  A class of scalar models with non-polynomial interaction, which naturally
admits an analytical resummation of the series of tadpole diagrams is studied
in perturbation theory. In particular, we focus on a model containing only one
renormalizable coupling that appear as a multiplicative coefficient of the
squared field. A renormalization group analysis of the Green functions of the
model shows that these are only approximated solutions of the flow equations,
with errors proportional to powers of the coupling, therefore smaller in the
region of weak coupling. The final output of the perturbative analysis is that
the renormalized model is non-interacting with finite mass and vanishing
vertices or, in an effective theory limited by an ultraviolet cut-off, the
vertices are suppressed by powers of the inverse cut-off. The relation with
some non-polynomial interactions derived long ago, as solutions of the
linearized functional renormalization group flow equations, is also discussed.
