One-loop Renormalization of Black Hole Entropy Due to Non-minimally Coupled Matter

TL;DR

Solodukhin analyzes one-loop quantum corrections to black hole entropy from non-minimally coupled matter using the Euclidean path integral and replica trick on α-sheeted cones. He proves that the ultraviolet divergences of entanglement entropy are absorbed by standard renormalization of gravitational couplings, deriving explicit renormalization formulas for $G$ and higher-curvature terms with non-minimal coupling $\xi$, and highlights how conformal coupling ($\xi=1/6$) simplifies the result. The work links entanglement entropy to Wald-type Noether entropy and discusses its equality to thermodynamic entropy in equilibrium, supporting a consistent picture where entropy divergences are handled by established gravitational renormalization. Overall, the paper provides a coherent framework for renormalizing black hole entanglement entropy in the presence of non-minimally coupled scalar fields, without invoking new renormalization prescriptions beyond the gravitational action.

Abstract

The quantum entanglement entropy of an eternal black hole is studied. We argue that the relevant Euclidean path integral is taken over fields defined on $α$-fold covering of the black hole instanton. The statement that divergences of the entropy are renormalized by renormalization of gravitational couplings in the effective action is proved for non-minimally coupled scalar matter. The relationship of entanglement and thermodynamical entropies is discussed.