118
4 Four-Dimensional Superfield Supersymmetry
The presence of such a function seems to make impossible finding the explicit oneloop Kählerian potential in the general case. It is clear that (x → ∞) → 1. Indeed,
I (s) =
1
16π 2 (
1
2s
−
M
2
2
e
s M
2 /4
(
∞
0
due
−su
2 −
M
2 /2
0
due
−su
2 )) (4.254)
Substituting su
2
= w
2 , we get
I (s) =
1
16π 2 (
1
2s
−
M
2
2
e
s M
2 /4
(
1
2
π
s
−
1
√
s
M
2
√
s/2
0
dwe
−w
2 )) =
=
1
16π 2 [
1
2s
−
M
2
2
e
s M
2 /4 1
2
π
s
(1 − (M
2
√
s/2)].
(4.255)
To evaluate this expression, we employ the asymptotics of the probability integral
(y) at large arguments [97]:
(y)| y→∞ = 1 −
1
π
e
−y
2
∞
k=0
(−1)
k
(k +
1
2
)
y 2k+1
.
(4.256)
We find that the term with k = 0 identically cancels the “usual” term
1
2s
in (4.254).
Taking into account only the M → ∞ dominant term (remind that the limit of very
high masses was studied earlier in [85]), one finds
I (s) =
1
16π 2 s 2 M 4 ,
(4.257)
which differs from the case M = 0 considered earlier where the analogue of this
function was proportional to
1
s
. One could note that such behavior of the heat kernel seems to be similar to that one occurring in the Wess-Zumino model [27, 72].
Nevertheless, the presence of a large mass in the denominator gives a hope that the
corrections to the effective action will be suppressed in a M → ∞ limit.
To give a description of the principal difference of new theory, we restrict ourselves
only to calculating the Kählerian effective potential in a new theory. Let its action be
S[, ¯
] =
d
8 z( − M
2
) ¯
+ (
d
6 zW (() + h.c.).
(4.258)
Here M is a large parameter related to the physical mass. To simplify the calculation
of the effective action in the theory, it is natural to represent the one-loop contribution
in the form of a functional integral over the unique unconstrained real scalar field as
we have done in the Sect. 4.8 for the Wess-Zumino model. Using the insertion of the
effective action of the free real scalar superfield whose classical action looks like
S v = −
1
16
d
8 zv D
α ¯
D
2 D α ( − M
2
)v,
(4.259)
4 Four-Dimensional Superfield Supersymmetry
The presence of such a function seems to make impossible finding the explicit oneloop Kählerian potential in the general case. It is clear that (x → ∞) → 1. Indeed,
I (s) =
1
16π 2 (
1
2s
−
M
2
2
e
s M
2 /4
(
∞
0
due
−su
2 −
M
2 /2
0
due
−su
2 )) (4.254)
Substituting su
2
= w
2 , we get
I (s) =
1
16π 2 (
1
2s
−
M
2
2
e
s M
2 /4
(
1
2
π
s
−
1
√
s
M
2
√
s/2
0
dwe
−w
2 )) =
=
1
16π 2 [
1
2s
−
M
2
2
e
s M
2 /4 1
2
π
s
(1 − (M
2
√
s/2)].
(4.255)
To evaluate this expression, we employ the asymptotics of the probability integral
(y) at large arguments [97]:
(y)| y→∞ = 1 −
1
π
e
−y
2
∞
k=0
(−1)
k
(k +
1
2
)
y 2k+1
.
(4.256)
We find that the term with k = 0 identically cancels the “usual” term
1
2s
in (4.254).
Taking into account only the M → ∞ dominant term (remind that the limit of very
high masses was studied earlier in [85]), one finds
I (s) =
1
16π 2 s 2 M 4 ,
(4.257)
which differs from the case M = 0 considered earlier where the analogue of this
function was proportional to
1
s
. One could note that such behavior of the heat kernel seems to be similar to that one occurring in the Wess-Zumino model [27, 72].
Nevertheless, the presence of a large mass in the denominator gives a hope that the
corrections to the effective action will be suppressed in a M → ∞ limit.
To give a description of the principal difference of new theory, we restrict ourselves
only to calculating the Kählerian effective potential in a new theory. Let its action be
S[, ¯
] =
d
8 z( − M
2
) ¯
+ (
d
6 zW (() + h.c.).
(4.258)
Here M is a large parameter related to the physical mass. To simplify the calculation
of the effective action in the theory, it is natural to represent the one-loop contribution
in the form of a functional integral over the unique unconstrained real scalar field as
we have done in the Sect. 4.8 for the Wess-Zumino model. Using the insertion of the
effective action of the free real scalar superfield whose classical action looks like
S v = −
1
16
d
8 zv D
α ¯
D
2 D α ( − M
2
)v,
(4.259)