Power series representation for Bosonic effective interactions

We develop a power series representation and estimates for an effective action of the form $$\log \frac{\int {\mathrm{e}}^{\mathcal{A}(\varphi ,\psi )}\mathrm{d}{\mu }_r(\varphi )}{\int {\mathrm{e}}^{\mathcal{A}(\varphi ,0)}\mathrm{d}{\mu }_r(\varphi )}.$$ Here, $\mathcal{A}(\varphi ,\psi )$ is an analytic function of the real fields $\varphi (x),\psi (x)$ indexed by $x$ in a finite set $X$, and $\mathrm{d}{\mu }_r(\varphi )$ is the product measure characterised by $$\int f(\varphi )\mathrm{d}{\mu }_r(\varphi )=\int f(\varphi )\prod _{x\in X}\chi (|\varphi (x)|\le r)\mathrm{d}\varphi (x).$$ Such effective interactions occur in the small field region for a renormlization group analysis. The customary way to analyse them is a cluster expansion, possibly preceded by a decoupling expansion. Using methods similar to a polymer expansion, we estimate the power series of effective interaction without introducing an artificial decomposition of the underlying space. This technique is illustrated by a model renormalization group flow motivated by the ultraviolet regime in many boson systems.