Instanton calculus for the self-avoiding manifold model

TL;DR

This paper develops instanton calculus methods to analyze the large order asymptotics of the self-avoiding manifold model, including UV divergence handling, large-d expansion, and IR divergence analysis, advancing understanding of membrane models.

Contribution

It introduces a systematic instanton calculus approach for the SAM model, including UV divergence renormalization and large-d expansion techniques.

Findings

Computed the functional determinant of fluctuations around instantons.

Identified UV and IR divergence issues and their resolutions.

Validated methods against known results for self-avoiding walk (D=1).

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 epsilon-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 epsilon->0 and d->infinity, as well as IR divergences when epsilon= 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 epsilon= 0. The consistency with the standard instanton-calculus results for the self-avoiding walk is checked for D = 1.