Chapter 6
Conclusions
In this book, we considered the superfield method in supersymmetric field theories,
both in three- and four-dimensional space-times. This method allows to preserve
the manifest supersymmetry at any step of calculations, and the calculations within
it turn out to be much more compact than within the framework of the component
approach.
We studied several examples of superfield theories and presented in details quantum calculations for these models. In three dimensions, our examples were a scalar
superfield theory and supersymmetric gauge theories. In four dimensions, we considered the Wess-Zumino model, the general chiral superfield model, the higherderivative chiral superfield model and the N = 1 SYM theory with chiral matter. In
all these theories we developed the supergraph technique, studied a general form of
a superfield effective action and calculated low-energy leading contributions to the
effective actions of these theories in one-, and in certain cases, two-loop approximations. It is natural to expect that the development of superfield quantum calculations
in other superfield models formulated in terms of N = 1 superfields is in principle
no more difficult. We also discussed the background field method in supergauge
theories.
Let us briefly discuss other applications and generalizations of the superfield
approach in the quantum field theory. In the last years the following most important
ways of applications of superfield supersymmetry were developed.
1. Studying of theories with extended supersymmetry. It is known that theories with
extended supersymmetry possess better renormalization properties, e.g. while
the N = 1 SYM theory is renormalizable, the N = 4 SYM theory is finite. The
most important examples of theories possessing extended supersymmetry are
N = 2 and N = 4 SYM theories. During last years numerous results in studying
of these theories were obtained (see e.g. [110, 111, 120] and references therein).
It is natural that the most adequate method for considering such theories must display manifest N = 2 supersymmetry. Such a method is a harmonic superspace
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Petrov, Quantum Superfield Supersymmetry, Fundamental Theories of Physics 202,
https://doi.org/10.1007/978-3-030-68136-4_6
147
Conclusions
In this book, we considered the superfield method in supersymmetric field theories,
both in three- and four-dimensional space-times. This method allows to preserve
the manifest supersymmetry at any step of calculations, and the calculations within
it turn out to be much more compact than within the framework of the component
approach.
We studied several examples of superfield theories and presented in details quantum calculations for these models. In three dimensions, our examples were a scalar
superfield theory and supersymmetric gauge theories. In four dimensions, we considered the Wess-Zumino model, the general chiral superfield model, the higherderivative chiral superfield model and the N = 1 SYM theory with chiral matter. In
all these theories we developed the supergraph technique, studied a general form of
a superfield effective action and calculated low-energy leading contributions to the
effective actions of these theories in one-, and in certain cases, two-loop approximations. It is natural to expect that the development of superfield quantum calculations
in other superfield models formulated in terms of N = 1 superfields is in principle
no more difficult. We also discussed the background field method in supergauge
theories.
Let us briefly discuss other applications and generalizations of the superfield
approach in the quantum field theory. In the last years the following most important
ways of applications of superfield supersymmetry were developed.
1. Studying of theories with extended supersymmetry. It is known that theories with
extended supersymmetry possess better renormalization properties, e.g. while
the N = 1 SYM theory is renormalizable, the N = 4 SYM theory is finite. The
most important examples of theories possessing extended supersymmetry are
N = 2 and N = 4 SYM theories. During last years numerous results in studying
of these theories were obtained (see e.g. [110, 111, 120] and references therein).
It is natural that the most adequate method for considering such theories must display manifest N = 2 supersymmetry. Such a method is a harmonic superspace
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Petrov, Quantum Superfield Supersymmetry, Fundamental Theories of Physics 202,
https://doi.org/10.1007/978-3-030-68136-4_6
147