A Mode-Sum Prescription for Vacuum Polarization in Odd Dimensions

TL;DR

This paper introduces a systematic mode-sum regularization method for calculating vacuum polarization of scalar fields in odd-dimensional black hole spacetimes, with closed-form regularization parameters applicable across dimensions.

Contribution

It provides the first general approach to regularized vacuum polarization in higher odd dimensions, with explicit formulas and applicability to various metrics.

Findings

Regularization parameters computed in closed form for arbitrary dimensions

Method applicable to any metric function $f(r)$ in odd dimensions

Plots of vacuum polarization in Schwarzschild-Tangherlini spacetime for dimensions 5 to 11

Abstract

We present a new mode-sum regularization prescription for computing the vacuum polarization of a scalar field in static spherically-symmetric black hole spacetimes in odd dimensions. This is the first general and systematic approach to regularized vacuum polarization in higher dimensions. Remarkably, the regularization parameters can be computed in closed form in arbitrary dimensions and for arbitrary metric function $f(r)$. In fact, we show that in spite of the increasing severity and number of the divergences to be regularized, the method presented is mostly agnostic to the number of dimensions. Finally, as an explicit example of our method, we show plots for vacuum polarization in the Schwarzschild-Tangherlini spacetime for odd $d=5,...,11$.