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We calculate the one loop beta functions for nonlinear sigma models in fourdimensions containing general two and four derivative terms. In the O(N) modelthere are four such terms and nontrivial fixed points exist for all N \geq 4.In the chiral SU(N) models there are in general six couplings, but only fivefor N=3 and four for N=2; we find fixed points only for N=2,3. In theapproximation considered, the four derivative couplings are asymptotically freebut the coupling in the two derivative term has a nonzero limit. These resultssupport the hypothesis that certain sigma models may be asymptotically safe.

One loop beta functions and fixed points in Higher Derivative Sigma Models