Critical value of symmetry breaking parameter in the phase transition of   decay rate
  Phase transition of decay rate from quantum tunneling to thermal activity regimes is investigated in (3+1)-dimensional field theories with symmetry breaking term $f\phi$. By applying the two independent criteria for the sharp first-order transition to the same model, the upper and lower bounds of critical value of the symmetry breaking parameter are obtained. Unlike two dimensional case continuum states of the fluctuation operator near sphaleron solution play an important role to determine the type of transition. 
Hungsoo Kim, Soo-Young Lee, Sahng-Kyoon Yoo, D. K. Park, Jae Kwan kim