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 $N\ge 4$. 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.