Masujima M. Applied Mathematical Methods in Theoretical Physics
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448 10 Calculus of Variations: Applications
(2) Landau gauge:KernelK
(Landau)
αµ,βν
(x − y) of the quadratic part of the gauge field
action functional I
gauge
[A
γµ
] in the Landau gauge, (10.10.37),
I
quad
gauge
[A
γµ
] =−
1
2
d
4
xd
4
yA
µ
α
(x)K
(Landau)
αµ,βν
(x − y)A
ν
β
(y)
is given by
K
(Landau)
αµ,βν
(x − y) = δ
αβ
−η
µν
∂
2
+ ∂
µ
∂
ν
δ
4
(x − y), (10.10.53a)
µ, ν = 0, 1, 2, 3; α, β = 1, ..., N, (10.10.53b)
which is the four-dimensional transverse projection operator and hence is not
invertible. The ‘‘free’’ Green’s function D
(Landau)
αµ,βν
(x − y) of the gauge field in the
Landau gauge satisfies the following equation:
δ
αβ
η
µν
∂
2
− ∂
µ
∂
ν
D
(Landau)
βν,γλ
(x − y) = δ
αγ
η
µ
λ
δ
4
(x − y). (10.10.54)
We Fourier-transform the ‘‘free’’ Green’s function D
(Landau)
αµ,βν
(x − y)as
D
(Landau)
αµ,βν
(x − y) = δ
αβ
D
(Landau)
µ,ν
(x − y)
= δ
αβ
d
4
k
(2π)
4
exp
)
−ik(x − y)
*
D
(Landau)
µν
(k). (10.10.55)
We have the momentum space Green’s function D
(Landau)
µν
(k) satisfying
−η
µν
k
2
+ k
µ
k
ν
D
(Landau)
νλ
(k) = η
µ
λ
. (10.10.56a)
We introduce the four-dimensional transverse projection operator
T
(k)by
T µν
(k) = η
µν
−
k
µ
k
ν
k
2
, k
T
(k) =
T
(k)k = 0,
2
T
(k) =
T
(k), (10.10.57)
and write D
(Landau)
νλ
(k)as
D
(Landau)
νλ
(k) =
T νλ
(k)A(k
2
). (10.10.58)
Multiplying by
T τµ
(k) on the left-hand side of (10.10.56a), we have
T τµ
(k)(−k
2
)
µν
T
(k)D
(Landau)
νλ
(k) =
T τλ
(k). (10.10.56b)
10.10 Path Integral Quantization of Gauge Field I 449
Substituting (10.10.58) into the left-hand side of (10.10.56b), and noting the
idempotence of
T
(k), we have
−k
2
A(k
2
)
T τλ
(k) =
T τλ
(k). (10.10.59)
From (10.10.59), we obtain
A(k
2
) =−
1
k
2
, (10.10.60)
D
(Landau)
νλ
(k) =−
1
k
2
η
νλ
−
k
ν
k
λ
k
2
. (10.10.61)
We now calculate the kernel M
(Landau)
αx,βy
(A
γµ
) of the Faddeev–Popov ghost in the
Landau gauge.
