124
4 Four-Dimensional Superfield Supersymmetry
where
S G F =
1
16ξ
d
8 ztr( ¯
D
2 V )(D
2 V )
(4.289)
is the gauge-fixing action [33].
We introduce the total gauge-ghost action
S total = S SY M + S G F + S G H
(4.290)
and the corresponding generating functional
Z [J, {η}] =
DV D{c} exp(i(S total +
d
8 ztr J V +
d
6 ztr(η
c
+ ηc) +
+
d
6
¯
z( ¯
η
¯
c
+ ¯
η ¯
c))).
(4.291)
Here {η} is the set of all sources: η, η
, ¯
η, ¯
η
. To develop a diagram technique we
must split the action (4.290) into a sum of the free (quadratic) part and vertices. It is
easy to see [2, 33] that
e
−gV D α e
gV
= g D α V −
1
2
g
2
[V, D α V ] +
1
6
g
3
[V, [V, D α V ]] + . . . . (4.292)
Therefore (4.274) looks like
S SY M =
d 8 z tr
1
16
V D α ¯
D 2 D α V +
1
16
g( ¯
D 2 D α V )[V, D α V ] −
−
1
64
g 2 [V, D α V ] ¯
D 2 [V, D α V ] −
1
48
g 2 ( ¯
D 2 D α V )[V, [V, D α V ]] + . . .
. (4.293)
And the ghost action is
S G H =
d 8 z tr
¯
c c − ¯
cc +
1
2
g( ¯
c − c )[V, c + ¯
c] +
g 2
12
(c − ¯
c )[V, [V, c − ¯
c]]
+
+ . . . .
(4.294)
These expressions are sufficient for one- and two-loop calculations.
The quadratic action, with the added gauge-fixing term, is
S 0 = −
1
2
d
8 z tr V
−
1
8
D
α ¯
D
2 D α +
1
16ξ
{D
2
, ¯
D
2
}
V +
d
8 ztr( ¯
c
c − ¯
cc
).
(4.295)