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

Abstract

By employing the $1/N$ expansion, we compute the vacuum energy~$E(\delta ϵ)$ 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~$\delta ϵ$. 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~$\delta ϵ$ 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~$S^1$, $R$, $RE(\delta ϵ)$ 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~$\Lambda \sim e^{-2\pi /{\lambda }_R}$, 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~${\lambda }_R$.

Comments: 24 pages, 3 figures. Fig. 2 was replaced by a correct one and corresponding descriptions in the text were corrected. A note added. The final version to appear in PTEP