Instanton Calculus for the Self-Avoiding Manifold Model

Abstract

We compute the normalisation factor for the large order asymptotics of perturbation theory for the self-avoiding manifold (SAM) model describing flexible tethered (D-dimensional) membranes in d-dimensional space, and the ε-expansion for this problem. For that purpose, we develop the methods inspired from instanton calculus, that we introduced in a previous publication (Nucl. Phys. B 534 (1998) 555), and we compute the functional determinant of the fluctuations around the instanton configuration. This determinant has UV divergences and we show that the renormalized action used to make perturbation theory finite also renders the contribution of the instanton UV-finite. To compute this determinant, we develop a systematic large-d expansion. For the renormalized theory, we point out problems in the interplay between the limits ε→ 0 and d→∞, as well as IR divergences when ε=0. We show that many cancellations between IR divergences occur, and argue that the remaining IR-singular term is associated to amenable non-analytic contributions in the large-d limit when ε=0. The consistency with the standard instanton-calculus results for the self-avoiding walk is checked for D=1.