  The neutral massless scalar quantum field $\Phi$ in four-dimensional
space-time is considered, which is subject to a simple bilinear
self-interaction. Is is well-known from renormalization theory that adding a
term of the form $-\frac{m^2}{2} \Phi^2$ to the Lagrangean has the formal
effect of shifting the particle mass from the original zero value to m after
resummation of all two-leg insertions in the Feynman graphs appearing in the
perturbative expansion of the S-matrix. However, this resummation is
accompanied by some subtleties if done in a proper mathematical manner.
Although the model seems to be almost trivial, is shows many interesting
features which are useful for the understanding of the convergence behavior of
perturbation theory in general. Some important facts in connection with the
basic principles of quantum field theory and distribution theory are
highlighted, and a remark is made on possible generalizations of the
distribution spaces used in local quantum field theory. A short discussion how
one can view the spontaneous breakdown of gauge symmetry in massive gauge
theories within a massless framework is presented.
