Perturbative calculation of field space entanglement entropy

TL;DR

The paper addresses how to perturbatively compute the field-space entanglement entropy between interacting quantum fields, a task previously thought to be plagued by divergences. It develops a careful replica-trick framework in which the limit $n \to 1$ is taken after including all relevant terms, and demonstrates that certain diagrams cancel divergences, yielding a finite, well-defined expansion for the entropy. Applying this to a two-field mass-mixing model $\mathcal{L}$ with off-diagonal coupling $-g\,\phi_1\phi_2$, the leading $O(g^2)$ entropy per unit volume is finite and involves a logarithmic dependence on $g$, consistent with prior nonperturbative results. This provides a practical perturbative method for analyzing field-space entanglement in quantum field theories and has potential applications to hidden sectors and dark matter models, where perturbative control is valuable.

Abstract

We present a general method for the perturbative calculation of the entanglement entropy between two interacting quantum fields. Previous attempts at calculating this quantity perturbatively have encountered a seemingly pathological divergence; we explain why this divergence is a result of improperly truncating a series expansion and give a prescription for avoiding this problem. We then apply our method to a simple example of two mass-mixing scalar fields.

Figures (2)

• Figure 1: Schematic view of the manifold $M_n$. A copy of one of the fields fields, $\phi$ say, lives on each of the blue sheets, while the copies of $\psi$ fields live on the red. For $\tau<0$, the $i$th $\phi$ couples to the $i$th $\psi$, but for $\tau>0$ it couples to the $i+1$st.
• Figure 2: The process contributing to the $n$th Rényi entropy at order $g^{2n}$. Red (blue) lines represent propagators of fields on the red (blue) sheets.