Exact form factors of the O(N) $\sigma$-model

TL;DR

This paper develops a comprehensive formula for calculating exact form factors in the O(N) sigma-model, verifies it against large N expansions, and explicitly computes low particle form factors for specific cases like O(3) and O(4).

Contribution

It introduces a general form factor formula for the O(N) sigma-model and confirms its accuracy through large N limit comparisons with Feynman graph expansions.

Findings

Large N form factors match Feynman graph expansions.

Explicit low particle form factors for O(3) and O(4) models.

Agreement between exact formulas and 1/N expansion results.

Abstract

A general form factor formula for the $O(N)\sigma$-model is constructed and applied to several operators. The large N limits of these form factors are computed and compared with the 1/N expansion of the $O(N)\sigma$-model in terms of Feynman graphs and full agreement is found. In particular, O(3) and O(4) form factors are discussed. For the $O(3)\sigma$-model several low particle form factors are calculated explicitly.