Large Orders for Self-Avoiding Membranes

TL;DR

This paper analyzes the large order behavior of perturbative expansions for self-avoiding membranes, revealing factorial growth and potential Borel summability, with implications for systematic expansions and calculations.

Contribution

It derives the large order behavior of the perturbative series for self-avoiding membranes and introduces a variational approximation for the instanton solution, enabling a systematic 1/d expansion.

Findings

Perturbative expansion exhibits factorial growth similar to polymers.

Instanton solution becomes exact in large bulk dimension d.

Epsilon-expansion likely Borel summable, affecting calculation methods.

Abstract

We derive the large order behavior of the perturbative expansion for the continuous model of tethered self-avoiding membranes. It is controlled by a classical configuration for an effective potential in bulk space, which is the analog of the Lipatov instanton, solution of a highly non-local equation. The n-th order is shown to have factorial growth as (-cst)^n (n!)^(1-epsilon/D), where D is the `internal' dimension of the membrane and epsilon the engineering dimension of the coupling constant for self-avoidance. The instanton is calculated within a variational approximation, which is shown to become exact in the limit of large dimension d of bulk space. This is the starting point of a systematic 1/d expansion. As a consequence, the epsilon-expansion of self-avoiding membranes has a factorial growth, like the epsilon-expansion of polymers and standard critical phenomena, suggesting Borel summability. Consequences for the applicability of the 2-loop calculations are examined.