Transcript Document

NGB and their parameters



Gradient expansion: parameters of the NGB’s

Dispersion relations for the gluons
Masses of the NGB’s
The role of the chemical potential for scalar
fields: BE condensation
1
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 2
Gradient expansion:
NGB’s parameters
Recall from HDET that in the CFL phase
D AB  B
dv A† iV  DAB
LD    
  (L  R)
*

4
 D AB iV  DAB 
and in the basis
D AB  D AdAB
9
1
 A
i 
(

)

A i 
2 A1
 A1,,8 D D
A
DA  
D  2D
 A  9
9
3
dv A† iV   D A  A
LD    
  (L  R)

4
 D A iV   
Propagator
dAB
SAB 
V  V   D 2A
Coupling to the U(1) NGB:
V 

 DA
i / f 
DA 

V 
i / f 
Ue , Ve
  f ,
  f 
 L  ei(  ) L ,  R  ei(  ) R
 i
U  e U,
 i
Ve V
4
Consider now the case of the U(1)B NGB. The
invariant Lagrangian is:
dv A†  iV  
LD     2
4
U DA
 U†2 D A  A

iV   
At the lowest order in 

0

dv A† 
L  
 DA
 2i 22
4
 2

f
 f
2i 22 
 2 
f
f
 A

0


generates 3-linear and 4-linear couplings
5
Generating functional:

Z[]   DD e
†
i

A(  ) 
†
2
2i
D
2

DA
1
A
A()  S0 
0 
1
2
f
f
 0 1
0 1
0  
, 1  


 1 0 
1 0
Z[]  (det[A()])
1/ 2
V 
S 
 D A
e
1
0
D A 
V  
1
Tr[log A(  )]
2
i
Seff []   Tr[log A( )]
2
6
  1 
2iD
2 2 D   
iTr[log A( )]  iTr log  S 1  S
 0  S 2 1    
f
f
  
  
2n 1
( 1)
 iTr logS   i 
n
n 1
1

 2iD

2 D
i 0  iS 2 i1 
 iS
f
f


2
n
At the lowest order:
Seff
 iS(y, x)2i(x) D

i
iS(x, y)2i(y) D
  dxdyTr 
i0
i0  
4
f
f


 iS(x, x)22 (x) D 
i
  dxTr 
i1 
2
2
f


7
Feynman rules

For each fermionic internal line
V 
idAB
iSAB  idABS(p) 

V  V   D 2A  D A
For each vertex a term iLint
DA 

V 

 For each internal momentum not constrained by
momentum conservation:

2
2 d
4

2
d  3 d d 0
4 
(2)
4 d 
 Factor 2x(-1) from Fermi statistics and spin.
A factor 1/2 from replica trick.

A statistical factor when needed.
8
+
iLeff
1 dv  2 D 2A
 iLI (p)  iL II (p)     3 2 
2 4 A  f 
2 2
2


V

(

p)

V



V

(

p)

V



2
D

2

2
A
 d 


D
(

p)D
(
)
D
(
)
A
A
A


DA ( )  V  V   D  i
2
A
Goldstone theorem:
LI (0)  LII (0)  0
Expanding in p/D:
9
9 1 dv
Leff (x)  2 2  (V   )(x)(V  )(x)
 f  2 4
2
0
0 
1 0


1
0 
0
0 
3

dv   
 4 V V  0 0  1 0 
3


1

0
0
0


3 
2
1 9 2 
2
2

Leff (x) 



v



0

2 2 

2  f 
 
1
9
2
v  , f  2
3

2
2

CFL
10
For the V NGB same result in CFL, whereas in 2SC
2
1
4

v 2  , f 2  2
3

2SC
With an analogous calculation:
Leff
2 (21  8log 2) 1 8
a 2
2
a 2


(


)

v
|

| 

0
2 2

36 F
2 a 1
2
1

(21  8log 2)
2
2
2
v  , F  FT 
2
3
36
11
Dispersion relation for the NGB’s
1
E
|p|
3
Different way of computing:
0|J |
a

b
 0 1 
 iFd p , p   p , p 
 3 
ab
Current
conservation:
1 2
pp  E  | p |  0
3
2
12
Masses of the NGB’s
QCD mass term:
 L M R  h.c.
M 
M
Z2 L
  (Y†X)T  e4iT
L masses  c  det[M]Tr[M 1  h.c.  c '  det()Tr[(M† ) 2 ] 
c"  Tr[M† ]Tr[M †]
13
Calculation of the coefficients from QCD
Mass insertion in QCD
Effective 4-fermi
3D
c  2 , c '  0, c"  0
2
2
Contribution to the vacuum energy
14
Consider:
LQCD
1 a a
 (iD   0 )   L M R   R M L  GG
4
Solving for
 ,L
  ,L as in HDET
1

i D    M 

2
0

 ,L
0
 ,R
like chemical potential
  i
1 †
† 0 
L D   iV  D 
MM g    ,L 
  ,L ( D  ) 2   ,L 
2
2


