Large-N CP(N-1) sigma model on a finite interval and the renormalized string energy

We continue the analysis started in a recent paper of the large-N two-dimensional sigma model, defined on a finite space interval L with Dirichlet (or Neumann) boundary conditions. Here we focus our attention on the problem of the renormalized energy density (x, Λ, L) which is found to be a sum of two terms, a constant term coming from the sum over modes, and a term proportional to the mass gap. The approach to E(x,Λ,L) → N Λ^2/(4 pi) at large LΛ is shown, both analytically and numerically, to be exponential: no power corrections are present and in particular no Lüscher term appears. This is consistent with the earlier result which states that the system has a unique massive phase, which interpolates smoothly between the classical weakly-coupled limit for LΛ → 0 and the “confined” phase of the standard model in two dimensions for LΛ → ∞.