# Numerical calculations of the decay rate of a false vacuum for a modified quartic potential

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## Abstract

The possibility of two or more vacuum states with different energy densities is a common feature of many potentials. The classical state is in stable equilibrium at any of these minima but is made unstable by quantum effects. The formalism that describes transitions from a false (higher energy density) vacuum to a true (lowest energy density) vacuum was developed decades ago. It suggests that false vacuum decay occurs via a quantum tunneling process, with the nucleation of a bubble of true vacuum, which, as it expands, transforms the metastable false vacuum into the more stable phase. The well-known procedure to calculate the tunneling rate uses an Euclidean Bounce solution. However, some potentials exist for which this method doesn't work. All these potentials are unbounded from below, and hence don't have a proper true vacuum. Instead of a single bounce, these potentials have either an infinite family of bounces with identical tunneling rates or no bounce solution. In this thesis, we investigate these phenomena. Since potentials are only known in finite regions of field space, we approximate the known unbounded potential with a series of potentials bounded from below and for which the tunneling rate is well-defined. We compute the change in the tunneling rate as the field space distance and the energy density distance between the true and false vacua increases. We find that neither our numerical calculation of a modified quartic potential, nor the method of "Pseudo-Bounce", provide a reliable path for the calculation of the decay rate of the false vacuum. If −λϕ⁴ is only an effective potential for small values of ϕ, our numerical analysis shows that we must know the actual potential to estimate both the decay path and the decay rate. Espinosa et al.'s approach never agrees with the numerical result.