†
 ,L
  L  R, M  M †   
15
Consider fermions at finite density:
L  i     ig

0

as a gauge field A0
Invariant under:
Define:
e
i(t)
,     (t)
1
1 †
†
XL 
MM , X R 
MM
2
2
Invariance under:
  ,L  L(t)  ,L ,   ,R  R(t)  ,R ,
X L  L(t)X L L† (t)  iL(t) 0 L† (t),
X R  R(t)X R R † (t)  iR(t) 0R † (t)
16
The same symmetry should hold at the level
of the effective theory for the CFL phase
(NGB’s), implying that
T
T
 MM 
M M
 0  0   0  i 
  i
 
 2 
 2 
†
†
The generic term in the derivative expansion
of the NGB effective lagrangian has the form
L NGB
n
m
p
  0  iMM / 2      M  q †r
F D 
    2  
D

 D  F 
2
2
†
2
17
L NGB
m
n
p
  0  iMM / 2      M  q †r
F D 
    2  
D

 D  F 
2
2
†
2
Compare the two contribution to quark
masses:
kinetic term
mass insertion
4
4
m
1
m
2 2
FD 2 2 2  2
D F

m 1 Dm
FD 2 2 
2
F F
F
2
2
2
2
2
Same order of magnitude for
F
mD
since
18
The role of the chemical
potential for scalar fields:
Bose-Einstein condensation

A conserved current may be coupled to the a
gauge field.

Chemical potential is coupled to a conserved
charge.

The chemical potential must enter as the
fourth component of a gauge field.
19
Complex scalar field:

 
       m             i         
L    0  i     0  i        m         
†

† 
2
2
†
†
2
†
2
†
2
†
†
†
0
0
breaks C
negative mass term
p2  (m 2   2 )  2Qp0  0 (Q  1)
Mass spectrum:

(E  Q)  m  | p |
2
For  < m
2
2
m P,P    m
20
At  = m, second order phase transition.
Formation of a condensate obtained from:
V   m         
2
2
†
2
2


m
†
 
2
Charge
density
†
2
  m 
V
 2
†

 2      m2 


Ground state = Bose-Einstein condensate
21
v
 m
1
2
i(x ) / v

(x)

v

h(x)
e
 
, v 


2

2
1
1

L2        h  h  v 2 h 2  2h 0
2
2
2
Mass spectrum
At zero momentum
2
 p2  2v 2
det 
 2iE
2iE 
0
2 
p 
M  M  2v  4
2
2
2
2
0
M2  0
M 2  62  2m2
22
At small momentum
E NGB
2  m2

|p|
2
2
3  m
E massive
2
2
9


m
2
 6 2  2m 2  2
|
p
|
6  m 2
 m
1
 2


2

3  m
3
2
v
2
NGB
2
23
Back to CFL. From the
structure
m P,P    m
m  
m m
2c

(m u  md )ms ,
2
2
F
m K 
ms2  m 2u
2c

(m u  ms )m d ,
2
2
F
m K0 ,K0 
2
d
2
u
ms2  md2
2c

(md  ms )m u
2
2
F
3D 2
c 2
2
2

(21  8log 2)
2
F 
362
First term from “chemical potential” like MM
kinetic term, the second from mass insertions
24
†
For large values of ms:
m  
m K 
2
s
2c
(m u  md )ms ,
2
F
m
2c

ms md , m K0 ,K0 
2
2
F
2
s
m
2c

ms m u
2
2
F
and the masses of K+ and K0 are pushed down.
For the critical value
ms crit
1/ 3
 12  3
 2 2
m u,d D 2  3.03 3 m u,d D 2 ,
masses vanish
 F 
25
ms crit  40  110 MeV
2
For larger values of ms these modes become
unstable. Signal of condensation. Look for a
kaon condensate of the type:
e
i 4
 1  (cos   1)  i 4 sin 
2
4
(In the CFL vacuum,  = 1) and substitute
inside the effective lagrangian
2



1 ms  2
2cms m
2
V( )  F   
(1  cos ) 
 sin  
2
F
 2  2 

negative contribution from
the “chemical potential”
positive contribution from
mass insertion
26
Defining
eff
2
s
m

,
2
m 
0 2
K
2cms m

2
F
 1 2

2
0 2
V()  F   eff sin   (m K ) (1  cos ) 
 2

2
with solution
m 

cos  
and hypercharge
density
0 2
K

2
eff
, eff  m
0
K
  m0  4 
V
K
2
nY  
 eff F 1  4 

eff
eff 


27
Mass terms break original SU(3)c+L+R to
SU(2)IxU(1)Y. Kaon condensation breaks this to
U(1)
1
1

Q   3 
8  , [Q, ]  0
2
3 

0

0
0

 

