  In this work we study the main properties and the one-loop renormalization of
a Yang-Mills theory in which the kinetic term contains also a fourth-order
differential operator; in particular, we add to the Yang-Mills Lagrangian the
most general contribution of mass dimension six, weighted with a dimensionful
parameter. This model is renormalizable; in the literature two values for the
beta function for the gauge coupling have been reported, one obtained using the
heat kernel approach and one with Feynman diagrams. In this work we repeat the
computation using heat kernel techniques confirming the latter result. We also
considered coupling with matter.
  We then study the supersymmetric extension of the model; this is a nontrivial
task because of the complicate structure of the higher-derivative term. Some
partial results were known, but a computation of the beta functions for the
full supersymmetric non-Abelian higher-derivative gauge theory was missing. We
make use of the (unextended) supersymmetric higher-derivative Lagrangian
density for the Yang-Mills field in six spacetime dimensions derived in
arXiv:hep-th/0505082; by dimensional reduction we obtain the N=1 and N=2
supersymmetric higher-derivative super-Yang-Mills Lagrangian in four spacetime
dimensions, whose beta function we evaluate using heat kernels. We also deduce
the beta function for N=4 supersymmetry.
