
4
Why CFL?
α
iL
0ψ ψ
α
iR
β
jL
β
jR
0ψ ψ
0  Δε εijC
αβC
0  Δε εijC
αβC
5
What happens going down with m? If m << ms, we get
3 colors and 2 flavors (2SC)
α
iL
0ψ ψ
β
jL
αβ3
0 = Δε εij
SU(3)c  SU(2)L  SU(2)R  SU(2)c  SU(2)L  SU(2)R
However, if m is in the intermediate region
we face a situation with fermions having
different Fermi surfaces (see later). Then
other phases could be important (LOFF, etc.)
6
Difficulties with lattice calculations
 Define euclidean variables: x 0  ix , x  x
4
E
0   ,
4
E
i
  i
i
 Dirac operator with chemical potential
m
E
m
E
D(m)   D  m
 At m  0
4
E
γ5D(0)γ5 = -D(0)
†
D(0) = -D(0)
 Eigenvalues of D(0) pure immaginary
 If l eigenvector of D(0),  5 l
eigenvector with eigenvalue - l
i
E
7
i
E
det[D(0)]   (l)(l) > 0
For m not zero the argument does not apply
and one cannot use the sampling method for
evaluating the determinant. However for
isospin chemical potential and two degenerate
flavors one can still prove the positivity.
For finite baryon density no lattice
calculation available except for small m and
close to the critical line (Fodor and Katz)
8
Hierarchies of effective
lagrangians
Integrating out
heavy degrees of
freedom we have two
scales. The gap D and
a cutoff, d above
which we integrate
out. Therefore:
two different
effective theories,
LHDET and LGolds 9
 LHDET is the effective theory of the fermions
close to the Fermi surface. It corresponds to the
Polchinski description. The condensation is taken
into account by the introduction of a mean field
corresponding to a Majorana mass. The d.o.f. are
quasi-particles, holes and gauge fields. This holds
for energies up to the cutoff.
 LGolds describes the low energy modes (E<< D), as
Goldstone bosons, ungapped fermions and holes
and massless gauge fields, depending on the
breaking scheme.
10
Effective theory at the
Fermi surface (HDET)
Starting point: LQCD
1 a am
LQCD  iD  Fm F  m 0, a  1, ,8
4
la
a a
m
a
Dm  m  igs AmT , D   m D , T 
2
at asymptotic m >> LQCD,
( p  m0 )(p)  0  (p0  m)(p)  a p(p)
(p0  m)2 | p |2  p0  E  m | p |
11
Introduce the projectors:
1  a  vF
E(p)
|p|
P 
, vF  v 

 pˆ
2
p ppF
p ppF
and decomposing:


mv F
p  mv F 
H   a   
H   ( 2m  a  ) 
 States  close to the FS
 States  decouple for large m
12
4
d p  ipx
e (p)
Field-theoretical version: (x)  
4
(2)
pm  mv m 
v m  (0, v), v  1,
v 
4 d.of.

,

0
m
m
Choosing
v p
 ( 0, )
  (  v)v

, ,v
0
Separation of light and heavy d.o.f.
light d.o.f. m  d  | p |  m  d,  d 
heavy d.o.f .
| p |  m  d, | p |  m  d,
 d
 d,
 d
13
Separation of light and heavy d.o.f.
heavy
light
light
heavy
d
14
Momentum integration for the light fields
d
dp
m
 (2)4  (2)4  dd d
4
2

d

dv m

4 
2
0
The Fourier decomposition becomes
dv  imvx
(x)  
e
 v ( x)
4
m2 d 2
 i x
 v (x)  
e
v ( )
2
 (2)
 v (
)  (p) 
For any fixed v, 2-dim theory
2
d
 (2)2
v1
v2
15
In order to decouple the states corresponding
to E 
Momenta from the Fermi sphere
dv  imvx
(x)   e
  (x)    (x)
4
2
2
m
d
 i x
  (x)  P v (x)  P 
e v ( )
2
 (2)
substituting inside LQCD and using
2
2
m
dv
d
4
†
†
d
x

(x)

