Masujima M. Applied Mathematical Methods in Theoretical Physics
Подождите немного. Документ загружается.
458 10 Calculus of Variations: Applications
expectation values of {ξ
α
(x)}
N−M
α=1
and {η
i
(x)}
n−(N−M)
i=1
equal to 0. From the global G
invariance of V
˜
φ(x)
,wehave
V
˜
φ(x)
= V
˜v +˜η(x)
, (10.11.19)
and find that V(
˜
φ(x)) is independent of {ξ
α
(x)}
N−M
α=1
. Without the gauge extension,
we had the ξ
α
(x)-dependence from the derivative term in L
scalar
1
2
∂
µ
˜
φ
T
(x)∂
µ
˜
φ(x) =
1
2
∂
µ
ξ
α
(x)∂
µ
ξ
α
(x) +···. (10.11.20)
Thus {ξ
α
(x)}
N−M
α=1
would be the massless Nambu–Goldstone boson with the gra-
dient coupling with the other fields. As in (10.11.17), after the gauge extension,
we have the freedom of the gauge transformation and are able to eliminate
Nambu–Goldstone boson fields {ξ
α
(x)}
N−M
α=1
completely. We employ the local phase
transformation and the nonlinear gauge transformation of the following form:
˜
φ
(x) = exp
−i
θ
α
/∈S
ξ
α
(x)θ
α
v
α
˜
φ(x) =˜v +˜η(x), (10.11.21a)
θ
γ
A
γµ
(x) = exp
−i
θ
α
/∈S
ξ
α
(x)θ
α
v
α
θ
γ
A
γµ
(x) + exp
i
θ
β
/∈S
ξ
β
(x)θ
β
v
β
×
i∂
µ
exp
−i
θ
β
/∈S
ξ
β
(x)θ
β
v
β
exp
i
θ
α
/∈S
ξ
α
(x)θ
α
v
α
.
(10.11.21b)
As a result of these local G transformations, we managed to eliminate {ξ
α
(x)}
N−M
α=1
completely and have the total Lagrangian density L
tot
as
L
tot
=−
1
4
F
γµν
(x)F
µν
γ
(x) +
1
2
1
∂
µ
+ iθ
α
A
αµ
(x)
˜v +˜η(x)
2
T
×
/
∂
µ
+ iθ
α
A
µ
α
(x)
˜v +˜η(x)
0
− V
˜v +˜η(x)
. (10.11.22)
After the gauge transformations, (10.11.21a) and (10.11.21b), we have the covariant
derivative as
∂
µ
+ iθ
α
A
αµ
(x)
( ˜v +˜η(x)) = ∂
µ
˜η(x) +i
(
θ
α
˜v
)
A
αµ
(x) + i
θ
α
˜η(x)
A
αµ
(x).
(10.11.23)
10.11 Path Integral Quantization of Gauge Field II 459
From the definitions of µ
2
α,β
, (10.11.12a) and (10.11.12b), we have
1
2
/
∂
µ
+ iθ
α
A
αµ
(x)
˜v +˜η(x)
0
T
/
∂
µ
+iθ
β
A
µ
β
(x)
˜v +˜η(x)
0
=
1
2
∂
µ
˜η
T
(x)∂
µ
˜η(x) +µ
2
α,β
A
αµ
(x)A
µ
β
(x)
+ ∂
µ
˜η
T
(x)i
(
θ
α
˜v
)
A
µ
α
(x)
+ ∂
µ
˜η
T
(x)i
θ
α
˜η(x)
A
µ
α
(x) +
˜η(x), θ
α
θ
β
˜v
A
αµ
(x)A
µ
β
(x)
+
1
2
˜η(x), θ
α
θ
β
˜η(x)
A
αµ
(x)A
µ
β
(x). (10.11.24)
We expand V(
˜
φ
(x)) around
˜
φ
(x) =˜v and obtain
V
˜v +˜η(x)
=
1
2
M
2
ij
˜η
i
(x) ˜η
j
(x) +
1
3!
f
ijk
˜η
i
(x) ˜η
j
(x) ˜η
k
(x)
+
1
4!
