60
4 Four-Dimensional Superfield Supersymmetry
Here K φ =
∂ K ((, ¯
)
∂
| =φ, ¯
= ¯
φ etc., the similar definitions are applied to derivatives of
W , i.e. all these functions depend on the scalar fields φ, ¯
φ only. We note that in the
higher-derivative theory (4.44) the auxiliary field F acquires a nontrivial dynamics.
The consequences of this fact, especially within the context of the spontaneous
supersymmetry breaking, are discussed in [64].
However, the theory (4.44), due to the presence of higher derivatives, involves
ghost states (a possible way for their eliminating is discussed in [64]), and the theory
(4.45) is non-renormalizable for general forms of K and W . We will study perturbative aspects of these theories within the superfield approach further.
4.2.2 Abelian Gauge Superfield Model
Now, let us go to the real scalar superfield case. The key feature of this superfield
consists in the fact that it allows for introducing the gauge symmetry on the superspace.
Indeed, let us consider the action of the real scalar superfield V (see e.g. [32]):
S =
1
2
d
8 zV (
D
α ¯
D
2 D α
8
)V.
(4.46)
This action is evidently gauge invariant with respect to the transformations:
V → V + i(( − ¯
),
(4.47)
where is a chiral superfield parameter, and ¯
is an antichiral one (to show the
invariance, we use the fact that ¯
D
2 D α = 0 for a chiral ). Further, to develop a
consistent quantum description, we must fix the gauge.
This action is constructed with use of the operator −
D
α ¯
D
2 D α
8
≡ 1/2 , where
1/2 = −
D
α ¯
D
2 D α
8
is a projecting operator possessing the property
n
1/2 = 1/2 for
any integer n ≥ 1. One can see that there are two projecting operators more, 0+ =
¯
D
2 D
2
16
and 0− =
D
2 ¯
D
2
16
satisfying the similar properties, i.e.
n
0± = 0± . One can
say that 0+ is a projector on the chiral space, 0− is a projector on the antichiral
space since acting of 0+ on any superfield produces the chiral superfield, and
of 0− —the antichiral superfield. The 1/2 projects on the so-called linear space
since, for any superfield , the new superfield 1/2 ≡ ˜
possesses the properties
D
2 ˜
= ¯
D
2 ˜
= 0, and such a superfield is called the linear superfield. However,
it is used very rarely, some results for it can be found in [23], and its interesting
application for formulating the Goldstino model is presented in [65].
It is straightforward to show that the projecting operators 1/2 , 0+ and 0−
satisfy the properties
4 Four-Dimensional Superfield Supersymmetry
Here K φ =
∂ K ((, ¯
)
∂
| =φ, ¯
= ¯
φ etc., the similar definitions are applied to derivatives of
W , i.e. all these functions depend on the scalar fields φ, ¯
φ only. We note that in the
higher-derivative theory (4.44) the auxiliary field F acquires a nontrivial dynamics.
The consequences of this fact, especially within the context of the spontaneous
supersymmetry breaking, are discussed in [64].
However, the theory (4.44), due to the presence of higher derivatives, involves
ghost states (a possible way for their eliminating is discussed in [64]), and the theory
(4.45) is non-renormalizable for general forms of K and W . We will study perturbative aspects of these theories within the superfield approach further.
4.2.2 Abelian Gauge Superfield Model
Now, let us go to the real scalar superfield case. The key feature of this superfield
consists in the fact that it allows for introducing the gauge symmetry on the superspace.
Indeed, let us consider the action of the real scalar superfield V (see e.g. [32]):
S =
1
2
d
8 zV (
D
α ¯
D
2 D α
8
)V.
(4.46)
This action is evidently gauge invariant with respect to the transformations:
V → V + i(( − ¯
),
(4.47)
where is a chiral superfield parameter, and ¯
is an antichiral one (to show the
invariance, we use the fact that ¯
D
2 D α = 0 for a chiral ). Further, to develop a
consistent quantum description, we must fix the gauge.
This action is constructed with use of the operator −
D
α ¯
D
2 D α
8
≡ 1/2 , where
1/2 = −
D
α ¯
D
2 D α
8
is a projecting operator possessing the property
n
1/2 = 1/2 for
any integer n ≥ 1. One can see that there are two projecting operators more, 0+ =
¯
D
2 D
2
16
and 0− =
D
2 ¯
D
2
16
satisfying the similar properties, i.e.
n
0± = 0± . One can
say that 0+ is a projector on the chiral space, 0− is a projector on the antichiral
space since acting of 0+ on any superfield produces the chiral superfield, and
of 0− —the antichiral superfield. The 1/2 projects on the so-called linear space
since, for any superfield , the new superfield 1/2 ≡ ˜
possesses the properties
D
2 ˜
= ¯
D
2 ˜
= 0, and such a superfield is called the linear superfield. However,
it is used very rarely, some results for it can be found in [23], and its interesting
application for formulating the Goldstino model is presented in [65].
It is straightforward to show that the projecting operators 1/2 , 0+ and 0−
satisfy the properties