(x)


v ( ) v ( )
2


 4 (2)
16
Proof:
2
2
2


m
dv F dv ' F d
d '
4
†
 d x (x)(x)      4 4 (2)2 (2)2
4 4
†
(2) d  '  mv ' mv   v ' ( ') v ( )
2
Using:
(2) d
4
 (2) d
2
2
4


'  mv ' mv  

'  4d  v ' v  2
m
2
One gets easily the result.
17
4
d
 x(iD  m 0 )  
dv †
 iV  D   † (2m  iV  D)   ( iD    h.c.) 
4
Vm  (1, v), Vm  (1,  v)
m  (  0 ,(v   )v),
m  m  m
   m   Vm† 
   m   Vm† 
   m      m 
   m      m 
Eqs. of motion:
iV  D   i 0 D   0
(2m  iV  D)   i 0 D    0
18
iV  D   0
At the leading
order in m:
  0
At the same order:
dv †
L D    iV  D 
4
Propagator:
1
V
1
(p0  m)  0  p  
V 1



m
p  m 0 (p0  m) 2  | p |2
2 V
V 1 0
  (1  a  v)  P  0
2 2
( T() )
( T(†) )
19
Integrating out the heavy d.o.f.
For the heavy d.o.f. we can formally
repeat the same steps leading to:
4
d
 x(iD  m 0 )  
dv †
 iV  D   † (2m  iV  D)   ( iD    h.c.) 
4
Eliminating the E- fields one would get the
non-local lagrangian:

dv  †
1
m †
LD  
 iV  D   P  
Dm D  

4 
2m  iV  D

1 m 
P  g   V V  V  Vm 
2
m
m
20
Decomposing
       h
one gets
L D  L D  L Dh  LhD
dv †
L D     iV  D 
4

dv  †
1
h
h
m
†
h
LD  
  iV  D   P  
Dm D      h 

4 
2m  iV  D

dv  h †
1
h
m h †
h 
L 
  iV  D   P  
Dm D    

4 
2m  iV  D

h
D
When integrating out the heavy fields
21
Contribute only if some gluons
are hard, but suppressed by
asymptotic freedom
This contribution from LhD gives the
bare Meissner mass
22
HDET in the condensed phase
Assume
 C
A
B
 D AB
(A, B collective indices)
due to the attractive interaction:
G
A B C† D†
LI   abab VABCDa  b a  b
4
*
VABCD  VCDAB
, VABCD  VBACD  VABDC
Decompose
23
L I  Lcond  Lint
G
G
CD* AT
B
Lcond  VABCD   C  VABCD  ABC†C D*
4
4
G
Lint   VABCD   AT C B   AB   C†C D*  CD* 
4
We define D AB 
Lcond
G
G *
CD
*
VCDAB , D AB  VCDABCD*
2
2
1 * AT B 1
 D AB C  D AB A †C B*
2
2
and neglect Lint . Therefore
24
LD  
dv 1
B
A†
B
*
AT
B
A†
B
 A†

iV

D



iV

D


D

C


D

C






AB 

AB 

4 2 AB
 (x)   (  v, x)
Nambu-Gor’kov basis
A



1
A

 
 A* 
2  C  
D AB  B
dv A† iV  DAB
LD    

*

4
 D AB iV  DAB 
V 
1
S( ) 
† 
(V  )(V  )  DD  D†
D 

V 

 D, D†   0
25

From the definition:
D
*
AB
G
C†
D*
  VABCD  C
2
one derives the gap equation (e.g. via functional
formalism)
D
dv m
 iGVABCD 
4 
2
*
AB
2
d
1
*
 (2)2 DCE DED


1
1

† 
DAB  (V  )(V  )  DD  AB
26
Four-fermi interaction one-gluon exchange inspired
3
a
m a
L I  G ml  l 
16
Fierz using:
8
2
(l )ab (l ) d  (3da dbd  dabd d )

3
a 1
a
a
( m )ab ( m )dc  2ac bd
m  (1, ), m  (1, )
G
LI   V( ai)(bj)( k)( d )iabj  k † d †
4
V( ai)(bj)( k)( d )  (3daddb  da dbd )dik d j
27
D ( ai)(bj)  ab3ijD
In the 2SC case
dv m
D  4iG 
4 
2
D
2
d
 (2)2
2
0

d
G
D
D    d
,
2
2
2 0
 D
2
D
2
m2
4 2

4 pairing fermions
28
G determined at T = 0. M, constituent mass ~ 400 MeV
L
3
dp
1
1  8G 
3
2
2
(2

