One loop beta functions and fixed points in higher derivative sigma models

We calculate the one loop beta functions of nonlinear sigma models in four dimensions containing general two-and four-derivative terms. In the O(N) model there are four such terms and nontrivial fixed points exist for all N >= 4. In the chiral SU(N) models there are in general six couplings, but only five for N - 3 and four for N = 2; we find fixed points only for N = 2, 3. In the approximation considered, the four-derivative couplings are asymptotically free but the coupling in the two-derivative term has a nonzero limit. These results support the hypothesis that certain sigma models may be asymptotically safe.