A Closer Look at Constrained Instantons

TL;DR

This paper demonstrates that conventional gauge-invariant constraints can be used consistently for constructing constrained instantons in theories with spontaneous symmetry breaking, resolving previous difficulties.

Contribution

It re-examines the asymptotic structure of constrained instantons, providing explicit constructions and numerical support to show the consistency of gauge-invariant constraints.

Findings

Conventional gauge-invariant constraints are compatible with constrained instanton solutions.

Explicit constructions of constrained instantons in massive φ^4 and Yang--Mills theories are provided.

Numerical solutions support the analytic matching and boundary condition satisfaction.

Abstract

Instantons play a crucial role in understanding non-perturbative dynamics in quantum field theories, including those with spontaneously broken gauge symmetries. In the broken phase, finite-size instanton-like configurations are no longer exact stationary points of the Euclidean action, in contrast to the symmetric phase. Non-perturbative effects in this setting are therefore typically studied within the constrained instanton framework. However, a previous study pointed out a possible difficulty in constructing consistent constrained instanton solutions based on conventional gauge-invariant constraints. In this work, we revisit the asymptotic structure of constrained instantons and re-examine the claimed difficulty. By carefully tracking the behavior of the solutions near the spatial origin and at infinity, we show that the required boundary conditions can be satisfied without encountering the inconsistency. We explicitly construct consistent constrained instantons in both massive $\phi^4$ theory and Yang--Mills theory with spontaneous symmetry breaking, and we support our analytic matching procedure with numerical solutions. Our results establish that conventional gauge-invariant constraints can be consistently employed in semiclassical computations when asymptotic expansions are treated properly.