  Perturbative analyses seem to suggest that fermions whose mass comes solely
from a Yukawa coupling to a scalar field can be made arbitrarily heavy, while
the scalar remains light. The effects of the fermion can be summarized by a
local effective Lagrangian for the light degrees of freedom. Using weak
coupling and large N techniques, we present a variety of models in which this
conclusion is shown to be false when nonperturbative variations of the scalar
field are considered. The heavy fermions contribute nonlocal terms to the
effective action for light degrees of freedom. This resolves paradoxes about
anomalous and nonanomalous symmetry violation in these models. Application of
these results to lattice gauge theory imply that attempts to decouple lattice
fermion doubles by the method of Swift and Smit cannot succeed, a result
already suggested by lattice calculations.