)
|
p
|

M
0
with
L md
For m  400  500 MeV, L  800 MeV, M  400 MeV
D2SC  33  88 MeV
Similar values for CFL.
29
Gap equation in QCD
3 1


cos

2

g
2 2
D(p0 ) 
dq
d(cos

)

0

2 
2
12
 1  cos   G /(2m )

1 1

 cos 

D(q 0 )
2
2

2 
1  cos   F /(2m )  q 02  D(q 0 ) 2

30
Hard-loop approximation
Fm
q0
 2 q0
G  mD
4
|q |
2
D
electric
magnetic
 2 
b  256 

 Nf 
4
5/ 2
g 5
|q |0
gm
m  Nf
22
2
2
2
D
For small momenta
magnetic gluons are
unscreened and dominate
giving a further
logarithmic divergence
 bm 
g
D(q 0 )
D(p0 ) 
dq 0 log 
 2
2 
2
18
|
p

q
|
0  q 0  D (q 0 )
 0
2
31
Results:
 g
 b 
D(p0 )  D 0 sin 
log   
 p0  
 3 2
D 0  2bme
3 2

2g

g2
2
c / g 
 1  (log(d / D))  D  de 
c


from the double log
To be trusted only for m > 10 GeV but, if
extrapolated at 400-500 MeV, gives values
for the gap similar to the ones found using a
4-fermi interaction.
5
However condensation arises at asymptotic
values of m.
32
Symmetries of
superconducting phases
Consider again the 3 flavors, u,d,s and the
group theoretical structure of the two
difermion condensate:
a
ia
 
b
jb
[(3c ,3L(R) )  (3c ,3L(R) )]S  (3 ,3
*
c
*
L(R)
)  (6c ,6L(R) )
33
implying in general
a
iL
 
b
jL
abI
a b
i j
b a
i j
 D ijI  D 6 (d d  d d ) 
a b
i j
b a
i j
a b
i j
b a
i j
 D( d d  d d )  D 6 (d d  d d ) 
a b
i j
b a
i j
 ( D  D 6 )d d  ( D 6  D )d d
In NJL case with cutoff 800 MeV,
constituent mass 400 MeV and m  400 MeV
D  85.3MeV, D6  1.3MeV
34
Original symmetry:
G QCD  SU(3)c  SU(3) L  SU(3) R  U(1) B  U(1) A
broken to
anomalous
GCFL  SU(3)cLR  Z2  Z2
# Goldstones
anomalous
3  8  1  1  8  8  8  1  1  17  1
massive
8 give mass to the gluons and 8+1 are true
massless Goldstone bosons
35
Notice:
 Breaking of U(1)B makes the CFL phase
superfluid
 The CFL condensate is not gauge invariant,
but consider
a
iL
b †
jL
a
iR
b †
jR
X  ab  (  ) , Y  ab  (  )
k

ijk
k

ijk
ij  (Y†X)ij   (Yaj )* X ia
a
 is gauge invariant and breaks the global part of GQCD. Also
det(X), det(Y)
break
U(1) B  U(1)A  Z2  Z2
36
U(1) A
is broken by the anomaly but induced by a 6fermion operator irrelevant at the Fermi
surface. Its contribution is parametrically
small and we expect a very light NGB
(massless at infinite chemical potential)
Spectrum of the CFL phase
Choose the basis:
lA
1 9
a A
 
(
l
)

A i 
2 A1
a
i
A  1, ,8 Gell  Mann matrices
2
l9  l0 
1
3
Tr l Al B   2dAB
37
1
1
i
a
 
( l A ) a i 
Tr l A 

2 ai
2
Inverting:
 