M
(Landau)
αx,βy
A
γµ
=
δF
α
A
γµ
(x)
δA
γµ
(x)
−D
adj
µ
γβ
δ
4
(x − y)
=−∂
µ
∂
µ
δ
αβ
+C
αβγ
A
γµ
(x)
δ
4
(x − y)
≡ M
L
α,β
A
γµ
(x)
δ
4
(x − y). (10.10.62)
The Faddeev–Popov ghost part of I
eff
[A
γµ
, c
α
, c
β
]isgivenby
I
ghost
eff
)
A
γµ
, c
α
, c
β
*
=
d
4
x c
α
(x)M
L
α,β
A
γµ
(x)
c
β
(x)
=
d
4
x ∂
µ
c
α
(x)
D
adj
µ
αβ
c
β
(x). (10.10.63)
From this, we find that the Faddeev–Popov ghost in the Landau gauge is the
massless scalar Fermion and that Green’s function D
(C)
L α,β
(x − y)isgivenby
D
(C)
L α,β
(x − y) = δ
αβ
d
4
k
(2π)
4
exp
)
−ik(x − y)
*
1
k
2
. (10.10.64)
The effective interaction Lagrangian density L
int
eff
is given by the following
expression and there emerges the oriented ghost–ghost–gauge coupling in the
internal loop,
450 10 Calculus of Variations: Applications
L
int
eff
= L
eff
− L
quad
eff
= L
gauge
F
γµν
(x)
+ c
α
(x)M
L
α,β
A
γµ
(x)
c
β
(x)
−
1
2
A
µ
α
(x)δ
αβ
η
µν
∂
2
− ∂
µ
∂
ν
A
ν
β
(x) − c
α
(x)δ
αβ
−∂
2
c
β
(x)
=
1
2
C
αβγ
∂
µ
A
αν
(x) − ∂
ν
A
αµ
(x)
A
µ
β
(x)A
ν
γ
(x)
−
1
4
C
αβγ
C
αδε
A
βµ
(x)A
γν
(x)A
µ
δ
(x)A
ν
ε
(x)
+ C
αβγ
∂
µ
c
α
(x)A
γµ
(x)c
β
(x). (10.10.65)
(3) Covariant gauge:KernelK
αµ,βν
(x − y;ξ ) of the quadratic part of the effective
action functional I
eff
[A
γµ
, c
α
, c
β
] with respect to the gauge field A
γµ
(x)inthe
covariant gauge, (10.10.38),
I
quad. in A
γ
µ
eff
)
A
γµ
, c
α
, c
β
*
=
d
4
x
L
gauge
F
γµν
(x)
−
1
2
F
2
α
A
γµ
(x)
quad. in A
γ
µ
≡−
1
2
d
4
xd
4
yA
µ
α
(x)K
αµ,βν
(x − y;ξ)A
ν
β
(y), (10.10.66)
is given by
K
αµ,βν
(x − y;ξ) = δ
αβ
1
−η
µν
∂
2
+ (1 −ξ )∂
µ
∂
ν
2
δ
4
(x − y), (10.10.67)
and is invertible for ξ>0. The ‘‘free’’ Green’s function D
(cov)
αµ,βν
(x − y;ξ )ofthe
gauge field A
γµ
(x) in the covariant gauge satisfies the following equation:
δ
αβ
1
η
µν
∂
2
− (1 −ξ )∂
µ
∂
ν
2
D
(cov)
βν,γλ
(x − y;ξ) = δ
αγ
η
µ
λ
δ
4
(x − y). (10.10.68)
We Fourier-transform the ‘‘free’’ Green’s function D
(cov)
αµ,βν
(x − y;ξ )as
D
(cov)
αµ,βν
(x − y;ξ) = δ
αβ
d
4
k
(2π)
4
exp
)
−ik(x − y)
*
D
(cov)
µ,ν
(k;ξ ). (10.10.69)
We have the momentum space Green’s function D
(cov)
µ,ν
(k;ξ ) satisfying
1
−η
µν
k
2
+(1 −ξ )k
µ
k
ν
2
D
(cov)
ν,λ
(k;ξ ) = η
µ
λ
. (10.10.70a)
We define the four-dimensional projection operators,
T
(k)and
L
(k), by
T µ,ν
(k) = η
µ,ν
−
k
µ
k
ν
k
2
,
L µ,ν
(k) =
k
µ
k
ν
k
2
, (10.10.71)
10.10 Path Integral Quantization of Gauge Field I 451
2
T
(k) =
T
(k),
2
L
(k) =
L
(k),
T
(k)
L
(k) =
L
(k)
T
(k) = 0. (10.10.72)
Equation (10.10.70a) is written in terms of
T
(k)and
L
(k)as
−k
2
µ,ν
T
(k) +ξ
µ,ν
L
(k)
D
(cov)
ν,λ
