  We explore the nature of running couplings in the higher derivative linear
and nonlinear sigma models and show that the results in dimensional
regularization for the physical running couplings do not always match the
values quoted in the literature. Heat kernel methods identify divergences
correctly, but not all of these divergences are related to physical running
couplings. Likewise the running found using the Functional Renormalization
Group does not always appear as running couplings in physical processes, even
for the case of logarithmic running. The basic coupling of the higher
derivative SU(N) nonlinear sigma model does not run at all at one loop, in
contrast to published claims for asymptotic freedom. At one loop we describe
how to properly identify the physical running couplings in these theories, and
provide revised numbers for the higher derivative nonlinear sigma model.
