160
The one
SU
(2) instanton solution centered at
z
μ
can be written explicitly in com
ponents as
A
cl
aμ
=
2
g
η
aμν
(
x

z
)
ν
(
x

z
)
2
+
R
2
,
(10.55)
where the symbol
η
aμν
is antisymmetric in
μ, ν
, defined as
η
aij
=
aij
,
i, j
= 1
,
2
,
3
,
η
ai
4
=

η
a
4
i
=
δ
ai
.
(10.56)
Note that the one antiinstanton solution would be obtained by replacing
η
aμν
with
¯
η
aμν
, defined by ¯
η
aij
=
η
aij
, ¯
η
ai
4
=

η
ai
4
.
The Euclidean action
S
E
is related to the Lorentzian one by
S
=
iS
E
.
The one
instanton solution has Euclidean action
S
cl
E
= 8
π
2
/g
2
. In background field gauge, the
action for the quantum fluctuations is
S
E
=
S
cl
E
+
Z
d
4
x
1
2
(
D
μ
A
0
aν
)
2
+
g
abc
F
μν
a
A
0
bμ
A
0
cν
+ (
D
μ
Φ)
*
(
D
μ
Φ) +
¯
ψγ
μ
D
μ
ψ
+ (
D
μ
¯
η
a
)(
D
μ
η
a
) +
· · ·
(10.57)
where we have exhibited only quadratic terms in the quantum fluctuations, which is
all we need to the oneloop computation. Note that the term proportional to (
D
μ
A
0
μ
)
2
from expanding the YangMills Lagrangian is canceled by the gauge fixing term. As
before, we will write
M
A
,
M
Φ
,
M
ψ
and
M
η
for the kinetic operators on
A
0
, Φ,
ψ
, and
(
η,
¯
η
) respectively. The result of the oneloop Gaussian integral is then
e

8
π
2
/g
2
Z
d
(zero modes)
det
0
M
ψ
det
0
M
η
p
det
0
M
A
det
0
M
Φ
,
(10.58)
where we have separated out the integration over zero modes of
A
0
,
Φ
, ψ, η
in the
instanton background: these modes are annihilated by the respective kinetic operator
M
; det
0
is the determinant taken over modes of nonzero eigenvalues only, defined using
a suitable (gauge invariant) regulator.
Explicitly, the kinetic operators are
M
A
A
0
aμ
=

D
2
A
0
aμ

2
g
abc
F
bμν
A
0
cν
,
M
ψ
=
γ
μ
D
μ
,
M
2
ψ
=
D
2

i
2
γ
μν
F
aμν
T
a
,
M
Φ
=
M
η
=

D
2
.
(10.59)
Here
T
a
are the generators of the
SU
(2) gauge group. Let us first calculate
D
2
in the
161
background of an instanton with
z
= 0 and
R
= 1,
D
2
= (
∂
μ

igT
a
A
cl
aμ
)(
∂
μ

igT
b
A
cl,μ
b
)
=

2
iT
a
η
aμν
{
x
ν
r
2
+ 1
, ∂
μ
} 
4
T
a
T
b
η
aμν
η
b
μ
ρ
x
ν
x
ρ
(
r
2
+ 1)
2
=
+
4
r
2
+ 1
T
a
η
aμν
(

ix
ν
∂
μ
)

4
r
2
(
r
2
+ 1)
2
T
2
,
(10.60)
where
r
2
=
x
μ
x
μ
,
T
2
=
T
a
T
a
. Note that the
SO
(4) rotation generator

ix
[
μ
∂
ν
]
can be
split into the
SU
(2)
×
SU
(2) generators
L
a
1
=

i
2
η
aμν
x
μ
∂
ν
,
L
a
2
=

i
2
η
aμν
x
μ
∂
ν
.
(10.61)
They obey
L
a
1
L
a
1
=
L
a
2
L
a
2
≡
L
2
=

1
8
(
x
μ
∂
ν

x
ν
∂
μ
)
2
.
(10.62)
Writing the Laplacian
in terms of
L
2
also, we have
D
2
=
1
r
3
∂
r
r
3
∂
∂r

4
L
2
r
2

8
r
2
+ 1
T
a
L
a
1

4
r
2
(
r
2
+ 1)
2
T
2
.
(10.63)
This expression can be generalized to the kinetic operator acting on fields of nonzero
spin,
M
=

1
r
3
∂
r
r
3
∂
∂r
+
4
L
2
r
2
+
8
r
2
+ 1
T
a
L
a
1
+
4
r
2
(
r
2
+ 1)
2
T
2
+
16
(
r
2
+ 1)
2
T
a
S
a
1
,
(10.64)
where
S
a
1
together with
S
a
2
are the
SU
(2)
×
SU
(2) generators associated with the
intrinsic spin of the field. The fermion
ψ
transforms in the representation (
1
2
,
0)
⊕
(0
,
1
2
)
of the
SU
(2)
×
SU
(2)
∼
SO
(4) rotation (Euclidean version of Lorentz) group, whereas
A
0
μ
transforms in the representation (
1
2
,
1
2
). Explicitly, acting on the spinor,
S
1
and
S
2
are
S
a
1
=

i
8
η
aμν
γ
μν
,
S
a
2
=
i
8
η
aμν
γ
μν
,
S
2
1
=
3
4
1

γ
5
2
,
S
2
2
=
3
4
1 +
γ
5
2
.
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 Summer '19
 Physics, Quantum Field Theory, φ, Lorentz, soft photon