A Fast Way to Compute Functional Determinants of Radially Symmetric Partial Differential Operators in General Dimensions

Recently the partial wave cutoff method was developed as a new calculational scheme for a functional determinant of quantum field theory in radial backgrounds. For the contribution given by an infinite sum of large partial waves, we derive explicitly radial WKB series in the angular momentum cutoff for $d=2,3,4$ and 5 ($d$ is the spacetime dimension), which has uniform validity irrespectively of any specific values assumed for other parameters. Utilizing this series, precision evaluation of the renormalized functional determinant is possible with a relatively small number of low partial wave contributions determined separately. We illustrate the power of this scheme in numerically exact evaluation of the prefactor (expressed as a functional determinant) in the case of the false vacuum decay of 4D scalar field theory.