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 $S_{\mathrm{BH}} = \mathcal{A}_\Sigma / 4 G_F$. The correction is proportional to $(1/6-\xi)$ and vanishes identically for conformal coupling.