A Mode-Sum Prescription for Vacuum Polarization in Even Dimensions

TL;DR

This paper introduces a new mode-sum regularization method for calculating vacuum polarization of scalar fields in even-dimensional black hole spacetimes, overcoming logarithmic singularity challenges and applicable to various fields and metrics.

Contribution

It provides the first systematic regularization scheme for vacuum polarization in even dimensions, with explicit formulas and broad applicability.

Findings

Regularization parameters computed in closed form for arbitrary even dimensions.

Applied method to Schwarzschild-Tangherlini spacetime for even dimensions 4 to 10.

Method extends to massive fields and nonvacuum spacetimes.

Abstract

We present a mode-sum regularization prescription for computing the vacuum polarization of a scalar field in static spherically-symmetric black hole spacetimes in even dimensions. This is the first general and systematic approach to regularized vacuum polarization in higher even dimensions, building upon a previous scheme we developed for odd dimensions. Things are more complicated here since the even-dimensional propagator possesses logarithmic singularities which must be regularized. However, in spite of this complication, the regularization parameters can be computed in closed form in arbitrary even dimensions and for arbitrary metric function $f(r)$. As an explicit example of our method, we show plots for vacuum polarization of a massless scalar field in the Schwarzschild-Tangherlini spacetime for even $d=4,...,10$. However, the method presented applies straightforwardly to massive fields or to nonvacuum spacetimes.