Dr Qiang Yin (Kyushu University)
Title
A new method for calculating the decay rate of the false vacuum at finite temperature by use of the saddle-point approximation
Abstract
Coleman and Callan utilized path integral and saddle point approximation to calculate the decay rate of the false vacuum state, which is related to the imaginary part of the energy eigenvalue[Coleman, 1977; Callan and Coleman, 1977]. The method was later extended by Affleck to evaluate the transition rate at finite temperature, which is proportional to the imaginary part of the free energy[Affleck, 1981].
In the calculation at finite temperature, classical solutions satisfying periodic boundary conditions were employed for the saddle point approximation. Additionally, fluctuations around these classical solutions were required be periodic as well. However, inconsistencies arose in this framework, particularly concerning the handling of the zero eigenvalue associated with certain classical solutions.
Our research introduces a novel calculation method that emphasizes a rigorous evaluation of the path integral derived from the trace in the free energy. This method incorporates three significant improvements over conventional approaches:
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The trace is explicitly computed by integrating over the endpoints of paths within the path integral.
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The saddle point approximation is performed in a functional space adhering to Dirichlet boundary conditions, instead of the periodic boundary conditions used in previous studies.
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Influence from shot solutions, as introduced by Andreassen et al., are incorporated into the calculation.
Our new approach eliminates the need for the so-called collective coordinate method to handle the zero eigenvalue in the calculation. Moreover, it provides a more intuitive way of calculating the trace in the path integral approach, avoiding the ambiguous application of periodic boundary conditions. Transition rates calculated with our method exhibit expected behavior over a broader range compared to those obtained using conventional approaches.
Furthermore, our research highlights several inconsistencies in Coleman and Callan’s original framework that require a more thorough explanation. By addressing these issues, our method contributes to a deeper understanding of the calculation of decay and transition rates in quantum systems.