  By employing the $1/N$ expansion, we compute the vacuum
energy~$E(\delta\epsilon)$ 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~$\delta\epsilon$. 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~$\delta\epsilon$ 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~$S^1$, $R$,
$RE(\delta\epsilon)$ 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~$\lambda_R$.
