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

TL;DR

This paper introduces an efficient method using radial WKB series to compute functional determinants in radial backgrounds across multiple dimensions, enabling precise calculations with fewer partial wave contributions.

Contribution

It derives explicit radial WKB series for partial wave sums in various dimensions, improving the efficiency and accuracy of functional determinant evaluations in quantum field theory.

Findings

Enables accurate evaluation of functional determinants with fewer partial waves

Provides explicit radial WKB series valid across dimensions 2 to 5

Demonstrates effectiveness in false vacuum decay calculations

Abstract

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.