Entanglement Entropy of Massive Scalar Fields: Mass Suppression, Violation of Universal mR Scaling, and Implications for Black Hole Thermodynamics

Abstract

We investigate the entanglement entropy of a massive scalar field using the spherical shell lattice model introduced by Das and Shankaranarayanan. A systematic numerical analysis is performed to study the dependence of the entropy on the field mass and on the size of the entangling region for both ground and excited states. For the ground state, we find that the entanglement entropy is exponentially suppressed by the field mass, reflecting the presence of a finite correlation length, while the geometric area-law scaling remains robust for all masses. For localized excited states, however, we uncover a qualitatively different behavior. The excess entropy does not exhibit universal scaling in the dimensionless variable mR. Instead, numerical results show that data points with identical mR but different (m,R) pairs do not collapse onto a single curve, demonstrating a clear violation of simple scaling. This breakdown is traced to the presence of an additional length scale associated with the finite width of the wave-packet excitation. This result identifies the coexistence of multiple infrared scales as a key feature of excited-state entanglement in massive quantum field theories. Mutual information provides an additional finite diagnostic of correlations in the chosen nested geometry. The numerical results show a strong dependence on the field mass, although the detailed behavior is sensitive to the geometric setup used in the calculation. These findings clarify how particle mass and excitation structure jointly determine entanglement properties, and suggest that the matter contribution to the generalized entropy in semiclassical gravity may depend on independent infrared parameters rather than on a single correlation scale. Implications for black hole entropy and the island formula are briefly discussed.

Figures (8)

• Figure 1: Ground state entanglement entropy vs. radius for different masses. For $m = 0$, the entropy grows with $R$, following the area law. For massive fields, the entropy is exponentially suppressed by the factor $e^{-mR}$. At $R = 10$, $S(m=0.5)/S(m=0) \approx 0.007$, and for $m \geq 1.0$ the entropy is less than $10^{-5}$, making the curves indistinguishable from zero on this linear scale.
• Figure 2: Comparison of ground-state and excited-state entanglement entropy for $m = 0.5$. The excited state exhibits a localized peak in entropy around the wave packet position.
• Figure 3: Excess entropy $\Delta S$ for different field masses. The excess entropy decreases with increasing radius and field mass, reflecting the suppression of long-range correlations in massive fields.
• Figure 4: Log–log plot of the entanglement entropy as a function of the subsystem radius $R$ for different field masses. The slope of the curves determines the scaling exponent $\alpha$ in the relation $S \propto R^\alpha$.
• Figure 5: Excited-state entropy $S_{\mathrm{exc}}$ at fixed radius $R=4$ as a function of the field mass. The entropy shows a non-monotonic dependence on the mass, reflecting the competition between localization of the excitation and suppression of long-range correlations.