Abstract
The 'thin-wall approximation' gives a simple estimate of the decay rate of an unstable quantum field. Unfortunately, the approximation is uncontrolled. In this paper I show that there are actually two different thin-wall approximations and that they bracket the true decay rate: I prove that one is an upper bound and the other a lower bound. In the thin-wall limit, the two approximations converge. In the presence of gravity, a generalization of this lemma provides a simple sufficient condition for non-perturbative vacuum instability.
Highlights
Metastable states may decay by quantum tunneling
For a given field potential VðφÞ, the instanton can be derived by numerically integrating the second-order Euclidean equations of motion. For those who lack either the computing power or the patience to solve the equations of motion, or who seek an intuitive understanding of the parametric dependence of the decay rate, Coleman showed that we can often use the “thin-wall” approximation
While the instanton is a bubble that smoothly interpolates from the false vacuum to near the true vacuum, the thin-wall approximation treats this transition as abrupt [1]; the decay exponent is approximated by ðV false σ4 − VtrueÞ3
Summary
Metastable states may decay by quantum tunneling. In the semiclassical (small ħ) limit, the most important contribution to the decay rate rate ∼ A exp1⁄2−B=ħ; ð1Þ is the tunneling exponent B. For a given field potential VðφÞ, the instanton can be derived by numerically integrating the second-order Euclidean equations of motion. While the instanton is a bubble that smoothly interpolates from the false vacuum to near the true vacuum, the thin-wall approximation treats this transition as abrupt [1]; the decay exponent is approximated by ðV false σ4 − VtrueÞ3
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