Entanglement Entropy of a Non-Minimally Coupled Self-Interacting Scalar across a Schwarzschild Horizon at $\mathcal{O}(\alpha )$

Abstract

We compute the first-order correction in the quartic coupling $\alpha$ to the entanglement entropy of a massive, non-minimally coupled scalar across the horizon of a four-dimensional Schwarzschild black hole, treating the non-minimal coupling $\xi$ as a free parameter. Combining the replica trick on the conical manifold ${\mathcal{M}}_n$ with heat-kernel methods in proper-time regularization, we obtain a closed-form expression for $\delta S^{(1)}(m,\alpha ,\xi )$. The bare correction exhibits a log-enhanced quadratic divergence ${\epsilon }^{-2}\ln (m^2{\epsilon }^2)$, arising from interference between bulk fluctuations and the distributional curvature at the tip; we show it is cancelled at $\mathcal{O}(\alpha )$ by the bulk mass counterterm. The residual $m^2\ln (m^2{\epsilon }^2)$ divergence renormalizes Newton's constant, preserving $S_{\mathrm{BH}}={\mathcal{A}}_{\Sigma }/4G_F$. The correction is proportional to $(1/6-\xi )$ and vanishes identically for conformal coupling.

v2: substantially revised; added mass-counterterm cancellation (Appendix A), corrected figures, updated bibliography, rewritten introduction and abstract