  The renormalization of entanglement entropy of quantum field theories is
investigated in the simplest setting with a $\lambda \phi^4$ scalar field
theory. The 3+1 dimensional spacetime is separated into two regions by an
infinitely flat 2-dimensional interface. The entanglement entropy of the system
across the interface has an elegant geometrical interpretation using the
replica trick, which requires putting the field theory on a curved spacetime
background. We demonstrate that the theory, and hence the entanglement entropy,
is renormalizable at order $\lambda$ once all the relevant operators up to
dimension-4 are included in the action. This exercise has a one-to-one
correspondence to entanglement entropy interpretation of the black hole entropy
which suggests that our treatment is sensible. Our study suggests that
entanglement entropy is renormalizable and is a physical quantity.
