  We develop a novel cluster expansion for finite-spin lattice systems subject
to multi-body quantum -- and, in particular, classical -- interactions. Our
approach is based on the use of ``decoupling parameters", advocated by Park
[34], which relates partition functions with successive additional interaction
terms. Our treatment, however, leads to an explicit expansion in a
$\beta$-dependent effective fugacity that permits an explicit evaluation of
free energy and correlation functions at small $\beta$. To determine its
convergence region we adopt a relatively recent cluster summation scheme that
replaces the traditional use of Kikwood-Salzburg-like integral equations by
more precise sums in terms of particular tree-diagrams [2]. As an application
we show that our lower bound of the radius of $\beta$-analyticity is larger
than Park's for quantum systems two-body interactions.
