  We investigate three-form gauge theories with higher derivative interactions
and their supersymmetric extensions in four space-time dimensions. For the
bosonic three-form gauge theories, we show that derivatives on the field
strength of the 3-form gauge field yield a tachyon as far as the Lagrangian
contains a quadratic kinetic term, while such the term with opposite sign gives
rise to a ghost. We confirm that there is neither a tachyon nor a ghost when
all higher derivative terms are given by functions of the field strength. For
this ghost/tachyon-free Lagrangian, we determine the boundary term necessary
for the consistency between the equation of motion and energy-momentum tensor.
For supersymmetric extensions, we present ghost/tachyon-free higher derivative
interactions of arbitrary order of the field strength and corresponding
boundary terms as well.
