104
4 Four-Dimensional Superfield Supersymmetry
discussed yet in scientific literature), one must note that the divergence in the twoloop chiral effective potential is caused by the one-loop divergent contributions to
the two- and three-point functions involving the gauge superfield. In the maximal,
N = 4 supersymmetric case, these divergences must mutually cancel (for the twopoint functions, both of gauge and chiral superfields, such a cancellation has been
explicitly shown in [23], see also [2]). It allows to conclude that in the “complete”,
N = 4 SYM theory, no such a nonlocality arises. Another way for interpretation of
this situation can be related with the concept of the Wilsonian effective action [79]
whose use naturally introduces an infrared cutoff parameter with a subsequent
replacement of the nonlocal factor ln(−
μ 2 ) in (4.202) by a constant ln(
2
μ 2 ), thus, this
expression acquires the usual local form of the chiral effective potential.
Chiral contributions to the effective action arise also in other theories describing
dynamics of chiral superfields. For example, in a general chiral superfield theory (see
the next section), the leading chiral contribution is also a chiral effective potential,
while in the higher-derivative field theory the leading chiral contribution is of second order in space-time derivatives of a chiral superfield (see Sect. 4.10), and these
corrections are finite.
We conclude that the presence of quantum contributions to the chiral effective
Lagrangian is indeed quite characteristic for theories including chiral superfields, as
it was claimed in the seminal paper [67].
4.9 General Chiral Superfield Model
In the previous section, we have studied the Wess-Zumino model, that is, the simplest example of a theory describing a dynamics of the chiral superfield. The natural
question is the possibility for constructing of the most generic model for this superfield. Such a model naturally arises within the context of the superstring theory, from
which viewpoint, low-energy effective field theory models represent themselves as
theories where the integration over massive string modes is carried out, and the extra
dimensions of the ten-dimensional background manifold, where the dynamics of the
superstring takes place, are compactified, so, the structure of this manifold has the
form M
4
× K where M
4 is the usual four-dimensional Minkowski space, and K is a
some six-dimensional compact space. As a result, after reducing to M
4 , one arrives
at the theory of chiral superfields described by the action [1]:
S[, ¯
] =
d
8 zK ((
i
, ¯
i
) + (
d
6 zW ((
i
) + h.c.).
(4.205)
We can use matrix notations via introduction of column vector
= {
i
}, after which
the consideration in the case of several chiral superfields is analogous to the case of
one chiral superfield (some extensions of this model involving the gauge superfields
are discussed in [80]). We can consider this theory for arbitrary functions K and W .
4 Four-Dimensional Superfield Supersymmetry
discussed yet in scientific literature), one must note that the divergence in the twoloop chiral effective potential is caused by the one-loop divergent contributions to
the two- and three-point functions involving the gauge superfield. In the maximal,
N = 4 supersymmetric case, these divergences must mutually cancel (for the twopoint functions, both of gauge and chiral superfields, such a cancellation has been
explicitly shown in [23], see also [2]). It allows to conclude that in the “complete”,
N = 4 SYM theory, no such a nonlocality arises. Another way for interpretation of
this situation can be related with the concept of the Wilsonian effective action [79]
whose use naturally introduces an infrared cutoff parameter with a subsequent
replacement of the nonlocal factor ln(−
μ 2 ) in (4.202) by a constant ln(
2
μ 2 ), thus, this
expression acquires the usual local form of the chiral effective potential.
Chiral contributions to the effective action arise also in other theories describing
dynamics of chiral superfields. For example, in a general chiral superfield theory (see
the next section), the leading chiral contribution is also a chiral effective potential,
while in the higher-derivative field theory the leading chiral contribution is of second order in space-time derivatives of a chiral superfield (see Sect. 4.10), and these
corrections are finite.
We conclude that the presence of quantum contributions to the chiral effective
Lagrangian is indeed quite characteristic for theories including chiral superfields, as
it was claimed in the seminal paper [67].
4.9 General Chiral Superfield Model
In the previous section, we have studied the Wess-Zumino model, that is, the simplest example of a theory describing a dynamics of the chiral superfield. The natural
question is the possibility for constructing of the most generic model for this superfield. Such a model naturally arises within the context of the superstring theory, from
which viewpoint, low-energy effective field theory models represent themselves as
theories where the integration over massive string modes is carried out, and the extra
dimensions of the ten-dimensional background manifold, where the dynamics of the
superstring takes place, are compactified, so, the structure of this manifold has the
form M
4
× K where M
4 is the usual four-dimensional Minkowski space, and K is a
some six-dimensional compact space. As a result, after reducing to M
4 , one arrives
at the theory of chiral superfields described by the action [1]:
S[, ¯
] =
d
8 zK ((
i
, ¯
i
) + (
d
6 zW ((
i
) + h.c.).
(4.205)
We can use matrix notations via introduction of column vector
= {
i
}, after which
the consideration in the case of several chiral superfields is analogous to the case of
one chiral superfield (some extensions of this model involving the gauge superfields
are discussed in [80]). We can consider this theory for arbitrary functions K and W .