134
4 Four-Dimensional Superfield Supersymmetry
noncommutative generalization) can arise only in subgraphs which are not associated
to the external W α , ¯
W ˙
α legs.
From formal viewpoint such a difference has the following origin. The superfield
strength W α by its construction contains three spinor derivatives, hence arising of any
superfield strength leg in the framework of the usual superfield description decreases
by three the number of the D-factors which could be converted to momenta (we
note that the use of the background field method allows to sum automatically the
infinite number of “usual” graphs and forbids an existence of supergraphs with a
superficial quadratic or linear divergence). As a result, the convergence of a supergraph is improved. It is essential in the noncommutative field theory since it means
that the only dangerous (quadratic and linear) infrared divergences could be generated by subgraphs of a given supergraph since each contribution to the effective
action is in worst case only logarithmically divergent. In part, it means that there
is no contradiction between the result of Bichl et al. [107] according to which the
separate contributions to the one-loop two-point function of the gauge superfield
in the U (1) NC SYM theory possess quadratic divergences, and only their sum is
free of dangerous UV/IR mixing, and the result of Zanon et al. [108] according to
which all one-loop contributions to the effective action in the same theory are free
of the dangerous UV/IR mixing. It is worth to notice that the calculations in the last
paper were carried out in the framework of the background field method. We should
mention also using of the background field method in the papers [108] devoted to
study of the one-loop effective action in various noncommutative SYM theories. An
important advantage of the background field method is that it allows to preserve the
gauge covariance at all steps of calculations.
However, the background field method has one disadvantage—presence of nontrivial commutators of background covariant derivatives makes all calculations to be
extremely difficult from the technical viewpoint even in the case of the absence of
the external chiral matter fields.
4.11.3 Proper-Time Method for the Supergauge Theories
Within supergauge theories, there exists a powerful tool allowing for the application
of the background field formalism in a manner based on the proper-time method.
This method has been developed in [74]. To illustrate it, let us consider the following
one-loop effective action of the gauge theories:
(1)
=
i
2
ln det(D
m
D m + W
α
D α + ¯
W ˙
α
¯
D
˙
α
+ ||
2
).
(4.334)
Actually, this expression emerges when one couples the SYM theory, whose action
is given by the sum of (4.324) and (4.326), to the external chiral matter. It has been
discussed in [110]. Here we illustrate how this expression can be evaluated using the
method developed in [74] where it was applied to obtain the contribution of the form
4 Four-Dimensional Superfield Supersymmetry
noncommutative generalization) can arise only in subgraphs which are not associated
to the external W α , ¯
W ˙
α legs.
From formal viewpoint such a difference has the following origin. The superfield
strength W α by its construction contains three spinor derivatives, hence arising of any
superfield strength leg in the framework of the usual superfield description decreases
by three the number of the D-factors which could be converted to momenta (we
note that the use of the background field method allows to sum automatically the
infinite number of “usual” graphs and forbids an existence of supergraphs with a
superficial quadratic or linear divergence). As a result, the convergence of a supergraph is improved. It is essential in the noncommutative field theory since it means
that the only dangerous (quadratic and linear) infrared divergences could be generated by subgraphs of a given supergraph since each contribution to the effective
action is in worst case only logarithmically divergent. In part, it means that there
is no contradiction between the result of Bichl et al. [107] according to which the
separate contributions to the one-loop two-point function of the gauge superfield
in the U (1) NC SYM theory possess quadratic divergences, and only their sum is
free of dangerous UV/IR mixing, and the result of Zanon et al. [108] according to
which all one-loop contributions to the effective action in the same theory are free
of the dangerous UV/IR mixing. It is worth to notice that the calculations in the last
paper were carried out in the framework of the background field method. We should
mention also using of the background field method in the papers [108] devoted to
study of the one-loop effective action in various noncommutative SYM theories. An
important advantage of the background field method is that it allows to preserve the
gauge covariance at all steps of calculations.
However, the background field method has one disadvantage—presence of nontrivial commutators of background covariant derivatives makes all calculations to be
extremely difficult from the technical viewpoint even in the case of the absence of
the external chiral matter fields.
4.11.3 Proper-Time Method for the Supergauge Theories
Within supergauge theories, there exists a powerful tool allowing for the application
of the background field formalism in a manner based on the proper-time method.
This method has been developed in [74]. To illustrate it, let us consider the following
one-loop effective action of the gauge theories:
(1)
=
i
2
ln det(D
m
D m + W
α
D α + ¯
W ˙
α
¯
D
˙
α
+ ||
2
).
(4.334)
Actually, this expression emerges when one couples the SYM theory, whose action
is given by the sum of (4.324) and (4.326), to the external chiral matter. It has been
discussed in [110]. Here we illustrate how this expression can be evaluated using the
method developed in [74] where it was applied to obtain the contribution of the form