Chapter 2
Effective Action and Loop Expansion:
General Formulation
Before entering the discussion of superfield theories, let us give the general definition
of the effective action and describe methods for its calculation. An effective action
is a key object of quantum field theory. Studying of an effective action allows to
investigate problems of vacuum stability, Green functions, spontaneous symmetry
breaking, anomalies and many other issues. In this chapter we present the description
of its general structure and manners of its calculation, which in the next chapters will
be applied to supersymmetric field theories. We follow the methodology described
in [34, 35].
An effective action (in particular, in a superfield theory) is defined, as usual, as a
generating functional of one-particle-irreducible Green functions. It is obtained as a
Legendre transform for the generating functional of connected Green functions:
[] = W [J ] −
dz J (z))(z).
(2.1)
Here [] is an effective action, dz denotes an integral over the corresponding space
or its subspace (in some cases, to abbreviate the notations we omit the sign of the integral together with the coordinate dependence of the fields, i.e. J ≡
dz J (z))(z)),
J (z) is the classical source, (z) =
δW [J ]
δ J (z)
is a so called mean (super)field, or, as in
the same, a background (super)field, which is an essentially classical object (in principle, one can consider a set of background (super)fields as well, so, in general, is
a column vector); in terms of the functional integral one has
=
Dφφe
i S[φ]+iφ J
Dφe i S[φ]+iφ J ,
where S[φ] is the classical action. This is exactly the definition of the vacuum
expected value (v.e.v.) in the path integral formalism, and W [J ] =
1
i
log Z [J ] is
a generating functional of the connected Green functions. It is easy to see that the
[] satisfies the equation
© 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_2
5
Effective Action and Loop Expansion:
General Formulation
Before entering the discussion of superfield theories, let us give the general definition
of the effective action and describe methods for its calculation. An effective action
is a key object of quantum field theory. Studying of an effective action allows to
investigate problems of vacuum stability, Green functions, spontaneous symmetry
breaking, anomalies and many other issues. In this chapter we present the description
of its general structure and manners of its calculation, which in the next chapters will
be applied to supersymmetric field theories. We follow the methodology described
in [34, 35].
An effective action (in particular, in a superfield theory) is defined, as usual, as a
generating functional of one-particle-irreducible Green functions. It is obtained as a
Legendre transform for the generating functional of connected Green functions:
[] = W [J ] −
dz J (z))(z).
(2.1)
Here [] is an effective action, dz denotes an integral over the corresponding space
or its subspace (in some cases, to abbreviate the notations we omit the sign of the integral together with the coordinate dependence of the fields, i.e. J ≡
dz J (z))(z)),
J (z) is the classical source, (z) =
δW [J ]
δ J (z)
is a so called mean (super)field, or, as in
the same, a background (super)field, which is an essentially classical object (in principle, one can consider a set of background (super)fields as well, so, in general, is
a column vector); in terms of the functional integral one has
=
Dφφe
i S[φ]+iφ J
Dφe i S[φ]+iφ J ,
where S[φ] is the classical action. This is exactly the definition of the vacuum
expected value (v.e.v.) in the path integral formalism, and W [J ] =
1
i
log Z [J ] is
a generating functional of the connected Green functions. It is easy to see that the
[] satisfies the equation
© 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_2
5