f
ijkl
˜η
i
(x) ˜η
j
(x) ˜η
k
(x) ˜η
l
(x). (10.11.25a)
We have
M
2
ij
=
∂
2
V
˜
φ(x)
∂φ
i
(x)∂φ
j
(x)
˜
φ(x)=˜v
, (10.11.25b)
f
ijk
=
∂
3
V
˜
φ(x)
∂φ
i
(x)∂φ
j
(x)∂φ
k
(x)
˜
φ
(x)=˜v
, (10.11.25c)
f
ijkl
=
∂
4
V
˜
φ(x)
∂φ
i
(x)∂φ
j
(x)∂φ
k
(x)∂φ
l
(x)
˜
φ(x)=˜v
. (10.11.25d)
From (10.11.24) and (10.11.25a), the total Lagrangian density L
tot
after the local G
transformation is
L
tot
=−
1
4
F
γµν
(x)F
µν
γ
(x) +
1
2
µ
2
α,β
A
αµ
(x)A
µ
β
(x) + ∂
µ
˜η(x)i
(
θ
α
˜v
)
A
µ
α
(x)
+
1
2
∂
µ
˜η(x)∂
µ
˜η(x) −
1
2
M
2
i,j
˜η
i
(x) ˜η
j
(x) +∂
µ
˜η(x)i
θ
α
˜η(x)
A
µ
α
(x)
+˜η(x)(θ
α
θ
β
˜v)A
αµ
(x)A
µ
β
(x) −
1
3!
f
ijk
˜η
i
(x) ˜η
j
(x) ˜η
k
(x)
+
1
2
˜η(x)
θ
α
θ
β
˜η(x)
A
αµ
(x)A
µ
β
(x) −
1
4!
f
ijkl
˜η
i
(x) ˜η
j
(x) ˜η
k
(x) ˜η
l
(x). (10.11.26)
From (10.11.26), we obtain as the quadratic part of L
tot
,
460 10 Calculus of Variations: Applications
L
quad
tot
=−
1
4
∂
µ
A
αν
(x) − ∂
ν
A
αµ
(x)
∂
µ
A
ν
α
(x) − ∂
ν
A
µ
α
(x)
+
1
2
µ
2
α,β
A
αµ
(x)A
µ
β
(x) +
1
2
∂
µ
˜η(x)∂
µ
˜η(x) −
1
2
M
2
i,j
˜η
i
(x) ˜η
j
(x)
+∂
µ
˜η(x)i(θ
α
˜v)A
µ
α
(x). (10.11.27)
Were it not for the last term in (10.11.27) which represents the mixing of the gauge
field and the scalar field, we can regard (10.11.27) as the ‘‘free’’ Lagrangian density
of the following fields:
(1) M massless gauge fields corresponding to the M unbroken generators,
T
α
∈ S, belonging to the stability group of the vacuum,
(2) (N − M ) massive vector fields corresponding to the (N − M) broken
generators, T
α
/∈ S, with the mass eigenvalues, ˜µ
2
(α)
, α = M + 1, ..., N,
(3) (n − (N − M)) massive scalar fields with the mass matrix,
M
2
i,j
.
The (N − M) Nambu–Goldstone boson fields {ξ
α
(x)}
N−M
α=1
get eliminated from
the particle spectrum by the gauge transformations, (10.11.21a) and (10.11.21b),
and absorbed as the longitudinal mode of the gauge fields which corresponds to the
(N − M) broken generators T
α
/∈ S.Thesaid(N − M)gaugefieldsbecomethe(N −
M) massive vector fields with two transverse modes and one longitudinal mode.
