Vacuum energy of the supersymmetric $mathbb{C}P^{N-1}$ model on $mathbb{R}times S^1$ in the $1/N$ expansion

By employing the $1/N$ expansion, we compute the vacuum energy~$E(deltaepsilon)$ of the two-dimensional supersymmetric (SUSY) $mathbb{C}P^{N-1}$ model on~$mathbb{R}times S^1$ with $mathbb{Z}_N$ twisted boundary conditions to the second order in a SUSY-breaking parameter~$deltaepsilon$. This quantity was vigorously studied recently by Fujimori et al. using a semi-classical approximation based on the bion, motivated by a possible semi-classical picture on the infrared renormalon. In our calculation, we find that the parameter~$deltaepsilon$ receives renormalization and, after this renormalization, the vacuum energy becomes ultraviolet finite. To the next-to-leading order of the $1/N$ expansion, we find that the vacuum energy normalized by the radius of the~$S^1$, $R$, $RE(deltaepsilon)$ behaves as inverse powers of~$Lambda R$ for~$Lambda R$ small, where $Lambda$ is the dynamical scale. Since $Lambda$ is related to the renormalized t~Hooft coupling~$lambda_R$ as~$Lambdasim e^{-2pi/lambda_R}$, to the order of the $1/N$ expansion we work out, the vacuum energy is a purely non-perturbative quantity and has no well-defined weak coupling expansion in~$lambda_R$.