Higher-Order Corrections to Instantons

TL;DR

This paper analyzes higher-order instanton effects in the double-well potential, providing analytical calculations and numerical verification of multi-instanton contributions to energy levels beyond perturbation theory.

Contribution

It introduces analytical methods for higher-order multi-instanton corrections and verifies their accuracy against numerical data and perturbative series.

Findings

Two-instanton contributions accurately describe energy differences.

Higher-order corrections match numerical eigenvalues.

Perturbative coefficients' factorial growth aligns with instanton calculations.

Abstract

The energy levels of the double-well potential receive, beyond perturbation theory, contributions which are non-analytic in the coupling strength; these are related to instanton effects. For example, the separation between the energies of odd- and even-parity states is given at leading order by the one-instanton contribution. However to determine the energies more accurately multi-instanton configurations have also to be taken into account. We investigate here the two-instanton contributions. First we calculate analytically higher-order corrections to multi-instanton effects. We then verify that the difference betweeen numerically determined energy eigenvalues, and the generalized Borel sum of the perturbation series can be described to very high accuracy by two-instanton contributions. We also calculate higher-order corrections to the leading factorial growth of the perturbative coefficients and show that these are consistent with analytic results for the two-instanton effect and with exact data for the first 200 perturbative coefficients.