Instantons in $\phi^4$ Theories: Transseries, Virial Theorems and Numerical Aspects

TL;DR

This paper investigates the numerical properties of instantons in $ abla$-vector $ abla$-theories, combining transseries and convergence techniques to accurately compute integrals relevant for loop corrections and large-order behavior.

Contribution

It introduces a novel numerical approach using transseries and convergence acceleration to evaluate instanton integrals in $ abla$-vector $ abla$-theories.

Findings

High-precision instanton integral values obtained.

Enhanced understanding of large-order perturbation growth.

Method improves accuracy of instanton-related calculations.

Abstract

We discuss numerical aspects of instantons in two- and three-dimensional $\phi^4$ theories with an internal $O(N)$ symmetry group, the so-called $N$-vector model. Combining asymptotic transseries expansions for large argument with convergence acceleration techniques, we obtain high-precision values for certain integrals of the instanton that naturally occur in loop corrections around instanton configurations. Knowledge of these numerical properties are necessary in order to evaluate corrections to the large-order factorial growth of perturbation theory in $\phi^4$ theories. The results contribute to the understanding of the mathematical structures underlying the instanton configurations.