Higher Derivative Sigma Models

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.