(k;ξ ) = η
µ
λ
. (10.10.70b)
Expressing D
(cov)
ν,λ
(k;ξ )as
D
(cov)
ν,λ
(k;ξ ) =
T ν,λ
(k)A(k
2
) +
L ν,λ
(k)B(k
2
;ξ ), (10.10.73)
and making use of (10.10.72), we obtain
−k
2
A(k
2
)
µ
T λ
(k) +
−ξ k
2
B(k
2
;ξ )
µ
L λ
(k) = η
µ
λ
. (10.10.74)
Multiplying by
T τ ,µ
(k)and
L τ ,µ
(k) on the left-hand side of (10.10.74), respec-
tively, we obtain
−k
2
A(k
2
)
T τ ,λ
(k) =
T τ ,λ
(k),
−ξ k
2
B(k
2
;ξ )
L τ ,λ
(k) =
L τ ,λ
(k),
(10.10.75)
from which, we obtain
A
k
2
=−
1
k
2
, B
k
2
;ξ
=−
1
ξ
1
k
2
, (10.10.76)
D
(cov)
ν,λ
(k;ξ ) =−
1
k
2
η
ν,λ
−
k
ν
k
λ
k
2
+
1
ξ
k
ν
k
λ
k
2
=−
1
k
2
η
ν,λ
−
1 −
1
ξ
k
ν
k
λ
k
2
. (10.10.77)
The parameter ξ shifts the longitudinal component of the covariant gauge Green’s
function. In the limit ξ →∞, the covariant gauge Green’s function D
(cov)
ν,λ
(k;ξ )
coincides with the Landau gauge Green’s function D
(Landau)
ν,λ
(k), and, at ξ = 1, the
covariant gauge Green’s function D
(cov)
ν,λ
(k;ξ ) coincides with the
t Hooft–Feynman
gauge Green’s function D
(Feynman)
ν,λ
(k),
D
(cov)
ν,λ
(k;∞) =−
1
k
2
η
ν,λ
−
k
ν
k
λ
k
2
= D
(Landau)
ν,λ
(k), (10.10.78a)
D
(cov)
ν,λ
(k;1) =−
1
k
2
η
ν,λ
= D
(Feynman)
ν,λ
(k). (10.10.78b)
We calculate the kernel M
(cov)
αx,βy
(A
γµ
) of the Faddeev–Popov ghost in the covariant
gauge.
M
(cov)
αx,βy
A
γµ
=
δF
α
A
γµ
(x)
δA
γµ
(x)
−D
adj
µ
γβ
δ
4
(x − y) =
9
ξM
(Landau)
αx,βy
A
γµ
=
9
ξM
L
α,β
A
γµ
(x)
δ
4
(x − y). (10.10.79)
452 10 Calculus of Variations: Applications
We absorb
√
ξ factor in the normalization of the Faddeev–Popov ghost field, c
α
(x)
and c
β
(x). The ghost part of the effective action functional is then given by
I
ghost
eff
)
A
γµ
, c
α
, c
β
*
=
d
4
x c
α
(x)M
L
α,β
(A
γµ
(x))c
β
(x)
=
d
4
x ∂
µ
c
α
(x)
D
adj
µ
αβ
c
β
(x), (10.10.80)
just like in the Landau gauge, (10.10.63). From this, we find that the Faddeev–Popov
ghost in the covariant gauge is the massless scalar Fermion whose Green’s function
D
(C)
L α,β
(x − y)isgivenby
D
(C)
L α,β
(x − y) = δ
αβ
d
4
k
(2π)
4
exp
)
−ik(x − y)
*
1
k
2
. (10.10.81)
We calculate the interaction Lagrangian density L
int
eff
in the covariant gauge. We
have
L
eff
= L
gauge
F
γµν
(x)
−
1
2
F
2
α
A
γµ
(x)
+ c
α
(x)M
L
α,β
A
γµ
(x)
c
β
(x),
(10.10.82)
L
quad
eff
=−
1
2
A
µ
α
(x)δ
αβ
1
−η
µν
∂
2
+ (1 −ξ )∂
µ
∂
ν
2
A
ν
β
(x) + c
α
(x)δ
αβ
−∂
2
c
β
(x).
(10.10.83)
From (10.10.82) and (10.10.83), we obtain L
int
eff
as
L
int
eff
= L
eff
− L
quad
eff
=
1
2
C
αβγ
∂
µ
A
αν
(x) − ∂
ν
A
αµ
(x)
A
µ
β
(x)A
ν
γ
(x)
−
1
4
C
αβγ
C
αδε
A
βµ
(x)A
γν
(x)A
µ
δ
(x)A
ν
ε
(x) + C
αβγ
∂
µ
c
α
A
γµ
(x)c
β
(x).