We call this mass-generating mechanism for the gauge fields as Higgs–Kibble
mechanism. We make the lists of the degrees of freedom of the matter-gauge
system before and after the gauge transformations, (10.11.21a) and (10.11.21b):
Before the gauge transformation Degrees of freedom
N massless gauge fields 2N
(N −M) Goldstone boson fields N − M
(n − (N − M)) massive scalar fields n − (N − M)
Total degrees of freedom n + 2N
and
After the gauge transformation Degrees of freedom
M massless gauge fields 2M
(N −M) massive vector fields 3(N − M)
(n − (N − M)) massive scalar fields n − (N − M)
Total degrees of freedom n + 2N
There are no changes in the total degrees of freedom, n + 2N.Beforethegauge
transformation, the local G invariance of L
tot
, (10.11.17), is manifest, whereas
after the gauge transformation, the particle spectrum content of L
tot
, (10.11.26), is
manifest and the local G invariance of L
tot
is hidden. In this way, we can give the
10.11 Path Integral Quantization of Gauge Field II 461
mass term to the gauge field without violating the local G invariance and verify that
the mass matrix of the gauge field is indeed given by µ
2
α,β
.
Path Integral Quantization of Gauge Field in R
ξ
-Gauge:Weshallemploythe
R
ξ
-gauge as the gauge-fixing condition, and carry out the path integral quantization
of the matter-gauge system described by the Lagrangian density, (10.11.26). We
can also eliminate the mixed term ∂
µ
˜η(x)i(θ
α
˜v)A
µ
α
(x) of the scalar fields ˜η(x)and
the gauge fields A
αµ
(x)inL
quad
tot
. R
ξ
-gauge is the gauge-fixing condition involving
Higgs scalar field and the gauge field , {˜η(x), A
γµ
(x)},linearly.Wewillusethe
general notation
{φ
a
}=
/
η
i
(x), A
γµ
(x)
0
, a = (i, x), (γµ, x). (10.11.28)
As the general linear gauge-fixing condition, we employ
F
α
({φ
a
}) = a
α
(x), α = 1, ..., N. (10.11.29)
We assume that
F
α
1
φ
g
a
2
= a
α
(x), α = 1, ..., N, {φ
a
} fixed, (10.11.30)
has the unique solution g(x) ∈ G. We parameterize the element g(x)inthe
neighborhood of the identity element of G by
g(x) = 1 +iε
α
(x)θ
α
+O
ε
2
, (10.11.31)
with ε
α
(x) arbitrary infinitesimal function independent of {φ
a
}. The Faddeev–Popov
determinant
F
[{φ
a
}] of the gauge-fixing condition, (10.11.29), is defined by
F
[{φ
a
}]
$
x
dg(x)
$
α,x
δ
F
α
1
φ
g
a
2
− a
α
(x)
= 1, (10.11.32)
and is invariant under the linear gauge transformation, (10.11.31),
F
)1
φ
g
a
2*
=
F
[
{
φ
a
}
]
. (10.11.33)
According to the first formula of Faddeev–Popov, the vacuum-to-vacuum transi-
tion amplitude of the matter-gauge system governed by the Lagrangian density,
L
tot
((10.11.26)), is given by
Z
F
(a
α
(x)) ≡
0, out
0, in
F,a
=
D[{φ
a
}]
F
[{φ
a
}]exp
[
iI
tot
[{φ
a
}]
]
×
$
α,x
δ
F
α
({φ
a
}
− a
α
(x)), (10.11.34)
462 10 Calculus of Variations: Applications
D
[
{φ
a
}
]
≡ D
)
η
i
(x)
*
D
%
A
γµ
(x)
'
, (10.11.35)
I
tot
[{φ
a
}] =
d
4
x L
tot
((10.11.26)). (10.11.36)
Since
F
[{φ
a
}]in
0, out
0, in
F,a
gets multiplied by δ(F
α
({φ
a
}) −a
α
(x)), it is suffi-
cient to calculate
F
[{φ
a
}]for{φ
a
} which satisfies (10.11.29). We will parameterize
g(x) as in (10.11.31) so that
F
[{φ
a
}] = DetM
F
({φ
a
}), (10.11.37)
{M
F
({φ
a
})}
αx,βy
=
δF
α
1
φ
g
a
(x)
