4.4 Feynman Supergraphs
75
We transport all spinor supercovariant derivatives to one propagator (here the terms
with spinor supercovariant derivatives moved to the external lines are omitted as the
irrelevant ones since they do not contribute to the divergent part [63]). As a result
we arrive at
16k
α ˙
γ
(k + p)
γ ˙
α
δ 12
¯
D
˙
β D
2 ¯
D ˙
γ D γ ¯
D
2 D
β
256
δ 12 .
(4.103)
We can use (4.13) several times. At the end we get
16k
α ˙
γ
(k + p)
γ ˙
α
(k + p) γ ˙
γ (k + p)
˙
βδ
δ 12
D δ ¯
D
2 D
β
32
δ 12 .
(4.104)
Equations (4.13) and (4.17) allow one to write
δ 12 D δ ¯
D
2 D
β
δ 12 = −8δ
β
δ δ 12 .
We substitute this expression in (4.102). Using the identity k
α ˙
β k γ ˙
β = δ
α
γ k
2 we obtain
the contribution from (4.102) in the form
k
α ˙
α
(k + p)
β ˙
β
(k + p)
2
δ 12 ,
which after integration over θ 2 leads to the following expression for I 3 :
I 3 = −4ξ
2
1
d
4
θ 1
d
4 p
(2π) 4
d
4 k
(2π) 4 (∂ α ˙
α σ)(− p, θ 1 )(∂ β ˙
β ¯
σ)( p, θ 1 ) ×
× k
α ˙
α
(k + p)
β ˙
β
(k + p)
2 G(k)G(k + p).
(4.105)
A detailed analysis carried out in [63] shows that this correction is divergent.
Calculation of corrections from supergraphs in other superfield theories is carried
out in a similar manner.
We demonstrated that supergraph technique is a very efficient method for considering quantum corrections in superfield theories whereas the component study
is much more complicated since one supergraph corresponds to several component
diagrams (it is amusing that the exact expression for the classical action of dilaton
supergravity occupies a whole page [63]). Since, as we saw in this section, the superfield theories display divergences, the next step of the development of the superfield
formalism consists in introducing renormalization.
4.5 Superficial Degree of Divergence. Renormalization
We found that divergent quantum corrections arise in superfield theories as well as
in usual ones. Therefore we face two problems:
(i) to classify possible divergences;
(ii) to develop a procedure of renormalization in superfield theories.
75
We transport all spinor supercovariant derivatives to one propagator (here the terms
with spinor supercovariant derivatives moved to the external lines are omitted as the
irrelevant ones since they do not contribute to the divergent part [63]). As a result
we arrive at
16k
α ˙
γ
(k + p)
γ ˙
α
δ 12
¯
D
˙
β D
2 ¯
D ˙
γ D γ ¯
D
2 D
β
256
δ 12 .
(4.103)
We can use (4.13) several times. At the end we get
16k
α ˙
γ
(k + p)
γ ˙
α
(k + p) γ ˙
γ (k + p)
˙
βδ
δ 12
D δ ¯
D
2 D
β
32
δ 12 .
(4.104)
Equations (4.13) and (4.17) allow one to write
δ 12 D δ ¯
D
2 D
β
δ 12 = −8δ
β
δ δ 12 .
We substitute this expression in (4.102). Using the identity k
α ˙
β k γ ˙
β = δ
α
γ k
2 we obtain
the contribution from (4.102) in the form
k
α ˙
α
(k + p)
β ˙
β
(k + p)
2
δ 12 ,
which after integration over θ 2 leads to the following expression for I 3 :
I 3 = −4ξ
2
1
d
4
θ 1
d
4 p
(2π) 4
d
4 k
(2π) 4 (∂ α ˙
α σ)(− p, θ 1 )(∂ β ˙
β ¯
σ)( p, θ 1 ) ×
× k
α ˙
α
(k + p)
β ˙
β
(k + p)
2 G(k)G(k + p).
(4.105)
A detailed analysis carried out in [63] shows that this correction is divergent.
Calculation of corrections from supergraphs in other superfield theories is carried
out in a similar manner.
We demonstrated that supergraph technique is a very efficient method for considering quantum corrections in superfield theories whereas the component study
is much more complicated since one supergraph corresponds to several component
diagrams (it is amusing that the exact expression for the classical action of dilaton
supergravity occupies a whole page [63]). Since, as we saw in this section, the superfield theories display divergences, the next step of the development of the superfield
formalism consists in introducing renormalization.
4.5 Superficial Degree of Divergence. Renormalization
We found that divergent quantum corrections arise in superfield theories as well as
in usual ones. Therefore we face two problems:
(i) to classify possible divergences;
(ii) to develop a procedure of renormalization in superfield theories.