Instability of the fixed point of theO(N) nonlinear σ-model in (2+ε) dimensions

We calculate the full dimension (canonical plus anomalous), ${\mathit{y}}_{2\mathit{s}}$, of an infinite number of O(N) invariant operators with 2s gradients. We find that for large (ε=d-2), ${\mathrm{y}}_{2\mathrm{s}}$=2(1-s)+ε{1+[${\mathrm{s}}^2$/ (N-2)][1+O(1/s)]}+[${\mathrm{\epsilon }}^2$ ${\mathit{s}}^3$/(N-2${)}^2$]2/3+O(1/s)]+O(${\mathrm{\epsilon }}^3$), correct to two-loop order. Thus, in two-loop order ${\mathit{y}}_{2\mathit{s}}$ grows even more rapidly than in one-loop order. Even if ε 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.