92
4 Four-Dimensional Superfield Supersymmetry
One should note that since D
2 v and ¯
D
2 v are not real superfields, we must impose
two gauge-fixing conditions, and (4.162) takes the form
e
i W v =
Dve
−
i
16
d
8 zvD
α ¯
D
2 D α v
δ(
1
4
D
2 v − ¯
φ)δ(
1
4
¯
D
2 v − φ)det F P , (4.163)
where
F P =
−
1
4
¯
D
2
0
0 −
1
4
D
2
is a Faddeev-Popov matrix. We note that W v is constant by constructing. We multiply
the left-hand and right-hand sides of (4.160) and (4.163) respectively, as a result we
arrive at
e i (1) +W v =
DφD ¯
φDv exp
i
2
φ ¯
φ
− 1
4
¯
D 2
− 1
4 D 2
¯
φ
¯
φ
−
i
16
vD α ¯
D 2 D α v
×
× δ(
1
4
D 2 v − ¯
φ)δ(
1
4
¯
D 2 v − φ)det F P .
(4.164)
Integrating over φ, ¯
φ with use of delta functions, we obtain
e
i
(1) +W v =
DφD ¯
φ exp
i
2
d
8 zv( −
1
4
¯
D
2
−
1
4
¯
D
2
)v
det F P .
(4.165)
However, W v and det F P are field-independent constants which are irrelevant for
our purposes and can be omitted. We also took into account that
1
16
{D
2
, ¯
D
2
} −
1
8
D
α ¯
D
2 D α = , hence the one-loop effective action is equal to
(1)
=
i
2
Tr log( −
1
4
¯
D
2
−
1
4
¯
D
2
)
(4.166)
Here as usual = m + λ, ¯
= m + λ ¯
. This one-loop effective action can be
expressed in the form of the Schwinger expansion:
(1)
=
i
2
Tr
∞
0
ds
s
exp(is( −
1
4
¯
D
2
−
1
4
¯
D
2
)),
(4.167)
or, after manifest writing the trace,
(1) =
i
2
d 8 z 1 d 8 z 2
∞
0
ds
s
δ 8 (z 1 − z 2 ) exp(is(−
1
4
¯
D 2 −
1
4
¯
D 2 ))e is δ 8 (z 1 − z 2 ).
(4.168)