110
4 Four-Dimensional Superfield Supersymmetry
action will involve the term (4.217). The situation does not essentially change if this
self-coupling is described by the generic function V (φ), instead of the cubic one,
and the corresponding contribution to the one-loop effective action can be obtained
through a straightforward replacement φ → V
(φ) in (4.217). At the same time, the
sector in which the light superfield φ is coupled to the heavy one , in the usual
case when the theory display divergences, certainly, among other terms, will yield
the contribution b
d
8 zφ ¯
φ ln
M
2
μ 2 , where b is also finite (this contribution is always
present if the corresponding supergraphs involve massive propagators), as we already
showed. Therefore, either this contribution increases with growth of M, or, if one
fixes the parameter μ imposing the condition M = μ (or M = bμ, with a constant b),
the one-loop contribution of the light fields only takes the form c
d
8 zg
2
φ ¯
φ ln
g
2 φ ¯
φ
M 2
which increases with growing of M. Actually we demonstrated that such increasing
contributions emerge in each theory involving heavy fields, if the result displays a
dependence on μ (i.e. if the theory involves divergences), which is common if the
theory does not involve higher derivatives or extended supersymmetry.
4.10 The Higher-Derivative Chiral Superfield Models
The next generalization for the models of the chiral superfield consists in including higher derivatives. Originally, a higher-derivative extension of a supersymmetric
theory has been introduced for the purposes of the regularization in [86]. Besides
of this, the higher-derivative terms have been implemented in gravity, first in [87]
where it was shown to achieve a renormalizable gravity. Moreover, it was found in
[88] that the higher-derivative additive modifications of the gravity action arise due
to the presence of the conformal anomaly of matter fields in a curved space. In [89],
a supersymmetric analogue of this anomaly and the corresponding additive modification of the supergravity action were obtained, and in [63] the quantum dynamics
of the dilaton superfield was studied.
Actually, the higher-derivative field theories are studied in different contexts,
including different gravity modifications which are intensively applied to explain
the cosmic acceleration [91], and the Horava-Lifshitz model of gravity [92]. In the
context of supersymmetry, the new wave of an interest to higher-derivative superfield
theories was recently called by the paper [93]. Treating the notorious problem of
arising of the ghosts whose presence is typical for the higher-derivative field theories,
we note that an attempt to solve this problem was carried out in [94], where the
hypothesis that contributions of ghosts can be decoupled in certain cases (so-called
“benign ghosts”) was discussed. At the same time, it is natural to treat the higherderivative theories as effective models for studying of the low-energy effects, see the
discussion in [95].
We start with the simplest example of a higher-derivative superfield theory [96]:
S[, ¯
] =
d
8 z ¯
+ (
d
6 zW (() + h.c.).
(4.218)
4 Four-Dimensional Superfield Supersymmetry
action will involve the term (4.217). The situation does not essentially change if this
self-coupling is described by the generic function V (φ), instead of the cubic one,
and the corresponding contribution to the one-loop effective action can be obtained
through a straightforward replacement φ → V
(φ) in (4.217). At the same time, the
sector in which the light superfield φ is coupled to the heavy one , in the usual
case when the theory display divergences, certainly, among other terms, will yield
the contribution b
d
8 zφ ¯
φ ln
M
2
μ 2 , where b is also finite (this contribution is always
present if the corresponding supergraphs involve massive propagators), as we already
showed. Therefore, either this contribution increases with growth of M, or, if one
fixes the parameter μ imposing the condition M = μ (or M = bμ, with a constant b),
the one-loop contribution of the light fields only takes the form c
d
8 zg
2
φ ¯
φ ln
g
2 φ ¯
φ
M 2
which increases with growing of M. Actually we demonstrated that such increasing
contributions emerge in each theory involving heavy fields, if the result displays a
dependence on μ (i.e. if the theory involves divergences), which is common if the
theory does not involve higher derivatives or extended supersymmetry.
4.10 The Higher-Derivative Chiral Superfield Models
The next generalization for the models of the chiral superfield consists in including higher derivatives. Originally, a higher-derivative extension of a supersymmetric
theory has been introduced for the purposes of the regularization in [86]. Besides
of this, the higher-derivative terms have been implemented in gravity, first in [87]
where it was shown to achieve a renormalizable gravity. Moreover, it was found in
[88] that the higher-derivative additive modifications of the gravity action arise due
to the presence of the conformal anomaly of matter fields in a curved space. In [89],
a supersymmetric analogue of this anomaly and the corresponding additive modification of the supergravity action were obtained, and in [63] the quantum dynamics
of the dilaton superfield was studied.
Actually, the higher-derivative field theories are studied in different contexts,
including different gravity modifications which are intensively applied to explain
the cosmic acceleration [91], and the Horava-Lifshitz model of gravity [92]. In the
context of supersymmetry, the new wave of an interest to higher-derivative superfield
theories was recently called by the paper [93]. Treating the notorious problem of
arising of the ghosts whose presence is typical for the higher-derivative field theories,
we note that an attempt to solve this problem was carried out in [94], where the
hypothesis that contributions of ghosts can be decoupled in certain cases (so-called
“benign ghosts”) was discussed. At the same time, it is natural to treat the higherderivative theories as effective models for studying of the low-energy effects, see the
discussion in [95].
We start with the simplest example of a higher-derivative superfield theory [96]:
S[, ¯
] =
d
8 z ¯
+ (
d
6 zW (() + h.c.).
(4.218)