Entanglement entropy from entanglement contour: higher dimensions

We study the $\mathit{\text{entanglement}}\mathit{\text{contour}}$ and \textit{partial entanglement entropy} (PEE) in quantum field theories in 3 and higher dimensions. The entanglement entropy is evaluated from a certain limit of the PEE with a geometric regulator. In the context of the $\mathit{\text{entanglement}}\mathit{\text{contour}}$, we classify the geometric regulators, study their difference from the UV regulators. Furthermore, for spherical regions in conformal field theories (CFTs) we find the exact relation between the UV and geometric cutoff, which clarifies some subtle points in the previous literature. We clarify a subtle point of the additive linear combination (ALC) proposal for PEE in higher dimensions. The subset entanglement entropies in the $\mathit{\text{ALC}}\mathit{\text{proposal}}$ should all be evaluated as a limit of the PEE while excluding a fixed class of local-short-distance correlation. Unlike the 2-dimensional configurations, naively plugging the entanglement entropy calculated with a UV cutoff will spoil the validity of the \textit{ALC proposal}. We derive the $\mathit{\text{entanglement}}\mathit{\text{contour}}$ function for spherical regions, annuli and spherical shells in the vacuum state of general-dimensional CFTs on a hyperplane.