Constrained instantons in scalar field theories

TL;DR

This paper develops a non-perturbative constrained-instanton method to compute vacuum decay rates in theories lacking ordinary instantons, by constraining the Euclidean path integral with a functional ξ[φ] and using a Lagrange-multiplier formulation. Applying the method to a massive scalar field with a negative φ^4 term, it demonstrates a robust two-branch structure of constrained solutions for both φ^3 and φ^6 constraints, where the upper branch yields genuine constrained instantons with a single negative mode and the lower branch yields action minima. The authors provide detailed numerical evidence, including the scaling relations, asymptotics, and a consistent counting of negative modes via the projection factor ν(ξ̄); they also discuss the implications for computing the vacuum decay rate and outline extensions to other theories. Overall, the constrained-instanton framework offers a practical pathway to evaluate non-perturbative decay rates in theories without standard saddles, with potential applications to electroweak vacuum stability and beyond.

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.

Figures (16)

• Figure 1: Solutions for the $\phi^3$ constraint for several values of $K_3\equiv \kappa/(m\lambda^{1/2})$.
• Figure 2: Constrained instantons for the $\phi^3$ constraint at different values of $K_3\equiv \kappa/(m\lambda^{1/2})$. The red and green dashed lines show the short and long distance fits given by eq. \ref{['eq:small-large-fits']}
• Figure 3: Left: The action $S$ of our solutions as a function of $K_3\equiv \kappa/(m\lambda^{1/2})$ for the $\phi^3$ constraint. The dashed red line denotes the value of the massless instanton action \ref{['eq:mlessact']}. The solid black line represents the solutions that contribute to the tunnelling rate, while the grey dashed line shows the ones that do not. Right: The constraint $\xi$ as a function of $K_3$ for the same solutions.
• Figure 4: The action as a function of the constraint $\bar{\xi}$ for the $\phi^3$ constraint. The black solid line corresponds to the solutions that contribute to the tunnelling rate. The dashed red line shows the massless instanton action.
• Figure 5: The projection prefactor $\nu$ as a function of $K_3\equiv \kappa/(m\lambda^{1/2})$ for the $\phi^3$ constraint. The red dashed line marks $\kappa = \kappa_\mathrm{crit}$.