Three-loop Correction to the Instanton Density. II. The Sine-Gordon potential

TL;DR

This paper calculates the three-loop quantum corrections to the instanton density in the Sine-Gordon potential using quantum field theory, confirming previous results and revealing the nature of diagram contributions.

Contribution

It provides a three-loop calculation of the instanton density for the Sine-Gordon potential using quantum field theory, aligning with resurgence-based results.

Findings

Three-loop correction matches previous resurgence calculations.

Jacobian-related diagrams give the largest contribution.

Individual diagrams contain irrational parts, but their sum is rational.

Abstract

In this second paper on quantum fluctuations near the classical instanton configurations, see {\em Phys. Rev. D \bf 92}, 025046 (2015) and arXiv:1501.03993, we focus on another well studied quantum-mechanical problem, the one-dimensional Sine-Gordon potential (the Mathieu potential). Using only the tools from quantum field theory, the Feynman diagrams in the instanton background, we calculate the tunneling amplitude (the instanton density) to the three-loop order. The result confirms (to seven significant figures) the one recently recalculated by G. V. Dunne and M. \"{U}nsal, {\it Phys. Rev. \bf D 89}, 105009 (2014) from the resurgence perspective. As in the double well potential case, we found that the largest contribution is given by the diagrams originating from the Jacobian. We again observe that in the three-loop case individual Feynman diagrams contain irrational contributions, while their sum does not.