(10.10.84a)
The generating functional Z
F
[J
γµ
, ζ
α
, ζ
β
](W
F
[J
γµ
, ζ
α
, ζ
β
]) of (the connected parts
of) the ‘‘full’’ Green’s functions in the covariant gauge is given by
Z
F
)
J
γµ
, ζ
α
, ζ
β
*
≡ exp
)
iW
F
)
J
γµ
, ζ
α
, ζ
β
**
≡
0, out
T
exp
i
d
4
z
/
J
γµ
(z)
ˆ
A
µ
γ
(z)
+
c
α
(z)ζ
α
(z) + ζ
β
(z)c
β
(z)
0, in
10.10 Path Integral Quantization of Gauge Field I 453
=
D[A
γµ
]D[c
α
]D[c
β
]exp
i
d
4
z
/
L
eff
+ J
γµ
(z)A
µ
γ
(z)
+
c
α
(z)ζ
α
(z) + ζ
β
(z)c
β
(z)
= exp
iI
int
eff
1
i
δ
δJ
µ
γ
, i
δ
δζ
α
,
1
i
δ
δζ
β
Z
F,0
)
J
γµ
, ζ
α
, ζ
β
*
.
(10.10.85)
We have I
int
eff
[A
γµ
, c
α
, c
β
]as
I
int
eff
)
A
γµ
, c
α
, c
β
*
=
d
4
x L
int
eff
(10.10.84a)
. (10.10.84b)
The generating functional Z
F,0
[J
γµ
, ζ
α
, ζ
β
]isgivenby
Z
F,0
)
J
γµ
, ζ
α
, ζ
β
*
=
D[A
γµ
]D[c
α
]D[c
β
]exp
i
d
4
z
/
L
quad
eff
((83))
+ J
γµ
(z)A
µ
γ
(z) +c
α
(z)ζ
α
(z) + ζ
β
(z)c
β
(z)
= exp
i
d
4
xd
4
y
−
1
2
J
µ
α
(x)D
(cov)
αµ,βν
(x − y;ξ )J
ν
β
(y)
−
ζ
β
(x)D
(C)
αβ
(x − y)ζ
α
(y)
. (10.10.86)
Equation (10.10.85) is the starting point of Feynman–Dyson expansion of the ‘‘full’’
Green’s function with the ‘‘free’’ Green’s functions and the effective interaction
vertices, (10.10.84a). Noting the fact that the Faddeev–Popov ghost appears only
in the internal loops, we might as well set ζ
α
(z) = ζ
β
(z) = 0inZ
F
[J
γµ
, ζ
α
, ζ
β
]and
define Z
F
[J
γµ
]by
Z
F
[J
γµ
] ≡ exp
)
iW
F
)
J
γµ
**
≡
0, out
T
exp
i
d
4
zJ
γµ
(z)
ˆ
A
µ
γ
(z)
0, in
=
D[A
γµ
]D[c
α
]D[c
β
]exp
i
d
4
z
/
L
eff
+J
γµ
(z)A
µ
γ
(z)
0
=
D[A
γµ
]
F
[A
γµ
]exp
i
d
4
z
L
gauge
F
γµν
(z)
−
1
2
F
2
α
A
γµ
(z)
+ J
γµ
(z)A
µ
γ
(z)
. (10.10.87)
This generating functional of the (connected part of) Green’s functions can be used
for the proof of the gauge independence of the physical S-matrix.
454 10 Calculus of Variations: Applications
10.11
Path Integral Quantization of Gauge Field II
We discuss the method with which we give a mass term to the gauge field without violating
gauge invariance. We write the n-component real scalar field φ
i
(x), i = 1, ..., n,as
˜
φ(x) =
φ
1
(x)
·
·
φ
n
(x)
,
˜
φ
T
(x) =
φ
1
(x), ... φ
n
(x)
, (10.11.1)
and express the globally G invariant matter field Lagrangian density as
L
matter
(
˜
φ(x), ∂
µ
˜
φ(x)) =
1
2
∂
µ
˜
φ
T
(x)∂
µ
˜
φ(x) −V(
˜
φ(x)). (10.11.2)
Here we assume the following three conditions:
(1) G is a semisimple N-parameter Lie group whose N generators we write as
T
α
, α = 1, ..., N.
(2) Under the global G transformation,
˜
φ(x) transform with the n-dimensional
reducible representation, θ
α
, α = 1, ..., N,
δφ
i
(x) = iε
α
(θ
α
)
ij
φ
j
(x), α = 1, ..., N, i, j = 1, ..., n. (10.11.3)
(3) V(
˜
φ(x)) is the globally G invariant quartic polynomial in
˜
φ(x)whosethe
global G invariance condition is given by
∂V
˜
φ(x)
∂φ
i
(x)
(θ
α
)
ij
φ
j
(x) = 0, α = 1, ..., N. (10.11.4)
We assume that the minimizing
˜
φ(x)ofV(
˜
φ(x)) exists and is given by the constant
vector ˜v,
∂V
˜
φ(x)
∂φ
i
(x)
˜
φ(x)=˜v
= 0, i = 1, ..., n. (10.11.5)
We differentiate the global G invariance condition, (10.11.4), with respect to φ
k
(x)
and set
˜
φ(x) =˜v, obtaining the broken symmetry condition,
∂
2
V
˜
φ(x)
∂φ
k
(x)∂φ
i
(x)
˜
φ(x)=˜v
(θ
α
)
ij
v
j
= 0, α = 1, ..., N, i, j, k = 1, ..., n.
