) = 0. So the Lagrangian looks like
L
=
1
2
(
∂w
)
2
+
1
2
w
2
(
∂R
a
4
)
2

λ
4
w
2

μ
2
λ
2
(7.8)
So all the interactions for the pions are hidden in the rotation matrix and it comes in with a derivative.
This means all the scattering amplitude are proportional to the momentum of the pions.
If we define
ζ
i
=
π
i
/
(
π
4
+
ν
) then we have
R
i
4
=
2
ζ
i
1 +
ζ
2
,
R
44
=
1

ζ
2
1 +
ζ
2
(7.9)
Under the unbroken diagonal
SU
(2) symmetry we have
ζ
→
α
×
ζ
but under the broken symmetry we
have
δ
ζ
=
(1

ζ
2
) + 2
ζ
(
·
ζ
)
(7.10)
34
Quantum Field Theory III
Lecture 7
Suppose we take
μ
2
→ ∞
and
λ
→ ∞
but
μ
2
/λ
fixed, then it is effectively making the Mexican hat
potential steeper, and it is harder and harder to get out of the minimum. In the limit we will have
σ
2
(
x
) +
π
2
(
x
)
→
ν
2
(7.11)
so in the above parametrization this means
w
(
x
) becomes a constant function. Then
S
=
σ
+
i
π
·
τ
becomes
ν
times a unitary matrix. We can write it as
S
=
νe
i
π
(
x
)
·
τ
/ν
=
ν
Σ(
x
)
(7.12)
If we expand, then we will get
ν
Σ =
ν
1 +
i
ν
π
·
τ

1
2
ν
2
(
π
·
τ
)
2
+
. . .
=
ν
+
i
π
·
τ

1
2
ν
π
2
+
. . .
(7.13)
Now by symmetry the theory is invariant under Σ
→
U
Σ(
x
)
W
†
. We want a low energy effective action
and want to find invariant terms under the above symmetry that we can put into the action. The terms
without derivatives can include
Tr ΣΣ
†
=
ν
2
(7.14)
But this is not very interesting because it is just a constant. Terms with derivatives can be
Tr (
∂
μ
Σ)(
∂
μ
Σ
†
)
,
h
Tr (
∂
μ
Σ)(
∂
μ
Σ
†
)
i
2
,
Tr (
∂
μ
∂
ν
Σ)(
∂
μ
∂
ν
Σ
†
)
, . . .
(7.15)
We can then write
L
eff
=
ν
2
4
Tr (
∂
μ
Σ)(
∂
μ
Σ
†
) +
c
1
h
Tr (
∂
μ
Σ)(
∂
μ
Σ
†
)
i
2
+
. . .
(7.16)
The only relevant term for low energies is the first term.
Let’s think what are the possible vacuum for this theory, where
G
is broken into
H
.
One possible
vacuum is just
φ
0
= (0
,
0
,
0
, ν
). Then
gφ
0
is also a valid vacuum. However
H
is the subgroup where
φ
0
is invariant and we have
hφ
0
=
φ
0
. So the space of vacua is the same as the coset space
G/H
, which is
exactly the quotient group. Say if
G
=
SO
(4) and
H
=
SO
(3), then the space of vacua is the space of unit
4vectors, which is just the sphere
S
3
. The effect of our introducing
σ
is to introduce a coordinate of the
3sphere, i.e. choosing a point of identity, and obtain a way of spatially disturbing the vauum.
In general for any
G/H
we can put coordinates on the manifold with
φ
a
. The Lagrangian will look like
L
=
1
2
g
ab
(
x
)
∂
μ
φ
a
(
x
)
∂
μ
φ
b
(
x
) +
. . .
(7.17)
The quantity
g
ab
can be considered as the metric over the manifold
M
=
G/H
.
This is what people
nowadays call the nonlinear sigma model.
Let’s add a symmetry breaking term
cσ
to the original sigma model Lagrangian, where
c
is a small
positive number. Now the potential is
V
=
λ
4
(
π
+
σ
2

ν
2
)
2

cσ
(7.18)
35
Quantum Field Theory III
Lecture 7
The minimum of this potential is at
π
= 0 because we want to maximize
cσ
, and the minimum is at
0 =
∂V
∂σ
=
λσ
(
σ
2

ν
2
)

c,
σ
≈
ν
+
c
2
λν
2
=
w
(7.19)
If we ignore fermions, the equation of motion for
σ
is just
σ
=

λσ
(
σ
2
+
π
2

ν
2
)

c
(7.20)
and for
π
we have
π
=

λ
π
(
σ
2
+
π
2

ν
2
)
(7.21)
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 Fall '05
 Quantum Field Theory, Representation theory, Lie group, Lie algebra, Root system