80
4 Four-Dimensional Superfield Supersymmetry
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
).
(4.113)
We see that this divergence has the form of pole part proportional to
1
. To cancel it
we must add to the initial kinetic term
S =
d
8 z(x, θ) ¯
(x, θ),
(4.114)
the counterterm
S countr = −
λ
2
32π 2
d
8 z(z) ¯
(z)
(4.115)
which corresponds to the replacement of
d
8 z ¯
in the classical action by
Z
d
8 z(z) ¯
(z) where
Z = 1 −
λ
2
32π 2
(4.116)
is the wave function renormalization.
The essential advantage of supersymmetric theories is the fact that the number of
counterterms in these theories is less than in their non-supersymmetric analogues. For
example, Wess-Zumino model is a supersymmetric generalization of φ
4 -theory, but
it involves only one renormalization constant corresponding to the renormalization
of kinetic term and no renormalization of couplings. The conclusion about absence
of divergent correction to the coupling λ (or as is the same—to the chiral potential)
is also called non-renormalization theorem since it is treated as a consequence of the
non-renormalization theorem discussed in the Sect. 4.4, following which, all loop
corrections are local in superspace involving only one integral over d
4
θ. However,
actually, the existence of the finite corrections to the chiral (holomorphic) potential
is not forbidden, for example, they arise in the massless Wess-Zumino model [25,
26, 72, 73]. In the Sect. 4.8 we discuss such corrections.
There are also some interesting properties of renormalization in superfield theories.
First, all tadpoles in the Wess-Zumino model (see Fig. 4.4) vanish. Indeed, such
a supergraph has a contribution proportional to D
2
δ 11 = δ 12 D
2
δ 12 = 0. The similar
situation can occur in other superfield models involving the Wess-Zumino model
as an ingredient. However, in theories including vertices proportional to an integral
over the whole superspace (e.g. dilaton supergravity) tadpole contributions are not
equal to zero [63].
Second, all contributions from vacuum supergraphs are proportional to
d
4
θc
(with c is a constant) and also vanish. However, this statement is not true for background dependent propagators. The methodology of background dependent propa-
4 Four-Dimensional Superfield Supersymmetry
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
).
(4.113)
We see that this divergence has the form of pole part proportional to
1
. To cancel it
we must add to the initial kinetic term
S =
d
8 z(x, θ) ¯
(x, θ),
(4.114)
the counterterm
S countr = −
λ
2
32π 2
d
8 z(z) ¯
(z)
(4.115)
which corresponds to the replacement of
d
8 z ¯
in the classical action by
Z
d
8 z(z) ¯
(z) where
Z = 1 −
λ
2
32π 2
(4.116)
is the wave function renormalization.
The essential advantage of supersymmetric theories is the fact that the number of
counterterms in these theories is less than in their non-supersymmetric analogues. For
example, Wess-Zumino model is a supersymmetric generalization of φ
4 -theory, but
it involves only one renormalization constant corresponding to the renormalization
of kinetic term and no renormalization of couplings. The conclusion about absence
of divergent correction to the coupling λ (or as is the same—to the chiral potential)
is also called non-renormalization theorem since it is treated as a consequence of the
non-renormalization theorem discussed in the Sect. 4.4, following which, all loop
corrections are local in superspace involving only one integral over d
4
θ. However,
actually, the existence of the finite corrections to the chiral (holomorphic) potential
is not forbidden, for example, they arise in the massless Wess-Zumino model [25,
26, 72, 73]. In the Sect. 4.8 we discuss such corrections.
There are also some interesting properties of renormalization in superfield theories.
First, all tadpoles in the Wess-Zumino model (see Fig. 4.4) vanish. Indeed, such
a supergraph has a contribution proportional to D
2
δ 11 = δ 12 D
2
δ 12 = 0. The similar
situation can occur in other superfield models involving the Wess-Zumino model
as an ingredient. However, in theories including vertices proportional to an integral
over the whole superspace (e.g. dilaton supergravity) tadpole contributions are not
equal to zero [63].
Second, all contributions from vacuum supergraphs are proportional to
d
4
θc
(with c is a constant) and also vanish. However, this statement is not true for background dependent propagators. The methodology of background dependent propa-