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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~$E(δε)$ 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~$δε$. 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~$δε$ 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~$S^1$, $R$, $RE(δε)$ behaves as inverse powers of~$ΛR$ for~$ΛR$ small, where $Λ$ is the dynamical scale. Since $Λ$ is related to the renormalized 't~Hooft coupling~$λ_R$ as~$Λ\sim e^{-2π/λ_R}$, 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~$λ_R$.