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

TL;DR

This paper computes the operator product expansions in the 2D O(N) non-linear sigma model, revealing how geometric structures like connections relate to short-distance singularities and temperature effects.

Contribution

It introduces a geometric framework for understanding operator product expansions in the 2D O(N) model and computes the connection perturbatively, highlighting temperature-dependent singularities.

Findings

Connection becomes free of singularities at zero temperature with proper normalization

Perturbative computation of the geometric connection in the O(N) model

Correlation functions have well-defined limits at zero temperature with normalization

Abstract

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.