106
4 Four-Dimensional Superfield Supersymmetry
. . .
Fig. 4.11 One-loop supegraphs contributing to Kählerian effective potential
Fig. 4.12 Obtaining the
background dependent
propagator (4.209) through
summation
=
+
.
.
.
.
+ . . .
D
2
¯
D
2
D
2
¯
D
2
D
2 ¯
D
2 D
2
¯
D
2
Fig. 4.13 A standard link in
a diagram contributing to the
one-loop effective potential
|
D
2
|
¯
D
2
We note that this propagator is valid for calculating not only of the Kählerian effective
potential but also of the chiral one. It can be obtained in two ways. The first way
consists in a summation over insertions (see Fig. 4.12).
Here dashed-and-dotted vertical line is the external field K ¯
− 1, and the horizontal simple line is the usual propagator − ¯
D
2
1 D
2
2
δ
8 (z 1 −z 2 )
16
: either if the external field
is constant or if it is chiral, we find that the sum of the contributions (with integrals
in internal points are assumed where it is necessary) above is
∞
n=0
−
¯
D
2 D
2
16
[−(K ¯
− 1)
¯
D
2 D
2
16
]
n
δ
8
(z 1 − z 2 ) = −
¯
D
2
1 D
2
2
16K ¯
(z 1 )
δ
8
(z 1 − z 2 ),
(4.210)
which proves (4.209).
In another manner, the result (4.209) can be proved through the definition of the
new chiral field φ
= K ¯
φ (indeed, in this case K ¯
is either constant or chiral,
so, φ
can be assumed to be chiral as well, so that the free action acquires the form
S f =
d
8 zφ
¯
φ which yields
(z 1 ) ¯
φ(z 2 ) = −
¯
D
2
1 D
2
2
16
δ
8
(z 1 − z 2 ), which, in its part,
implies (4.209).
A supergraph of the structure depicted at Fig. 4.11, with 2n legs represents itself
as a ring containing n links of the form given by Fig. 4.13.
Carrying out the calculations in the same way as in the previous section (see [82,
85] for the details), we find that the total contribution of all these diagrams after
D-algebra transformations, summation, integration over momenta and subtraction
of divergences is equal to
4 Four-Dimensional Superfield Supersymmetry
. . .
Fig. 4.11 One-loop supegraphs contributing to Kählerian effective potential
Fig. 4.12 Obtaining the
background dependent
propagator (4.209) through
summation
=
+
.
.
.
.
+ . . .
D
2
¯
D
2
D
2
¯
D
2
D
2 ¯
D
2 D
2
¯
D
2
Fig. 4.13 A standard link in
a diagram contributing to the
one-loop effective potential
|
D
2
|
¯
D
2
We note that this propagator is valid for calculating not only of the Kählerian effective
potential but also of the chiral one. It can be obtained in two ways. The first way
consists in a summation over insertions (see Fig. 4.12).
Here dashed-and-dotted vertical line is the external field K ¯
− 1, and the horizontal simple line is the usual propagator − ¯
D
2
1 D
2
2
δ
8 (z 1 −z 2 )
16
: either if the external field
is constant or if it is chiral, we find that the sum of the contributions (with integrals
in internal points are assumed where it is necessary) above is
∞
n=0
−
¯
D
2 D
2
16
[−(K ¯
− 1)
¯
D
2 D
2
16
]
n
δ
8
(z 1 − z 2 ) = −
¯
D
2
1 D
2
2
16K ¯
(z 1 )
δ
8
(z 1 − z 2 ),
(4.210)
which proves (4.209).
In another manner, the result (4.209) can be proved through the definition of the
new chiral field φ
= K ¯
φ (indeed, in this case K ¯
is either constant or chiral,
so, φ
can be assumed to be chiral as well, so that the free action acquires the form
S f =
d
8 zφ
¯
φ which yields
(z 1 ) ¯
φ(z 2 ) = −
¯
D
2
1 D
2
2
16
δ
8
(z 1 − z 2 ), which, in its part,
implies (4.209).
A supergraph of the structure depicted at Fig. 4.11, with 2n legs represents itself
as a ring containing n links of the form given by Fig. 4.13.
Carrying out the calculations in the same way as in the previous section (see [82,
85] for the details), we find that the total contribution of all these diagrams after
D-algebra transformations, summation, integration over momenta and subtraction
of divergences is equal to