(10.11.6)
10.11 Path Integral Quantization of Gauge Field II 455
We expand V(
˜
φ(x)) around
˜
φ(x) =˜v and choose V( ˜v) = 0 as the origin of energy.
V
˜
φ(x)
=
1
2
˜
φ(x) −˜v
T
M
2
˜
φ(x) −v
+ higher order term. (10.11.7a)
We set
M
2
i,j
≡
∂
2
V
˜
φ(x)
∂φ
i
(x)∂φ
j
(x)
˜
φ
(x)=˜v
, i, j = 1, ..., n, (10.11.7b)
where M
2
is the mass matrix of the n-component real scalar field. We express the
broken symmetry condition, (10.11.6), in terms of the mass matrix M
2
as
M
2
i,j
(
θ
α
˜v
)
j
= 0, α = 1, ..., N, i, j = 1, ..., n. (10.11.8)
We let the stability group of the vacuum, i.e., the symmetry group of the ground
state
˜
φ(x) =˜v,betheM-dimensional subgroup S ⊂ G.Whenθ
α
is the realization
on the scalar field
˜
φ(x) of the generator T
α
belonging to the stability group S,this
θ
α
annihilates the ‘‘vacuum’’ ˜v
(θ
α
˜v) = 0forθ
α
∈ S, (10.11.9)
and we have the invariance of ˜v expressed as
exp
i
θ
α
∈S
ε
α
θ
α
˜v =˜v, stability group of the vacuum. (10.11.10)
As for the M realizations θ
α
∈ S on the scalar field
˜
φ(x)oftheM generators T
α
∈ S,
the broken symmetry condition, (10.11.8), is satisfied automatically due to the
stability condition, (10.11.9), and we do not get new information from (10.11.8).
As for the remaining (N − M) realizations θ
α
/∈ S on the scalar field
˜
φ(x)ofthe
(N − M) broken generators T
α
/∈ S which break the stability condition, (10.11.9),
we get the following information from (10.11.8), namely, the mass matrix
M
2
i,j
has the (N − M) nontrivial eigenvectors belonging to the eigenvalue 0
θ
α
˜v = 0, θ
α
/∈ S. (10.11.11)
Here we show that {θ
α
˜v, θ
α
/∈ S} span the (N − M)-dimensional vector space. We
define the N × N matrix µ
2
α,β
by
µ
2
α,β
≡ (θ
α
˜v, θ
β
˜v) ≡
n
i=1
(θ
α
˜v)
†
i
(θ
β
˜v)
i
, α, β = 1, ..., N. (10.11.12a)
456 10 Calculus of Variations: Applications
From the Hermiticity of θ
α
,wehave
µ
2
α,β
=
˜v, θ
α
θ
β
˜v
=
n
i=1
v
∗
i
θ
α
θ
β
˜v
i
, (10.11.12b)
µ
2
α,β
− µ
2
β,α
=
˜v,
)
θ
α
, θ
β
*
˜v
= iC
αβγ
˜v, θ
γ
˜v
= 0, (10.11.13)
i.e., we know that µ
2
α,β
is a real symmetric matrix. The last equality of (10.11.13)
follows from the antisymmetry of θ
α
. Next we let ˜µ
2
α,β
be the restriction of µ
2
α,β
to the subspace {θ
α
˜v, θ
α
/∈ S}.The ˜µ
2
α,β
is the real symmetric (N − M) ×(N − M )
matrix and diagonalizable. We let the (N − M) ×(N − M) orthogonal matrix that
diagonalizes ˜µ
2
α,β
to be O, and let ˜µ
2
α,β
to be the diagonalized ˜µ
2
α,β
.
