Consequences of vacuum instability in quantum field theory

When the effective potential of a quantum field theory has nonglobal minima, the question arises as to whether they may correspond to the vacuum state. The conventional response is negative because of vacuum instability and also on the basis of an analogy with many-body theory where the effective potential is compared to the Helmholtz free energy for which certain convexity properties may be proved using Van Hove's theorem. In pursuing an alternative possibility, it is pointed out here that when a nonglobal minium of the potential occurs, the field equations can sustain a pseudoparticlelike classical solution in four-dimensional Euclidean space-time and a corresponding soliton solution in three-dimensional space. The existence of these solutions is not inconsistent with Derrick's theorem which assumes that the vacuum is an absolute minimum. The solutions are interpreted physically in Minkowski space as vacuum bubbles created by quantum tunneling from the metastable vacuum to one of lower energy. Thus explicit calculations of the decay probability are rendered feasible. A simple theory where the phenomenon occurs is the skewed Goldstone model, which is studied in detail. Extension to Higgs-Kibble gauge theories is straightforward. As an example, in the Weinberg-Salam theory the lower limit on the Higgs mass derived by Weinberg and Linde, and the limit derived by Gildener and Weinberg for the modified version incorporating a dimensional transmutation mechanism, can be significantly modified. For the original version of the theory, the lower limit is changed from 4.91 GeV to about 3.5 GeV, for mixing angle ${\theta }_W=35\mathrm{}°$. For this new range of masses the vacuum is totally secure, for practical purposes, against spontaneous bubble formation by vacuum fluctuations.