Entanglement Entropy in (3+1)-d Free $U(1)$ Gauge Theory

TL;DR

The paper shows that for a free $U(1)$ gauge theory in $3+1$ dimensions, the entanglement entropy defined via the extended Hilbert space and computed with a replica-trick path integral yields a universal logarithmic term with coefficient $C=a={-31\over 90}$, matching the $a$-anomaly. It further demonstrates that the extractable portion of entanglement, accessible through local operations, has a different logarithmic coefficient $D={-16\over 90}$, due to boundary massless scalar modes on the region boundary. The analysis relies on careful continuum limits, gauge fixing with Faddeev–Popov determinants, and a smoothing of the conical singularity, linking the universal log term to conformal anomalies and boundary dynamics. These results clarify discrepancies in the literature and have implications for gauge theories, holography, and the interpretation of operational entanglement measures in gauge contexts.

Abstract

We consider the entanglement entropy for a free $U(1)$ theory in $3 + 1$ dimensions in the extended Hilbert space definition. By taking the continuum limit carefully we obtain a replica trick path integral which calculates this entanglement entropy. The path integral is gauge invariant, with a gauge fixing delta function accompanied by a Faddeev-Popov determinant. For a spherical region it follows that the result for the logarithmic term in the entanglement, which is universal, is given by the $a$ anomaly coefficient. We also consider the extractable part of the entanglement, which corresponds to the number of Bell pairs which can be obtained from entanglement distillation or dilution. For a spherical region we show that the coefficient of the logarithmic term for the extractable part is different from the extended Hilbert space result. We argue that the two results will differ in general, and this difference is accounted for by a massless scalar living on the boundary of the region of interest.