4.5 Superficial Degree of Divergence. Renormalization
79
Here we used 2L − 2P = 2 − 2V . However, any vertex gives contribution −2 to the
term −2V and l V c − 1 or n V t + m V t to terms of ω involving summations. It is evident
that either l V c − 1 or n V t + m V t is not less than 2 since either l V c ≥ 3 or n V t + m V t ≥ 2.
Hence in a general case
V t
(n V t + m V t ) +
V c
(l V c − 1) − 2V ≥ 0, thus the number of
divergent structures is not restricted, and the theory is non-renormalizable. This is
quite natural since the constants K i j (for i + j > 2) and W l (for l ≥ 4) have negative
mass dimensions.
The next problem consists in introducing a regularization scheme. The most natural way to do it in field theories, including supersymmetric ones, is the dimensional
regularization. It can be implemented as usual: any integral
d
4 k
(2π) 4
1
(k 2 + m 2 ) N
is replaced by its extension to (4 + )-dimensional space-time.
μ
−
d
4+ k
(2π) 4+
1
(k 2 + m 2 ) N ,
so, all divergences are described by poles in , as in the usual field theory (no more
than
1
L for a L-loop correction).
However, there are some peculiarities. First of all, within the component description any supersymmetric action includes spinors and hence γ-matrices which are well
defined if and only if the dimension of the space-time is integer. Therefore we must
use some modification of the dimensional regularization called dimensional reduction. According to it, all objects which are well-defined only for specific dimensions,
for example integer ones (such as spinors and Dirac γ matrices) are evaluated at these
dimensions (in our case—at the dimension equal to 4), and integrals over momenta—
at arbitrary dimensions. At the same time, the dimensional reduction leads to some
difficulties in calculation of higher loop corrections since many supergraphs involve
contractions of essentially four-dimensional objects, such as Levi-Civita tensor
abcd ,
with d-dimensional objects, and such contractions need additional definitions. As a
result the ambiguities frequently arise. However, such a situation is observed only
beyond two loops. The detailed discussion of different problems related with applying the dimensional regularization in supersymmetric field theories is presented in
[69].
The technique for renormalization in superfield theories is quite analogous to that
one in common QFT. It is carried out via introducing the corresponding counterterms.
Example. Let us consider the one-loop contribution to the kinetic term in the WessZumino model. The corresponding supergraph is given by Fig. 4.1 (see above), its
contribution (4.97) yields
79
Here we used 2L − 2P = 2 − 2V . However, any vertex gives contribution −2 to the
term −2V and l V c − 1 or n V t + m V t to terms of ω involving summations. It is evident
that either l V c − 1 or n V t + m V t is not less than 2 since either l V c ≥ 3 or n V t + m V t ≥ 2.
Hence in a general case
V t
(n V t + m V t ) +
V c
(l V c − 1) − 2V ≥ 0, thus the number of
divergent structures is not restricted, and the theory is non-renormalizable. This is
quite natural since the constants K i j (for i + j > 2) and W l (for l ≥ 4) have negative
mass dimensions.
The next problem consists in introducing a regularization scheme. The most natural way to do it in field theories, including supersymmetric ones, is the dimensional
regularization. It can be implemented as usual: any integral
d
4 k
(2π) 4
1
(k 2 + m 2 ) N
is replaced by its extension to (4 + )-dimensional space-time.
μ
−
d
4+ k
(2π) 4+
1
(k 2 + m 2 ) N ,
so, all divergences are described by poles in , as in the usual field theory (no more
than
1
L for a L-loop correction).
However, there are some peculiarities. First of all, within the component description any supersymmetric action includes spinors and hence γ-matrices which are well
defined if and only if the dimension of the space-time is integer. Therefore we must
use some modification of the dimensional regularization called dimensional reduction. According to it, all objects which are well-defined only for specific dimensions,
for example integer ones (such as spinors and Dirac γ matrices) are evaluated at these
dimensions (in our case—at the dimension equal to 4), and integrals over momenta—
at arbitrary dimensions. At the same time, the dimensional reduction leads to some
difficulties in calculation of higher loop corrections since many supergraphs involve
contractions of essentially four-dimensional objects, such as Levi-Civita tensor
abcd ,
with d-dimensional objects, and such contractions need additional definitions. As a
result the ambiguities frequently arise. However, such a situation is observed only
beyond two loops. The detailed discussion of different problems related with applying the dimensional regularization in supersymmetric field theories is presented in
[69].
The technique for renormalization in superfield theories is quite analogous to that
one in common QFT. It is carried out via introducing the corresponding counterterms.
Example. Let us consider the one-loop contribution to the kinetic term in the WessZumino model. The corresponding supergraph is given by Fig. 4.1 (see above), its
contribution (4.97) yields