Metastability of quantum fields in a thermal bath

We investigate the metastability of scalar fields in quantum field theories at finite temperature, focusing on a detailed understanding of the bounce solution. At finite temperature, the bounce solution depends on two variables: the Euclidean time <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>τ</a:mi></a:math> and the spatial radial distance <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>r</c:mi></c:math>, and it is periodic in the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>τ</e:mi></e:math> direction. We propose a novel method to determine the bounce that describes transitions in a thermal bath, suitable for numerical calculations. Two types of bounces exist for transitions in a thermal bath: <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>τ</g:mi></g:math>-dependent and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>τ</i:mi></i:math>-independent bounces. We apply our method to compute these bounces in several models, including both thin-wall and thick-wall scenarios, to examine their properties. Specifically, we evaluate the critical temperature below which the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>τ</k:mi></k:math>-independent bounce becomes destabilized due to fluctuations, rendering it irrelevant. We demonstrate that in the thick-wall case, the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>τ</m:mi></m:math>-dependent bounce smoothly transitions into the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>τ</o:mi></o:math>-independent one as temperature increases, whereas in the thin-wall case, the transition between the two types of bounces is discontinuous.