Path integral predictions for pre-asymptotic false vacuum decay

Abstract

When tunneling occurs out of generic initial states, a significant fraction of probability is lost at early times during which the dynamics is governed by excited resonance states. However, first-principles analyses based on path integrals have only captured the leading asymptotic behavior during which the tunneling rate is dominated by the false vacuum contribution. In this work, we discuss the behavior in the pre-asymptotic regime from a first-principles path integral perspective. We demonstrate how the relevant expressions can be evaluated systematically through semi-classical methods in the recently developed steadyon picture. This approach allows one to trace the role of the relevant physical scales, making transparent the underlying assumptions and approximations and offering a clear path to establishing a systematically improvable framework to evaluate tunneling rates non-perturbatively.

Figures (3)

• Figure 1: (a): A $1$-dimensional potential $V(x)$ (inspired by the potential used in Refs. Andreassen:2016cvxAndreassen:2016cff) with a barrier separating a false vacuum $\mathcal{F}$ from a true vacuum $\mathcal{R}$. For an example initial wavefunction $\psi(t=0)$, time evolution causes it to tunnel through the barrier, where the mass is fixed to $m = 6\cdot10^3$ in natural units ($\hbar = 1$). (b): The time evolution of the probability $P_\mathcal{F}(t)$ of finding the particle in $\mathcal{F}$ is well-described by a 2-state resonant expansion, up to corrections at early times due to contributions from higher energy resonant states. At large times, the decay of $P_\mathcal{F}(t)$ approaches an exponential decay described by a single resonant state, shown in the inset figure.
• Figure 2: At $t=0$, the particle is fully localised within the well $\mathcal{F}$. $x_*$ denotes the point beyond which its wave function decays exponentially.
• Figure 3: Assigning the factor $1-i \epsilon$ to the time variable rather than the Hamiltonian allows to interpret the steadyon as a solution along a diagonal contour in the complex time plane. Its projections on the real- and imaginary-time axis, respectively, yield the familiar classical and Euclidean-time solutions, respectively.