72
4 Four-Dimensional Superfield Supersymmetry
Fig. 4.1 Contributions to
the two-point function of the
chiral superfield in the
Wess-Zumino model
−
− D
2
¯
D
2
This observation was firstly made in [67], its consequences will be studied further.
Now let us study evaluation of contributions from supergraphs. The algorithm of it,
after rewriting all vertices as integrals over the whole superspace, is the following one.
1. We start with one of loops. If the number of D-factors in this loop is equal to 4 we
turn to the step 2. If it is more than 4, superfluous D-factors can be transported
to external lines or another loops via integration by parts, and some of them are
converted into internal momenta via identities D
2 ¯
D
2 D
2
= 16
2
, {D α , ¯
D ˙
β } =
2i∂ α ˙
β . As a result we stay with exactly 4 D-factors. If the number of D-factors
is less than 4, then the contribution from the entire supergraph is equal to zero.
2. We shrink this loop into a point using Eq. (4.17) and integrate over one of d
4
θ
via the delta function which is free of derivatives.
3. This procedure is repeated for next loops.
4. We integrate over internal momenta.
So, this algorithm, besides of an usual integration over internal momenta, involves
additional steps, that is, D-algebra transformations. The best way to study evaluating
of supergraphs consists in considering some examples.
Example 4.1 One-loop supergraph in the Wess-Zumino model (Fig. 4.1).
The contribution of this supergraph is equal to
I 1 =
λ
2
2
d
4
θ 1 d
4
θ 2
d
4 p
(2π) 4 p, θ 1 ) ¯
p, θ 2 )δ 12
¯
D
2
1 D
2
2
16
δ 12 ×
×
d
4 k
(2π) 4
1
(k 2 + m 2 )((k + p) 2 + m 2 )
.
(4.94)
The number of D-factors is just 4. D-algebra transformations are trivial: we use
identity (4.17) and write δ 12
¯
D
2
1 D
2
2
16
δ 12 = δ 12 . The free delta function δ 12 allows us to
integrate over d
4
θ 2 , afterwards, we denote θ 1 = θ. As a result we get
I 1 =
1
2
λ
2
d
4
θ
d
4 p
(2π) 4 p, θ) ¯
p, θ)
d
4 k
(2π) 4
1
(k 2 + m 2 )((k + p) 2 + m 2 )
.
(4.95)
The integral over k can be calculated via dimensional regularization, the result for it is
d
4 k
(2π) 4
1
(k 2 + m 2 )((k + p) 2 + m 2 )
=
1
16π 2 (
1
−
1
0
dt log
p
2 t (1 − t) + m
2
μ 2
).
(4.96)
4 Four-Dimensional Superfield Supersymmetry
Fig. 4.1 Contributions to
the two-point function of the
chiral superfield in the
Wess-Zumino model
−
− D
2
¯
D
2
This observation was firstly made in [67], its consequences will be studied further.
Now let us study evaluation of contributions from supergraphs. The algorithm of it,
after rewriting all vertices as integrals over the whole superspace, is the following one.
1. We start with one of loops. If the number of D-factors in this loop is equal to 4 we
turn to the step 2. If it is more than 4, superfluous D-factors can be transported
to external lines or another loops via integration by parts, and some of them are
converted into internal momenta via identities D
2 ¯
D
2 D
2
= 16
2
, {D α , ¯
D ˙
β } =
2i∂ α ˙
β . As a result we stay with exactly 4 D-factors. If the number of D-factors
is less than 4, then the contribution from the entire supergraph is equal to zero.
2. We shrink this loop into a point using Eq. (4.17) and integrate over one of d
4
θ
via the delta function which is free of derivatives.
3. This procedure is repeated for next loops.
4. We integrate over internal momenta.
So, this algorithm, besides of an usual integration over internal momenta, involves
additional steps, that is, D-algebra transformations. The best way to study evaluating
of supergraphs consists in considering some examples.
Example 4.1 One-loop supergraph in the Wess-Zumino model (Fig. 4.1).
The contribution of this supergraph is equal to
I 1 =
λ
2
2
d
4
θ 1 d
4
θ 2
d
4 p
(2π) 4 p, θ 1 ) ¯
p, θ 2 )δ 12
¯
D
2
1 D
2
2
16
δ 12 ×
×
d
4 k
(2π) 4
1
(k 2 + m 2 )((k + p) 2 + m 2 )
.
(4.94)
The number of D-factors is just 4. D-algebra transformations are trivial: we use
identity (4.17) and write δ 12
¯
D
2
1 D
2
2
16
δ 12 = δ 12 . The free delta function δ 12 allows us to
integrate over d
4
θ 2 , afterwards, we denote θ 1 = θ. As a result we get
I 1 =
1
2
λ
2
d
4
θ
d
4 p
(2π) 4 p, θ) ¯
p, θ)
d
4 k
(2π) 4
1
(k 2 + m 2 )((k + p) 2 + m 2 )
.
(4.95)
The integral over k can be calculated via dimensional regularization, the result for it is
d
4 k
(2π) 4
1
(k 2 + m 2 )((k + p) 2 + m 2 )
=
1
16π 2 (
1
−
1
0
dt log
p
2 t (1 − t) + m
2
μ 2
).
(4.96)