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 A1
A1,,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
Ue , Ve
f ,
f
L ei( ) L , R ei( ) R
i
U e U,
i
Ve 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 22
4
2
f
f
2i 22
2
f
f
A
0
generates 3-linear and 4-linear couplings
5
Generating functional:
Z[] DD 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
2iD
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
2iD
2 D
i 0 iS 2 i1
iS
f
f
2
n
At the lowest order:
Seff
iS(y, x)2i(x) D
i
iS(x, y)2i(y) D
dxdyTr
i0
i0
4
f
f
iS(x, x)22 (x) D
i
dxTr
i1
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
pp 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 e4iT
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 GG
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 ig
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
mD
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 ) 2Qp0 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 2h 0
2
2
2
Mass spectrum
At zero momentum
2
p2 2v 2
det
2iE
2iE
0
2
p
M M 2v 4
2
2
2
2
0
M2 0
M 2 62 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
362
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
g22 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
182 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
31
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 ( D0)
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 d1
2
For constant D,
Ginzburg-Landau
expanding up to D6
42