64
4 Four-Dimensional Superfield Supersymmetry
Let us shortly describe the prescriptions for the introduction of Green functions in
a usual field theory (see e.g. [36]). We start with the classical action S[φ] representing
itself as a local space-time functional of n (super)fields φ
i (i = 1, . . . , n) forming an
(iso)vector
φ. Let
φ 0 = {φ
i
0 } be some background. Our action S[φ] is assumed to be
an analytic functional, i.e. it can be expanded into power series in a neighborhood of
the
φ 0 :
S[φ] = S[
φ 0 ] +
∞
n=2
1
n!
S i 1 ...i n (φ − φ 0 )
i n . . . (φ − φ 0 )
i 1 .
(4.60)
The term with n = 2 is called a quadratic (or linearized) action:
S 0 =
1
2
˜
φ
i S i j [
φ 0 ] ˜
φ
j
.
(4.61)
Here and further ˜
φ
i
= φ
i
− φ
i
0 . Terms with n ≥ 3 are called interaction terms S int ,
so, the whole action is rewritten as
S[φ] = S[
φ 0 ] + S 0 [ ˜
φ
i
;
φ 0 ] + S int [ ˜
φ
i
;
φ 0 ].
(4.62)
The Green function G
i j is determined on the base of the linearized action as
S i j [
φ 0 ]G
jk
= δ
k
i ; G
i j S jk [
φ 0 ] = δ
i
k .
(4.63)
We assume that the generating functional of Green functions is, as usual, defined in
the form
Z [J ] = N
Dφ exp(
i
(S[φ] + J φ)).
(4.64)
Here J is an essentially classical source, and N is a normalization factor.
The propagators can be obtained on the base of the generating functional as
φ(z 1 ) . . . φ(z n ) =
1
i
δ
δ J (z 1 )
. . .
1
i
δ
δ J (z n )
N
Dφ exp(
i
(S[φ] + J φ)).
(4.65)
We can calculate the path integral (4.64). To do it, after relabelling ˜
φ → φ in (4.62),
we expand the complete action as S[φ] = S 0 [φ] + S int [φ] with S 0 [φ] =
1
2
dzφφ
is a quadratic action, and be some operator. Of course, the path integration is a quite
formal operation being well-defined only for the Gaussian integral and expressions
derived from it. However, this is not a problem since both in the standard and in the
superfield cases we need only Gaussian integrals. As usual,
4 Four-Dimensional Superfield Supersymmetry
Let us shortly describe the prescriptions for the introduction of Green functions in
a usual field theory (see e.g. [36]). We start with the classical action S[φ] representing
itself as a local space-time functional of n (super)fields φ
i (i = 1, . . . , n) forming an
(iso)vector
φ. Let
φ 0 = {φ
i
0 } be some background. Our action S[φ] is assumed to be
an analytic functional, i.e. it can be expanded into power series in a neighborhood of
the
φ 0 :
S[φ] = S[
φ 0 ] +
∞
n=2
1
n!
S i 1 ...i n (φ − φ 0 )
i n . . . (φ − φ 0 )
i 1 .
(4.60)
The term with n = 2 is called a quadratic (or linearized) action:
S 0 =
1
2
˜
φ
i S i j [
φ 0 ] ˜
φ
j
.
(4.61)
Here and further ˜
φ
i
= φ
i
− φ
i
0 . Terms with n ≥ 3 are called interaction terms S int ,
so, the whole action is rewritten as
S[φ] = S[
φ 0 ] + S 0 [ ˜
φ
i
;
φ 0 ] + S int [ ˜
φ
i
;
φ 0 ].
(4.62)
The Green function G
i j is determined on the base of the linearized action as
S i j [
φ 0 ]G
jk
= δ
k
i ; G
i j S jk [
φ 0 ] = δ
i
k .
(4.63)
We assume that the generating functional of Green functions is, as usual, defined in
the form
Z [J ] = N
Dφ exp(
i
(S[φ] + J φ)).
(4.64)
Here J is an essentially classical source, and N is a normalization factor.
The propagators can be obtained on the base of the generating functional as
φ(z 1 ) . . . φ(z n ) =
1
i
δ
δ J (z 1 )
. . .
1
i
δ
δ J (z n )
N
Dφ exp(
i
(S[φ] + J φ)).
(4.65)
We can calculate the path integral (4.64). To do it, after relabelling ˜
φ → φ in (4.62),
we expand the complete action as S[φ] = S 0 [φ] + S int [φ] with S 0 [φ] =
1
2
dzφφ
is a quadratic action, and be some operator. Of course, the path integration is a quite
formal operation being well-defined only for the Gaussian integral and expressions
derived from it. However, this is not a problem since both in the standard and in the
superfield cases we need only Gaussian integrals. As usual,