A
B
A
1
D 

i
j
abI
T
  (l A )a (l B )b D ijI  Tr    l AIl BI  
2
2  I

(I )ab  abI
T

g
 I I  g  Tr[g]
for any 3x3 matrix g
I
We get
 
A
B
quasi-fermions
 D AdAB
 A1,,8 D D
A
DA  
D  2D
 A  9
9
A (p)  (v  )2  D 2A
38
gluons
Expected
m g F
2
g
2
2
NG coupling constant
but wave function renormalization effects
important (see later)
NG bosons
Acquire mass through quark masses
except for the one related to the
breaking of U(1)B
39
NG boson masses quadratic in mq since the
approximate invariance
(Z2 ) L  (Z2 ) R :  L( R )   L( R )
and quark mass term:
 L M R  h.c.
M  M
Notice: anomaly breaks (Z2 ) L  (Z2 ) R
through instantons, producing a chiral
condensate (6 fermions -> 2), but of
order (LQCD/m)8
40
In-medium electric charge
Dm  m  ig T   iQAm
a
m a
The condensate breaks U(1)em but leaves
invariant a combination of Q and T8.
CFL vacuum:
Define:
QSU(3)c
X Y d
i
a
i
a
i
a
2
 1 1 2

T8  diag   ,  ,    Q
3
 3 3 3
Q  QSU(3)c  1  1  Q  Q  1  1  Q leaves invariant the
ground state
Q X  X Q  Qabdbi  dajQ ji  0
41
Eigs(Q)  0, 1 Integers as in the Han-Nambu
model
Am  Am cos   Gm sin 
g  Am sin   Gm cos 
8
m
rotated fields
new interaction:
gsg8mT8  1  eAm1  Q  eQAm  gs GmT
2 e
gs
tan  
, e  e cos , g s 
cos 
3 gs
3
 cos2  Q  1   sin 2  1  Q 
T

2 
42
The rotated “photon” remains massless,
whereas the rotated gluons acquires a mass
through the Meissner effect
A piece of CFL material for massless quarks
would respond to an em field only through NGB:
“bosonic metal”
For quarks with equal masses, no light modes:
“transparent insulator”
For different masses one needs non zero
density of electrons or a kaon condensate
leading to massless excitations
43
In the 2SC case, new Q and B are conserved
1
1
 2 1
Q  Q 1 
1  T8   ,    1  1  (1, 2, 2)
6
3
 3 3
2
1 1 1 1 1 2
B  B
T8   , ,    , ,    (0,0,1)
3
 3 3 3  3 3 3
Q
u , a  1,2
B
1/2
0
da , a  1,2
-1/2
0
1
1
0
1
a
3
u
3
d
44
Spectrum of the 2SC phase
Remember
D ( ai)(bj)  ab3ijD
No Goldstone bosons

a
i

3
i
a  1, 2 gapped
ungapped
SU(3)c  SU(2)c

8  3  5 massive gluons
(equal gap)
Light modes:
3i  3 gluons (M  0)
45
2+1 flavors
It could happen
m
m  ms , m
m u , md
m u , m d , ms

Phase transition expected
m u , m d  m  ms
E F  m  p2F  M 2  pF  m2  M 2
M1 > M 2
The radius of
the Fermi
sphere decrases
with the mass
46
pF1  m2  ms2 , pF2  m
Simple model:
 unpair
 pair  2
p F1
3
dp
2
3
(2

)
0
pFcomm

0
3
dp
(2)3

pair
p Fcomm
 pair   unpair


p F2
p2  ms2  m  2 
0

p2  ms2  m  2
pFcomm

0
 0  p Fcomm
1
4
2 2

m

4
D
m 
2  s
16
d 3p
| p | m 
3 
(2)
d 3p
m2D 2
| p | m  
3 
(2)
42
2
s
m
m
4m
condensation
energy
Condensation if:
ms2
m>
2D
47
Notice that the transition must be first
order because for being in the pairing phase
2
s
m
D>
0
2m
(Minimal value of the gap to get condensation)
48
Effective lagrangians
 Effective lagrangian for the CFL phase
 Effective lagrangian for the 2SC phase
49
Effective lagrangian for the
CFL phase
NG fields as the phases of the condensates in
the (3, 3) representation
X  ab 
k

