Operator expansion in the σ model

In the two-dimensional σ model with the symmetry group O(N) at large N we investigate the operator expansion for the Green function for x2→0 (here, n(x) is the dynamical field of the σ model). As a preliminary we formulate in the framework of the 1/N expansion a renormalization scheme which assumes the introduction of an intermediate scale μ2, and we factorize the regions of large ( p>μ) and small ( p<μ) Euclidean momenta. We show that the definition of the composite operators and the coefficient functions which appear in the operator expansion is unique only in the leading approximation in the parameter 1/N, when solutions corresponding to an extremum of the action dominate. The corrections of order f(μ2)/N [here f(μ2) is the effective coupling constant at the point μ2] to the operators and the corrections --m2x2[ f(μ2)/N] (here m2 is the scale factor of the σ model) to the coefficient functions depend decisively on the way of factorizing the regions of large and small momenta. In view of the analogy between the σ model and quantum chromodynamics (QCD) this result indicates the limitations on the accuracy of the QCD sum-rule method