2
δε
β
(y)
g=1
, α, β = 1, ..., N, (10.11.38)
F
α
({φ
g
a
}) = F
α
({φ
a
}) +M
F
({φ
a
})
αx,βy
ε
β
(y) +O(ε
2
). (10.11.39)
We now consider the nonlinear gauge transformation g
0
parameterized by
ε
α
(x;{φ
a
}) =
1
M
−1
F
({φ
a
})
2
αx,βy
λ
β
(y) (10.11.40)
with λ
β
(y) arbitrary infinitesimal function independent of {φ
a
}. Under this nonlin-
ear gauge transformation, we have
(1) I
tot
[{φ
a
}] is gauge invariant,
(2)
D[{φ
a
}]
F
[{φ
a
}] = gauge invariant measure, (10.11.41)
(3) The gauge-fixing condition F
α
({φ
a
}) gets transformed into
F
α
1
φ
g
0
a
2
= F
α
({φ
a
}) + λ
α
(x) + O
λ
2
. (10.11.42)
Since the value of the functional integral remains unchanged under the change of
the function variables, the value of Z
F
(a
α
(x)) remains unchanged. With the choice
as
λ
α
(x) = δa
α
(x), α = 1, ..., N, (10.11.43)
we have
Z
F
(a
α
(x)) = Z
F
(a
α
(x) + δa
α
(x)) or
d
da
α
(x)
Z
F
(a
α
(x)) = 0. (10.11.44)
SincewefindthatZ
F
(a
α
(x)) is independent of a
α
(x), we can introduce an arbitrary
weighting functional H[a
α
(x)] for Z
F
(a
α
(x)) and path-integrate with respect to
a
α
(x), obtaining as the weighted Z
F
(a
α
(x))
Z
F
≡
$
α,x
da
α
(x)H[a
α
(x)]Z
F
(a
α
(x))
10.11 Path Integral Quantization of Gauge Field II 463
=
D
[
{φ
a
}
]
F
[
{φ
a
}
]
H
)
F
α
({φ
a
})
*
exp
[
iI
tot
[{φ
a
}]
]
. (10.11.45)
As the weighting functional H[a
α
(x)], we use the quasi-Gaussian functional,
H [a
α
(x)] = exp
−
i
2
d
4
xa
2
α
(x)
(10.11.46)
and obtain Z
F
as
Z
F
=
D[{φ
a
}]
F
[{φ
a
}]exp
i
d
4
x {L
tot
((10.11.26)) −
1
2
F
2
α
({φ
a
})}
=
D[{φ
a
}]D[c
α
]D[c
β
]exp
i
d
4
x
L
tot
((10.11.26)) −
1
2
F
2
α
({φ
a
})
+
c
α
(x)M
F
({φ
a
(x)})c
β
(x)
. (10.11.47)
Since the gauge-fixing condition, (10.11.29), is linear in {φ
a
(x)},wehave
{M
F
({φ
a
})}
αx,βy
= δ
4
(x − y)M
F
({φ
a
(x)})
α,β
. (10.11.48)
Summarizing the results, we have
0, out
0, in
F
=
D[{φ
a
}]D[c
α
]D[c
β
]exp[iI
eff
[{φ
a
}, c
α
, c
β
]] (10.11.49a)
with the effective action functional I
eff
[{φ
a
}, c
α
, c
β
]givenby
I
eff
[{φ
a
}, c
α
, c
β
] =
d
4
x L
eff
({φ
a
(x)}, c
α
(x), c
β
(x)). (10.11.49b)
The effective Lagrangian density L
eff
({φ
a
(x)}, c
α
(x), c
β
(x)) is given by
L
eff
({φ
a
(x)}, c
α
(x), c
β
(x)) = L
tot
({φ
a
(x)};(10.11.26)) −
1
2
F
2
α
({φ
a
(x)})
+
c
α
(x)M
F
({φ
a
(x)})
α,β
c
β
(x). (10.11.49c)
R
ξ
-gauge: In order to eliminate the mixed term ∂
µ
˜η(x)i(θ
α
˜v)A
µ
α
(x)inthequadratic
part of the Lagrangian density L
quad
tot
in (10.11.27), we choose
F
α
({φ
a
(x)}) =
9
ξ
∂
µ
A
µ
α
(x) −
1
ξ
˜η(x)i
(
θ
α
˜v
)
, ξ>0. (10.11.50)
We have the exponent of the quasi-Gaussian functional as
1
2
F
2
α
{φ
a
(x)}
=
ξ
2
∂
µ
A
µ
α
(x)
2
−˜η(x)i
(
θ
α
˜v
)
∂
µ
A
µ
α
(x) −
1
2ξ
1
(
θ
α
˜v
)
˜η(x)
2
2
(10.11.51)
464 10 Calculus of Variations: Applications
so that the mixed terms add up to the four divergence,
L
quad
tot
−
1
2
F
2
α
({φ
a
(x)})
mixed
= ∂
µ
˜η(x)i(θ
α
˜v)A
µ
α
(x) +˜η(x)i(θ
α
˜v)∂
µ
A
µ
α
(x)
= ∂
µ
/
˜η(x)i(θ
α
˜v)A
µ
α
(x)
0
. (10.11.52)
Namely, in the R
ξ
-gauge, the mixed term does not contribute to the effective
action functional I
eff
[{φ
a
}, c
α
, c
β
]. We obtain the Faddeev–Popov determinant in
the R
ξ
-gauge from the transformation laws of A
αµ
(x)and ˜η(x).
