4.11 Supergauge Theories
129
in the cancellation of the logarithmically divergent contributions. This condition is
satisfied for an appropriate gauge group, in particular, SU (N ). The tadpole graphs
(for the massless chiral superfield) give contributions proportional to the integral
d
4 k
(2π) 4
1
k 2 which vanishes within the dimensional regularization, hence they are identically equal to zero. This mechanism explaining the vanishing of divergences is
discussed, e.g. in [23, 102], where it is shown to be caused by the N = 2 superconformal symmetry. The most important example of such theories is the N = 4 SYM
theory (4.305) which is equivalent to (4.304) with only one pair of hypermultiplets
Q, ˜
Q which are transformed under the adjoint representation of the Lie algebra. This
theory is the first known example of a finite supersymmetric theory without higher
derivatives (see e.g. [2, 23]).
4.11.2 Background Field Method
The approach to studying of supergauge theories described above is very useful for
consideration of quantum corrections in the sector of chiral superfields , Q, ˜
Q only.
To study contributions depending on gauge superfields we must develop a method
allowing to preserve the manifest gauge invariance at any step, whereas earlier we
have obtained contributions in terms of the superfield V which are in general case
not gauge invariant. Therefore we should introduce an approach in which external
lines are background strengths W α , ¯
W ˙
α and their gauge covariant derivatives. This
method was developed in [66] (see also [106] and references therein), here we give
its description.
The problem of calculation of the effective action in the SYM theory characterized
by the action (4.274) is much more complicated than in scalar superfield theories.
The main difficulties are the following ones. First, the nonpolynomiality of the action
(4.274) implies in an infinite number of vertices which seems to result in infinite
number of types of divergent quantum corrections (such a situation is treated in
many cases as a non-renormalizability of the theory), second, it is easy to see that
the usual background-quantum splitting V → V 0 + v where V 0 is a background field
and v is a quantum field, cannot provide manifest gauge covariance of the quantum
corrections. Really, because of the nonpolynomiality of W α (4.275), to get a covariant
quantum correction (which by definition must be expressed in terms of the strengths
W α , ¯
W ˙
α which are the only objects transforming in a covariant way under the gauge
transformations unlike of the superfield V itself) we need to summarize an infinite
number of supergraphs with different numbers of external V 0 legs (to the best of
our knowledge, this summation never has been performed). The background field
method provides an efficient solution for these problems.
The starting point of this method is the nonlinear background-quantum splitting
for the gauge superfield V defining the action (4.274) [66]:
e
gV
→ e
g e
gv e
g ¯
.
(4.312)
129
in the cancellation of the logarithmically divergent contributions. This condition is
satisfied for an appropriate gauge group, in particular, SU (N ). The tadpole graphs
(for the massless chiral superfield) give contributions proportional to the integral
d
4 k
(2π) 4
1
k 2 which vanishes within the dimensional regularization, hence they are identically equal to zero. This mechanism explaining the vanishing of divergences is
discussed, e.g. in [23, 102], where it is shown to be caused by the N = 2 superconformal symmetry. The most important example of such theories is the N = 4 SYM
theory (4.305) which is equivalent to (4.304) with only one pair of hypermultiplets
Q, ˜
Q which are transformed under the adjoint representation of the Lie algebra. This
theory is the first known example of a finite supersymmetric theory without higher
derivatives (see e.g. [2, 23]).
4.11.2 Background Field Method
The approach to studying of supergauge theories described above is very useful for
consideration of quantum corrections in the sector of chiral superfields , Q, ˜
Q only.
To study contributions depending on gauge superfields we must develop a method
allowing to preserve the manifest gauge invariance at any step, whereas earlier we
have obtained contributions in terms of the superfield V which are in general case
not gauge invariant. Therefore we should introduce an approach in which external
lines are background strengths W α , ¯
W ˙
α and their gauge covariant derivatives. This
method was developed in [66] (see also [106] and references therein), here we give
its description.
The problem of calculation of the effective action in the SYM theory characterized
by the action (4.274) is much more complicated than in scalar superfield theories.
The main difficulties are the following ones. First, the nonpolynomiality of the action
(4.274) implies in an infinite number of vertices which seems to result in infinite
number of types of divergent quantum corrections (such a situation is treated in
many cases as a non-renormalizability of the theory), second, it is easy to see that
the usual background-quantum splitting V → V 0 + v where V 0 is a background field
and v is a quantum field, cannot provide manifest gauge covariance of the quantum
corrections. Really, because of the nonpolynomiality of W α (4.275), to get a covariant
quantum correction (which by definition must be expressed in terms of the strengths
W α , ¯
W ˙
α which are the only objects transforming in a covariant way under the gauge
transformations unlike of the superfield V itself) we need to summarize an infinite
number of supergraphs with different numbers of external V 0 legs (to the best of
our knowledge, this summation never has been performed). The background field
method provides an efficient solution for these problems.
The starting point of this method is the nonlinear background-quantum splitting
for the gauge superfield V defining the action (4.274) [66]:
e
gV
→ e
g e
gv e
g ¯
.
(4.312)