False vacuum decay of excited states in finite-time instanton calculus

Abstract: Extracting information about a system's metastable ground state energy employing functional methods usually hinges on utilizing the late-time behavior of the Euclidean propagator, practically impeding the possibility of determining decay widths of excited states. We demonstrate that such obstacles can be surmounted by working with bounded time intervals, adapting the standard instanton formalism to compute a finite-time amplitude corresponding to excited state decay. This is achieved by projecting out the desired resonant energies utilizing carefully chosen approximations to the excited state wave functions in the false vacuum region. To carry out the calculation, we employ unconventional path integral techniques by considering the emerging amplitude as a single composite functional integral that includes fluctuations at the endpoints of the trajectories. This way, we explicitly compute the sought-after decay widths, including their leading quantum corrections, for arbitrary potentials, demonstrating accordance with traditional WKB results. While the initial starting point of weighting Euclidean propagator contributions according to their endpoints using false vacuum states has been proposed earlier, we find several flaws in the published evaluation of the relevant amplitudes. Although we show that the previous proposition of employing a sequential calculation scheme -- where the functional integral is evaluated around extremal trajectories with fixed endpoints, weighted only at a subsequent stage -- can lead to the desired goal, the novel composite approach is found to be more concise and transparent.