δA
αµ
(x) =−
D
adj
µ
ε(x)
α
, α = 1, ..., N; δ ˜η(x) = iε
α
(x)
1
θ
α
( ˜v +˜η(x))
2
.
(10.11.53)
{M
F
({φ
a
})}
αx,βy
≡
δF
α
1
φ
g
a
(x)
2
δε
β
(y)
g=1
≡ δ
4
(x − y)M
F
({φ
a
(x)})
α,β
= δ
4
(x − y)
9
ξ
−∂
µ
D
adj
µ
α,β
−
1
ξ
µ
2
α,β
−
1
ξ
˜v, θ
α
θ
β
˜η(x)
.
(10.11.54)
We absorb
√
ξ in the normalization of the Faddeev–Popov ghost fields
{c
α
(x), c
β
(x)}.
We then have the Faddeev–Popov ghost Lagrangian density as
L
ghost
(c
α
(x), c
β
(x)) ≡ c
α
(x)M
F
({φ
a
(x)})
α,β
c
β
(x)
= ∂
µ
c
α
(x)∂
µ
c
α
(x) −
1
ξ
µ
2
α,β
c
α
(x)c
β
(x)
+ C
γαβ
∂
µ
c
α
(x)A
µ
γ
(x)c
β
(x) −
1
ξ
c
α
(x)( ˜v, θ
α
θ
β
˜η(x))c
β
(x).
(10.11.55)
From L
tot
((10.11.26)), the gauge-fixing term, −
1
2
F
2
α
({φ
a
(x)}), (10.11.51), and
L
ghost
((10.11.55)), we have the effective Lagrangian density L
eff
((10.11.49c)) as
L
eff
((10.11.49c)) = L
tot
((10.11.26)) −
1
2
F
2
α
({φ
a
(x)}) +L
ghost
((10.11.55))
= L
quad
eff
+ L
int
eff
. (10.11.56)
We have L
quad
eff
as
10.11 Path Integral Quantization of Gauge Field II 465
L
quad
eff
=−
1
2
A
αµ
(x)
)
δ
αβ
1
−η
µν
∂
2
+ (1 −ξ )∂
µ
∂
ν
2
− µ
2
α,β
η
µν
*
A
βν
(x)
+
1
2
η
i
(x)
−δ
ij
∂
2
−
M
2
ij
+
1
ξ
(
θ
α
˜v
)
i
(
θ
α
˜v
)
j
η
j
(x)
+
c
α
(x)
−δ
αβ
∂
2
−
1
ξ
µ
2
α,β
c
β
(x). (10.11.57)
We have L
int
eff
as
L
int
eff
= L
eff
− L
quad
eff
=
1
2
C
αβγ
A
βµ
(x)A
γν
(x)
∂
µ
A
ν
α
(x) −∂
ν
A
µ
α
(x)
−
1
4
C
αβγ
C
αδε
A
βµ
(x)A
γν
(x)A
µ
δ
(x)A
ν
ε
(x) + ∂
µ
˜η(x)i
θ
α
˜η(x)
A
µ
α
(x)
+˜η(x)
θ
α
θ
β
˜v
A
αµ
(x)A
µ
α
(x) −
1
3!
