4.4 Feynman Supergraphs
71
The propagators in momentum space look like
φ(1) ¯
φ(2) =
i
k 2 + m 2 δ 12 ;
(4.91)
φ(1)φ(2) =
im D
2
4k 2 (k 2 + m 2 )
δ 12 ;
V (1)V (2) = −
i
k 2 δ 12 .
(4.92)
Here 1, 2 are numbers of arguments (actually we must write V (1) ≡ V (−k, θ 1 ) and
V (2) ≡ V (k, θ 2 ) etc.), and δ 12 is the purely Grassmannian delta function defined
earlier. The D-factors are introduced as above. Note, however, that spinor derivatives depend after Fourier transform on a momentum of a propagator with which they
are associated. The external superfields also can be represented in the form of the
Fourier integral. Each propagator is parametrized by a momentum, and any vertex
corresponds to an integration over d
4
θ, to multiplication by a corresponding coupling
and a delta function over incoming momenta multiplied by (2π)
4 . As usual, contribution of any supergraph includes integration over all momenta and a combinatoric
factor which is defined by the same rules as in standard quantum field theory.
An essentially new feature of superfield theories consists in the presence of the Dfactors. To evaluate D-algebra we can transport them from one propagator to another
via integration by parts, then, we make use of the identity (4.17) in appropriate
situations.
We can prove the following non-renormalization theorem.
The final result for the contribution of any supergraph should have the form of
one integral over d
4
θ [43].
Proof Let us consider the supergraph with L loops, V vertices and P propagators.
Any vertex contains an integration over d
4
θ, i.e. there are V such integrations. Then,
due to (4.91) any propagator carries a delta function over Grassmannian coordinates,
i.e. there are P delta functions. Then, in any loop we can reduce the number of delta
functions by one using Eq. (4.17), i.e. there are P − L independent delta functions.
As a result we can carry out P − L integrations from V ones corresponding to
vertices, and after D-algebra transformations we stay with V − (P − L) integrations.
And, due to the famous topological identity, V − (P − L) = 1, therefore the result
contains one integration over d
4
θ. The theorem is proved.
This theorem means that all quantum corrections are local in θ-space. It is frequently treated naively as a proof of absence of chiral corrections which are proportional to an integral over d
2
θ. However, such an interpretation is wrong since any
contribution in the form of an integral over the chiral subspace can be rewritten as
an integral over the total superspace using the identity
d
6 z f (() =
d
8 z(−
D
2
4
) f (().
(4.93)
71
The propagators in momentum space look like
φ(1) ¯
φ(2) =
i
k 2 + m 2 δ 12 ;
(4.91)
φ(1)φ(2) =
im D
2
4k 2 (k 2 + m 2 )
δ 12 ;
V (1)V (2) = −
i
k 2 δ 12 .
(4.92)
Here 1, 2 are numbers of arguments (actually we must write V (1) ≡ V (−k, θ 1 ) and
V (2) ≡ V (k, θ 2 ) etc.), and δ 12 is the purely Grassmannian delta function defined
earlier. The D-factors are introduced as above. Note, however, that spinor derivatives depend after Fourier transform on a momentum of a propagator with which they
are associated. The external superfields also can be represented in the form of the
Fourier integral. Each propagator is parametrized by a momentum, and any vertex
corresponds to an integration over d
4
θ, to multiplication by a corresponding coupling
and a delta function over incoming momenta multiplied by (2π)
4 . As usual, contribution of any supergraph includes integration over all momenta and a combinatoric
factor which is defined by the same rules as in standard quantum field theory.
An essentially new feature of superfield theories consists in the presence of the Dfactors. To evaluate D-algebra we can transport them from one propagator to another
via integration by parts, then, we make use of the identity (4.17) in appropriate
situations.
We can prove the following non-renormalization theorem.
The final result for the contribution of any supergraph should have the form of
one integral over d
4
θ [43].
Proof Let us consider the supergraph with L loops, V vertices and P propagators.
Any vertex contains an integration over d
4
θ, i.e. there are V such integrations. Then,
due to (4.91) any propagator carries a delta function over Grassmannian coordinates,
i.e. there are P delta functions. Then, in any loop we can reduce the number of delta
functions by one using Eq. (4.17), i.e. there are P − L independent delta functions.
As a result we can carry out P − L integrations from V ones corresponding to
vertices, and after D-algebra transformations we stay with V − (P − L) integrations.
And, due to the famous topological identity, V − (P − L) = 1, therefore the result
contains one integration over d
4
θ. The theorem is proved.
This theorem means that all quantum corrections are local in θ-space. It is frequently treated naively as a proof of absence of chiral corrections which are proportional to an integral over d
2
θ. However, such an interpretation is wrong since any
contribution in the form of an integral over the chiral subspace can be rewritten as
an integral over the total superspace using the identity
d
6 z f (() =
d
8 z(−
D
2
4
) f (().
(4.93)