  We consider the large $N$ interacting vector $O(N)$ model on a sphere in
$4-\epsilon$ Euclidean dimensions. The Gaussian theory in the UV is taken to be
either conformally or non-conformally coupled. The endpoint of the RG flow
corresponds to a conformally coupled scalar field at the Wilson-Fisher fixed
point. We take a spherical entangling surface in de Sitter space and compute
the entanglement entropy everywhere along the RG trajectory. In $4$ dimensions,
a free non-conformal scalar has a universal area term scaling with the
logarithm of the UV cutoff. In $4-\epsilon$ dimensions, such a term scales as
$1/\epsilon$. For a non-conformal scalar, a $1/\epsilon$ term is present both
at the UV fixed point, and its vicinity. For flow between two conformal fixed
points, $1/\epsilon$ terms are absent everywhere. Finally, we make contact with
replica trick calculations. The conical singularity gives rise to boundary
terms residing on the entangling surface, which are usually discarded.
Consistency with our results requires they be kept. We argue that, in fact,
this conclusion also follows from the work of Metlitski, Fuertes, and Sachdev,
which demonstrated that such boundary terms will be generated through quantum
corrections.
