Critical value of symmetry breaking parameter in the phase transition of decay rate

The phase transition of the decay rate from quantum tunneling to thermal activity regimes is investigated in (3 + 1)-dimensional field theories with symmetry breaking term fF. By applying two independent criteria for the sharp first-order transition to the same model, the upper and lower bounds of the critical value of the symmetry breaking parameter are obtained. Unlike the two-dimensional case continuum states of the fluctuation operator near the sphaleron solution play an important role in determining the type of transition.