f
ijk
η
i
(x)η
j
(x)η
k
(x)
+
1
2
˜η(x)
θ
α
θ
β
˜η(x)
A
αµ
(x)A
µ
β
(x) −
1
4!
f
ijkl
η
i
(x)η
j
(x)η
k
(x)η
l
(x)
+C
αβγ
∂
µ
c
α
(x)c
β
(x)A
µ
γ
(x) −
1
ξ
c
α
(x)
˜v, θ
α
θ
β
˜η(x)
c
β
(x). (10.11.58)
From L
quad
eff
((10.11.57)), we have the equations satisfied by the ‘‘free’’ Green’s
functions
/
D
(A
)
αµ,βν
(x − y), D
(η)
i,j
(x − y), D
(C)
α,β
(x − y)
0
,
of the gauge fields, Higgs scalar fields, and the Faddeev–Popov ghost fields,
{A
αµ
(x);η
i
(x);c
α
(x);c
β
(x)},as
−
)
δ
αβ
1
−η
µν
∂
2
+∂
µ
∂
ν
− ξ∂
µ
∂
ν
2
−µ
2
α,β
η
µν
*
D
(A
)
βν,γλ
(x − y)
= δ
αγ
η
µ
λ
δ
4
(x − y), (10.11.59a)
−δ
ij
∂
2
−
M
2
ij
+
1
ξ
(
θ
α
˜v
)
i
(
θ
α
˜v
)
j
D
(η)
j,k
(x − y) = δ
ik
δ
4
(x − y), (10.11.59b)
−δ
αβ
∂
2
−
1
ξ
µ
2
α,β
D
(C)
β,γ
(x − y) = δ
αγ
δ
4
(x − y). (10.11.59c)
We Fourier-transform the ‘‘free’’ Green’s functions as
D
(A
)
αµ,βν
(x − y)
D
(η)
i,j
(x − y)
D
(C)
α,β
(x − y)
=
d
4
k
(2π)
4
exp
)
−ik(x − y)
*
D
(A
)
αµ,βν
(k)
D
(η)
i,j
(k)
D
(C)
α,β
(k)
. (10.11.60)
466 10 Calculus of Variations: Applications
We have the momentum space ‘‘free’’ Green’s functions satisfying
−
δ
αβ
k
2
η
µν
−
k
µ
k
ν
k
2
+ ξ
k
µ
k
ν
k
2
− µ
2
α,β
η
µν
D
(A
)
βν,γλ
(k) = δ
αγ
η
µ
λ
,
(10.11.61a)
δ
ij
k
2
−
M
2
ij
+
1
ξ
(
θ
α
˜v
)
i
(
θ
α
˜v
)
j
D
(η)
j,k
(k) = δ
ik
, (10.11.61b)
δ
αβ
k
2
−
1
ξ
µ
2
α,β
D
(C)
β,γ
(k) = δ
αγ
. (10.11.61c)
We obtain the momentum space ‘‘free’’ Green’s functions as
D
(A
)
αµ,βν
(k) =−
η
µν
−
k
µ
k
ν
k
2
1
k
2
−µ
2
α,β
−
1
ξ
k
µ
k
ν
k
2
1
k
2
− µ
2
ξ
α,β
=−η
µν
1
k
2
− µ
2
α,β
−
1
ξ
−1
k
µ
k
ν
1
k
2
−µ
2
·
1
k
2
−µ
2
ξ
α,β
,
(10.11.62a)
D
(η)
i,j
(k) = (1 −P)
i,k
1
k
2
− M
2
k,j
−
(
θ
α
˜v
)
i
1
µ
2
·
1
k
2
− µ
2
ξ
α,β
θ
β
˜v
j
=
1
k
2
− M
2
i,j
−
(
θ
α
˜v
)
i
1
ξ
·
1
k
2
·
1
k
2
−µ
2
ξ
α,β
θ
β
˜v
j
,
(10.11.62b)
D
(C)
α,β
(k) =
1
k
2
− µ
2
ξ
α,β
. (10.11.62c)
Here P
i,j
is the projection operator onto the (N − M)-dimensional subspace spanned
by Nambu–Goldstone boson.
