One-loop corrections to the instanton transition in the Abelian Higgs model: Gel'fand-Yaglom and Green's function methods

TL;DR

This paper compares Green's function and Gel'fand-Yaglom methods for calculating one-loop corrections to instanton transitions in the Abelian Higgs model, addressing challenges with nontrivial topology.

Contribution

It introduces modifications to the Gel'fand-Yaglom method to handle nontrivial topological backgrounds and compares it with Green's function approaches in this context.

Findings

Both methods yield consistent numerical results.

Modified Gel'fand-Yaglom method effectively handles topological backgrounds.

Comparison clarifies advantages and limitations of each approach.

Abstract

The fluctuation determinant, the preexponential factor for the instanton transition, has been computed several years ago in the Abelian Higgs model, using a method based on integrating the Euclidean Green' function. A more elegant method for computing functional determinants, using the Gel'fand-Yaglom theorem, has been applied recently to a variety of systems. This method runs into difficulties if the background field has nontrivial topology, as is the case for the instanton in the Abelian Higgs model. A shift in thre effective centrifugal barriers makes the s-wave contribution infinite, an infinity that is compensated by the summation over the other partial waves. This requires some modifications of the Gel'fand-Yaglom method which are the main subject of this work. We present here both, the Green' s function and the Gel'fand-Yaglom method and compare the numerical results in detail.