From the mass gap in O(N) to the non-Borel-summability in O(3) and O(4) sigma-models

TL;DR

This paper analytically derives the mass gap in O(N) sigma models and examines the large-order behavior of their weak coupling expansions, revealing sign-oscillation for large N but positivity for N=3 and 4.

Contribution

It provides an analytical derivation of the mass gap and analyzes the asymptotic series behavior for different N values in O(N) sigma models.

Findings

Large N series sign-oscillates, as expected from solutions.

N=3 and N=4 series are sign-positive.

Insights into non-Borel-summability in specific models.

Abstract

We give an analytical derivation of the mass gap of the O(N) sigma models and investigate a large-order behavior of the weak coupling asymptotic expansion for the energy. For sufficiently large N the series is sign-oscillating, which is expected from the large N solution of the sigma model. However, for N=3 and N=4 the series are sign positive.