Bounce solutions and the transition to thermal hopping in φ⁴ theory

Abstract

Scalar field theory with an asymmetric potential is studied at all temperatures for φ⁴ theory with φ³ symmetry breaking. The equations of motion are solved numerically to obtain O(4) symmetric and O(3) cylindrical symmetric bounce solutions. The bounce solution which interpolates between the zero-temperature and high-temperature solutions is also obtained numerically, using a multigrid method. These solutions control the rates for tunneling from the false vacuum to the true vacuum by bubble formation. The range of validity of the thin-wall approximation is investigated. An analytical solution for the bounce is presented, which reproduces the action in the thin-wall as well as the thick-wall limits. The nature of the transition from quantum tunneling at low temperatures to thermal hopping at high temperatures is investigated with cubic symmetry breaking. It is found that, for a small value of the symmetry-breaking coupling f, the transition is first order. For higher values of f, the transition continues to be first order, though weakly so.