4.9 General Chiral Superfield Model
105
By this definition, there is no higher derivatives in the classical action. Therefore the
theory with the action (4.205) is the most general theory without higher derivatives
describing a dynamics of a chiral superfield. There are a lot of phenomenological
applications of this model in string theory (see [80] and references therein). In a general case this theory is non-renormalizable. However, non-renormalizable theories
are naturally treated within the framework of the effective field theory approach [81].
Therefore all integrals over momenta effectively involve natural cutoff through the
condition p M String where p is momentum, and M String = 10
17 GeV ∼ 10
−2 M Pl
is a characteristic string mass.
The effective action in the theory, just as in the Wess-Zumino model, can be
presented as a series in supercovariant derivatives D A = (∂ a , D α , ¯
D ˙
α ) in the form
(4.147)–(4.151) being described by the same objects, that is, Kählerian effective
potential K e f f , auxiliary fields’ effective potential F e f f which we do not consider in
this section, and the chiral effective potential W e f f .
The one-loop contribution to the effective action is totally determined by the
quadratic part of expansion of
1
S[ ¯
+
√
¯
φ, , +
√ φ] in quantum fields φ, ¯
φ
which looks like
S 2 =
1
2
d
8 z
φ ¯
φ
K K ¯
K ¯
K ¯
¯
φ
¯
φ
+ [
d
6 z
1
2
W
φ
2
+ h.c.] (4.206)
and defines the propagators, and the higher terms of this expansion define the vertices.
Here K ¯
=
∂
2 K ( ¯
,,)
∂∂ ¯
, K =
∂
2 K ( ¯
,,)
∂ 2
, etc, W
=
d
2 W
d 2 .
The corresponding matrix Green function is again represented by the form (4.156).
In the case when all derivatives of superfields , ¯
are omitted (that is just the case
of the Kählerian effective potential), it satisfies the equation
W
−
1
4
K ¯
¯
D
2
−
1
4
K ¯
D
2
¯
W
G ++ (z 1 , z 2 ) G +− (z 1 , z 2 )
G −+ (z 1 , z 2 ) G −− (z 1 , z 2 )
= −
δ + 0
0 δ −
.
(4.207)
The solution of this equation looks like
G =
1
K
2
¯
− W
¯
W
¯
W
1
4
K ¯
¯
D
2
1
4
K ¯
D
2
W
δ + 0
0 δ −
.
(4.208)
Now we turn to studying of quantum contributions to the Kählerian effective potential
depending only on superfields , ¯
but not on their derivatives.
The one-loop Feynman supergraphs contributing to the Kählerian effective potential are given by Fig. 4.11.
Here, bold external lines correspond to alternating W
and ¯
W
. The bold internal
lines are the background dependent φ ¯
φ-propagators of the form
G 0 ≡ ≡φ ¯
φ = −i
¯
D
2 D
2
16K ¯
δ
8
(z 1 − z 2 ).
(4.209)
105
By this definition, there is no higher derivatives in the classical action. Therefore the
theory with the action (4.205) is the most general theory without higher derivatives
describing a dynamics of a chiral superfield. There are a lot of phenomenological
applications of this model in string theory (see [80] and references therein). In a general case this theory is non-renormalizable. However, non-renormalizable theories
are naturally treated within the framework of the effective field theory approach [81].
Therefore all integrals over momenta effectively involve natural cutoff through the
condition p M String where p is momentum, and M String = 10
17 GeV ∼ 10
−2 M Pl
is a characteristic string mass.
The effective action in the theory, just as in the Wess-Zumino model, can be
presented as a series in supercovariant derivatives D A = (∂ a , D α , ¯
D ˙
α ) in the form
(4.147)–(4.151) being described by the same objects, that is, Kählerian effective
potential K e f f , auxiliary fields’ effective potential F e f f which we do not consider in
this section, and the chiral effective potential W e f f .
The one-loop contribution to the effective action is totally determined by the
quadratic part of expansion of
1
S[ ¯
+
√
¯
φ, , +
√ φ] in quantum fields φ, ¯
φ
which looks like
S 2 =
1
2
d
8 z
φ ¯
φ
K K ¯
K ¯
K ¯
¯
φ
¯
φ
+ [
d
6 z
1
2
W
φ
2
+ h.c.] (4.206)
and defines the propagators, and the higher terms of this expansion define the vertices.
Here K ¯
=
∂
2 K ( ¯
,,)
∂∂ ¯
, K =
∂
2 K ( ¯
,,)
∂ 2
, etc, W
=
d
2 W
d 2 .
The corresponding matrix Green function is again represented by the form (4.156).
In the case when all derivatives of superfields , ¯
are omitted (that is just the case
of the Kählerian effective potential), it satisfies the equation
W
−
1
4
K ¯
¯
D
2
−
1
4
K ¯
D
2
¯
W
G ++ (z 1 , z 2 ) G +− (z 1 , z 2 )
G −+ (z 1 , z 2 ) G −− (z 1 , z 2 )
= −
δ + 0
0 δ −
.
(4.207)
The solution of this equation looks like
G =
1
K
2
¯
− W
¯
W
¯
W
1
4
K ¯
¯
D
2
1
4
K ¯
D
2
W
δ + 0
0 δ −
.
(4.208)
Now we turn to studying of quantum contributions to the Kählerian effective potential
depending only on superfields , ¯
but not on their derivatives.
The one-loop Feynman supergraphs contributing to the Kählerian effective potential are given by Fig. 4.11.
Here, bold external lines correspond to alternating W
and ¯
W
. The bold internal
lines are the background dependent φ ¯
φ-propagators of the form
G 0 ≡ ≡φ ¯
φ = −i
¯
D
2 D
2
16K ¯
δ
8
(z 1 − z 2 ).
(4.209)