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

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.