Vacuum Polarization in Quantum-Corrected and Effective Black Hole Geometries: a High Performance Approach

Abstract

We compute the renormalized scalar vacuum polarization $\langle ϕ^2 \rangle$ of a massive, non-minimally coupled quantum field in the Hartle--Hawking state exterior to a family of static, spherically symmetric quantum-corrected, effective and regular black holes: the Kazakov--Solodukhin quantum-deformed black hole, the two effective loop-quantum-gravity geometries of Zhang--Lewandowski--Ma--Yang, the renormalization-group improved Schwarzschild black hole of Bonanno--Reuter, and the Bardeen regular black hole. We adapt the extended point-splitting mode-sum formalism, implemented as a high-performance code, and present the exterior profile of the vacuum polarization over a grid of the quantum-deformation parameter, the field mass $μ=mM$ and the curvature coupling $ξ$. The five geometries display qualitatively distinct horizon responses. In all five geometries the deformation imprint is governed by the sign of the background Ricci scalar through the DeWitt--Schwinger curvature term linear in $(ξ-1/6)\mathcal{R}$, and is strongly suppressed at conformal coupling. We verify this both in magnitude and in the $ξ$-dependence. In the near-extremal regime of the two geometries with inner horizons we find a sign change of the horizon polarization at light field mass -- absent for Reissner--Nordström at matched temperature and removed by conformal coupling -- identifying a genuinely quantum-geometric, de-Sitter-core-driven regime. Every result reduces to the Schwarzschild value in the classical limit, validating the calculation.