4.9 General Chiral Superfield Model
109
K = φ ¯
φ + ¯
; W =
M
2
2
+
λ
2
φ
2
+
g
3!
φ
3
.
(4.213)
Let us, for the sake of the simplicity, consider the situation when the heavy superfield
is a purely quantum one, and the light superfield φ is a purely background one. In
this case the quadratic action of looks like
S 2 =
d
8 z ¯
+ (
1
2
d
6 z(M + λφ))
2
+ h.c.)
(4.214)
It is evident that this action identically replays the quadratic action of the quantum fields in the usual Wess-Zumino model after the background-quantum splitting
(besides of (4.206), see also the previous section of this review and [27]). So, one can
repeat all calculations performed for the Wess-Zumino model, and the renormalized
one-loop contribution to the low-energy effective action is given by the result (4.211),
where K ((, ,) = ¯
. Explicitly, the one-loop Kählerian effective potential in our
theory is
K
(1)
= −
1
32π 2 (M + λφ)(M + λ ¯
φ) ln
(M + λφ)(M + λ ¯
φ)
μ 2
.
(4.215)
We expand it in power series in
1
M
, with introducing a new parameter μ
= αμ, where
α is a some real number:
K
(1)
= −
1
32π 2 λ
2
φ ¯
φ ln
M
2
μ 2 + O(
1
M
).
(4.216)
This expression involves the term ln
M
2
μ 2 . It is clear that this term is not suppressed
at M → ∞ but instead of this, increases as M grows. Moreover, if one suggests
that the light field φ is not purely external but also possesses a nontrivial quantum
dynamics, the contribution to one-loop Kählerian effective potential generated by
Feynman diagrams constructed of light propagators is
K
(1)
φ = −
1
32π 2 g
2
φ ¯
φ ln
g
2
φ ¯
φ
μ 2 ,
(4.217)
and if we fix the normalization parameter as μ = αM, we see that in this case, instead
of the contribution (4.216), the (4.217) will grow as M → ∞. The same situation
can be verified as well for other models involving light and heavy fields in different
situations (presence of nonminimal couplings, nontrivial quantum and background
parts both for light and heavy fields etc.) [85].
To argue the existence of the contributions increasing with growth of M in a
general case, one can notice as well the following facts. First, if the theory involves
a cubic self-coupling of the light (massless) chiral superfield φ with a coupling g,
as a consequence of integrating over light quantum fields the one-loop effective
109
K = φ ¯
φ + ¯
; W =
M
2
2
+
λ
2
φ
2
+
g
3!
φ
3
.
(4.213)
Let us, for the sake of the simplicity, consider the situation when the heavy superfield
is a purely quantum one, and the light superfield φ is a purely background one. In
this case the quadratic action of looks like
S 2 =
d
8 z ¯
+ (
1
2
d
6 z(M + λφ))
2
+ h.c.)
(4.214)
It is evident that this action identically replays the quadratic action of the quantum fields in the usual Wess-Zumino model after the background-quantum splitting
(besides of (4.206), see also the previous section of this review and [27]). So, one can
repeat all calculations performed for the Wess-Zumino model, and the renormalized
one-loop contribution to the low-energy effective action is given by the result (4.211),
where K ((, ,) = ¯
. Explicitly, the one-loop Kählerian effective potential in our
theory is
K
(1)
= −
1
32π 2 (M + λφ)(M + λ ¯
φ) ln
(M + λφ)(M + λ ¯
φ)
μ 2
.
(4.215)
We expand it in power series in
1
M
, with introducing a new parameter μ
= αμ, where
α is a some real number:
K
(1)
= −
1
32π 2 λ
2
φ ¯
φ ln
M
2
μ 2 + O(
1
M
).
(4.216)
This expression involves the term ln
M
2
μ 2 . It is clear that this term is not suppressed
at M → ∞ but instead of this, increases as M grows. Moreover, if one suggests
that the light field φ is not purely external but also possesses a nontrivial quantum
dynamics, the contribution to one-loop Kählerian effective potential generated by
Feynman diagrams constructed of light propagators is
K
(1)
φ = −
1
32π 2 g
2
φ ¯
φ ln
g
2
φ ¯
φ
μ 2 ,
(4.217)
and if we fix the normalization parameter as μ = αM, we see that in this case, instead
of the contribution (4.216), the (4.217) will grow as M → ∞. The same situation
can be verified as well for other models involving light and heavy fields in different
situations (presence of nonminimal couplings, nontrivial quantum and background
parts both for light and heavy fields etc.) [85].
To argue the existence of the contributions increasing with growth of M in a
general case, one can notice as well the following facts. First, if the theory involves
a cubic self-coupling of the light (massless) chiral superfield φ with a coupling g,
as a consequence of integrating over light quantum fields the one-loop effective