Functional Determinants for False Vacuum Decay

Abstract

We derive simple expressions to regularise functional determinants from fluctuations of fields with spin 0, 1/2, and 1. These are important for the precise dimensionful determination of false vacuum decay rates. We work in $D = 4$ Euclidean dimensions and use familiar Feynman diagrammatic techniques with a double expansion in interactions and masses, together with dimensional regularisation in momentum space. We Fourier transform to coordinate space and end up with a simple regularisation prescription in terms of single integrals over the Euclidean radius of field-dependent masses and their derivatives. Our results apply to models with an arbitrary scalar potential and with any number of scalars, fermions, gauge bosons and associated ghosts. We exemplify this approach on the Standard Model with a streamlined calculation of the renormalisation and isolation of divergences in fluctuation determinants.