The Case of the Disappearing Instanton

TL;DR

The paper analyzes how instantons, the Euclidean solutions mediating vacuum tunneling, can disappear as the potential is varied, distinguishing smooth and abrupt losses of tunneling. It provides a unified semiclassical framework explaining abrupt disappearances via annihilation with higher-action solutions carrying extra negative modes, and shows these events occur in both flat-space and gravitational contexts, including cusp behavior when Hawking–Moss solutions participate. Through simple 1-field and 2-field examples and a gravitational treatment, the authors connect these disappearances to saddle-point structure and catastrophe theory, with a detailed examination of the 6D Einstein–Maxwell landscape. The results have implications for eternal inflation and landscape transitions, highlighting that even when a direct instanton vanishes, alternative pathways or nonperturbative effects may still enable transitions, albeit with altered rates. Overall, the work provides a coherent account of how tunneling channels can abruptly vanish or smoothly fade, enriching our understanding of vacuum dynamics in quantum field theory and gravity.

Abstract

Instantons are tunneling solutions that connect two vacua, and under a small change in the potential, instantons sometimes disappear. We classify these disappearances as smooth (decay rate goes to 0 at disappearance) or abrupt (decay rate not equal to 0 at disappearance). Abrupt disappearances mean that a small change in the parameters can produce a drastic change in the physics, as some states become suddenly unreachable. The simplest abrupt disappearances are associated with annihilation by another Euclidean solution with higher action and one more negative mode; higher-order catastrophes can occur in cases of enhanced symmetry. We study a few simple examples, including the 6D Einstein-Maxwell theory, and give a unified account of instanton disappearances.

Figures (19)

• Figure 1: As $V_C$ is raised, the $A \rightarrow C$ tunneling rate $\Gamma$ goes smoothly to zero as the $AC$ instanton disappears. The disappearance is denoted by a star. In flat spacetime, the instanton disappears when the vacua become degenerate.
• Figure 2: As a new minimum is lowered, the direct $A \rightarrow C$ tunneling rate $\Gamma$ jumps abruptly to zero as the $AC$ instanton disappears. This happens after $B$ becomes degenerate with $A$, but before $B$ becomes degenerate with $C$.
• Figure 3: Tunneling from $A$ to $C$ proceeds by bubble nucleation. In the semiclassical description the field makes a quantum jump: a zero-energy bubble of true vacuum $C$ appears in the false vacuum $A$. The bubble then classically grows, completing the transition.
• Figure 4: A cross-section through the center of the bubble. The field value interpolates from close to the true vacuum at $\rho = 0$ to exactly the false vacuum as $\rho \rightarrow \infty$. In the thin-wall limit this profile sharpens to a step function.
• Figure 5: The undershoot/overshoot argument guarantees that an instanton solution exists. A particle released too low undershoots and ends up at $\phi > A$. A particle released too high overshoots and ends up at $\phi < A$. By continuity, a particle released at the critical height asymptotically come to rest at $A$. This particle's trajectory gives the field profile of the instanton.