INSTABILITY OF THE FIXED-POINT OF THE O(N) NONLINEAR SIGMA-MODEL IN 2+ EPSILON DIMENSIONS

We calculate the full dimension (canonical plus anomalous), y2s, of an infinite number of O(N) invariant operators with 2s gradients. We find that for large s (epsilon = d - 2), y2s = 2(1 - s) + epsilon{1 + [s2/(N - 2)][1 + O(1/s)]} + [epsilon2s3/(N - 2)2][2/3 + O(1/s)] + O(epsilon3), correct to two-loop order. Thus, in two-loop order y2s grows even more rapidly than in one-loop order. Even if epsilon is arbitrarily small, one can always find, to two-loop order, operators with positive full dimension by choosing s sufficiently large. We argue that the conventional analysis of this problem may be inadequate.