ijk
a
iL
 
b *
jL
, Y  ab 
k

ijk
a
iR
 
b *
jR
Quarks and X, Y transform as
 L  ei( ab )gc LgTL ,  R  ei( ab )gc R gTR
gc  SU(3)c , g L,R  SU(3) L,R , eia  U(1) B , e ib  U(1) A
T 2i( ab )
L
X  gc Xg e
T 2i( ab )
R
, Y  gc Yg e
50
Since
X, Y  U(3)
The number of NG fields is
# X  # Y  (1  8)  (1  8)  18
8 of these fields give mass to the gluons. There
are only 10 physical NG bosons corresponding to
the breaking of the global symmetry (we consider
also the NGB associated to U(1)A)
SU(3)L  SU(3)R  U(1)V  U(1)A
SU(3)LR  Z2  Z2
51
Better use fields belonging to SU(3). Define
2i( )
ˆ
X  Xe
and
dX  det(X)  e
transforming as
6i(  )
2i( )
ˆ
Y  Ye
dY  det( Y)  e
6i( )
T
ˆ
ˆ
X  gc XgL
T
ˆ
ˆ
Y  gc YgR
a
  b
The breaking of the global symmetry can be
described by the gauge invariant order parameters
j
i
 ˆ
ˆ
ˆ
ˆ
  (Ya ) * X a    Y X, d X , d Y
i
j
52
, dX, dY are 10 fields describing the physical NG
bosons. Also

R
  g g
T
L
shows that T transforms as the usual chiral field.
Consider the currents:
ˆ mX
ˆ †  X(
ˆ m X
ˆ†X
ˆ †g m )  X
ˆ mX
ˆ †  gm
J mX  XD
ˆ mY
ˆ †  Y(
ˆ m Y
ˆ†Y
ˆ †g m )  Y
ˆ m Y
ˆ †  gm
J m  YD
Y
gm  igsgma T a
JmX,Y  gcJmX,Yg†c
53
Most general lagrangian up to two derivatives
invariant under G, the space rotation group O(3)
and Parity (R.C. & Gatto 1999)
P : (X  Y,J X  J Y ,   ,   )
2
2
FT2  0
F
1
1
2
2
0
0
0 2
T


L   Tr  J X  J Y   aT Tr  J X  J Y     0     0  



 2
4
4
2
2
2
2
2
2
2
v
2
FS2 
F
v

0
0
S




 Tr J X  J Y
 aS Tr  J X  J Y  
 






 2


4
4
2






2
FT2  ˆ ˆ † ˆ ˆ † 2 
FT2  ˆ ˆ † ˆ ˆ †
L   Tr X 0X  Y 0 Y
 aT Tr X 0X  Y 0 Y  2g 0  




4
4
2
FS2  ˆ ˆ † ˆ ˆ † 2 
FS2  ˆ ˆ † ˆ ˆ †
 Tr XX  YY
 aS Tr XX  YY  2g  




4
4
2
2
2
2
v
1
1
v
2
2


   0     0   
 

54
2
2
2
2




Using SU(3)c gauge invariance we can choose:
ˆ Y
ˆ e
X
†
ia Ta / FT
Expanding at the second order in the fields
1
L    0
2

a 2
2
v 2
2
2
2
1
1
v
v
2
2
a

   0     0   
 
 

2
2
2
2
2
2
F
v 2  S2
FT
2
Gluons acquire Debye and Meissner masses (not
the rest masses, see later)
m a g F , m a g F a v g F
2
D
2 2
T s T
2
s
2 2
S s S
2 2 2
S
s T
55
Low energy theory supposed to be valid at
energies << gap. Since we will see that gluons
have masses of order D they can be decoupled




2
FT2  ˆ ˆ † ˆ ˆ † 2 
FT2  ˆ ˆ † ˆ ˆ †
L   Tr X 0X  Y 0 Y
 aT Tr X 0X  Y 0 Y  2g 0  




4
4
2
FS2  ˆ ˆ † ˆ ˆ † 2 
FS2  ˆ ˆ † ˆ ˆ †
 Tr XX  YY
 aS Tr XX  YY  2g  




4
4
2
2
v 2
1
1
v 2
2
2
   0     0   
 

2
2
2
2




1 ˆ ˆ† ˆ ˆ†
gm   X m X  Y m Y
2


56
we get the gauge invariant result:
~ -lagrangian
2
T


F

2




L NGB 
Tr  t  t    v Tr   
4
1
1
2
2
2
2
2
2
 ( t )  v  |  |  ( t )  v  |  |
2
2




 ˆ
ˆ
Y X
57