Critical curvature of large-$N$ nonlinear $O(N)$ sigma model on $S^2$

TL;DR

This paper analyzes the nonlinear $O(N)$ sigma model on a curved sphere, identifying a critical curvature threshold for mass generation that depends on the gravitational coupling, revealing a new mathematical constant.

Contribution

It provides an analytical determination of the critical curvature in the large-$N$ limit, generalizing the concept of the Euler-Mascheroni constant.

Findings

Existence of a critical curvature $R_c$ for mass generation.

Critical curvature $R_c$ is a function of the gravitational coupling $\xi$.

Mass generation occurs only when $R<R_c$.

Abstract

We study the nonlinear $O(N)$ sigma model on $S^2$ with the gravitational coupling term, by evaluating the effective potential in the large-$N$ limit. It is shown that there is a critical curvature $R_c$ of $S^2$ for any positive gravitational coupling constant $\xi$, and the dynamical mass generation takes place only when $R<R_c$. The critical curvature is analytically found as a function of $\xi$ $(>0)$, which leads us to define a function looking like a natural generalization of Euler-Mascheroni constant.