4.10 The Higher-Derivative Chiral Superfield Models
119
one can show that the one-loop effective action corresponding to the theory (4.258)
can be expressed through the following Schwinger representation
(1)
=
i
2
Tr
ds
s
exp[is(( − M
2
) +
1
4
¯
D
2
+
1
4
¯
D
2
)].
(4.260)
Since we restrict ourselves here to the Kählerian part of the effective action, we can
express the one-loop effective action as
(1)
=
i
2
Tr
d
8 z
ds
s
exp[is(
1
4
¯
D
2
+
1
4
¯
D
2
)]e
is(−M
2 )
δ
8
(z − z
)| z=z .
(4.261)
The relevant terms from the operator exp(is(
1
4
¯
D
2
+
1
4
¯
D
2
)) again have the form
(4.229), and, after Wick rotation s = it, the one-loop Kählerian effective action looks
like
(1)
K = −i
d
4
θd
4 x 1
dt
t
1
[cosh(t
¯
) − 1]e
−t(−M
2 )
δ
4
(x 1 − x 2 )| x 1 =x 2 .
(4.262)
Expanding the hyperbolic cosine in series in , after Wick rotation we find
(1)
K =
d
4
θd
4 x 1
dt
t
∞
n=0
1
(2n + 2)!
(t
2
¯
)
n+1
n V (t; x 1 , x 2 )| x 1 =x 2 . (4.263)
Here the function V (t; x 1 , x 2 ) can be read off from the (4.249). As we already noted,
this expression can be found in a closed form, for M = 0, only in the limit M → ∞.
It follows from (4.249) that
n V (t; x 1 , x 2 ) =
1
t n (
d
d(M 2 )
)
n V (t; x 1 , x 2 ),
(4.264)
so that, after taking x 1 = x 2 ,
n V (s; x 1 , x 2 )| x 1 =x 2 =
(−1)
n
(n + 1)!
16π 2 (M 2 t) n+2 .
(4.265)
Taking all together, we find
(1)
K =
1
32π 2
d
8 z
dt
M 2 t 2
∞
n=0
(−1)
n
(n + 1)!
(2n + 2)!
t ¯
M 2
n+1
.
(4.266)
This expression is similar to Eq. (4.176) obtained for the Wess-Zumino model. As a
result, we have