  Entanglement entropy (EE) in interacting field theories has two important
issues: renormalization of UV divergences and non-Gaussianity of the vacuum. In
this letter, we investigate them in the framework of the two-particle
irreducible formalism. In particular, we consider EE of a half space in an
interacting scalar field theory. It is formulated as $\mathbb{Z}_M$ gauge
theory on Feynman diagrams: $\mathbb{Z}_M$ fluxes are assigned on plaquettes
and summed to obtain EE. Some configurations of fluxes are interpreted as
twists of propagators and vertices. The former gives a Gaussian part of EE
written in terms of a renormalized 2-point function while the latter reflects
non-Gaussianity of the vacuum.
