Vacuum energy of the supersymmetric CP^N-1 model on R × S¹ in the 1/N expansion

By employing the 1/N expansion, we compute the vacuum energy E(δε) of the two-dimensional supersymmetric (SUSY) CP^N-1 model on R × S¹ with 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 S¹, R, RE(δε) behaves as inverse powers of Λ R for Λ R small, where Λ is the dynamical scale. Since Λ is related to the renormalized 't Hooft coupling Λ_R as Λ∼ 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,.