Renormalizations and Operator Expansion in σ-Model

Abstract

The operator expansion (OPE) is studied for the Green function n(0) n(x)) at x2 → 0 (n(x) is the dynamical field of σ-model) in the framework of the two-dimensional σ-model with the O(N) symmetry group at large N. As a preliminary step we formulate the ronormalization scheme which permits introduction of an arbitrary intermediate scale μ2 in the framework of 1/N expansion and discuss factorization (separation) of small (p < μ ) and large (p > μ ) momentum region. It is shown that definition of composite local operators and coefficient functions figuring in OPE is unambiguous only in the leading order in 1/N expansion when dominant are the solutions with exrtemum of action. Corrections of order f(μ2)/N (here /(μ2) is the effective interaction constant at the point μ2) in composite operators and coefficient functions essentially depend on factorization method of high and low momentum regions. It is shown also that contributions to the power corrections of order m2x2/(μ2)/N in the Green function (here m is the dynamical mass-scale factor in σ-model) arise simultaneously from two sources: from the mean vacuum value of the composite operator n ∂2n and from the hard particle contributions in the coefficient function of unite operator. Due to the analogy between σ-model and QCD the obtained result indicates theoretical limitations to the sum rule method in QCD.