4.3 Generating Functional and Green Functions for Superfields
65
Dφ exp(
i
(S[φ] + J φ)) =
Dφ exp(
i
(
1
2
φφ + S int [φ] + J φ)) =
= exp(
i
S int (
i
δ
δ J
))
Dφ exp(
i
(
1
2
φφ + J φ)).
(4.66)
Since this integral is Gaussian-like, we immediately obtain
Dφ exp(
i
(
1
2
φφ + J φ)) = exp(−
i
2
J (
)J )det
−1/2
(
).
(4.67)
In these expressions, the integration over the space-time is assumed where it is
necessary. One concludes that all dependence of this expression on the sources is
concentrated in the term exp(−
i
2
J (
)J ). Constructing of Feynman diagrams from
Eqs. (4.65), (4.66), (4.67) is quite straightforward.
Let us apply this approach to a superfield theory. Our example is the Wess-Zumino
model [32], and a consideration of other theories is exactly analogous. We do not
address here the specifics of gauge theories in which one must introduce gauge fixing
and ghosts, since afterwards all proceeding is just the same.
The action of the Wess-Zumino model with chiral sources is
S J [, ¯
; J, ¯
J ] =
d
8 z ¯
+ (
d
6 z(
λ
3!
3
+
m
2
2
+ J ) + h.c.). (4.68)
As usual, conjugated terms to chiral superfields are antichiral ones. This action can
be rewritten in terms of integrals over chiral and antichiral subspaces only:
S J [, ¯
; J, ¯
J ] =
d
6 z(
1
2
(−
¯
D
2
4
) ¯
+
λ
3!
3
+
m
2
2
+ J ) + h.c. (4.69)
The generating functional is
Z [J, ¯
J ] =
DD ¯
exp(i S J [, ¯
; J, ¯
J ]).
(4.70)
It is convenient to follow the approach developed in [33]. Using it, the action S J
(4.69) can be represented in matrix form
S J =
1
2
dz 1 dz 2
(z 1 ) ¯
(z 1 )
m − 1
4
¯
D 2
− 1
4 D 2 m
δ + (z 1 − z 2 )
0
0
δ − (z 1 − z 2 )
×
×
(z 2 )
¯
(z 2 )
+
d 6 z(z)J (z) +
d 6 ¯
z ¯
(¯ z) ¯
J (¯ z) +
λ
3!
(
d 6 z 3 + h.c.). (4.71)
In this expression, integration in all terms is assumed with taking into account the corresponding chirality, e.g.
dz 1 dz 2 (z 1 )mδ + (z 1 − z 2 ))(z 2 ) ≡
d
6 z 1 d
6 z 2 (z 1 )
mδ + (z 1 − z 2 ))(z 2 ), etc. We see that the operator determining the quadratic part
of the action (see (4.66), (4.67)) looks like
65
Dφ exp(
i
(S[φ] + J φ)) =
Dφ exp(
i
(
1
2
φφ + S int [φ] + J φ)) =
= exp(
i
S int (
i
δ
δ J
))
Dφ exp(
i
(
1
2
φφ + J φ)).
(4.66)
Since this integral is Gaussian-like, we immediately obtain
Dφ exp(
i
(
1
2
φφ + J φ)) = exp(−
i
2
J (
)J )det
−1/2
(
).
(4.67)
In these expressions, the integration over the space-time is assumed where it is
necessary. One concludes that all dependence of this expression on the sources is
concentrated in the term exp(−
i
2
J (
)J ). Constructing of Feynman diagrams from
Eqs. (4.65), (4.66), (4.67) is quite straightforward.
Let us apply this approach to a superfield theory. Our example is the Wess-Zumino
model [32], and a consideration of other theories is exactly analogous. We do not
address here the specifics of gauge theories in which one must introduce gauge fixing
and ghosts, since afterwards all proceeding is just the same.
The action of the Wess-Zumino model with chiral sources is
S J [, ¯
; J, ¯
J ] =
d
8 z ¯
+ (
d
6 z(
λ
3!
3
+
m
2
2
+ J ) + h.c.). (4.68)
As usual, conjugated terms to chiral superfields are antichiral ones. This action can
be rewritten in terms of integrals over chiral and antichiral subspaces only:
S J [, ¯
; J, ¯
J ] =
d
6 z(
1
2
(−
¯
D
2
4
) ¯
+
λ
3!
3
+
m
2
2
+ J ) + h.c. (4.69)
The generating functional is
Z [J, ¯
J ] =
DD ¯
exp(i S J [, ¯
; J, ¯
J ]).
(4.70)
It is convenient to follow the approach developed in [33]. Using it, the action S J
(4.69) can be represented in matrix form
S J =
1
2
dz 1 dz 2
(z 1 ) ¯
(z 1 )
m − 1
4
¯
D 2
− 1
4 D 2 m
δ + (z 1 − z 2 )
0
0
δ − (z 1 − z 2 )
×
×
(z 2 )
¯
(z 2 )
+
d 6 z(z)J (z) +
d 6 ¯
z ¯
(¯ z) ¯
J (¯ z) +
λ
3!
(
d 6 z 3 + h.c.). (4.71)
In this expression, integration in all terms is assumed with taking into account the corresponding chirality, e.g.
dz 1 dz 2 (z 1 )mδ + (z 1 − z 2 ))(z 2 ) ≡
d
6 z 1 d
6 z 2 (z 1 )
mδ + (z 1 − z 2 ))(z 2 ), etc. We see that the operator determining the quadratic part
of the action (see (4.66), (4.67)) looks like