4.3 Generating Functional and Green Functions for Superfields
67
We can introduce two-point free propagators:
G ++ (z 1 , z 2 ) =
1
i 2
δ
2 Z 0 [J ]
δ J (z 1 )δ J (z 2 )
| J =0 = i(−
1
4
)
2 ¯
D
2
1
¯
D
2
2 K ++ (z 1 , z 2 )
G +− (z 1 , z 2 ) =
1
i 2
δ
2 Z 0 [J ]
δ J (z 1 )δ ¯
J (z 2 )
| J =0 = i(−
1
4
)
2 ¯
D
2
1 D
2
2 K +− (z 1 , z 2 )
G −− (z 1 , z 2 ) =
1
i 2
δ
2 Z 0 [J ]
δ ¯
J (z 1 )δ ¯
J (z 2 )
| J =0 = i(−
1
4
)
2 D
2
1 D
2
2 K −− (z 1 , z 2 ). (4.78)
Here K +− (z 1 , z 2 ) = K −+ (z 1 , z 2 )= −
1
−m 2 δ
8
(z 1 − z 2 ), K ++ (z 1 , z 2 ) =
m D
2
4(−m 2 )
δ
8
(z 1 − z 2 ), K −− (z 1 , z 2 ) =
m ¯
D
2
4(−m 2 )
δ
8
(z 1 − z 2 ).
There is an alternative way to obtain the Green functions [43]. Indeed, the
quadratic part of the action (4.69) can be rewritten as an integral over whole superspace:
S J [, ¯
; J, ¯
J ] =
d
8 z
¯
+
m
2
(−
D
2
4
)) +
m
2
¯
(−
¯
D
2
4
) ¯
+ (−
D
2
4
)J + ¯
(−
¯
D
2
4
) ¯
J
,
(4.79)
which has the matrix form
S J [, ¯
; J, ¯
J ] =
1
2
d
8 z
(z) ¯
(z)
−m
D
2
4
1
1 −m
¯
D
2
4
(z)
¯
(z)
+
+
d
8 z
(z)(−
D
2
4
)J (z) + ¯
(¯ z)(−
¯
D
2
4
) ¯
J (¯ z)
.
(4.80)
Then, since
−m
D
2
4
1
1 −m
¯
D
2
4
−1
=
1
− m 2
m ¯
D
2
4
m D
2
4
,
(4.81)
after the functional integration, we arrive at the expression similar to (4.76), that is,
Z [J, ¯
J ] = exp
i
λ
3!
d
6 z
1
i
δ
δ J (z)
3 + h.c.
(det
−1/2 ˜
) exp(−
i
2
˜
A[J, ¯
J ]),
(4.82)
where ˜
=
−m
D
2
4
1
1 −m
¯
D
2
4
, and the argument of the exponential looks like