Abstract
In a previous paper, we developed a power series representation and estimates for an effective action of the form
ln
[
∫
e
f
(
α
1
,
…
,
α
s
;
z
∗
,
z
)
d
μ
(
z
∗
,
z
)
/
∫
e
f
(
0
,
…
,
0
;
z
∗
,
z
)
d
μ
(
z
∗
,
z
)
]
. Here,
f
(
α
1
,
…
,
α
s
;
z
∗
,
z
)
is an analytic function of the complex fields
α
1
(
x
)
,
…
,
α
s
(
x
)
,
z
∗
(
x
)
,
z
(
x
)
indexed by
x
in a finite set
X
and
d
μ
(
z
∗
,
z
)
is a compactly supported product measure. Such effective actions occur in the small field region for a renormalization group analysis. We illustrate the technique by a model renormalization group flow motivated by the ultraviolet regime in many boson systems.