  We study several problems related to the construction and the use of
effective Lagrangians by considering an extension of the standard model that
includes a heavy scalar singlet coupled to the leptonic doublet. Starting from
the full renormalizable model, we build an effective field theory by
integrating out the heavy scalar. A local effective Lagrangian (up to operators
of dimension six) is obtained by expanding the one-loop effective action in
inverse powers of the heavy mass. This is done by matching some Green functions
calculated with both the full and the effective theories.
  Using this simple example we study the renormalization of effective
Lagrangians in general and discuss how they can be used to bound new physics.
We also discuss the effective Lagrangian after spontaneous symmetry breaking,
and the use of the standard model classical equations of motion to rewrite it
in different forms. The final effective Lagrangian in the physical basis is
well suited to the study of the phenomenology of the model, which we comment on
briefly. Finally, as an example of the use of our effective field theory, we
consider the leptonic flavour-changing decay of the $Z$ boson in the effective
theory and compare the results obtained with the full model calculation.
