  The loop vertex expansion (LVE) is a constructive technique which uses only
canonical combinatorial tools and no space-time dependent lattices. It works
for quantum field theories without renormalization. Renormalization requires
scale analysis. In this paper we provide an enlarged formalism which we call
the multiscale loop vertex expansion (MLVE). We test it on what is probably the
simplest quantum field theory which requires some kind of renormalization,
namely a combinatorial model of the vector type with quartic interaction and a
propagator which mimicks the power counting of $\phi^4_2$. An ordinary LVE
would fail to treat even this simplest superrenormalizable model, but we show
how to perform the ultraviolet limit and prove its analyticity in the Borel
summability domain of the model with the MLVE.