(

,

)

(K
,
K
)

(K
,
K
)  ()
SU(2)I U(1) Y
breaking through the doublet
as in the SM
Only 2 NGB’s from K0, K+
instead of expected 3 (see
Chada & Nielsen 1976)
28
Chada and Nielsen theorem: The number of
NGB’s depends on their dispersion relation
I.
If E is linear in k, one NGB for any
broken symmetry
II.
If E is quadratic in k, one NGB for any
two broken generators
In relativistic case always of type I, in the
non-relativistic case both possibilities arise,
for instance in the ferromagnet there is one
NGB of type II, whereas for the
antiferromagnet there are two NGB’s of type I
29
Dispersion relations for the
gluons
The bare Meissner mass
The heavy field contribution comes
from the term

†
h
 D 
2

2  iV  D
h   P 
†
h
D D 
2  iV  D
 h
1  
P  g   V V  V  V 
2


30
Notice that the first quantized hamiltonian
is:
g2
2
2
H  p  gA  eA0 | p | gA0  gv  A 
|
A
|

(v

A)


2|p|
Since the zero momentum
propagator is the density one gets
3
2


d
p
1
(p

A)
2
2
g  2  Nf  
Tr  A 
3
2 
spin
(2) 2 | p | 
|p| 
|p|
2 2
g22 1
1
g

a
a
2
a
a
2
Nf
A  A  m BM  A  A , m BM  N f

2
2
6 2 a
2
6
a
31
Gluons self-energy
Vertices from
a 
 a
igA J
Consider first 2SC for the unbroken gluons:
2 2
g

00
2
 ab (p)  dab
|
p
|
,
2 2
18 D
kl
kl,self
kl
 ab
(p)   ab
(p)   ab
(p) 
from m2BM
2 2
2
2 2
2 2


g

p
g

g

kl
kl
kl
2
0
 dabd
1


d
d

d
d
p
,
ab
ab
2 
2 
2
2 2 0
3  6D 
3
18 D
2 2
g

0k
0 k
 ab (p)  dab
p
p
2 2
18 D
32

Bare Meissner mass cancels out the
constant contribution from the s.e.

All the components of the vacuum
polarization have the same wave function
renormalization
1  a 1  a b 1 a a
k a a
a a
L   Fa F   ab A A    E i E i  Bi Bi   E i E i
4
2
2
2
g 2 2
k
182 D 2
Dielectric constant  = k+1, and magnetic
permeability  =1
1
D
v


g
33
Broken gluons
a
00(0)
- ij(0)
1-3
0
0
4-7
3mg2/2
mg2/2
8
3mg2
mg2/3
2 2
g

2
mg 
2
3
34
But physical masses depend on the wave
function renormalization
g 2 2
 2
D
Rest mass defined as the energy at zero
momentum:
m R  2 D, a  4,5,6,7
g
mR 
, a 8

The expansion in p/D cannot be trusted, but
numerically
mR  0.9D, a  4,5,6,7
35
In the CFL case one finds:
2 2
g

2
2 2
mD 
(21

8log
2)

g
F
2
36
2 2
2
g

11
2
1
m


m2M  2    log 2    D
  36 27
2
3
from bare Meissner mass
Recall that from the effective lagrangian
we got:
m2D  T g2 FT2 , m2M  S v 2g 2 FT2
implying 
S
parameters.
 T  1
and fixing all the
36
mD
g 
16

, 1 
7  log 2 
We find: m R 
2 2 
216 D 
3
31

2
2

m R  1.70D
Numerically
mR  1.36D
37
Different quark masses
We have seen that for one massless flavors
and a massive one (ms), the condensate may be
disrupted for
ms2

2D
The radii of the Fermi spheres are:
2
m
pF1  2  ms2    s , p F2  
2
As if the two quarks had different chemical
potential (ms2/2)
38
Simulate the problem with two massless
quarks with different chemical potentials:
 u    d, d    d
 u  d
 u  d

, d 
2
2
Can be described by an interaction hamiltonian
H I  d 3
†
Lot of attention in normal SC.
39
HI changes the inverse propagator
D
 V   d 3

S 
*

D
V   d 3 

1
0
and the gap equation (for spin up and down
fermions):
dv  2 d 2
D  ig 
4   (2) 2 (
D
2

d
)

0
2
 D2
This has two solutions:
a) : D  D 0 , b) : D  2d D 0  D
2
2
0
40
Grand potential:
 dD 0 2 dg 


2 
 D0  g 

H

g
g
dg
     2 | D |2
g

D0

2 dD 0
 D  0  
D

2 D0 ( D0)
D0
Also:
 2
0 (d)  0 (0)   d
2
Favored solution
D  D0
41

 D  0 (d)    2d 2  D 02 
Also:
4
First order transition to the normal state at
D0
d  d1 
2
For constant D,
Ginzburg-Landau
expanding up to D6
42