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 CP(N-1) 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 $\mathcal{E}(x,\mathrm{\Lambda },L)$ 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 $\mathcal{E}(x,\mathrm{\Lambda },L)\to \frac{N}{4\pi }{\mathrm{\Lambda }}^2$ at large $L\mathrm{\Lambda }$ is shown, both analytically and numerically, to be exponential: no power corrections are present and in particular no L"uscher 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 $L\mathrm{\Lambda }\to 0$ and the “confined” phase of the standard CP(N-1) model in two dimensions for $L\mathrm{\Lambda }\to \mathrm{\infty }$.