˜µ
2
α,β
=
O ˜µ
2
O
α,β
=
(Oθ)
α
˜v,(Oθ)
β
˜v
= δ
αβ
·˜µ
2
(α)
. (10.11.14)
The diagonal element ˜µ
2
(α)
is given by
˜µ
2
(α)
=
(Oθ)
α
˜v,(Oθ)
α
˜v
=
n
i=1
(Oθ)
α
˜v
†
i
(Oθ)
α
˜v
i
. (10.11.15)
From the definition of the stability group of the vacuum and the definition of the
linear independence in the N-dimensional vector space spanned by the realization
θ
α
, we obtain the following three statements:
(1)
∀θ
α
/∈ S :(Oθ)
α
˜v = 0.
(2) (N − M ) diagonal elements, ˜µ
2
(α)
,of ˜µ
2
α,β
are all positive.
(3) (N − M ) θ
α
’s, θ
α
/∈ S, are linearly independent.
Hence we understand that {θ
α
˜v : ∀θ
α
/∈ S} span the (N − M)-dimensional vector
space. In this way, we obtain the following theorem.
Goldstone’s theorem: When the global symmetry induced by (N − M)
generators T
α
corresponding to {θ
α
: θ
α
/∈ S} is spontaneously broken
(θ
α
˜v = 0) by the vacuum ˜v, the mass matrix
M
2
i,j
has (N − M)eigen-
vectors {θ
α
˜v : θ
α
/∈ S} belonging to the eigenvalue 0, and these vectors
{θ
α
˜v : θ
α
/∈ S} span the (N − M)-dimensional vector space of Nambu–Goldstone
boson (massless excitation). The remaining n − (N − M)scalarfieldsare
massive.
From the preceding argument, we find that N × N matrix µ
2
α,β
is the rank
(N − M) and has the (N − M) positive eigenvalues ˜µ
2
(α)
and the M zero eigenvalues.
Now, we rearrange the generators T
α
in the order of the M unbroken generators
10.11 Path Integral Quantization of Gauge Field II 457
T
α
∈ S and the (N − M) broken generators T
α
/∈ S.Wegetµ
2
α,β
in the block
diagonal form
µ
2
α,β
=
0, 0,
0, ˜µ
2
α,β
, (10.11.16)
where the upper diagonal block corresponds to the M-dimensional vector space of
the stability group of the vacuum, S, and the lower diagonal block corresponds to
the (N − M)-dimensional vector space of Nambu–Goldstone boson.
ThepurposeoftheintroductionoftheN × N matrix µ
2
α,β
is twofold.
(1) {θ
α
˜v : θ
α
/∈ S} span the (N − M)-dimensional vector space.
(2) The mass matrix of the gauge fields to be introduced by the application of
Weyl’s gauge principle in the presence of spontaneous symmetry breaking
is given by µ
2
α,β
.
Having disposed of the first point, we move on to the discussion of the second
point.
Higgs–Kibble mechanism: We now extend the global G invariance which is
spontaneously broken by the ‘‘vacuum’’ ˜v of the matter field Lagrangian density,
L
matter
˜
φ(x), ∂
µ
˜
φ(x)
, to the local G invariance with Weyl’s gauge principle. The
total Lagrangian density L
tot
of the matter-gauge system after the gauge extension
is given by
L
tot
= L
gauge
F
γµν
(x)
+ L
scalar
˜
φ(x), D
µ
˜
φ(x)
=−
1
4
F
γµν
(x)F
µν
γ
(x) +
1
2
/
∂
µ
+ iθ
α
A
αµ
(x)
˜
φ(x)
0
T
i
×
/
∂
µ
+ iθ
β
A
µ
β
(x)
˜
φ(x)
0
i
− V
˜
φ(x)
=−
1
4
F
γµν
(x)F
µν
γ
(x) +
1
2
/
∂
µ
˜
φ
T
(x) − i
˜
φ
T
(x)θ
α
A
αµ
(x)
0
i
×
/
∂
µ
˜
φ(x) +iθ
β
A
µ
β
(x)
˜
φ(x)
0
i
− V
˜
φ(x)
. (10.11.17)
We parameterize the n-component scalar field
˜
φ(x)as
˜
φ(x) = exp
i
θ
α
/∈S
ξ
α
(x)θ
α
v
α
˜v +˜η(x)
. (10.11.18)
We remark that the (N − M) ξ
α
(x)’s correspond to the (N − M) broken generators
T
α
/∈ S,andthe ˜η(x)hasthe(n − (N − M)) nonvanishing components and is
orthogonal to Nambu–Goldstone direction, {θ
α
˜v : θ
α
/∈ S}.Wehavethevacuum