Entanglement Entropy for Screened Interactions via Dimensional Mapping to Harmonic Oscillators

We investigate interaction-induced corrections to entanglement entropy by mapping a screened Yukawa-type interaction to an effective harmonic oscillator system with controlled anharmonic perturbations. Starting from a one-dimensional interaction V(x) = -g² e^{-αm x}/x, we reformulate the problem in terms of a four-dimensional radial oscillator, where the finite screening length generates a systematic hierarchy of polynomial interactions in the radial coordinate. This mapping enables a controlled Rayleigh-Schrodinger perturbative treatment of the ground-state wavefunction and an explicit spectral analysis of the reduced density matrix. Working in the weak-screening regime, we compute the leading non-Gaussian correction arising from the quartic interaction ρ⁴, which appears at order α² in the expansion of the Yukawa-like potential. We obtain closed analytic expressions for the resulting small eigenvalues of the reduced density matrix and evaluate their contribution to the von Neumann entanglement entropy. We show that the entropy receives analytic corrections at order α², originating both from explicit anharmonic state-mixing effects and from the implicit α dependence of the Gaussian width parameter. Our results clarify the distinct roles of harmonic renormalization and genuinely non-Gaussian interactions in generating entanglement, establish a systematic power-counting and normalization scheme for higher-order ρ²ⁿ perturbations, and provide a transparent oscillator-based framework for computing entanglement entropy in weakly interacting low-dimensional and field-theoretic systems.