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arXiv:2511.08669

Path integral predictions for pre-asymptotic false vacuum decay

Joshua Lin, Bruno Scheihing-Hitschfeld, Thomas Steingasser

hep-thhep-phquant-phSubmitted Nov 11, 2025• Announced Nov 13, 2025
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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.

Comments
6 pages, 2 figures

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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.
Recent developments in the understanding of real-time path integrals led to the development of the ``steadyon picture'' for the semi-classical calculation of quantum tunneling rates. We discuss tunneling out of a generic localized initial state in this picture and present its application for the important example of a resonance state in a one-dimensional point particle potential. We find that the steadyon picture indeed reproduces existing results obtained using the WKB method. Our analysis furthermore demonstrates how applying this picture to physical states naturally addresses open conceptual questions regarding this framework. Finally, we perform a numerical study for a specific potential. We demonstrate in particular the existence of regimes in which the tunneling rate is dominated by higher resonances, rather than the false vacuum, as well as their importance.
In the study of quantum-mechanical tunneling processes, numerous approaches have been developed to determine the decay rate of states initially confined within a metastable potential region. Virtually all analytical treatments, however, fall into one of two superficially unrelated conceptual frameworks: the resonant-state approach and the instanton method. Whereas the concept of resonant states and their associated decay widths is grounded in physical reasoning by capturing the regime of uniform probability decay, the instanton method lacks a comparably clear physical interpretation. We demonstrate the equivalence of the two approaches, revealing that the contour-deformation prescription in the functional integral put forward by Callan and Coleman directly corresponds to the outgoing Gamow--Siegert boundary conditions defining resonant states.
When a boundary value problem in classical mechanics has no solution, the corresponding quantum propagator may be asymptotically dominated by locally flawed classical paths. We indicate how this assertion can be made precise and show how one calculates the propagator under these circumstances. The essential tools in our arguments will be the use of neutralizers and the path decomposition formula for path integrals. With this method the propagator for barrier penetration is expressed as an integral over real tunneling time, defined as the interval between first striking the barrier and last emerging from it. 14 refs., 3 figs.
We study the decay of general initial states out of a metastable potential well in quantum mechanics. We provide a closed-form expression for the probability current that tunnels through the barrier in terms of the resonant states into which the initial state can be decomposed. All ingredients are computed analytically to first subleading order in the semiclassical limit. Specializing to a coherently-oscillating state, we derive an equivalent semiclassical approximation to the current via the path integral, which circumvents the decomposition into resonant states. We test the accuracy of the approximation by comparing it to a numerical solution of the Schrödinger equation.
Metastable decay exhibits a familiar exponential regime bracketed by early-time deviations and late-time power-law tails. We adopt the real-time, flux-based definition of the decay rate in the spirit of Andreassen et al.\ direct method and present a complete analysis of one-dimensional quantum-mechanical resonance models. We show that the kernel admits a universal pole--plus--branch decomposition and use it to define two computable time scales: a dawn time, when a single resonant contribution starts dominating and exponential decay sets in, and a twilight time, when the branch-cut tail overtakes exponential decay. The latter can be expressed in closed form via the Lambert WWW function, making its parametric dependence manifest without fitting. For square, modified square, and Pöschl--Teller barriers we obtain simple thick-barrier formulas, clarify the relation ΓT=TtransΓT = T_{\text{trans}}ΓT=Ttrans​ between the decay rate ΓΓΓ, oscillation period TTT, and transmission probability TtransT_{\text{trans}}Ttrans​, and indicate how our spectral picture can be naturally extended to quantum field theoretic vacuum decay.
We revisit the path integral description of quantum tunneling and its generalization to excited states. For clarity, we focus on the simple toy model of a point particle in a double-well potential, for which we perform all steps explicitly. Instead of performing the familiar Wick rotation from physical to imaginary time—which is inconsistent with the requisite boundary conditions when treating tunneling from excited states—we regularize the path integral by adding an infinitesimal complex contribution to the Hamiltonian, while keeping time strictly real. We find that this gives rise to a complex stationary-phase solution, in agreement with recent insights from Picard-Lefshetz theory. We then show that there exists a class of analytic solutions for the corresponding equations of motion, which can be made to match the appropriate boundary conditions in the physically relevant limits of a vanishing regulator and an infinite physical time. We provide a detailed discussion of this nontrivial limit. We find that, for systems without an explicit time-dependence, our approach reproduces the picture of an instantonlike solution defined on a finite Euclidean-time interval. Lastly, we discuss the generalization of our approach to broader classes of systems, for which it serves as a reliable framework for high-precision calculations.