P
i,j
=
θ
α
,θ
β
/∈S
(
θ
α
˜v
)
i
1
µ
2
α,β
θ
β
˜v
†
j
, i, j = 1, ..., n, (10.11.63a)
P
i,j
θ
γ
˜v
j
=
θ
γ
˜v
i
,
θ
γ
˜v
†
i
P
i,j
=
θ
γ
˜v
†
j
, θ
γ
/∈ S . (10.11.63b)
The R
ξ
-gauge is not only the gauge that eliminates the mixed term of A
αµ
(x)and
˜η(x) in the quadratic part of the effective Lagrangian density, but also the gauge
that connects the Unitarity gauge (ξ = 0), the
t Hooft–Feynman gauge (ξ = 1)
and the Landau gauge (ξ →∞) continuously in ξ .
(1) Unitarity gauge (ξ = 0):
D
(A
)
αµ,βν
(k) =−
η
µν
−
k
µ
k
ν
µ
2
·
1
k
2
− µ
2
α,β
, (10.11.64a)
D
(η)
i,j
(k) =
1
k
2
− M
2
i,j
+
1
k
2
(
θ
α
˜v
)
i
·
1
µ
2
αβ
·
θ
β
˜v
j
, (10.11.64b)
D
(C)
α,β
(k) ∝ ξ −→ 0, infinitely massive ghost. (10.11.64c)
10.11 Path Integral Quantization of Gauge Field II 467
(2)
t Hooft–Feynman gauge (ξ = 1):
D
(A
)
αµ,βν
(k) =−η
µν
1
k
2
− µ
2
α,β
, (10.11.65a)
D
(η)
i,j
(k) =
1
k
2
− M
2
i,j
−
(
θ
α
˜v
)
i
1
k
2
1
k
2
− µ
2
αβ
θ
β
˜v
j
, (10.11.65b)
D
(C)
α,β
(k) =
1
k
2
− µ
2
α,β
, massive ghost at k
2
= µ
2
. (10.11.65c)
(3) Landau gauge (ξ →∞):
D
(A
)
αµ,βν
(k) =−
η
µν
−
k
µ
k
ν
k
2
1
k
2
−µ
2
α,β
, (10.11.66a)
D
(η)
i,j
(k) =
1
k
2
− M
2
i,j
, (10.11.66b)
D
(C)
α,β
(k) =
δ
αβ
k
2
, massless ghost at k
2
= 0. (10.11.66c)
We have the generating functional of (the connected part of) the ‘‘full’’ Green’s
functions as
Z
F
)
{J
a
}, ζ
α
, ζ
β
*
≡ exp
)
iW
F
)
{J
a
}, ζ
α
, ζ
β
**
≡
0, out
T
exp
i
d
4
z{J
a
(z)
ˆ
φ
a
(z)
+
c
α
(z)ζ
α
(z) +ζ
β
(z)c
β
(z)}
0, in
=
D[φ
a
]D[c
α
]D[c
β
]exp
i
d
4
z
1
L
eff
{φ
a
(z)}, c
α
(z), c
β
(z)
+ J
a
(z)φ
a
(z) +c
α
(z)ζ
α
(z) + ζ
β
c
β
(z)
2
= exp
iI
int
eff
1
i
δ
δJ
a
, i
δ
δζ
α
,
1
i
δ
δζ
β
Z
F,0
)
{J
a
}, ζ
α
, ζ
β
*
.
(10.11.67)
Z
F,0
[{J
a
}, ζ
α
, ζ
β
]andI
int
eff
[{φ
a
}, c
α
, c
β
]are,respectively,givenby
Z
F,0
)
{J
a
}, ζ
α
, ζ
β
*
=
D[φ
a
]D[c
α
]D[c
β
]exp
i
d
4
z
/
L
quad
eff
((10.11.57))
+ J
a
(z)φ
a
(z) + c
α
(z)ζ
α
(z) +ζ
β
(z)c
β
(z)