Constrained instantons in scalar field theories

TL;DR

This paper develops a comprehensive method for calculating vacuum decay rates in scalar field theories lacking traditional instanton solutions, by utilizing constrained instantons and analyzing their properties numerically.

Contribution

It extends the concept of constrained instantons into a complete computational framework for theories without saddle points, demonstrated on a scalar field with negative quartic interaction.

Findings

Identified two solution branches with distinct properties.

Numerically solved field equations revealing a two-branch structure.

Clarified the role of constrained instantons in vacuum decay.

Abstract

Instantons, localised saddle points of the action, play an important role in describing non-perturbative aspects of quantum field theories, for example vacuum decay or violation of conservation laws associated with anomalous symmetries. However, there are theories in which no saddle point exists. In this paper, we revisit the idea of constrained instantons, proposed initially by Affleck in 1981, and develop it into a complete method for computing the vacuum decay rate in such cases. We apply this approach to the massive scalar field theory with a negative quartic self-interaction using two different constraints. We solve the field equations numerically and find a two-branch structure, with two distinct solutions for each value of the constraint. By counting the negative modes, we identify one branch of solutions as the constrained instantons and the other as the minima of the action subject to the constraint. We discuss their significance for the computation of the vacuum decay rate.