4.10 The Higher-Derivative Chiral Superfield Models
121
So, if M = 0, the situation is more complicated. Nevertheless, a possible solution
in this case is presented in the paper [99]. Indeed, a typical structure of the one-loop
Kählerian effective potential is given by the integral:
K
(1)
=
1
2
μ
d
4− k
(2π) 4−
1
k 2 log F(k
2
),
(4.273)
where F(k
2
) is a function of momenta whose form is determined by the quadratic
action of quantum superfields. The simplest case is F(k
2
) = k
2
+ g
2
¯
taking place
in the Wess-Zumino model, see (4.182), and in supergauge theories, see [75, 100].
As we noted in previous sections, calculation of this integral is straightforward. In
[99], it was noted that in higher-derivative theories with 2n derivatives, one can write
F(k
2
) = (k
2
+ A 1 )(k
2
+ A 2 ) · · · (k
2
+ A n ), where A 1 . . . A n are the roots of F(k
2
)
taken with opposite signs. In this case, the integral (4.273) can be easily taken, and
the result will be a some function of roots A 1 . . . A n , an example is presented in
[99]. However, while the integral over momenta for this representation of F(k
2
)
is simple, the expressions for the roots A 1 , . . . A n are very complicated already at
n = 3, implying in a very involved result of integrating (4.273), and certainly are
more involved at higher values of n.
We considered the one-loop effective potential for two different versions of higherderivative chiral superfield models. It turns out that, in the case when the mass term is
purely chiral (a similar situation with the mass term takes place in the Wess-Zumino
model), the theory is finite. At the same time, if the mass term arises in the general
Lagrangian (that is the situation considered in [93]), the theory displays divergences
in the limit of an infinite mass though being super-renormalizable. We note, however,
that the equivalence of the higher-derivative theory of the chiral superfield and the
theory without higher derivatives but with an extended number of chiral superfields
described in [93] occurs only in the case when the mass term belongs to the general
Lagrangian (that is, the second case considered in the section). Therefore, the presence of these divergences can be considered as a sign in favour of the equivalence
established in [93]. Indeed, the expression (4.268), after relabelling
M 2 → , identically reproduces the one-loop Kählerian effective potential in the Wess-Zumino
model, with = m + λ. We can therefore conclude that the higher-derivative
theory (4.258), in the limit M → ∞, yields the same quantum contribution to the
effective potential as the Wess-Zumino model.
4.11 Supergauge Theories
This section is a brief review of results obtained for supergauge theories. Unfortunately, the restricted volume of this review does not allow to discuss all essential
results of last years in this area hence we only give here main ones.
121
So, if M = 0, the situation is more complicated. Nevertheless, a possible solution
in this case is presented in the paper [99]. Indeed, a typical structure of the one-loop
Kählerian effective potential is given by the integral:
K
(1)
=
1
2
μ
d
4− k
(2π) 4−
1
k 2 log F(k
2
),
(4.273)
where F(k
2
) is a function of momenta whose form is determined by the quadratic
action of quantum superfields. The simplest case is F(k
2
) = k
2
+ g
2
¯
taking place
in the Wess-Zumino model, see (4.182), and in supergauge theories, see [75, 100].
As we noted in previous sections, calculation of this integral is straightforward. In
[99], it was noted that in higher-derivative theories with 2n derivatives, one can write
F(k
2
) = (k
2
+ A 1 )(k
2
+ A 2 ) · · · (k
2
+ A n ), where A 1 . . . A n are the roots of F(k
2
)
taken with opposite signs. In this case, the integral (4.273) can be easily taken, and
the result will be a some function of roots A 1 . . . A n , an example is presented in
[99]. However, while the integral over momenta for this representation of F(k
2
)
is simple, the expressions for the roots A 1 , . . . A n are very complicated already at
n = 3, implying in a very involved result of integrating (4.273), and certainly are
more involved at higher values of n.
We considered the one-loop effective potential for two different versions of higherderivative chiral superfield models. It turns out that, in the case when the mass term is
purely chiral (a similar situation with the mass term takes place in the Wess-Zumino
model), the theory is finite. At the same time, if the mass term arises in the general
Lagrangian (that is the situation considered in [93]), the theory displays divergences
in the limit of an infinite mass though being super-renormalizable. We note, however,
that the equivalence of the higher-derivative theory of the chiral superfield and the
theory without higher derivatives but with an extended number of chiral superfields
described in [93] occurs only in the case when the mass term belongs to the general
Lagrangian (that is, the second case considered in the section). Therefore, the presence of these divergences can be considered as a sign in favour of the equivalence
established in [93]. Indeed, the expression (4.268), after relabelling
M 2 → , identically reproduces the one-loop Kählerian effective potential in the Wess-Zumino
model, with = m + λ. We can therefore conclude that the higher-derivative
theory (4.258), in the limit M → ∞, yields the same quantum contribution to the
effective potential as the Wess-Zumino model.
4.11 Supergauge Theories
This section is a brief review of results obtained for supergauge theories. Unfortunately, the restricted volume of this review does not allow to discuss all essential
results of last years in this area hence we only give here main ones.