4.4 Feynman Supergraphs
73
Fig. 4.2 Two-loop vacuum
contribution in the
Wess-Zumino model
−
− D
2
¯
D
2
−
|
D
2
¯
D
2
As a consequence, contribution of this supergraph takes the form
I 1 =
1
2
λ 2
d 4 θ
d 4 p
(2π) 4 p, θ) ¯
p, θ)
1
16π 2 (
1
−
1
0
dt log
p 2 t (1 − t) + m 2
μ 2
).
(4.97)
We see that I 1 diverges logarithmically contributing to the renormalization of the
kinetic term ¯
It should be noted that in general, using of the dimensional regularization in
superfield theory in higher loops possesses some peculiarities, in fact, in some cases
it is ambiguous [69].
Example 4.2 Two-loop vacuum supergraph in the Wess-Zumino model (Fig. 4.2).
The contribution of this supergraph is equal to
I 2 =
λ
2
6
d
4 kd
4 l
(2π) 8
d
4
θ 1 d
4
θ 2 (−
¯
D
2
1
4
)δ 12
¯
D
2
1 D
2
2
16
δ 12 (−
D
2
2
4
)δ 12 ×
×
1
(k 2 + m 2 )(l 2 + m 2 )((k + l) 2 + m 2 )
.
(4.98)
First we do D-algebra transformations: we can write
(−
¯
D
2
1
4
)δ 12
¯
D
2
1 D
2
2
16
δ 12 (−
D
2
2
4
)δ 12 = δ 12
¯
D
2
1 D
2
2
16
δ 12
¯
D
2
1 D
2
2
16
δ 12 .
Then we use Eq. (4.17) two times:
δ 12
¯
D
2
1 D
2
2
16
δ 12
¯
D
2
1 D
2
2
16
δ 12 = δ 12 .
As a result we can integrate over θ 2 using the delta function. We get
I 2 =
λ
2
6
d
4 kd
4 l
(2π) 8
d
4
θ 1
1
(k 2 + m 2 )(l 2 + m 2 )((k + l) 2 + m 2 )
. (4.99)
This integral vanishes in the standard case since it is proportional to an integral over
d
4
θ from a constant. However, if we suppose that the mass m is not a constant but
θ-dependent superfield, this contribution will not be equal to zero. Namely this case
is studied when the effective action is considered, and one uses the effective mass