Operator Product Expansions in the Two-Dimensional O(N) Non-Linear Sigma Model.pdf

a r X i v : h e p - t h / 9 4 0 6 0 0 7 v 1 3 J u n 1 9 9 4 UCLA/94/TEP/23 Operator Product Expansions in the Two-Dimensional O(N) Non-Linear Sigma Model? Hidenori SONODA? and Wang-Chang SU? Department of Physics, UCLA, Los Angeles, CA 90024–1547, USA The short-distance singularity of the product of a composite scalar field that deforms a field theory and an arbitrary composite field can be expressed geometrically by the beta functions, anomalous dimensions, and a connection on the theory space. Using this relation, we compute the connection perturbatively for the O(N) non-linear sigma model in two dimensions. We show that the connection becomes free of singularities at zero temperature only if we normalize the composite fields so that their correlation functions have well-defined limits at zero temperature. June 1994 ? This work was supported in part by the U.S. Department of Energy, under Contract DE- AT03-88ER 40384 Mod A006 Task C. ? sonoda@physics.ucla.edu ? suw@physics.ucla.edu 1. Introduction In refs. [1], [2], and [3] it was found that the singularities of a conjugate field that deforms a field theory and an arbitrary composite field admit a geometrical expression. This paper is a continuation of the study of the geometrical structure. The simple example of the four-dimensional φ4 theory has been examined in ref. [2]. In the present paper we will study a more non-trivial example of the two-dimensional non-linear sigma model in some detail. Let us briefly summarize the geometrical structure obtained in refs. [1], [2], and [3]. We consider a finite dimensional theory space with local coordinates gi(i = 1, ..., N), which are nothing but the parameters of a renormalized euclidean field theory in D-dimensions. Let the scalar field conjugate to gi be Oi. The conjugate fields form a basis of the tangent vector bundle of the theory space. The linearly independent composite fields {Φa}g make a basis of an infinite dimensional vector bundle. We denote the renorma