  This is a continuation of our previous works on entanglement entropy (EE) in
interacting field theories. In arXiv:2103.05303, we have proposed the notion of
$\mathbb{Z}_M$ gauge theory on Feynman diagrams to calculate EE in quantum
field theories and shown that EE consists of two particular contributions from
propagators and vertices. As shown in the next paper arXiv:2105.02598, the
purely non-Gaussian contributions from interaction vertices can be interpreted
as renormalized correlation functions of composite operators. In this paper, we
will first provide a unified matrix form of EE containing both contributions
from propagators and (classical) vertices, and then extract further
non-Gaussian contributions based on the framework of the Wilsonian
renormalization group. It is conjectured that the EE in the infrared is given
by a sum of all the vertex contributions in the Wilsonian effective action.
