  We introduce a "renormalized entanglement entropy" which is intrinsically UV
finite and is most sensitive to the degrees of freedom at the scale of the size
R of the entangled region. We illustrated the power of this construction by
showing that the qualitative behavior of the entanglement entropy for a
non-Fermi liquid can be obtained by simple dimensional analysis. We argue that
the functional dependence of the "renormalized entanglement entropy" on R can
be interpreted as describing the renormalization group flow of the entanglement
entropy with distance scale. The corresponding quantity for a spherical region
in the vacuum, has some particularly interesting properties. For a conformal
field theory, it reduces to the previously proposed central charge in all
dimensions, and for a general quantum field theory, it interpolates between the
central charges of the UV and IR fixed points as R is varied from zero to
infinity. We conjecture that in three (spacetime) dimensions, it is always
non-negative and monotonic, and provides a measure of the number of degrees of
freedom of a system at scale R. In four dimensions, however, we find examples
in which it is neither monotonic nor non-negative.
