Power series representation for Bosonic effective interactions

We develop a power series representation and estimates for an effective action of the form $$ \log\frac{\int{\rm e}^{\mathcal{A}(\phi,\psi)}{\rm d}\mu_r(\phi)}{\int{\rm e}^{\mathcal{A}(\phi,0)}{\rm d}\mu_r(\phi)}. $$ Here, $ \mathcal{A}(\phi,\psi) $ is an analytic function of the real fields $ \phi(x),\psi(x) $ indexed by $ x $ in a finite set $ X$, and $ {\rm d}\mu_r(\phi) $ is the product measure characterised by $$ \int f(\phi){\rm d}\mu_r(\phi)=\int f(\phi)\prod_{x\in X}\chi(|\phi(x)|\le r){\rm d}\phi(x). $$ 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.