Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Analytic First-Order Entanglement Entropy of Schwarzschild Black Holes with Interacting Scalars

Florin Manea

TL;DR

This work derives a closed-form first-order correction to the entanglement entropy for a self-interacting scalar field across a Schwarzschild horizon, using the replica trick and heat-kernel methods on a conical manifold. It yields explicit expressions for the correction $δS^{(1)}$ and a renormalized Newton constant $G_F$, revealing a log-enhanced quadratic divergence that is absorbed into the gravitational coupling, leaving a finite Bekenstein–Hawking entropy $S_{BH} = A_{Σ}/(4 G_F)$. The results show how the correction grows with light fields and strong interactions, but is suppressed by curvature and vanishes at leading order for conformally coupled fields ($ξ=1/6$). This analytic framework clarifies the direct link between quantum field interactions, entanglement across horizons, and the renormalization of gravitational couplings within an effective-field-theory interpretation of gravity.

Abstract

We compute the first-order correction to the entanglement entropy of a scalar field with quartic self-interaction across the horizon of a Schwarzschild black hole. The key result is a fully explicit analytic expression for both the renormalized Newton constant and the corresponding correction to the entropy. Using the Euclidean action, the replica trick, and heat-kernel methods, we obtain a closed-form expression that depends directly on the horizon area, the scalar mass, the interaction strength, and the curvature coupling. The correction exhibits a characteristic logarithmically enhanced quadratic divergence, which is absorbed into the renormalized gravitational coupling. Once this renormalization is performed, the entropy takes the standard Bekenstein-Hawking form with a finite, physically meaningful value. Our result makes it possible to study in detail how the correction varies with model parameters: light fields and strong interactions increase the effect, curvature suppresses it, and conformally coupled fields show no leading-order contribution. Overall, this work provides a clear analytic demonstration of how quantum field interactions modify black hole entropy through the renormalization of Newton's constant.

Analytic First-Order Entanglement Entropy of Schwarzschild Black Holes with Interacting Scalars

TL;DR

This work derives a closed-form first-order correction to the entanglement entropy for a self-interacting scalar field across a Schwarzschild horizon, using the replica trick and heat-kernel methods on a conical manifold. It yields explicit expressions for the correction and a renormalized Newton constant , revealing a log-enhanced quadratic divergence that is absorbed into the gravitational coupling, leaving a finite Bekenstein–Hawking entropy . The results show how the correction grows with light fields and strong interactions, but is suppressed by curvature and vanishes at leading order for conformally coupled fields (). This analytic framework clarifies the direct link between quantum field interactions, entanglement across horizons, and the renormalization of gravitational couplings within an effective-field-theory interpretation of gravity.

Abstract

We compute the first-order correction to the entanglement entropy of a scalar field with quartic self-interaction across the horizon of a Schwarzschild black hole. The key result is a fully explicit analytic expression for both the renormalized Newton constant and the corresponding correction to the entropy. Using the Euclidean action, the replica trick, and heat-kernel methods, we obtain a closed-form expression that depends directly on the horizon area, the scalar mass, the interaction strength, and the curvature coupling. The correction exhibits a characteristic logarithmically enhanced quadratic divergence, which is absorbed into the renormalized gravitational coupling. Once this renormalization is performed, the entropy takes the standard Bekenstein-Hawking form with a finite, physically meaningful value. Our result makes it possible to study in detail how the correction varies with model parameters: light fields and strong interactions increase the effect, curvature suppresses it, and conformally coupled fields show no leading-order contribution. Overall, this work provides a clear analytic demonstration of how quantum field interactions modify black hole entropy through the renormalization of Newton's constant.

Paper Structure

This paper contains 10 sections, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Entanglement entropy
  3. Von Neumann Entropy
  4. Background Geometry
  5. Replica Trick Review
  6. Free Theory
  7. Quartic Interaction
  8. Four-Dimensional Case
  9. Renormalization of Newton's constant
  10. Conclusions

Figures (1)

  • Figure 1: Left: $\delta S^{(1)}$ grows with increasing interaction strength $\alpha$ and decreases with curvature coupling $\xi$. Middle: $\delta S^{(1)}$ is largest for light scalar fields ($m$ small) and strong interactions ($\alpha$ large). Right: $\delta S^{(1)}$ is suppressed by curvature ($\xi$) and enhanced for lighter fields. The UV regulator $\epsilon$ was set to $10^{-10}$