4.2 Field Theory Models in the Four-Dimensional Superspace
63
S =
1
64g 2 tr
d
6 zW
α W α = −
1
16g 2 tr
d
8 z(e
−gV D
α e
gV
) ¯
D
2
(e
−gV D α e
gV
).
(4.58)
The theory described by the expression (4.58) is called the N = 1 SYM theory.
Its action is essentially non-polynomial, however, its quadratic part reproduces the
Abelian expression (4.46). In principle, one can expand this action in power series in
V and impose the Wess-Zumino gauge which allows to eliminate all V
3 and higher
terms as well as in the Abelian case (for the discussion of the noncovariant gauges
see [39]). However, the covariant gauges are much more convenient for studying this
theory.
Treating the component content, one must note that the nonpolynomial gauge
transformations (4.56) allow to eliminate the lower components of the superfield V .
However, the non-Abelian theory (4.58) is not free but nontrivially self-coupled. We
suggest that the components of the non-Abelian strength W α are again given by the
expression (4.52), with the only modification that f αβ is now a non-Abelian stress
tensor. Therefore, the action of the SYM theory, after carrying out the procedures
similar to those ones realized above is rewritten in components as
S =
1
4g 2 tr
d
4 x(−
1
2
f
αβ f αβ − i ¯
λ
˙
α
∇ ˙
αβ λ
β
+
1
2
D
2
),
(4.59)
Here ∇ ˙
αβ = ∂ ˙
αβ + i A ˙
αβ is a gauge covariant space-time derivative.
The details of the supercovariant description of the N = 1 SYM theory will be
given in the Sect. 4.11, where the perturbative approach for it is discussed.
4.3 Generating Functional and Green Functions
for Superfields
Now our aim consists of describing a method for calculating a generating functional
and Green functions for superfields and a subsquent application of this method to
calculation of superfield quantum corrections, i.e. in development of the Feynman
supergraph technique. We note that during last years the activity in development of
nonperturbative methods in superfield quantum theory stimulated by the paper [28]
essentially increased. Nevertheless, the importance of the results obtained through
the perturbative approach continues to be principal.
The generalization of the path integral method for a superfield theory turns out
to be quite straightforward but a bit formal. Actually, a generating functional is
defined in terms of a path integral which is well-defined only for some special cases.
However, the case of the Gaussian path integral is, first, well-defined both in a
standard field theory and in a superfield theory, second, sufficient for the development
of the superfield perturbation technique.
63
S =
1
64g 2 tr
d
6 zW
α W α = −
1
16g 2 tr
d
8 z(e
−gV D
α e
gV
) ¯
D
2
(e
−gV D α e
gV
).
(4.58)
The theory described by the expression (4.58) is called the N = 1 SYM theory.
Its action is essentially non-polynomial, however, its quadratic part reproduces the
Abelian expression (4.46). In principle, one can expand this action in power series in
V and impose the Wess-Zumino gauge which allows to eliminate all V
3 and higher
terms as well as in the Abelian case (for the discussion of the noncovariant gauges
see [39]). However, the covariant gauges are much more convenient for studying this
theory.
Treating the component content, one must note that the nonpolynomial gauge
transformations (4.56) allow to eliminate the lower components of the superfield V .
However, the non-Abelian theory (4.58) is not free but nontrivially self-coupled. We
suggest that the components of the non-Abelian strength W α are again given by the
expression (4.52), with the only modification that f αβ is now a non-Abelian stress
tensor. Therefore, the action of the SYM theory, after carrying out the procedures
similar to those ones realized above is rewritten in components as
S =
1
4g 2 tr
d
4 x(−
1
2
f
αβ f αβ − i ¯
λ
˙
α
∇ ˙
αβ λ
β
+
1
2
D
2
),
(4.59)
Here ∇ ˙
αβ = ∂ ˙
αβ + i A ˙
αβ is a gauge covariant space-time derivative.
The details of the supercovariant description of the N = 1 SYM theory will be
given in the Sect. 4.11, where the perturbative approach for it is discussed.
4.3 Generating Functional and Green Functions
for Superfields
Now our aim consists of describing a method for calculating a generating functional
and Green functions for superfields and a subsquent application of this method to
calculation of superfield quantum corrections, i.e. in development of the Feynman
supergraph technique. We note that during last years the activity in development of
nonperturbative methods in superfield quantum theory stimulated by the paper [28]
essentially increased. Nevertheless, the importance of the results obtained through
the perturbative approach continues to be principal.
The generalization of the path integral method for a superfield theory turns out
to be quite straightforward but a bit formal. Actually, a generating functional is
defined in terms of a path integral which is well-defined only for some special cases.
However, the case of the Gaussian path integral is, first, well-defined both in a
standard field theory and in a superfield theory, second, sufficient for the development
of the superfield perturbation technique.