Case of the disappearing instanton

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 →0 at disappearance) or abrupt (decay rate ≠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 six-dimensional Einstein-Maxwell theory, and give a unified account of instanton disappearances.